Sylvester Papers

This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world’s books discoverable online. Ithassurvivedlongenoughforthecopyrighttoexpireandthebooktoenterthepublicdomain. Apublicdomainbookisonethatwasneversubject tocopyrightorwhoselegalcopyrighttermhasexpired. Whetherabookisinthepublicdomainmayvarycountrytocountry. Publicdomainbooks are our gateways to the past, representing a wealth of history, culture and knowledge that’s often difficult to discover. Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book’s long journey from the publisher to a library and finally to you. Usage guidelines Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the public and we are merely their custodians. Nevertheless,this work is expensive, so in order to keep providing this resource,we have taken steps to prevent abuse by commercialparties, including placing technical restrictions on automated querying. We also ask that you: +Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for personal, non-commercial purposes. +Refrainfromautomatedquerying DonotsendautomatedqueriesofanysorttoGoogle’ssystem: Ifyouareconductingresearchonmachine translation,opticalcharacterrecognitionorotherareaswhereaccesstoalargeamountoftextishelpful,pleasecontactus. Weencouragethe use of public domain materials for these purposes and may be able to help. +Maintainattribution TheGoogle“watermark”youseeoneachfileisessentialforinformingpeopleaboutthisprojectandhelpingthemfind additional materials through Google Book Search. Please do not remove it. +Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other countries. Whether a book is still in copyright varies from country to country, and we can’t offer guidance on whether any specific use of any specific book is allowed. Please do not assume that a book’s appearance in Google Book Search means it can be used in any manner anywhere in the world. Copyrightinfringement liability can be quite severe. About Google Book Search Google’s mission is to organize the world’s information and to make it universally accessible and useful. Google Book Search helps readers discovertheworld’sbookswhilehelpingauthorsandpublishersreachnewaudiences. Youcansearchthroughthefulltextofthisbookontheweb athttp://books.google.com/ 1 JamesJoseph Sylvester STANFORDUNIVERSITY LIBRARIES Digitized byC.Ie 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I MATHEMATICAL PAPERS I.onllon: C.J.CLAY ANDSONS, CAMBRIDGE UNIVERSITY PRESSWAREHOUSE, AVEMARIA LANE. 81",DI1I: so.WELLINGTON STtl.EET. I.ttp)i,: F.A.BROCKHAUS. _rill11m:THEMACMILLAN COMPANY. IIDMball anb€&Iretta: MACMILLAN AND CO.• LTD. [AllRig-I.tsrmrvm']II I I i I \ I I ! I .\ I \ I I \, THECOLLECTED MATHEMATICAL PAPERS OF JAMES JOSEPH SYLVESTER:: F.R.S.,D.C.L., LL.D.,Sc.D., Honorary Fellow of St John'sCollege,Cambridge j Sometime Professor at University. College, London jat the University of Virginia; atthe Royal Military Academy, Woolwich; at theJohnsHopkins University, Baltimore andSavilian Professor in the University of Oxford VOLUME I (1837-1853) .... .... Cambridge AttheUniversity Press 19°4 U:.:.............•...•.............··........................·............· ..............· ........-........·..........................·...............•.·....... -:..................·......·...·.........·....-........-........-.................·.........:.... ..'........... .. ........... ..................".."-..:». ._... ...................... .... ...... --..' ........PRINTED BY1-AND C. F. CLAY, ATTHEUNIVERSITY PRESS• PREFATORY NOTE. THEobject aimed atin this volume hasbeen topresenta faithful record ofthecourse of theauthor's thought, Withoutsuchadditions asrecent developments ofthesubjectstreatedofmighthave afforded,and withoutany alterations otherthanthatconsiderable numberinvolved in theattemptto makethealgebraical symbols readasthewriterintended. While, for the reader's convenience, theauthor's references to his own papers have been accompanied by cross references to thepages ofthisvolume, placed in square brackets. Byfar thelongestpaperinthevolume is No. 57, "Onthe Theory of the Syzygetic Relations of twoRational Integral Functions, comprising an application totheTheory of Sturm's Functions," and to this many of the shorterpapersinthevolume are contributory. Thevolume contains also Sylvester's dialytic method ofelimination (No.9,etc.), his Essayon Canonical Forms (No. 34),and early investigations intheTheoryofInvariants (Nos. 42, 43, etc.), Itcontains alsocelebrated theorems astoDeterminants (Nos.37,39,48, etc.)andinvestigations astotheTransformation ofQuadratic Forms(the LawofInertia,No.47,and the recognition of the Invariant factors of a matrix, Nos. 22, 24, 36). Afulltable ofcontents is prefixed. H.F.BAKER. STJOHY'S CoLLEGE, CAKBRIDGE • .April,1904. TABLEOFCONTENTS 1.AnalytWal development ofFreenete optWal theoryofcrystals (Philosophical Magazine 1887, 1888) 2.Onthe motion andrestoffluids (Philosophical Magazine 1838) 3.Onthemotionandrestofrigidbodies (Philosophical Magazine 1889) 4.Ondefinitedoubleintegration, 8'ltpplementary to aformerpaperon themotionandrest offluids. (philosophical Magazine 1889) 5.OnanextensionofSirJohnWilson'stheorem to all numbers whatever . (Philosophical Magazine 1838) 6. Note to the foregoing (Philosophical Magazine 1889) 7. Onrational derivation fromequations of coexistence, that is to say, a newand extended theoryofelimination, PartI. (Philosophical Magazine 1889) 8.Onderivation ofcoexistence, PartII.,being the theory ofsimultaneous simple homo­ geneousequations (Philosophical Magazine 1840) 9.Amethodofdetermining by mere inspection thederivatives fromtwo equations of anydegree (Philosophical Magazine 1840) 10. Note oneiimimation (Philosophical Magazine 1840)PAGES 1-27 28-32 33-35 36-38 39 39 40-46 47-53 54-57 58 21.)CONTENTS. 11.OntherelationofSturm'sauxiliary functions to theroots ofanalgebraic equation (Plymouth BritishAssociation Report1841) 12.Examples ofthedialyticmethodofelimina­ tionasappliedtoternarysystemsof equations (Cambridge Mathematical Journal 1841) 13.lntroduction to an essayon theamountand di8tribution ofthemultiplicity ofthe rootsofan algebraic equation (Philosophical Magazine 1841) 14.Anewandmoregeneraltheoryofmultiple roots (Philosophical Magazine 1841) 15.On a linear method ofeliminating between double,treble,andothersystemsof algebraic equations. (Philosophical Magazine 1841) 16.Menwiron thedialyticmethodofelimina­ tion,PartL. (Philosophical Magazine 1842) 17.Elementary researches intheanalysisof combinatorial aggregation (Philosophical Magazine 1844) 18.Ontheexistenceofabsolute criteria forde­ termining therootsofnumerical equations (Philosophical Magazine 1844) 19.Anaccountofa discovery inthe theory of numbers relative to theequation Ax'+B]/+Ow=Dxyz (Philosophical Magazine 1847) 20.Ontheequation in numbers Ax'+B]/+Ow=Dxyz, andits associate systemofequations (Philosophical Magazine 1847) Onthegeneralsolution, in certain cases,of theequation x'+'!I+w=Mxyz (Philosophical Magazine 1847) I J I,vii PAGES 59, 60 61-65 66-68 69-74 75-85 86-90 91-102 103-106 107-109 1l0-1l3 114-118 viii CONTENTS. PAGES 119-13722.Ontheintersections, contacts,andothercor- relations oftwoconicsexpressed by indeterminate coordinates (Cambridge and DublinMathematical Jonmal 1850) 23.Aninstantaneou» demonstration ofPascals theorem by the method ofindeterminate coordinates 138 (Philosophical Magazine 1800) 24.Ona newclassoftheorems inelimination betweenquadratic functions (Philosophical Magazine 1850) 25.Additions to the articles'Onanewclassof theorems,' and'OnPascalstheorem,' (Philosophical Magazine 1850) 26.Onthe solution ofa81Jstemofequations in whichthreehomoqeneous quadratic func­ tionsofthreeunknoum quantities are relpectively equatedto numerical multiples ofafourthnon-homogeneous functionof the same . (Philosophical Magazine 1850) 27.Onaporismatic propertyoftwoconicshaving withoneanothera contact ofthethird order (Philosophical Magazine 1850) 28.Ontherotationofarigidbody abouta fixed point (Philosophical Magazine 1850) 29.Ontheintersections oftwoconics (Cambridge andDublinMathematical Journal1851) 30.Oncertaingeneralproperties ofhomoqeneoue functions (Cambridge andDublinMathematical Journal 1851) 31.Replyto Professor Boole's observations on a theorem contained inlast November numberofthisJournal. (Cambridge and DublinMathematical Journal1851)139-144 145-151 152-154 155, 156 157-161 162-164 165-180 181-183 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.CONTENTS. Sketchofamemoironelimination, trans­ formation andcanonical forms (Cambri~e andDublinMathematical Journal 1861) Onthe general theoryofassociated alge­ braiealforms. (Cambridge andDublinMathematical Journal 1851) Anessayoncanonical forms,supplement to a sketch ofamemoironelimination, transformation andcanonical forms (George Bell, FleetStreet,1851) Explanation ofthecoincidence ofa theorem givenbyMrSylvester intheDecember numherofthisJournal withone stated byProfessor DonkinintheJunenumber ofthe same (Philosophical Magazine 1851) Anenumeration ofthe contacts oflinesand surfaceeofthe second order . (Philoeophical Magazine 1851) Onthe relation betweentheminordeter­ minantsoflinearlyequivalent quadratic functions (Philosophical Magazine 1851) Note on quadratic functions andhyper­ determinants (Philosophical Magazine 1851) Onacertainfundamental theoremofde­ terminants (Philosophical Magazine 1851) Extensions ofthedialyticmethodofelimina­ tion (Philosophioal Magazine 1851) Ona remarkable discovery in the theory of canonicalforms andofhyperdeterminants (Philosophical Magazine 1851) Ontheprinciples ofthecaleulu«offorms. (Cambridge and Dublin Mathematical Journal1852) Ontheprinciples ofthecalculu«offorms. (Cambridge and Dublin Mathematical Journal 1852)ix PAGES 184-197 198-202 203-216 217,218 219-240 241-250 [Seep,647below.] 251 252-255 256-264 265-283 284-327 328-363 x CONTENTS. 44.Surunepr()jJriete nouvelle dereq'uation qui sert adeterminer leainegaliUs seculaires desplanete« (Nouvelles Annales deMa~hema~iques 1862) 45. On a remarkable theorem inthe theory of equal roots andmultiple points (philosophical Magazine 1862) 46.Observations on anewtheoryofmultiplicity (Philosophical Magazine 1862) 47.Ademonstration ofthe theorem thatevery homogeneous quadratic polynomial is reducible by realorthogonal substitutions to thefarmofasumofpositive and negative squares (Philosophical Magazine 1862) 48.OnStaudt's theorems concerning the contents ofpolygons andpolyhedrons, witha note on a new andresem1Jling classof theorems . (Philosophical Magazine 1862) 49. On a simplegeometricalproblem, iU'ltstrating a conjectured principle inthetheoryof geometrical method (Philosophical Magazine 1852) 50. On the expression ofthequotients which appearintheapplication ofSturm's methodto thediscovery ofthe real roots ofanequation (HullBritishAssociation Report1853) 51. On a theorem concerning the combination ofdeterminants (Cambridge and DublinMailiematical Journal 1853) 52. Note on the calculu« offorms (Cambridge andDublinMathematical Journal 1863) 53. On the relation between the volume ofa tetrahedron andtheproductofthe six­ teenalgebraical valuesofitssuperficies (Cambridge andDublinMathematical Journal 1853)PAGES 364-366 367-369 370-377 378-381 382-391 392-395 396-398 399-401 402, 403 404-410 CONTENTS. xi PAGES 54.On thecalculusof'forms,otherwise thetheory ofinvariants. 411-422 (Cambridge andDublinMathematical Journal 181)8) 55.Theoreme surleelimitesdesracine«reelle« desequations algebriques 423 (NouvelleB AnnaleB deMath~tiqueB 1868) 56.Nouvelle methode pourtrouoer 'I,melimite superieure et une limiteinjerieure des racinesreelleed'uneequation alglbrique queleonque 424-428 (NouvelleB Annalea deMathematiquea 1868) 57.On a theory ofthe81Jzygetic relationsoftwo rational integral functions, comprising anapplwation to the theory ofSturm's functions, andthatofthegreatestalge- braical common measure 429-586 (Philosophical TranBactionB oftheRoyalSocietyofLondon 1868) 58.On theconditions necessary andsujJicient to besatiJIfiedinorderthatafunction of anynumberofvariables maybelinearly equivalent to afunction ofanyless numberofvariables 587-594 (Philosophical Magazine 181)8) 59.OnNrGayley's impromptu demonstration ofthe rulefordetermining atsightthe degreeofanysymmetrical function of therootsojanequation expressed in termsofthecoefficients . 595-598 (Philosophical Magazine 1858) 60.Aproofthatalltheinvariants to acubic temaruformarerational functions of Aronhold'« invariants andofa cognate theoremforbiquadratic binaryforms.599-608 (Philosophical Magazine 181)8) 61.Ona remarkable modification ofSturm's theorem 609-619 (Philosophical Magazine 181)8) xii CONTENTS. 62.Noteon aremarkable modification ofSturm's theorem, andon anewruleforfinding superior andinferior limitsto theroots ofanequation (Philosophical Magazine 1858) 63. On the new ruleforfindingsuperior and inferior limitstotherealrootsofany algebraical equation (Philosophic&l Magazine 1858) 64. Note on the newruleoflimits (Philosophical Magazine 1868) 65. The algebraical theoryofthesecUlarin­ equality determinantive equation gene­ ralised (Philosophical Magazine 1858) 66.On,theexplicitvaluesofSturm'squotients (Philosophical Magazine 1858) 67. On a fundamental ruleinthealgorithm of continued fractions. (Philosophical Magazine 1858) 68.Onageneralisation oftheLagrangian theoremofinterpolation .(Philosophical Magazine 1858) NOTEONSYLVESTER'S THEOREMS ONDETERMINANTS INTHISVOLUME.PAGES 620-626 627-629 630-633 634-636 637-640 641-644 645, 646 647-650 1. ANALYTICAL DEVELOPMENT OF FRESNEL'S OPTICAL THEORY OF CRYSTALS. [Philosophical Magazine, XI.(1837),pp.461-469, 537-541; XII.(1838),pp.73-83,341-345.] THE following is,I believe, thefirst successful attempt toobtainthe fulldevelopment ofFresnel's Theory of Crystals bydirectgeometrical methods. Hitherto littlehasbeen done beyond finding and investigating theproperties of the wave surface, asubjectcertainly curious and interesting, butnotof chief importance forordinary practical purposes. MrKelland, in a most valuable contribution to theCambridge Philosophical Transactions", hasincidentally obtained thedifference of thesquaresofthevelocities of a planefront in termsoftheangles made by itwiththeoptic axes. I have obtained each of the velocities separately, andinaform precisely thesame forbiaxalasforuniaxalcrystals. I have also assigned in mylastproposition the place of the linesof vibration intermsofthelikequantities, andthatinashaperemarkably convenient fordetermining theplaneofpolarization when the rayisgiven. Foratfirstsightthereappearsto be some ambiguity inselecting whichof thetwolines ofvibration isto be chosen when thefrontisknown.IfPbe theperpendicular fromthecentreofthesurface of elasticity let fall upon thefront, lj"~theanglesmade by the front with the optic planes, e1,ellthe anglesbetweenitsdueline ofvibration andtheoptic axes, I have shown that cosel=/(bt -p2•s~~'t),J(b'-p2sin")Va2-~Sill~cose2=al-~•sint1' 80thatalldoubtiscompletely removed. The equation preparatory to obtaining the wave surface is found in Prop. 6 by common algebra, without anyuseoftheproperties of maxima and minima, and various othercurious relations arediscussed. Without the most careful attention to preserve pure symmetry, the expressions could never have been reduced to theirpresentsimpleforms. 8.•BeeLond.andEdinb.Phil.Mag.Vol.:II:.p, 886. 1 2 Analytical Development of [1 ANALYTICAL REDUCTION OFFRESNEL'S OPTICAL THEORY OFCRYSTALS. IndexofContents. InProposition 1,aplane front withinacrystalbeing given, thetwo lines ofvibration areinvestigated. InProposition 2 it is shown thattheproductof the cosines of theinclina­ tions of one of theaxes ofelasticity tothetwo lines of vibration, is to the same for eitherotheraxis ofelasticity inaconstant ratiofor the same crystal; and the two lines of vibration are proved to be perpendicular toeachother. InProposition 3,aline ofvibration being given, thefront to which it belongs is determined; anditis proved thatthereis only one such, and consequently any line of vibration hasbutoneotherlineconjugate to it. InProposition 4, certainrelations areinstituted between the positions of, and velocities due to, conjugate lines. InProposition 5, the angles made by the front with theplanes of elasticity are found in terms of thevelocities only. InProposition 6, the above is reversed. InProposition 7, the position of theplanes in which the two velocities are equal (viz. the.optic planes) is determined. InProposition 8, the position of a front in respect to theoptic axes is expressed in termsofthevelocities. InProposition 9, the problem is reversed, and itis shown thatifVI'Vt bethetwo normal velocities with which any front can move perpendicular to itself, and ~,~theangles which itmakes with the optic planes, then VII=at(sin~t~y+c2(cos£1;"-ly, Vt2=a2(sin£1;£2)'+et(cos£1;"Y. Inthe10ththe angle made by a line of vibration with the axes or elasticity is expressed in termsofthetwo velocities of the front to which it belongs. Inthe11thProposition the velocity due to any line of vibration is ex­ pressed in termsof the angles which it makes with the optic axes, viz. 'Ifl-b2=(a"-c")cos101coses- Inthe12thProposition f},e.areseparately expressed in termsof£1'". Inthe Appendix I have given the polar or ratherradio-angular equation to the wave surface, from which thecelebrated proposition of the rayflowsas animmediate consequence. 1J Fresnel:sOpticalTheoryofOrystals. 3 PROPOSITION 1. If l»+my+nz=° (a) betheequation to a given front,todetermine thelinesofvibration therein. (1)nmY Zx TItisclearthatifai,y,zbeanypointin one of theselines,theforce actingon aparticleplacedtherewhen resolved intotheplanemusttendto thecentre. Consequently theline of force ate,y,z'mustmeettheperpen­ diculardrawnuponthefront from theorigin. Now theequation tothis perpendicular is (2)andtheforcesactingatai,y,zarea'lx,bty,CSzparalleltoe,y,s,sothatthe equation totheline of force is X-xY-yz:» a'lx=biy-=C"i" From(2)weobtain Henceb'lyX-a2xY=(b2-as)xy c2zY-bt.llZ=(e2- b'l)yz a2xZ-&zX=(a'l-cS)zx.(3) (4) (5) (1Jt-a')xyn+(c'l-bt)yzl+(as-&)sam. =b'lg(nX-lZ)+c'z(lY-mX)+aSx(mZ-nY); but byequations (1) lZ-nX=0,mX-lY=0,nY-mZ=° therefore n l m .(bt-as)-+(&-bt)-+(a'l-c2)-=0.z a: y(b) Also we have nz+le+my=0 (a) therefore (b'-as)n'l+(c2-bt)l"+nl(c2-b2)~+(b2-as)~)=(a2-&)mS or Z)'l1 z(c'-bt)!-+-{(c2-b2)l'l+W-a2)n'l_(a 2-cr)m'l)-+(b2-al)=O.\xnl a; And inlikemannerinterchanging b,y.m.with c, Z,n (b'-c')('!L)S+~{(b2_&)l"+(c2_as)111,2_(a'l-b2)n'l1'!l+(c'-as)=O..eml x 1-2 4 Analytical Development of Henceif('!h,~)(~,~)bethetwo systems of valuesof'!1.,~,thena;a;leta:2 a:a:[1 (y='!h~=~)(Y=~X~=~)Xa;'Xa:JXlet'_, arethetwo lines of vibration required. PROPOSITION 2. Bylastproposition itappearsthat '!Nh&_a2 .x;~=b2-&(0) a.nd (d) therefore therefore a;a:,+YJ!lt+ZJZt=O. Andtherefore thetwo lines of vibration areperpendicular to eachother. N.B.Equations (0)and(d)mustnot be overlooked. PROPOSITION 3. .Alineofvibration isgiven(thatis'!b,:iaregiUfm)andthe position of a:Ja:l thefrontistobedetermined. Letlo:+my+nz=0 bethefrontrequired, thenle,+myI+nZI=0, and l m n(b2-02)-+(&-a2)-+(a2-b2)-=O. a:l YI Zl Eliminating nweget l(a2-b2)~-(1J2-C2)~)+m(at-b2)'!b-(02-a2)~)=0 Zl a:l Zl YI therefore a;(a2-b2)Ylt-(0'-at)Zitm="ih(bt-&)Zit-(at-b2)a:1' a;at(a:12+yJt+Zit)-(ata:l:l+btYl2+o'Z12) =~b2(a:J2+Yl2+Z12)-(ata;t+b'y/+~z~i) . 1] FretmdsOpticalTheoryofOrystals. 5 Ifnow we make a;,.1+'!Il+$II=l ala;,.I+bl'!/JI+&811=VII andtherefore andin likemanner therefore (al-VII)a;,.X+ (bl-vl)'!II'!I+(&-VII)ZI8=0 istheequation required. PROPOSITION 4. !..•!havingeachonly one value. shows thatonly one front correspondsmn tothegiven line of vibration. LetXI''!II,ZI'VIcorrespond to a;,.,'!IlJZlJtilfor theconjugate line of vibration, thentheequation tothefront may be expressed likewise by (al-VII)x.x+(lJ2-VII)'!Is'!!+(&-VII)ZsZ=0, 80that (al-VII)a;,.(lJ2-VII)'!II(&-VII)ZI (al-vs')Xs=(bl-vs')'!II=(el-vl)$1. PROPOSITION 5. =(at-t!}I) (al-VII)a;,.x.+(lJ2 -VII)(bl-VII)'!IJ'!/I+ (&-VII)(&-VII)ZIZI' Now. by Proposition 2,TofindCIJ.4>,V'theangle8madebythefrootwiththeplanesofelasticit'!l interm8oft!}.VI' Bythelastproposition (al-VIIYa;,.1 (COSllf)l= (al_t!}I)la;,.1 +(lJ2-vl)'y~1+(&-VIIYZII (al-VII)(al-vl)a;,.:xs 6 therefore (COB00'"Analytical Development of [1 =(-;-a--,--2--VI""'""2)--;-(a--=-.---v.=2)--;-(c-=-2----.-:b.cc-)~+(b· -vl')(b'-vl)(a2-(2)+(&-VI')(&-v.2)(1J2-a') (a'-v12)(a2-vt)(c2-b2) =a4(&-b')+b4(a2-(2)+c4(a2-b') (a2-vl)(a2-vt) =(al-1J2)(a2-(2). Similarly, (b'-VI')(b2-VII) (COB4»2=(bS_a2)(bl_&), (&-VI')(&-vl) (cos'ir'"=(&_a2)(&_bS). PROPOSITION 6. Bythelastproposition (cos00)" a2• 1 a"-Vl2=(a'-b")(a2-c')-V••(a'-b")(a2-&) (coscP)2 b' •1- -v-----b'-VI'-(b'-a')(b2-&)s>(b2-a2)(b2-c') (cosy)' c2• 1 a'-Vl2=(c'-b2)(&-a2)-V2.(&.=.a2)(Cs-b") therefore Justinthesameway (cos Cc>)2+0~scP)2+~os'ir)2=0 a2-vlb2-vlc'-v.2' 80thatVI"V2"arethetworootsoftheequation (cos00)2(coscP)"(cos'ir)"_0 22+b' •.•+&v"-' a-v -'u- - COR.Hencetheequation tothewavesurfacemaybeobtained bymaking (cos00)x+(COB4»Y+(cos'ir)z=V, 1J Freenete OpticalTheoryofOrystals. 7 or, if we plesse", /(a'u'-I)(a' -v') /(bIU'J.-1)(b'J.-v') V(a''':''bi)(a'J.-c'J.j·x+V (b'J.-a'J.)(lJi-c'J.)·Y /(cIu'J.-1)(e'-vi) +V(c'J.-a'J.)(e'_b').Z=1. PROPOSITION 7. By Prop. 4, a;(v12-a')YI(VI'-~) ZI(VI2- C ') ~(v.'J.-a')=Y.(lJl~~) =z,(v.'-e')' Hencewhen VI=v.we have, generally speaking,(0) Now~=~=~. Xsy,Z. XlX'J.+YlY'J.+ZlZ.=0; therefore Xl'+YI'+Zl'J.would=0, which is absurd. Theonlycasetherefore when VIcan=1.12is when one of those termsof .0be 0 h hXIZI0 dequation()comes0-:tussuppose VI=b,thenwe ave - =-=-0' an X'J.Z. 1.~XlYlwe can no ongerlnler-=-. X2Y2 Letnow(CcJl,~,"'I)(CcJ., 4>""..)be the two systemsofvalueswhich CcJ,4>, yassumewhen VI=1.12=b,thenapplying theequation of Prop. 5 we have Ja' - b'cosCcJl=-2--'a-c cos4>1=0 Jb2-C2 COS"'I=a2-c'J./a'_b'J. cosCcJ.=Va'J._e' cos4>,=0 sothatbmustcorrespond tothemean axis. I"Seebelow,p.27. En.] 8 Analytical Development of PROPOSITION 8.[1 '-1,~beingtheanglesmadeby thefrontwiththeopticplanes,tofind '-1,'-Iintermsof~,VI' Byanalytical geometry cos£1=cosCo)•cos~+cos4>.cosc/>J+cosV.cosVI _J(VII-a2)(v 21-al)la'-hi -(a2- bl)(a'-&)'Va'-cl +J(VII-&)(VII-&)1&-b' (&-al)(&-bl).V&-al _-v'{(VII-al)(VII-a')}+-v'{(VII-&)(V21-&)} - al- & andsimilarly cos12=cosCo)•cosCo)I+cos4>.cos~+cos,y.cosVI -v'{(v1'-at)(VII-al)}--v'{(VII-cl)(VII-&)} - al- & PROPOSITION 9. Bythelastproposition (VII-al)(vll-al)-(VII-as)(VII-as) cos'-1.cos£...J= (a2_as)' _(a4-04)-(al-&)(~I+VII) - (a 2-(;2)'--.-. _ (a2+as)-(VII+VII) -(al- &) therefore Again, (sin"..)2.(sin~)I=1-(C08 '-1)2-(cos ~)'+(cos£1)2(cos'-1)1 =1 _ 2(VII-al)(vl-a')+(VII-as)(vl-&) . (al-&)1 +(al+&)'-2(al+&)(V]I+V12)+(VI'+vl)2 (at_&)2 1] Fresnel8 OpticalTheoryofCry8tals. 9 therefore but therefore I(.~+~)I....('I+11)1 =aBill.-+,,-COB--2 2 Thusforuniaxalcrystalswhere ~1+~=1800 oo Fig.1.COR.Hence we may reduce the discovery of thetwo fronts into which aplanefrontisrefracted onenter- ingacrystaltothefollowing trigo­ nometrical problem. Letaspherebe described about anypointintheline in which the airfrontintersects theplaneof in­ cidence. LetthegreatcirclePII denotethelatterplane,IFthe former, 0..4..,OCalsogreatcircles, theplanesof single velocity. Sup­ poseIGHtobeone of the refracted frontsintersecting 0..4..,OCinGandH,then (al+c')-(al-c')cos(G+H)(sinPIF)' 2(vel. in air)' =(sinP1GB)'· Thedoublesign will give riseto two positions of therefracted frontIGH. Thepropositions whichfollowareperhapsmore curious thanimmediately useful, 10 Analytical Development of PROPOSITION 10.[1 Todetermine theportionofalineofvibration 'I.ntermsofthe two velocities ofitscorresponding front. We have heretodetermine thequantities 'l!2,~(of Prop. 1) in termsof II;II; VI>VI>or onputting XI~+YI~+Zl~=1,Xl'YI,Zlareto be found in termsofVI'v~. By Prop. 3 and by Prop. 5 l~:rn~:n~::(~-~)(a'-VI~)(a~-VI~) :(~-a')(b~-VII)(b~-V~~) :(a~-~)(&-Vl~)(c~-f)l~); therefore Let«,{J,'Ybetheanglesmadebythegivenline ofvibration withthe elasticaxes,then XI~ (COS«\2=-c:--~---: J II;~+:'h~+Zl2 dividedby (bl-cl)(al-vl)(~-VI')(&-VII)+(c~-a')(b~-VI~)(~- VI~)(a~-VI~) +(a~-~)(c~-VI~)(a~-VI')(b~-VI") andtherefore (bl-~)(al-VI~)(b~-vJ~)(c~-VII) (VI"-V,")(al-b2)(b"-0')(c'-a") (whereitis to be observed thatthereduction ofthedenominator is simply theeffect of a vastheapoftermsdisappearing undertheinfluence of contactwiththemagiccircuit(a"-~),(b~-cl) ,(~- a~),asimplerinstance of which wasseen inProposition 5). In factthecoefficient of v4.~ =(~-cl)+(cl-a~)+(a~-~) =0 1] FreenetsOpticalTheoryojOrystals. =(C2+lJ2).(c'-1>') +(a2+c').(a2-c') +(1J2+a2).(b2-a2) =(c4-/)4)+(a·-c4)+(b·-a·) =0.11 Theterminwhichneither VInorV,enters =a2b2c'{(1J2-c2)+(c2_a2)+(a2_b2)} =0. Thecoefficient of - VI2=a2.(b·-c4)+b2.(c4-a·)+dl•(a·-b.) andthatof Vol=1>'c2.(c'-b2)+c"a2.(a2-02)+a21>'.(bl-a2) eachofwhich Hence and2VI2-b2(a'-V,2)(c'_VI2)(cosa)=-- VI2-vl.(a2-b2)(a2-c'), inlikemanner(cos{3)'=&c. v12-b2(c'-V22)(a2-VI2) (cos'Y)'=Vl--V,2' (c'_b2)(c'_a2). PROPOSITION 11. el,ellbeingtheanglesbetweenany tine ofvibration and theopticaxes, requiTed thevelocity dueto that line in termsof1:11€2' Byanalytical geometry, cos€I=cosa.cosc/>t+cos'Y.cos,yl cos1:2=cosa.cosc/>t-cos'Y.cos,yl therefore cos1:1,cos€2=(cosa)"(cosc/>t)'-(cos'Y)'(cos'Ih)2 _vl-1>'{(a2-v2").(c2-V12) -(c'-v:).Ca2-VI2) } - VI2-v,'. (a2-c')' VI2-b2(a2-c')(vl-V12) =VI2-V 2'"---(of---c'J)"- 1J2-vl'J =a2-c" Hence VI2=b'-(a2-c'J)cos€Icos1:2J andinlikemanner, fortheconjugate line ofvibration V'J2=1>'-(a'J-&)cos1:/cos1:2" 12 Analytwal Development oj PROPOSITION 12. (cosEI'f+(cosEI)'=2(cosa.)l.(cos~'f+2(cos'Y)'.(cos'1h'f =2vll-lr{(al - VII).(cl-VII)+(c1-VII).(al-VII)} VII-vl (al-c1'f[1 but by Prop. 9 therefore multiplied by 2(al-c1'f(cos~)2 (sin'I-?)I+(cosT)I(sin~~P)2 (a2-c2)' b2-VI(Ic1)'I . {(sin'1)'+(sin~)"}a-SID~.sm" andwe have seen that lr-VIIcosEIcosEI=al_cl thereforeJ(lr-VII)sin~+sin~cosEI+cos1:1=-2--2' ./(•.)a-cvsm'I' sln~ therefore and in like manner cosE/=/{~-=_V:1.s~n~}Va2-cISID~ cosEI'=/{b:-V:1 •~n,,}Va-a sms, where VI>VIforthesakeofneatness areleftU'MtbfW'88Sed intermsof~,II' 1] Fremels OptkalTheoryofOrystals. 13 Thisis thesimplest form by which theposition ofthelines ofvibration canbe denoted. CoR.Fromthelastproposition itappearsthat cos1:1sin£1 --=-.-~cos1:2sm~ Fig. 2.pHencewe mayconstruct geometrically forthetwo planes of polarization. LetI, Kbetheprojections ofF the two optic axesonasphere,E theprojection ofthenormaltothe front,Ptheprojection of one line ofvibration; then cosPK sinKE cosPI=sinIE. DrawFEGthecircle of which Pisthepole,meetingPK,PIpro­ ducedinGandF. ThencosPK=sinKG, and COBPI=sinIF, therefore sinKG sinKE sinFI...sinIE therefore sinKGsinIF sinKE=sinIE therefore sinKEG=sinIEF therefore KEG=IEF or180o-IEF.ButPEF=PEG, therefore EPbisects eithertheangleIEKor thesupplement to it. Thesetwo positions of EPgive the two planes of polarization. The construction is thesameasthatgivenin Mr Airy's tracts,andoriginally proposed, I believe,by Mr MacCullagh. 14 .Analytical Development of ADDENDUM.[1 Ifintheequation of Prop. 6, viz. (cosCI)I+~os<p)1+(cost)'=0 al-vtb2-v'e2-'If . 1 1 1 1wechangea,b,C,'Vinto-,-b'-,-,andconsider vto bethelengtha e'V of alinedrawnperpendicular totheplane cosCI)•t£+cos<p.y+cosV.z=0, theequation totheextremity thereofmustbe aIr'(cosCI))'b'r'(cosep)'elr(cosV)' a'-r'+b'......r'+&-r' where CI),ep,Vdenotetheanglesbetween theradiusvectorr,andtheaxes ofe,y,Z,sothattheequation may bewritten a2afJb2y'e'z2 al-r'+l)2~~TJ+&-r'=0, whichiathatofthewavesurface. Butwe have seen that 'If=&{COBC1i~)r+a'{Sin('1i'I)r, therefore theequation tothewavesurfacemay bewritten ('1±'I)'(.'1±'")'1 ,cos~ sm-2- T2=c'+a' where'11'"denotetheanglesbetween theradiusvectorvandthetwolines whichwould be theopticaxesifa,b,e werechanged into~,-bi,!sothatac ifebetheinclination ofeithertothemeanaxisofelasticity TheselinesIshallcall by way of distinction theprimeradii.". *Upon the authority ofProfessor AiryIhaveappropriated the term optic axes to the lines normaltothefrontsof single velocity. IJ Fresnels OpticalTheoryofOrystals. 15 COR. 1.Ifrl,r2bethetwovaluesof rcorresponding tothesamevalues of'1.~we have !_~=!{(cos't-~)2_(cos'I+~)I} rllrllr;2 2 2 1{(.'I-~)2(.'t+':I)I} +-Sill- -Sill--a2 2 2 (1 1). .=as-assinII'sin'2' whichprovesthecelebrated problem oftworayshavinga common direction inacrystal. COR. 2. Theintersection of anyconcentric spherewiththewavesurface isfound by makingrconstant. Hence 11±~becomes constant, andthere­ foreT't±r':l=constant. Hencethecurveofintersection isthelocus of points,thesum or difference of whose distances from two poles when measured bythearcs ofgreatcircles is constant; thepolesbeingthepoints inwhichtheprimeradiipiercethesphere. Inthreecasesthesespherico-ellipses orspherico-hyperbolas becomegreat circles: (1)When't±':I=theanglebetween thetwo poles, in which case the curveofintersection isthegreatcircle which comprises thetwo poles. (2)When't-III=0,whenthelocus is agreatcircleperpendicular to theformerandbisecting theanglebetween theopticaxes. (3)When'I+~=180',whenthelocus is a greatcircleperpendicular tothetwo above, andbisecting thesupplemental anglebetween thetwo axes. Various otherproperties may be withthegreatest simplicity deduced fromtheradio-angular equation. Thehurryofthepress leaves me time onlytosubjointhefollowing PROPOSITION. Tofindtheinclination oftheradiusvector to the tangentplane,intermsof tAeangluwhichtheradiu.svectormakeswiththeprimeradii. Let0bethecentreofthewave surface, OA,OBthetwoprimeradii, OPanyradiusvector. LetOP=v,POA='11POB=~,andletthein­ clination oftheplanesPOA,POB=,.,.; 1(sin'I~-'2r (COS~l;_~Y then r al+ r;2, (takingonlythepositivesignforthesakeofbrevity). 16 Analytical Development of [1 qpa=90°,rpb=90°, drawqmperpendicular toph,thenpm=8",.andtherefore pm pm 8", pq=sinpqm=sinapb=sinfI. Inlikemanner 8'1pr=8lnfl..q---r b Fig.8.LetOQ.ORbethetwoadjacent radiivectores, 80assumed that QOA=POA,QOB=POB+8"" ROB=POB.ROA=POA+8~, andletp, q.r,a.bbetheprojections ofP,Q.R.A,Bon asphereof which 0isthecentre,thenit is clear that NowtheangleQPO -1r .POQ -1.!:.:J!!L.=tan.OQ-OP=tan.dr' d",·81 2 also therefore dr(11).- =ir!- - -sm(~+,,),rd" c'al therefore cotQPO=~(~-~t)sin('1+",)sinfl.. Inlikemanner cotRPO=~(~-~)sin(tt+,,)sinfl., therefore Fig.4.0'QPO=RPO. Alsoitisclearthatrpq=apb=JJ..And tofindtheinclination ofOPtoRPQ.we have only to describe asphereof which Pisthe centre,andintersecting PQ.PR,POinfl, R',O'. ThenR'(Y([=fl..and (YQ'=O'R'=cot-1{~(~-1)sin('1+",)sinfl.}. 1J Fresnei's Optical TheoryofOrystals. 17 DrawaNperpendicular toR'([,thenaNmeasures theinclination of theradiusvectortothetangentplane", And therefore therefore andthereforeIYO'N='!. ~2' cos~=tanaN.cotO'Q', cotO'N=cot0'([, c08~2 coto»=!r.(:2-~)sin~.sin('I+",). LetAOBtheanglebetween theoptic axes =2e,thenbymeretrigonometry J(,,--~)(~-"') .,.,. sine+-2 sine---2- sm"2= sin~.sin'" ' therefore thetangent oftheinclination between theradiusvectorand thenormal (1 1). JSin(e+"-;"')sin(e-'I~'2) =!r.I-:iSill('I+~). i. •a,,- Sill'I'sm'" Inlikemannerthetangent oftheinclination between thesameradius vectorandthenormalattheotherpointofthewave-surface piercedby~t J.(~+",).(~+~)_2(11 . sm6+-2-SIn6--2--l(r l)1-I)sm('I-~)' ,. ,aC SIn'I'SIn'2 Wemay, in thesameway, find theinclination ofthetangent plane toeitheroftheprimeradii,andtotheplanewhichcontains themboth, intermsof'Iand"';theformer by aremarkably elegantconstruction; butthefinalexpressions do notpresentthemselves underthesamesimple aspect. Ifwe call 4>theanglebetween theray and thefront, we may still furtherreducebysubstituting forritsvalues in termsof'I,'"and we shallobtain /{.('I+",). ('I+"') } xVsine+2-sme-2.cosec'I.cosec",. • O' is theprojection or the ray and R'O'orthetangent plane.Therefore O'Nbeing perpendicular toIrq'represents theirinclination. ~ 2 18 Analytkal Development of [1 And if '71"1''71"2betheinclinations ofthenormaltothetwoprimeradii, itmaybe shown that cos'71"1=cosr/Jsinl.t+sinr/Jcos'1sin~, cos'71"2=cosr/JsinIII±sinr/Jcos~sin~. CoR.1.Foruniaxal crystals ~=90°and '1+~=1800 ,sothatthe tangent oftheinclination ofnormaltoradiusvector =tr.(~-~) sin2,for onepoint, and =0 fortheother. COR.2.Foreverypointinthecircular sectionwhichpassesthrough thepoles sin ~=0,andfortheothertwocircularsections l.t±l2=0 or 180°. Therefore everypointinthethreecircularsectionsisanapse. CoR.3.Whenanearly=c,;-~is very small;andtherefore thea c normalandradiusvector very nearlycoincide. CoR.4.Referring to fig.4we seethat()'NbisectstheangleR'O'Q'. NowR'O,([0arerespectively perpendicular totheplanespassingthrough 0'andtheopticaxes;andtherefore themeridian planeaswemaytermit, thatis,theplanecontaining boththerayandthenormal, always bisects theangleformedbythetwoplanesdrawnthrough themyandthetwo optic axes. CoR.5.When Andtherefore r/Jassumes theform~,whichindicates thattheextremities ofthefourprimeradiiaresingular points. Inconcluding forthepresentitbehoves me to statethatonestephas beenomitted intheforegoing paper-, viz.theactualperformance ofthe eliminations whichleadtotherectilinear equation tothewave-surface. ButMrArchibald Smith's elegant andbriefMemoir in theCambridge Philosophical Transactionst oflastyearleavesnothingto bedesiredfurther onthathead. [*See below, p. 27. En.] [tVol.VI.AlsoPhil.Mag.April, 1838, p. SM.En.] IJ Fresnel:s Optical Theory ofCrystals. 19 ThatIhavenotexhibited itin its proper place (Prop. 6) arises only from myrespecttotheprinciple ofliterarypropriety. Withthisimportant blank supplied theAnalytical Theory may be pronounced to be complete. Forall errors andimperfections inwhatprecedes my excuse mustbe pressof time and atotalwant of the materials to bederivedfromconsulting worksof reference. SincewritingtheaboveIhavehadanopportunity ofreadingthepaperofour livingLaplaceinserted Il.BpartoftheThirdSupplement to hisSystemofRaysin theTramactwns oftheRoyalIrishAcademy, in which theprincipal foregoing resultsareobtained by aid of amore refined and transcendental analysis. Thenatureofthefoursingular pointsistherediscussed andtheexistence of fourcirclesofplanecontactdemonstrated. Theformermaybeveryea.silyshownthus:when L1isverysmall"2=2e-~costit verynearly,'"denoting theinclination oftheplanein which eisreckoned tothe planeinwhich L,isreckoned. Hence (1 )1 1(1 1) 1(11)- = - -+ - - - - - - cos {2e-I(cos .f,+ I)}r2alc'2a~tfJ 1 'I'- 1(11)1(11)1(11). =--+-- ----cos2e----- slD2e(cos·f'+I)~2a~c'2a'CZ 2aZtfJ 'I'- 1 1=[j2-lracJ{(a'-bZ)W-tfJ)}(costit±1)~, therefore {I(bl)1(bl)1} r=b1+"2(costlt±l) I-a- ;;0-1 11' Taketitconstant andlettheabscissee andordinates bereckoned respectively along andperpendicular totheprimeray. Then ~='tnearly,andr=J(y"+x")=x,x or,ifwechangetheorigintotheotherextremity oftheprimeray, 80thattheequation becomes 2-2 20 Analytieal Development of [1 Henceateachsingular pointthesurfaceistouched byacone,theequation to thegenerating lineofwhichisgivenbytheabove,theextreme anglebetweenit andtheprimeraybeing cot-1 [J{ (1-~)(~-I)}J Whenb=a,'"always=iandtheconereturnsintoaplane. QOQ=OP+d1..P8e,., Let8~=8",thenitillclearthatOQ=OR,and theintersection ofthetwoplanesperpendicula.r to OQ,ORistherefore alineperpendicular tothe planeQOR,and tothelinewhichbisectstheangle QOR.o Fig. 5. InfactifwedrawQT,RTperpendicular toOQ,ORrespectively intheplane QOR,theintersection inquestion passesthroughTandisperpendicular toOT; alsoAgain,letus suppose thattheposition ofanyperpendicular fromthecentre is given, andthatofthecorrespondiug radiusvectorrequired. LetOA,OB-denotewhatwehavetermedtheopticaxes,butwhichitwill be moreagreeable toanalogytotermtheprimeperpendiculars fromcentre,andlet OPbethegivennormal. TakeOQ, OR contiguous perpendiculars fromcentre inplanesPOQ, ROP, perpendicular toPOA,POBrespectively, thentheinclination ofthetwoformerwill be thesameasthatofthetwolatter,andmaybe termed p.- LetLlJ"nowdenotetheanglesPOA,POBrespectively, then QOA=LlJ QOB='".!+~, ROA=~+8LlJROB=". Theraywill befoundbyjoining0withtheintersection ofthreeplanesdrawn atP, Q, R, perpendicular toOP, OQ, OR, respectively. Now from Prop.9itappearsthat OP=J{at(sin~;")"+cf(cos~1;It)t}, usingonlyonesignforthesskeofsimplicity, which we may do by throwing the ambiguity uponthewayinwhich L1or'tismeasured, also L OT=OQ.sec(!ROQ) =OQ tothefirstorderofsmallness . ..OA,onarenotexpressed in the figure. 1] Fresnel:sOpticalTheoryofCrystals. Nowitiseasytosee(just88on p.16)that21 andalso8LtROP=-.-, SlOP. QOP=~sinp.' therefore ROP=QOPandtherefore POTisperpendicular toQOR. Hencetheproblem isreducedtofindingLtheintersection of twolinesTL,PL drawninthesameplanePOT. NowbecauseOTL,OPLareeachrightangles,acirclemay bemadetopass throughL, T, P, O. Hencetheangle PLO=PTO=tan:"OP_xPOT OT-UP and8~OPx-.-cos!p._ta_lOPxPOR.cosip. ~__Ismp.-n -....nd.OP.. - d.OP.. ' --o~ --o~ d'-l. dL.J OL=OP.secPUL. Alsotheposition oftheplanePOLisknown,andtherefore theradiusis completely determined inmagnitude andposition. Itmaybeworthwhilealsotoremarkthattheaboveconstructions enableus toforma.seriesofequations between themagnitude oftheradiusanditsincli­ nationstothetwoprimeperpendiculars. Infact,if we call '11'1771"2thetwoinclinations inquestion cos71"1=cosPOLcos11±sinPOLsin11'sin~, cos71"2=cosPOLcos'-I.+sinPOLsinL.J.sin;, andofcourseifwecalltheanglebetween thetwoprimenormals 2E J.(EI,.+~)•(£'I,.+L.J)sin!!:=SlO+~. S.lO h--2- . 2 sinI,.sme, CoR.1.When 11or'-1.=0,tanPOLassumes theform~which may be interpreted analogously tothemethodused inthereverseproblem, butmay be moreelegantly illustrated by CoR.2.Whichisthatthemeridian planePOT(thatis,theplaneinwhich bothnormalandradiuslie)bisectstheangleformedbyROP,QOP,andtherefore 22 Analytical Development of [1 Fig. 7.M Fig. 6.thatformedbytheplanesdrawnthrough thenormalandthetwoprimenormals towhichthesetwoareperpendicular. Now we havefound(Cor. 4, page 18), thatitalsobisectstheangleformedby thetwoplanespassing through the r radiusandthetwoprimeradii.Hence whentherayis given, we may find bytheeasiestgeometry thenormal andthetangentplane,andmceversd. Thussuppose(N,N/)(R,K)tobe theprojections oftheprimeperpen­ diculars andprimeradiionasphere concentric withthewavesurface. Letn betheprojection ofany givenperpendicular onthesame sphere;joinnN,nN';bisectNnN'bynM,whichwill bethemeridian plane. DrawfromR',R'TVperpendicular tonMandmakeR'T=TV.ProduceRV tomeetMninr,thenRrM=R'rM,and therefore ristheprojection oftheradius. Justinthesameway when risgivenwe may find n. Now suppose n tocometoN,then theposition ofthemeridian planenM becomes indeterminate, andrfromapoint becomes alocus,subjecttothecondition thatR'rN=RrN.FromrdrawrD perpendicular torN. ThenitisclearthatbecauserNbisectsRrR' sinRDsinRrsinRN sinR'D=sinR'r-sinR'N' andtherefore DisafixedpointandNDafixedlength,and cosrND=tanrN.cotND; therefore theprojection ofthelocus of ruponaplanedrawnatNperpendicular tothelinejoiningNwiththecentreaisgivenby theequation p=ON.cotND.cos6, Nbeingtheoriginandtheprojection ofNDtheprimeradius; whichisthe equation toa.circlepassingthroughB,andwhosediameter =ONcotND. Henceattheextremity ofeachprimeperpendicular thetangent planemeets thesurfaceinacirclepasaingthroughthatextremity andwhoseradius=ibcota, abeingtobe found from theequation sin(2E+a)sin(E+e)sma.=sin(E-e)' thatis tan(E+a)==(tanE)Icote. 1] Freenei« OpticalTheoryofOrystals. 23 Justinthesame way itmaybeshownthatthetraceoftheperpendiculars to thetangent planesofthesurfaceatthepointwhereitispiercedbyanyprime radiusuponaplaneperpendicular tothatradiusatitsextremity, is alsoacircle passing through it, andcurvedinanopposite direction fromthecircleofplane contactnearestto it. Hencetheenveloping coneatthesepointsmaybedescribed asbeingperpen­ diculartothecircularcone, formed by drawing linesfromthecentretotheabove described circle;thatiseverygenerating lineoftheone will beperpendicular to thegenerating linewhichitmeetsoftheother. Moregenerally iteasilya.ppears from fig. 6 thatifaseriesofgreatcircles (representing meridian planes)betakenintersecting thegreatcircleNRR'N' ina fixed point,aplaneperpendicular totheradiuspassing through tha.t point,willintersect thecone of raysaswellasthecone ofperpendiculars corre­ sponding tothosemeridian planes,in twocircles. Sothatthereexistanindefinite number ofcircular cones of rayscorresponding tocircular cones of perpendiculars touching eachotherinalinelyingintheplanecontaining theextreme axes,and havingtheircircularsections perpendicular tothatline. Thecuspsareexplained bythecone ofraysdegenerating intoarightline,and t.hecirclesofplanecontactbythecone ofperpendiculars sodegenerating. Furthermore Iobserve inconclusion thatwhenarayisgivenitfollows from thegeneralgeometrical construction abovethattherewillbetwomeridian planes according aswetakeRwithR',or with apoint180·fromR',andconsequently thesetwoplaneswillbeperpendicular toeachother. Andsimilarly whenanormalisgiventherewillbetwomeridian planesper­ pendicular toeachother. Thustheplanespassingthrough anyradiusandthetwonormalsatthepoints whereitpiercesthewavesurface,areperpendicular toeachother,asarealsothe twoplanespassingthroughanynormalanditstwocorresponding radii. Moreover aglanceatfig. 2 will show thatthetwolinesofvibration corresponding to anyfrontlierespectively inthetwomeridian planespassing through theperpendicular tothatfrontor, inotherwords,theintersection of aplanedrawnthrough eitherraybelonging toafrontperpendicular thereunto isalwaysalineofvibration inthatfront. Thishasbeennoticed,Ithink,bySirWilliam Hamilton fortheparticular case ofthesingular points. Astwofrontsbelong to everyray,so tworayspertaintoeveryfront.And fromwhathasbeensaidaboveitappearsthatthetwolinesofvibration inany frontaretheprojections ofitstworaysuponitsownplane. 24 Analyti.eal Development oj NOTE1.[1 Inthepaperabove,itisshownthatthemeridian plane,thatis, theplanecontaining therayandnormal, alwayspassesthrough aline ofvibration inthecorresponding point.Nowtheline of force calledinto actionbyadisplacement intheline ofvibration clearlylies inthisvery plane;fortheresolved partofitlies intheline ofvibration itself. Harmony andanalogy concurinsuggesting thatas two of thesefour linesareperpendicular to each other,soarealsotheothertwo, or in otherwords,thattherayisalwaysperpendicular tothedirection of unresolved force. Thefollowing investigation verifiesthisconjecture. Letx,y,zbethecoordinates of apointtakenatdistance unityfromthe originandin anylineofvibration; thenthecosines of theanglesmadeby theline of force withtheaxesareasu'lx:b'ly:c!zrespectively. Letubetheinclination between thelineofvibration andtheline of force,then u2x.x+b!y.Y+c'lz.z a2x"+b"y!+c2z! cos0)- --------~----=~-:-=-+----=--:-:::--v!(u·x"+b·y2+c·z2)(x"+y2+Zo)}v(a.x"+b4.y'l+~z'l)' Let thenv(u·w+by+~z'l)=P, pt=~(sec0»2. Nowleta,f),roybetheanglesofinclination between thecoordinate planesandthefrontin which theline ofvibration lies,andAsomequantity to bedetermined. I have shown in Prop.3thatif Acosa=(u'l-v')x, thenwill Acosf)=(b2-v2)y, and Acosroy=(&-vi)z; therefore A2=u·x2+b·y2+c.z2-2v'l(u'lx2+b"y'l+&Z2)+~=P2_~. Again, therefore Now1 P'-~1_l(t)'lv-.C-;('---se-c-O)-)! .=-~-V4co0)• 1J Freenets Optical TheoryofOrystals. 25 Andin MrSmith'sinvestigation oftheform ofthewave surface (already alludedto-)bygreatgoodfortuneI findreadyto myhand (cos(Z)2(cosfJ)2(cos'Yy 1 (a2-'II"+(b2-V2)2+(c'-v2)2=v2(T2-v2) ' Tbeingtheradiusvectortothepointwhosetangent planeisparallel to thepointinquestion. Hence 'l! v2p' (cot CI)Y='II(~_'II)=r2_'II=r2_pP Pbeingthelengthoftheperpendicular fromthecentreuponthetangent plane,forp=v. Hence(cot CI)f=the8quareofthecotangent oftheanglebetween radius vectorandnormal. Or,inotherwords, the line of force is as much inclined totheline of vibration astheray is to thenormal. Nowthenormal is perpendicular totheline ofvibration, and all four lineslie in one plane. Therefore the ray is perpendicular totheline of force. Q.E. D. I maybeallowed to conclude thislongpaperwithasummary of some of themostremarkable consequences which I have extricated fromFresnel's hypothesis. (1)Thetwomeridian planescorresponding to anygivenradiusare perpendicular to each otherj-, (2)So arethetwocorresponding toanygivennormal. (3)Everymeridian planebisectstheangle formed by two planesdrawn through theradiusand the two primeradii. (4)Italsobisectstheangleformed by two planesdrawnthrough the normalandthetwoprimenormals. (5)Eachmeridian planecontains one line of vibration andthecorre­ sponding line of force. (6)Theray isperpendicular totheline of force. Alltheseconclusions, exceptthefourth, are, I believe, original. •Seeabove, p. 18. tIhavedefined themeridian planetobethatwhichcontains radiusvectorandnormal belonging tothesamepoint. 26 Analytical Development of [1 (Cl)tThetheoryofexternal andinternal conicalrefraction followsimmediately asaparticular consequence fromthethirdandfourthcombined asalready shown;thesamepropositions alsoenableus to draw atangentplanetoany pointofthewavesurfaceby mere Euclidean geometry. May not some of theseconclusions servetosuggest to physical inquirers thequestion, Has thetheorybeenstartedfromthemostnaturalpointof view? NOTE2.Investigation- ofthe Wal'e Surface. Sincetheappearance ofthepreceding parts,I havesucceeded incom­ pletingtheself-sufficiency of mymethod bydeducing theequation tothe wavesurfacefromtheexpressions given in Prop. 5 for theanglesbetween afrontandtheprincipal planesintermsofitstwo velocities. Ifthese anglesbe00,rp,y,andthetwovelocities VI>v,we found J(all-v/)(a'l_v,') cos00=(a'l_6'i-(a2_c"), /(b'-VIII)(lr-v,') cosrp=V(b'-a')(b"-c"), /(0'-VI')(0'-VII') cosY=V(0'-a"Ho'-lr). Letthetangentplanetothewavesurfacebewritten cos(oJ•oX+cosrp.y+cosy..z=I, VI VI VI then Let(fJ) •Thisinvestigation supplies thestepwhichMrTovey WB.Bdesirous shouldappearinthe MagaeiM. [Phil.Mag.March, 1838, p. 261. En.] tInlieu ofVIwemightwritev.inthedenominator without affecting the result. /{(~-1)(all-VI')}008W_VVI':I:Observe, that--J{"IJi(.~},andso on for the rest."1 (a- )a-c 1] Fresnel:sOpticalTheoryofOrystals. 27 thenequation (ry)becomes A~IV+BTJY+O~=0, (1) andequation (fJ) Aa'Blf0& TIV+~ y+-yz=O, (2) andequation (a)maybewrittenundertwo forms, viz. (ai-V,i)A~IV+(bi-v,')B'TJY+(&-V,i)O'z=1, (3) or (a'_1)~IV+(b' -1)~y+(~-l)Qz=l. (4) Vii ~ VI' 'TJ VI' , From(1) From(2) From(3)and(1) From(2)and(4)Aa'Bb' 0&TIV+---;qy=--(z. A(a'-c'HIV+B(bl-&)TJY=1. A(a'-&)~+B(l;-c')'!L=c'. ~ 'TJ(5) (6) (7) (8) From(5)and(6) (JJ&z2-B'lfy'-A'a'w=AB:xy(a'i+b'~)' (9) From(7)and(8) cS-Bs(If-&)'y'-A'(a'-c')2w=AB:xy(i+~)x(a'-c')(lf-&).(10) From(9)and(10) -AB(a'-bS)(as-c')(bS-c'):xyf=a'&-(a'-&)(b'-&)(}Ic'z' 'TJ -{a'(bS-&)2-If(as-&)(If-&»)B'y' -{al(al-&)'-as(a'-&)(If-&»)A'w=a'c'-c'z'-&y'-al:r;2.(11) From(11),interchanging (a,IV,~)with(b,y,"1)we have ABW-a')(If-&)(a'-&)IVyi=If&-c'z'-c'x"-bSy'.(12) Finally,from(11)and (12) we have {a2cS-(a'-e')w-e'(w+y'+z'»){b'e'-(b'-&)y'-cI(w+y'+Zl») =(a'-cI)(b'-c')wy', thatis(IV'+y'+z·)(a'w+b'y'+&z')-a'(b'+&):r;2 -If(a'+c')y'-&(If+a')Z'+a'b'&=0 theequation required. 2. ONTHEMOTION AND REST OF FLUIDS. [Philosophical Magazine, XIII.(1838),pp.449-453.] M.OSTROGRADSKY'S memoir onthissubjectinserted intheScientific Memoirs seems to have excitedmuchattention, andhasbeen made the occasion of some au notations • by adistinguished writerinthePhilosophical Magazine. MrIvory'srecentpapersinthesameperiodical muststillmore tendtoinvestwitha newinterest all such speculations. Itseems to me desirable therefore topresentthetheoryof fluids in all thesimplicity of whichitissusceptible. Iconsider a fluidasacollection ofparticles subjectto some law of relativeposition otherthanthatofrigidity. Theseparticles bytheirmutual actionsmaintain theconnections ofthesystem. As to thelaw of force between themwe know nothing; butIassumeitis ageneralprinciple ofnature,thatfor each instantoftimethesum oftheinternal actions (reckoned by theproduct of eachparticleintothesquareofthespace due totheinternal forceactingonit)is aminimum. This in fact is Gauss's principle ofleastrestraint. We may if we please splitthisprinciple into twoparts;thatis to say, assumethattheinternal systemof forces is always such as if actingalone would keep thefluid atrest;andthenagainassume thatanyequilibrating systemof forces mustbesubjecttothelaw ofvirtual velocities. I sayassume,becauseitisimpossibleapriorito prove this. Lagrange's so-called demonstration isunworthy of his name, and(albeit sanctioned bythepowerful oral authority of anex-Cambridge Professor) contrary aliketo sense and honesty.Itisbettertherefore atonce to proceed lIpon Gauss's principle. Itmighteasily be shown thatthisis in effecttantamount in all cases to D'Alembert's andLagrange's principles combined. Beforeentering upontheinvestigation I may call attention to onepoint ofgreatanalytical interest, andrelating to the difficult subject ofthe algebraical sign, viz. thatifthedensityof apoint(x,y)in anycircumscribed b d b h.dv. dv h h .space e expresse y t equantity dx+dyso tatt e mass 18 JJd.xdy(:)+.Ifds:dy(~;), [*Phil.Mag.Ma.y,1838,p.385.En.] 2] On the Motion andRestofFluids. 29 J(udy+vdx),, thatisifwepieasethatisnotequivalent to / J(11.:~+v:)ds, (wheresis-forclearness' sakeand to avoid doublelimitstakenanelementof thebounding curve)asatfirstsightitmightappearto be,butis in fact equalto I shalldemonstrate thispointinthenextnumber- oftheMagazine. Itatfirstcausedme some troubleinconducting theannexed inquiry. Ishallalsotakeoccasion atsomeothertimetoreverttoanew species (asIbelieve) ofpartialdifferential equations; thatis to say, where there arefewer of themthanoftheprincipal variables, which may becalled therefore Indeterminate PartialDifferential Equations. Acomplete solution ofone of these appearsinthesubjoined Investigation. Forthesake ofsimplicity Itakeanincompressible fluid. The method isnowisedifferent forafluid ofvaryingdensity. LetAx,li.y,Azbe anydisplacement undergone byaparticleatthe pointe,y,zparalleltotheaxesx,y.zrespectively; itiseasilyshownthat toMtisfythecondition ofinvariability of mass we musthave d::+dt:+~~z=0. (<<) Onerelation between 11.,v.wthevelocities paralleltoe,y.zisobtained immediately byputtinguSt,vSt,w8t,forli.x.li.y,Az,which gives du dv dw _0 dx+dy+dz- , (1) asusual. Again,ifX,Y,Zbetheimpressed forces, and Xl.YI>ZItheinternal forcesactingon anyparticleparalleltotheaxes, we have dudududu Xl+X=dt+dx11.+dy v+dz w, (2) dv dv dv dv YI+Y=dt+dx11.+dy v+dzW, (3) dw dw dw dwZl+Z=dt+dx11.+dy v+dz w, (4) fromthemeregeometry ofthequestion. [" p. 86, below. En.] 30 On the Motion andRestofFluids. [2 Finally,Gauss'sprinciple teachesnsthat JJJdxdydz {XI~XI+YI~YI+ZI~Zll=0. (fJ) Now d(X~!I)+~~-tXI) +d(Z+ZI) dx dy dz =(du.)lI+(c!V)lI+(dw)lI+2{~~dw+dwdu.+~_"!'dV} dxdy dz dz dydo:dzdyde' asappearsfrom the equations (1), (2), (3), (4);and hence d~X,+~~XI+d~Z,=0 dedydz ' thecomplete solutionof which, free from thesign ofintegration, is d,yd4> ~XI=dy-dz' ~Y_dCIJ_~,y 1-dzde ' dcpdCIJ ~ZI=dX-dy' CIJ,cp,,ybeinganythreeindependent functions ofe,y.z. Onsubstituting thesevalues in equation (fJ)weobtain JJJdxdydz{XI~t-YI~~}+IJJdxdydz{YI:- ZI~} +JJJdxdydz{ZI~-XI~!}=O. Thismay beputundertheform JdzJJda:dy{~(,yX I)-d:(,yYI) } +1da:Ifdydz{:z(ClJY ,)-d~(ClJZ,)} +Jdy.lfded»ill:(cpZI)-t(cpx1) } -Jffda:dydz.,y(~~l-a;:l) -fJfda:dydz. CIJ(~~l_ ~~l) -fffda:dydz. cp(~l-dJzl)=o. 2J On theMotionandRestofFluid« 31 Hereitmustberemembered thatw,4>,y.areperfectly independent of each other. Alsothevaluesofthethreefirstwrittenquantities dependuponthe valuesofXl.YltZlatthebounding surface; the values of thethreelast­ writtendependuponthegeneralvalues ofXl'YltZl'Itiscleartherefore thateachsystemofthreeequations and each member ofeachsystemmust beseparately zero. Thethreelatterequations give dXI_dYI=01dydo; dYI -dZ.1=Of dz dy . dZI_dXI=° do;dz Thethreeformerrequirethatfor each sectionofthesurfaceparallelto theplanexy J'"(X,dz+Y,dy)~0,I foreachsectionparalleltoyz. n.l Iw(Yldy+Zldz)=0, , foreachsectionparalleltoza;"j fep(Zldz+Xldo;)=° andtheseequations are to hold good whatever'fr,4>,wmay be. Fromthe equations (ry)wederive X,do;+Yldy+Zldz=dj, (5) fromequations (~)weobtain f=constant for allpointsin anysectionofthebounding surfaceparallel totheplaneofxy, f=constant for allpointsin any section of thebounding surfaceparallel totheplaneofyz, f=constant for allpointsin any section of thebounding surfaceparallel totheplaneofza: Now bydrawing through all thepointsin aplaneparalleltoxy,planes paralleltoyz,we may cover thewholesurface; hencefisconstant all over thesurfacebounding thefluid. •Seeremarkatintroduction. 32 OntheMotionandRestofFluids. [2 -Therefore X1dx+Y1dy+Zldz=0, (6) for allvariations ofdx,dy, dztakenuponthesurface. Theequations (1, 2, 3,4,5, 6) are coincident withthoseobtained bythe usualmethod jwiththisdifference, thatXl'YI,Zlheretaketheplaceof dp dp dp -do;'-dy'-dz . Thusthenwehaveobtained alltheconditions requisite fordetermining themotionof fluids from theuniversal principle ofleastconstraint conjoined withthespecificcharacter ofthesysteminquestion. General Remarks. Inthecaseofequilibrium, thatis inthecasewherenoparticle moves, we have XI+X=O, YI+l=O, ZI+Z=O. HenceXdx+Ydy+Zdz is a complete differential alwaysandzero for thesurface. Theaboveresultshavebeenobtained upontheprinciples ofthediffer­ entialcalculus, andthecontinuity oftheforceshasbeentacitlyassumed. Ifnow we were to suppose forcesoffinitemagnitude (ascompared withthe wholesumactingupontheentiresystem) to beappliedto alayerofsingle particles or to alayerof athickness ofthesameorderofmagnitude asthe distances between theparticles themselves, (whichhasbeentreatedasan infinitesimal) itwouldappearthatourresultswould be no longerapplicable, justinthesamemanner asitwould be erroneous toapplytheprinciple ofvis-viva(forexample) without modification, tothecaseofimpulsive forces, because we had deduced itbythecalculus inthecase of themotion beingcontinuous. Hencetheaboveequations oughtnotstrictlytoapply tothemotionorrestofafluidcontained betweenphysical surfaces; forthe pressure afforded bythesesurfaces, whatever itsactualvalue may be, we knowaprioriiscommensurable withthewholeamount offorceactingon thefluidjbuttheimmediate application ofthispressure (aliasrepulsive force) is confined to thebounding layerof fluidparticles, oratmostextends toadistance bearing alowratiotothedistances between theparticles themselves. Accordingly, tothenon-applicability oftheequations for free fluids to th.e caseoffluidsconfinedattheboundaries, andto anindependent investigation upontheminimum principle forthisclass ofproblems, itis,thatI look for thetrueexplanation ofthephenomena ofcapillary attraction (vulgarly so called). 3. ONTHEMOTION AND REST OF RIGIDBODIES. [Philosophical Magazine, XIV.(1839), pp. 188-190.] INthesubjoined investigation, which,asfarasIknow, is my own, Iapplythesamemethodto rigid asinthepreceding paperIappliedto fluid systems. Lete,y,zbethecoordinates ofanyparticlein a rigid body; :e',y',z'the coordinates ofsome otherparticle,and let ~=:e+~ i=y+~ z'=z+l Call.:1:e,.:1y,tutheincrements whiche,y,zreceiveafterthelapse of a smallinterval oftime;sothattermsin which theyenterin two or more dimensions may beneglected. Then .:1(x')=.:1a:+d.:1a:h+d.:1a:k+d~'Cl+P, d:cdy dz A(./)_AdAyhd.:1ykd.:1ylQU:I-uy+d:c+dy+dz+ , A(')_A_d.:1zhd.:1zkd.:1zlR Uz-I.U+d:c+dy+dz+ , P,Q,Rcontaining binaryandhighercombinations ofh,k, l,which we shall haveno occasion to express. Atthecommencement oftheintervalthesquareddistance ofthetwo particles was(:e'-:e)'+(y'-y)'+(z'-z)';attheend oftheinterval the distance squaredis (:e'-:e+.:1(x')-.:1a:)'+(y'-Y+.:1(y')-Ay)'+(z'-z+.:1(z') -.:1z)', andthesetwo expressions mustbethesame by the conditions ofrigidity whatever h, k,andlmaybe;thatis h"+k'+lJ=(h+dt:h+d~:ek+d:;l+p)I +(k+d.:1yh+d~'!!.k+d.:1yl+Q)" d:cdy dz (d.:1zdAzdtu).+l+d:cIt+d!l k+dzl+R , forallvaluesofh,k,andt. 8. 3 34 On the Motion andRestofRigidBodies. [3 (1) (2) (3) (h) (j) (k)(e)(d) thereforeHencerejecting infinitesimals of the second order andequating to zero separately thecoefficients of h',/(;2,l',andofkl, lh;hk,wehave dAx_ 0 (a) dAydAz_Oda:- • dz+dy- . dAy_ 0 (b) dAz+dAx=0dy- . da:dz. dAz dAxdAy-dz-=O.(0) dy+Ck-=o.(f) Bydifferentiating (d),(e),(f)withrespecttoe,e,yrespectively, and substituting from(a),(b),(0),weobtain d'Ay_ 0d'Az_ 0d'~- 0dz2-,d:J;2-,dyl- . Bydifferentiating thesamewithrespecttoy,z,xrespectively, andpro­ ceedingasbefore, we have d'Az d'Ax0d'Ay_0 dy'-=0,de'=,da:'- • Thus,then,wehave d~=0,d'Ax=0d'Ax=0 u.;.c dy''dZ'' 'i~-0d'Ay0~~y-0dy-,dz'=,da:'- , dAz=0~.~-O d'Az=0dz'da:'-,dy', Ax=.A.+By+Cz, (0) ~=D+&+~ ~ Az=G+Hx+Ky, (q) .A,B,C,D, E, F, being constant foragiveninstantoftime;between which byvirtueoftheequations (d),(e),(f),we havetherelations E+K=~ H+C=~ B+F=Q Ifwecallu, v,wthethreecomponent velocities of theparticles atx,y,Z parallel to thethreeaxes, and Xl'YI•ZIJthethreeinternalforces,it is at once seen thatu, v, w,asalsoAXl,AYI,AZImustbesubjecttothesame equations aslimitAx,Ay,Azjsothat u=a+'YY-f3z, v=b+az-'YX, W=0+f3x-ay, AXI=~+'YlY-f3lz, AYI=bl+alz-'YlX, AZI=01+f3lx-aly· 3] OntheMotionandRestofRigidBodies. AlsoifX,Y,Zbetheimpressed forces,wehave duXI+X=dt' dv YI+Y=dt' . dw ZI+Z= dt '35 (4) (5) (6) (7-12)Andby Gauss's principle, callingmthemassof theparticleate,y,z, AIm(XII+YII+Zl)=O. Henceequating separately to zerothecoefficients of ai'bl,~and of III.f1t,"/1in thequantityIm(XIAX I+YIAYI+ZIAZ 1)wehave ImXI=O ImYI=0 ImZI=0 Im(ZIY-Ylz)=0 Im(Xlz-Zlx)=0 Im(Ylx-XIy)=0 Lastly,wehavetheequations da;(13)It=dt' dy(14)v=dt' ds(15) '":«: Fromthefifteenequations marked(1) to (15), themotionmaybedeter­ minedby assigning theposition of eachparticleatthe end of thetimetin termsofitsthreeinitialcoordinates, itsthreeinitialvelocities, and the initialvaluesoftheninequantities Imte, Imyz, Imy, Imzx, Imz, Imx'!j, Inthecase ofrestK,= -X,YI= -Y,ZI= -Z,andtheequations (7) to(12)inclusively taken,expresstheconditions ofequilibrium. Theequations (0),(p), (q), which have been obtained from conditions purelyg«rmetrical, establish thewell-known butinteresting andnotobvious fact,thatanysmallmotion of arigid body maybeconceived asmadeup ora motion of translation andamotionaboutoneaxis. 3-2 4. ONDEFINITE DOUBLE INTEGRATION, SUPPLEMENTARY TO A FORMER PAPER ONTHEMOTION AND REST OFFLUIDS. [PhiUJsophical Magazine, XIV.(1839), pp. 298-300.] INapaperonFluidswhichappeared intheDecember Number ofthis Magazine, I had occasionto remark, thatthemass of an area havingatthe point(x,y)adensity:+~;could be expressed by thesimple formula lbeing the length,anddsanelement ofthebounding curve:thismay be thoughttorequiresomeexplanation. oLp LM Fig.I.c oI 0'01------4'L.-..-~=+_¥ll--~(1)LetAPBqrepresent anyoval;PpL,QqMany two contiguous ordinates cuttingthe curve in Pp,Qq respectively, AC,BDthe twoextreme tangents parallel toOy,andpthe densityatanypoint(x,y).The ex­ pressionffpdxdywill serve to denote themassof the oval areaAPBq,and the limits may be twice taken,thatis, (i) the two values of ycorresponding toanyone ofx;and (ii) thetwo values of xcorresponding toCandD. Thismethodis in fact tantamount to takingthesum of the columns Pp qQ;butthisis notnecessary, for APBqmay be considered asthealgebraical sum ofthemixtilinear area APQBDC, andthemixtilinear areaBDCApq, or (ifanylineO'C'D'be drawnparalleltoOCLMD) ofAPQBD'C' andBD'G'Apq. Thusthenthemass=Jlk(Jpdy), fpdy beingleftindeterminate, andthe extremity ofxtravelled round from CtoD,andbackagainfromDtoC. 4] OnDefiniteDouble Integration. 37 This will be betterexpressed by transforming the variable, and summing withrespectto some quantity, suchasthearcofthecurve, which contin­ uously increases, or if we please,withrespecttoe,theanglesubtending any pointtakenwithinthe curve. Themassisthen =±J:..de {(Jpdy) ~}; alwaysremembering thatnoconstant need be added to fpdy,andthatthe doubtful sign arises from thechoice of ways in which emay bemeasured roundIftheareabenot included by one line;butby several, asfor example, by acurve and arightline,theaboveintegral, if broken up into asmanypartsasthereare breaches of continuity, will still apply. (2)Letus suppose thatwe have two areasexactlycoinciding with, and overlapping one another; butthedensityoftheoneat(x,y) to be p,and of theotherp', Letthemassof the first betreatedasthesum of columns paralleltoOy, andthatofthesecondasthe sum ofcolumns paralleltoOx. The one will be represented by ±J:"de(Jpdy) ~' theotherwill berepresented by fo"dy±2trde (fpdx)de' andthesum ofthetwo, orthejointmass,by "0 dxfO Idy±.II"de {fpdy} de ±z"de {fpdx}de' Solong as these two operations areperformed separately, thedoubtful signsmay be preserved in each term, because 8need not be travelled round inthesamedirection for thetwosummations; butif we perform the second integration conjointly for thetwo masses, theirsum fo{dx Idy}=..±I!1fde(fpdy)de±?(Jpdx)de ' themarkofinterrogation denoting thatone or the other,butnoteitherofthe signs±must be used, and thequestion is,which? Thiswillbeanswered by takingdifferent points in thebounding line whichmaybecontinuous or not. Now every line returning intoitself, whethercontinuous or not, will naturally divide with respectof any given 38 OnDefiniteDoubleIntegration. [4 system of axes,intoatmostfourparts,or sets of parts;two in which dtxand dybothincrease or bothdecrease, and two in which one increases andthe otherdecreases. p! PI?" /I P3 P, VV<,~ a: Fig.II.TakePuP"p.,P4,any points in thefourquadrants respectively, it will be observed that, AtPIthepcolumnentersad­ ditively,andthep'columnsubtrac­ tively. AtP,both columns areadditive. Atp.thep'column is additive andthepcolumnsubtractive. AtP4both columns entersubtree­ tively. Again, reckoning round in the direction ofthearrows,oy At Pua:andyareboth increasing. AtP":r;isincreasing andydecreasing. Atp.,a:andybothdecrease. AtP4,a:isdecreasing andyincreasing. Thus when fpdyandfp'dtxareaffected with thesamesigns,dtxand dyareof opposite signs;and when fpdy,Jp'dtxare of opposite signs, dtxand dyare of the samesign. Henceitappearsthatthemassofthearea,whosedensityat(a:,y)is p+p',iscapableofbeingrepresented by ±J:so{(Jpdy)::-(fp'dtx):~}. 5. ONANEXTENSION OFSIRJOHNWILSON'S THEOREM TOALLNUMBERS WHATEVER. [Philosophical Magazine, XIII.(1838),p.454.] THEannexed original theoreminnumbers will serve asapendant to theelegant discovery announced bytheever-to-be-lamented and com­ memorated Horner", with his dyingvoice,in your valued pages ]. THEOREM. IfNbe anynumberwhatever and Ph Pt,ps......p. beallthenumbers lessthanNand prime to it, theneither Pl.]Jg.PS p.+1, or else PI.Pt.pap.-I, isamultiple ofN. 6. NOTETOTHEFOREGOING. [Philosophical Magazine, XIV.(1839),pp.47, 48.] IHAVEto apologize for calling" original" (inthelastNumber of the Magazine) thetheorem ofnumbers which I termed" apendant toHorner's theorem." ThisMrIvoryhasdone me thehonourto inform me may be found in Gauss's Disquisitiones ArithmetiCOJ, p. 76.AsHorner's extension ofFermat's theorem suggested thisextension ofSirJohnWilson's to me, soIconcluded thathadthisextension ofWilson's been known to theworld it would naturally havesuggested his toHorner. Noacknowledgment of thiskindhavingbeen made, I took itforgrantedthatthetheorem Igave wasnew.Undoubtedly had Mr Hornerbeen aware of Gauss's theorem he would have mademention of it. Itakethisopportunity ofaddingthatmyacquaintance with Gauss's principle: hasnot been derived from thestudyof his works, butfroma casualstatement ofitin anEnglish work,Dynamics, by MrEarnshaw, of StJohn'sCollege, Cambridge . •Horller'l proofilhighlyvaluableala novelandhighlyingenious form ofreaaoning, but 1mtheorem maybededuoedwithinfinitely moreeaaeandbrevityfromFermat'. thanhelIe8IIl8 tohaftbeenaware of. [tPlril.Mag.Vol.XLp.466.,ED.] [tSee p,28above. ED.] \ 7. ONRATIONAL DERIVATION FROMEQUATIONS OFCOEXIST­ ENCE,THATISTO SAY, A NEWANDEXTENDED THEORY OFELIMINATION-. PARTI. [Philosophical Ma.gazine, xv.(1839),pp.428-435.] ANYnumber of equations existing atthesame time and havingthe samequantities repeated, may betermedequations ofcoexistence: inthe presentpaper weconsider only the caseoftwoalgebraical equations: x'"+Cltxm-1+~x"'~+ +am=0, W'+b1a;n-l+b2Xn~+ + bn=O. The above being"equations of coexistence," xiscalled"therepeating term." Ifwe suppose the equation cox"+C1Xr-l+~.x"-2+ + Cr=0 to becapableof being deduced from thetwo above,and, therefore, necessarily implied by them, this will be called "aParticular Derivative" fromthe equations of coexistence, of the rthdegree,(rbeing supposed less thanm andnt,and the coefficientsbeing rationalfunctions of the coefficientsof the equations of coexistence). There will be an indefinite numberin general of such derivatives, and the form involving arbitrary quantities which includes them allis called "thegeneralderivative of therthdegree." Any"Particular Derivative," in which thetermsare all integral, numerically aswellasliterally speaking, is called an "Integral Derivative." That"Integral Derivative" of any given degree in which the literal partsofthecoefficients areof thelowestpossibledimensionsi, andthe numerical partsaillowastheycanbe made,is called the"PrimeDerivative" [*TheresultsofthisandBOrnefollowing paperswere repeated, with demonstrations, in the paper"On a Theory of the Syzygetic Relations of tworational integralfunctions ccimprising an application tothe Theory of Sturm's Functions, andthatof thegreatestAlgebraical Common Measure," Phil.Tram.RoyalSoc.Vol. cxr.m., PartLpp.407-548,1858. See below Section D. Art.(16)ofthatpaper.ED.] tThisrestriction uponthevalue of ris not_ntially requisite, andisonlyintroduced to keep the attention fixedupon the particular objectsofthisfirstPart. tOfcoursethedimensions of the coefficients in the equations ofcoexistence aretobe understood 1108denotedby the indices subscribed. \ I 7JOnRational Derivation fromEquations ofCoexistence. 41 of that degree. So thatthereisnothing leftambiguous intheprime derivative save thesign. The"Derivative by succession" is thatparticular derivative which is obtained by performing upontheequations of coexistence the process commonly employed for thediscovery of the greatest common measure, and equating thesuccessive remainders to zero. To express the productofthesuma formed by addingeach of one row of quantities to each of another row, we simply write theone row above the other; anotation clearly capable of extension to anynumberof rows, which wouldnot bethecaseif we spoke of differences insteadofsums-. THEOREM 1. Let~,b«,...hm,be the roots of one equation of coexistence, k«,let,...kn, therootsoftheother. The generalderivative oftherthdegree is repre­ sentedby I(SR(~, ~,~...hr){(x-~)(x-~) ...(x-hr)}x{~k~','is;':kJ)=0, SR(h1,~,Its...hr)denoting anysymmetrical rational(integral orfractional) functionofh),Its...i;; {hr+),hr+2...i;} - k),-k2•••-kn beingto beinterpreted asaboveexplained, and~of course including as manytermsasthereare ways of puttingnthingsrandrtogether], A formtantamount totheabove, and which may be substituted forit. isitsanalogue, !(SR(khka...kr){(x-k)(x-let)...(x-kr)}x{~t',~t'::.~hm})=o. Whenr=0thetheorem gives simply {~'i;.,.s;1=0, - k),-ka'" -knJ andiscoincident withthatgiven by Bezout in his Theory of Elimination . • Thewiderviews which I have attained sincewritingthe above, and which will bedeveloped inafuturepaper,lead me torequestthatthisnotation maybeconsidered only astemporary. IIwouldhaftbeen more in aooordance withtheseviewstohave used the two rows todenote prodlldllofdiBerences thanof sums. Butachangenow in the text would beveryapttocause errorsinprinting. tThegeneralderivative may clearly beexpressed also by the sum of any two particular cIeriTati_ a1fected respectively witharbitrary rational coefficients. Theequivalency of an arbitraryjuftetioft totwoarbitrary multiplien isveryremarkable, andanalogoul towhatOOCUR intheselntionofcertaindifterential equations. 42 OnRational Derivation Subsidiary TMorem (A). Ifh,.,~...It".betheroots oftheequation tJ!"+~x'''-i+lZta:"'-l++a",=0,[7 andif thenem+~em-l+~em-s+......+a",-u=0, I h,.r __1_dI(er+i) (h,.-~)(h1-h,)...(hi -It".)-r+1du ' ubeingmadezeroafterdifferentiation. COR.IfR(hi)denoteanyintegral rationalfunction ofhi>then R(h,.) I(h~-~~)(h,.-"::/,;j...(h,.-hm) isalwaysintegral and iszerowhenthedimensions ofR(h,.)fallshortof (m-l). Subsidiary Theorem (B). ISR(hi'~...hT) {h,.,~...b; } -hTH,-hr+s...-It". can beexpressed bythesumofterms,each of which is theproductofseries oftheform R(h,.)~-------------- - -(hi-~)(h,.-hs)...(h,.-It".), itisalwaysintegral, andwhenthedimensions ofthenumerator fallshortof (m-r)ritvanishea ". Subsidiary Theorem (C). Theonlymodesofsatisfying theequation I{f(hi> ~...hr)xSR(h,.,hs...hr)}=0, forallforms of thelatterfactorsshortof(m-r)(n-r)dimensions, areto putf(h,.,hs...hr)=0, or else constant •Iimayberemarked alsoinpasaing,thatanyterminthenumerator whiohoonWns 41111 onepowernotgreaterthanm-lIrmaybeneglected andthrownout ofcaloulation. Moreover, ananalogous propoBitionmay bestatedoffractions inthedenominaton of whioh anynumber ofroWl!arewrittenoneundertheother;seethefirstnote,page41. 7] fromEquatio'fUJ ofCoexistence. THEOREM 2. Byvirtueofthesubsidiary theorem (B), the two equations43 THEOREM 3. And byvirtueofthesubsidiary theorem (0),thetwo above equations arethe"PrimeIntegerDerivatives," andareexactlyidentical with each other. CoR.1.The leading coefficient of the"primederivative" oftherth degreeisalways of (m-r)(n-r) dimensions. CoB.2.IfP,.betheprimederivative oftherthdegree and if (X=O,Y=O)bethetwoequations of coexistence, and)..,.,,.,... the two"prime constituents ofmultiplication" tothesaidderivative, thatis if)..,.and,.,... satisfytheequation )..,.X+,.,.,.Y=P,.,thenthecoefficient of theleading termsin)..,.and in,.,...isof(m-r-1)(n-r-1)dimensions. THEOREM 4. The"PrimeDerivative" of any given degree is anexactfactorof the ..derivative by succession,"of thesamedegree. The quotient resulting from strikingoutthisfactor iscalled"thequotient ofsuccession." THEOREM 5. If~,L1,L.,&c.,betheleadingcoefficients of thederivatives occurring first,second,third,&c., in order aftertheequations of coexistence, and if Ql,QI.~,&c.,represent the first, second, third,"quotients ofsuccession" reckoned in thesameorder,then Ql=I, 1 QI=LIt' LJ4 Q'=L~P LI4 Q4=L14L." 44 and ingeneralOnRational Derivation [7 CoR.Hence, in place of Sturm's auxiliary functions, we may substitute thefunctions derivedfromtheequations ofcoexistence(;x=0,ag=0) according toTheorem 2, dueregardbeinghadto the sign. Scholium. Hitherto ithasbeensupposed thatthevaluesofthecoeffi­ cientsintheequations ofcoexistence areindependent of oneanother,but particular relations may besupposed toexistwhich shall cause theleading termsgivenbyTheorem 2 to vanish, givingrise toanormal orsingular primes,astheymay be called, of the degreerof fewer than(m-r)(n-r) dimensions. Thetheoryof this,thefailingcase(so to say), is highlyinterest­ ing, and I have alreadydiscovered thelaw offormation forthequotients of succession on thesupposition ofanynumberof primes vanishing consecutively; butIforbearto vexthepatience of myreaderfurther,themore so, as I hope soon to be able to presentacomplete memoir, with all thestepshere indicated filled up, and numerous important additions, (theperfectimage of which thisisbutaroughmould), as homage to thelearnedandillustrious society which has latelydone me thehonour of admitting meintoitsranks. Whythishasnotalreadybeen done mustbe excused, by thefact ofthe theoryhavingsuggested itselfabroadin theintervals of sicknees]. Yetthus muchwill Iaddingeneralterms,namely,thatasmanyprimesasvanish consecutively, so many unitsmustbeaddedtotheindex2 oftheaccessions *Thattheappearanee of theindex4 maynotstartle,let myreaderbearin mind thatthere arewhatmaybetermedsecondary derivatives of succession for every degree appearing inthe prooesaof encceesivedivision. tTheprimederivatives mustbecapableofyielding aninternal evidence of the truthof Sturm'stheorem. Inf&Qt,for the case of allthe roots being possible, alittleconsideration will servetoshowthattheleadingtermofeaohprime derivative of the equation{Ix:}=0will consistofaseries of fractions, eaohof which fractions is,numuically speaking, of the .am<!.iyn. ZThereflections whichSturm's memorable theorem hadoriginally excited,were revived by happening tobepresentatasittingof theFrenchInstitute, wherealetterwasreadfromthe Minister of Public Instruction, requesting anopinionupon the expediency of forming tablesof elimination between twoequations ashigh8Sthe 5th or 6th degree containing onerepeating term.Theofferwasrejected, on the groundof theexcessive labourthatwould be required. Ithinkthatthis has been very muchoverrated; andprobably manywillbe of the sameopinion whohavedwelt upon the factthatnonumerical quantity will occur in the resulthigherthan thehighestindexof therepeating term.Woulditnotredound tothehonourofBritishscience thatsomepainstaking ingenious personmouldgirdhimselfto thetask?and would not thisbe aproperobjecttomeetwithencouragement from the Scientific A88ooiationof GreatBritain? 7] fromEquations ofCoexistence. 45 received in thenumerator anddenominator ofthesubsequent quotient; andinthequotient afterthat,it is not thesquareoftheleadingtermof thepenultimate prime,-but theproductofthistermby theleadingterm ofthatanormal prime of thesame degree which hasthelowest dimen­ sious,-that findsitsway into thenumerator. The rest of theformation remaining undisturbed, unlessanduntila new failure have takenplace. NO'fEONSTURM'S THEOREM. When one of the equations of coexistence is thedifferential coefficient withrespectto therepeated termoftheother, the prime derivatives given inTheorem 2which coincide in this casewithSturm'sauxiliary functions reducedtotheirlowestterms,may beexhibited underanintegralaspect. LetSPDintimate thatthesquared productofthedifferences is to be taken of the quantities whichfollowit. Let~indicatethesum of the quantities to which itis prefixed. SIthesum ofthebinaryproducts. S,thesum oftheternaryproducts, and so on LethI'~...hflbetheroots ofany equation. ThenSturm'slastauxiliary function may bereplaced by SPD(h 1,~...~). Thelastbutone may be replaced by !SPDo; s.:hfl-1)IX+ss,o;~...~-1)SPD(~,h'J'"hn-1). Theonepreceding by ISPD(hl>~...h..-t):r;2+IS1(h],~...hn-.Sl)SPD(~,i,...~)IX +181o;~...hn-l)SPD(~,~...1tn-t), andso on. ThusthenSturm's rule for determining theabsolute numberof real rootsinanequation isbasedwholly and solely upon thefollowing ALGEBRAICAL PROPOSITION. Iftherebenquantities, real and imaginary, theimaginary onesentering inpairs,asmanychanges ofsign asthereare intheterms !.8PD(h], ~), ISPD(~,~, ~), ISPD(~, ~hn-1), ISPD(~, ~hn) , 80many in numberare these pairs. 46 OnRational Derivation fromEquations ofCoexistence. [7 Query(I).Istherenoproposition applicable toanynquantities whatever ? Query (2). Is therenofaintlyanalogous proposition applicable tohigher powersthanthesquares? Query (3). Seeingthatin forming thecoefficients in theequation of thesquaresofthedifferences, we passfromnfunctions of therootsto n11,;1andnotnfunctions, of theirsquareddifferences, does not anatural passagetotheformer lie throughnfunctions ofthesquareddifferences? Inotherwords, may not thequantities ISPD(It,.,h,...hn),&c.,serveas naturaland valuable intermediaries between thecoefficients of anequation involving simple quantities and the coefficients of theequation involving the squaresoftheirdifferences ? P.S. In thenextpartItrustto be able to presentthereaders of this Magazine withadirectandsymmetrical methodofeliminating anynumber of unknown quantities between anynumberofequations of any degree, by anewlyinvented process of symbolical multiplication, and the use of com­ poundsymbols of notation. Imustnot omit to statethattheconstituents ofmultiplication x,.and fI-rexplained in Cor. 2 to Theorem 3 are equal to theexpression (kl,k2•••kn-r-l ) . - hI' - h<j... -hmI(o;-kl)(o;-~)...(o;-kn-r-l) k1.k)' (1,~•••n-r-l -len-,....-kn anditsanalogue respectively. 8. O~DERIVATION OFCOEXISTENCE. PARTII.BEING THE THEORY OF SIMULTANEOUS SIMPLE HOMOGENEOUS EQUATIOXS. [Philosophical Magazine. XVI.(18400),pp.37-43.] Art.(1). We shall have constant occasion in thispapertodenote different quantities by the same letteraffected with different subscribed numerical indices. Suchaletteris to betermeda"Base." Everycharacter consisting of abaseandaninferior index, thisindex iscalledanargument ofthebase,namely, thefirst, second, or nth argument, according as1, 2, or in generaln,be thenumbersubscribed. Art. (2). Iuse the symbol PDto denote the productof the differences of thequantities to which it is prefixed (each being to be subtracted from eachthatfollows); thus PD(a, b,c)indicates (b-a)(c-a)(c-b). PD(0,a,b,c)indicates abc(b-aHc-aHc-b). PD(0,a,b,c...l)indicates abc'"lxPD(a, b, c...l). Art. (3). ForwantofA.bettersymbol I use the Greek letter ~to denote thattheproductof factors to which it is prefixed is to be effected aftera certain symbolical manner. This I shall distinguish asthezeta-icproduct. The symbol ~will never be prefixed exceptto factors, each of which is madeupof one or more terms, consisting solely of lineararguments of different bases, thatis,characters bearingindices below butnone above. Iamtherebyenabled to give thisshortrule for zeta-ic multiplication: "Imagine alltheinferior indices to become superior, so thateachargument istransformed into apowerofitsbase;multiply according to therules of ordinary algebra;afterthemultiplication has been donefullyoutdepress allthe indices into theiroriginalposition; theresultisthezeta-icproduct-." •Itu._roelynecessary toaddthatananalogous interpretation maybeextended to any ze1a.iefunction whatever. Thus nal+~)'=c;+~bl +b" a,a. tcos(aJ=1-1.-2 +1.2.3.4'&c. 48 OnDerivation ojCoexistence. [8 Thusforexample ~(ar,b,)isthesameassimplyarb"but~(ar,a,) represents notara,butar+,. So in like manner ~{(aA-bl;)(al-b",)l =aA+1-aAb",-bl;al+-bm+1;, ~{(~-bl)(~-cl)(bl- cl)} =thedepressedproduct of(a-b)(a-c)(b-c) =thedepressed value ofat(b-c)+b·(c-a)+c'(a-b), thatis,=~bl-~Cl+b.c1-b.~+c.al-c.bl• Art.(40).We shall have occasion inthispartto combine the two symbols ~,PD:thuswe shall use ~PD(albl)todenote ~(bl-~), ~PD(fLtblCl)todenote ~{(bl-~)(Cl-al)(c,-bl)l. Art. (5). Forthesake of elegance of dictionI shall in futuresometimes omittoinserttheinferiorindexwhen it is unity;butthereadermust alwaysbearin mindthatit is tobeunderstood thoughnot expressed. I shallthusbeable to speak of thezeta-icproductof such and such bases mentioned by name. Art. (6). We are not yet come to thelimitofthepowers of our notation. Thezeta-icproduct of the sum of arguments will consist of thesum of products ofarguments, eachargument being(asIhavedefined) made up of abaseandaninferiorindex. Now we may imagine each index of every term ofthezeta-icproductafteritisfullyexpanded to beincreased ordiminished byunity,oreachatthesametimeto beincreased ordiminished by 2,or each ingeneralto be increased or diminished byr.I shalldenotethisalteration byaffixingan1·withthepositive or negative sign tothe];Thus nfLt-~)(~-cl)beingequal toa,-~C1+blCt-blall ~-'t'l(~-~)(a,.-Ct)is equal to a,-11,<;+b.<;-b.Ut, ~-l(al- bl)(~-cl)is equal to al-aoco+boco-boUo. Inlikemanner ~.fD(a,b,c)indicating b.~-b.cl+c.bl-c.~+a,Cl-a,bl> ,tzrPD(a,b,c)indicates I shall in generaldenote ~+rPD(a, b, c...l)actually expanded asthe .zeta-icproductofa,b.,c,...1initsrthphase. 8J OnDerivation ofCoexistence. Art. (7). General Properties ofZeta-icProducts ofDifferences.49 Iftherebemadeoneinterchange intheorderofthebases to which tisprefixed, thezeta-icproduct, inwhatever phaseitbetaken,remains unaltered inmagnitude, butchanges itssign. Ifinanyphaseof azeta-icproducttwo ofthebases be made to coincide, theexpansion vanishes. LetJIbe used, agreeably totheordinary notation, todenotethesumof thequantities towhichitisprefixed, J2todenotethesumofthebinary products, fsoftheternaryones,andso on. ThusletJl(a:tblCI)orJl(a,b,c)indicate ay+b,+C1, andJs(a:tblcl)orJ2(a,b,c)indicate a:tbl+a:tCI+blOt, andJs(alb1cl)orJa(a,b,c)indicate ayb1c1, weshallbeablenowtostatethefollowing remarkable proposition connecting theseveralphasesofcertain:thesamezeta-icproducts. Art.(8).Leta,b,e,...l,denoteanynumber ofindependent bases, say (n-1);butletthearguments of each base be periodic, andthenumber of termsineachperiodthesameforeverybase,namelyn,sothat Cr=Cr+n=Cr-n'Cn=Co=C-n, r beinganynumberwhatever. Then ~_IPD(0,a, b, c l) =~lfl(a, b,c l) tpD(0,a, b,c l)}, ~-'J,PD(0,a, b,Cl)=~{f2(a,b,Cl)~PD(O,a, b,Cl)J, ~-rPD(0,a,b,C...l)=~lJr(a,b,C...l)~PD(0,a,b,C•••l)}. Thisproposition admitsof agreatgeneralization ",butwe have now all that isrequisite forenabling us toarriveataproposition exhibiting underone coupd'ceileverycombination andeveryeffect of everycombination thatcan possiblybe made with anynumberofcoexisting equations ofthefirstdegree, containing anynumber ofrepeated, or to use theordinary language of analysts, (variable or)unknown quantities. s.•SeethePostscript tothispaperforonespecimen. 4 50 On Derivation ofCoexistence. [8 Forthesakeofsymmetry Imakeeveryequation homogeneous; sothat toeliminate nrepeated terms,no more thannequations will berequired. Inlikemannertheproblem ofdetermining nquantities fromnequations will be hererepresented bythecase in which we have to detennine the ratiosof(n+1)quantities fromnequations. Art. (9). Statement oftheEquations ofCoexistence. Lettherebeanynumber of bases (a,b,c...i),andasmanyrepeated terms(z,y,Z•..t),andletthenumber ofequations beanywhatever, sayn. Thesystemmay berepresented bythetypeequation arx+bry+CrE+...+irt=0, in which rcantakeup allintegervaluesfrom - 00to+co,Thespecific numberofequations givenwill berepresented bymakingthearguments of eachbaseperiodic, sothat a;=a"n+r,br=b",,+r,c,=C",,+r>...ir=i",,+r, p.beinganyintegerwhatever. Art.(10).Combination ofthegivenEquati01ls.-Leading Theorem. Takef,.q,...kasthearbitrary basesof newandabsolutely independent butperiodic arguments, havingthesameindexofperiodicity (n)asa,b,c...l, andbeinginnumber(n'-I),thatis, one fewer thanthereareunitsinthat index. Thennmber ofd1'jfe1'ing arbitrary coustents thusmanufactured is n(n-I). LetAx+By+Cz+...+Lt=0 bethegeneralprimederivative fromthe givenequations, thenwe maymake A=~PD(O,a,f,gk), B=~PD(0,b,f,gk), C=,PD(O,e,f,gk), L=~PD(OJl.],g ...k). Art.(11).COB.1.Inferences fromtheLeading Theorem. Letthenumber ofequations, or, which is thesamethiug,theindexof periodicity (n),bethesameasthenumber ofrepeated terms(e,y.z...t), thenonerelation existsbetween thecoefficients: thisis found by making the(n-1)new bases coincide with(n-I)outoftheold bases. We get accordingly, astheresultofelimination, ~PD(0,a,b,c'"i)=O. 8J OnDerivation ofCoexistence. 51 Art.(11).Con.2.Letthenumberofequations be one more thanthat ofthegivenbases,therewillthenbe twoequations ofcondition. These arerepresented bypreserving one new arbitrary base,asx.Theresultof elimination beinginthiscase ~PD(0,a,b,c...l,X)=O. Eaample. Theresultofeliminating between alx+bly=0, a,a;+baY=0, UaX+b,y=0, is~PD(0,a,b,X)=0,thatis ~~~-~~~+~~~-~~Ua+~~Ua-~~~=~ fromwhichwe infer, seeingthatAs,~,~areindependent, ba~-blu,=0, b.~-baa.=0, bla.- b.~=0, anytwoof which implythethird. Inlikemanner, ingeneral, ifthenumber ofequations exceedin any mannerthenumber of bases or repeated terms,theruleis tointroduce 80 manynewandarbitrary bases as together withtheold bases shall make up theDumber ofequations, andthenequatethezeta-icproductofthediffer­ encesofzero,theold bases andthenew bases, to nothing. Art.(12). COR.3.Letthenumberofequations beonefewerthanthe number (71)ofbasesorrepeated terms;thenumber ofintroduced bases in thegeneraltheorem is here (71-2).Makethese(71-2)basesequalseverally tothebaseswhich in thetypeequation are affixed to s, u...t,then 0=0, D=O, L=O, and wehaveleftsimply ~PD(0,a,c.d...ki)(JJ+~pD(0,b,c,d...ki)Y=O. Inlikemannerwe may make to vanish all butAand0,andthusget ~PD(0,a,b,d...ki)x+~PD(0,e,b,d...ki)z=0, 4-2 52 andsimilarly HenceOnDerivation ofCoexistence, ,PD(0,a, b...k)IX+,PD(O,b,c...i) t=O. /X ~PD(0,b,cl) Y ~PD(a, 0,c i) z ~PD(a, b,Ol) areseverally as[8 ~PD(a,b,c...0). Thisisthesymbolical representation asaformula oftheremarkable methoddiscovered byCramer, perfected byBezoutanddemonstrated by Laplace forthesolution ofsimultaneous simpleequations. Art.(13).COR.4.Inlikemanner ifthenumber ofrepeated terms betwogreater thanthenumber ofequations, wehavefortherelation between anythreeofthem,takenatpleasure, forinstance, e,y,z, ~PD(0,a,d...i)/X+~PD(O, b, d...i)y+~PD(O,e,d...i)z=O. And in like mannerwe mayproceed, however muchin excess thenumber ofrepeated terms(unknown quantities) isoverthenumber ofequations. Art.(14).Subcorollary toCorollary 3. Iftherebeanynumberofbases(a,b,c...i),andanyothertwo fewer in number(j,9...k) 'PD(a,j,g k)x~PD(b,c i) } +,PD(b,j, gk)x,PD(a, ci) ~.~~~.~~:.~ .~.:::.~~.~..~~~.~~:.~..:.~~...).~0., +~PD(i,j,g...k)x,PD(a, b, c... aformulathatfromitsverynaturesuggests andprovesa wideextension of itself. Inconclusion I feelmyselfboundtostatethatthe.principal substance ofCorollaries (I),(2)and(3) may be found in Garnier's Analyse Aigebrique, inthechapterheaded"Developpement deIll.TheoriedonneeparM.Laplace, &c."ButI am not awareofhavingbeenanticipated eitherinthefertile notation whichservestoexpressthemnor inthegeneraltheorems towhich it hasgivenbirth. P.S. I shallcontent myselfforthepresent withbarelyenunciating atheorem, one of a class destined itseemstotheauthortoplaynosecondary partinthedevelopment of some of themostcuriousandinteresting points ofanalysis. *ThecrOBSis used to denoteordinary algebraical multiplication. 8] OnDerivation ofCoexistence. 53 Lettherebe(11-1)basesa,b,C•••l,andletthearguments of eachbe "recurrents ofthenthorder-,"thatis tosaylet (27T£) (27T£) (27T£) (27T£)a,=4>cosn'b,=,ycosn'C,=Xcosn-,l,=Q)cosn. LetR,.denotethatanysymmetrical function oftherthdegreeisto be taken of thequantities in aparenthesis which come afterit,andlet~ indicate any function whatever. Thenthezeta-icproduct ~{~RT(a,b,c... l)x~p~PD(O, a,b.C...l») isequal to theproductofthenumber R.,{r'27T .27T)(47T_I(.47T)(67T(1'67T)\.C08n-+,.J(- l )slDn,cos-n+'V-l)slDn-'cosn+,.J- )SlDn, ......,(COS2(n:~)7T+,.J(_1) sin 2(n:1)7T)}, multiplied bythezeta-icphase ~j>-1'~PD(0,a,b,C...'l)!! • Iamindebted forthistermtoProfessor De Morgan, whose pupil I mayboastto have been. Iba~ethesanction alsoof hisauthority, andthatofanotherprofound analyst,my colleague MrGraves,for the use of the arbitrary termszeta-ie,zeta-ically. I takethisopportnnity of retracting thesymbolSPDused in my lastpaper, the letterShavingnomsaning except for Englishreaders. Isubstitute for itQDP,whereQrepresents theLatinwordQuadratus. On lOmefutureoccasion Ishallenlarge upon anewmethodofnotation, whereby the language of analysismay berendered much more expressive, depending essentially npon the use of similar figuresinserted withinoneanother, andcontaining numbers orletters,acoording as quantities oroperationa aretobe denoted. Thissystem to be carriedout would requirespeoia.!but very simpleprinting types to be founded for the purpose. Inthenextpartofthispaperaneasyand,ymmetrical modewillbe given of representing any polynomia.! eitherinitsdevelopable or expanded form. 9. AMETHOD OFDETERMINING BYMEREINSPECTION THE DERIVATIVES FROMTWOEQUATIONS OFANYDEGREE. [Philosophical Magazine, XVI.(1840), pp. 132-135.] LETtherebe twoequations, one ofthenth,theotherofthemthdegree ina;iletthecoefficients of thefirstequation bean,Cln-lJan-I'"au,each power of a;havingacoefficient attached to it,Clnbelonging toa;nandallto theconstant term. In likemannerletb""bm-l'"bobethecoefficients of thesecondequation. Ibeginwith .ARuleforabsolutely eliminating a;. Formoutofthe(a)progression of coefficients m lines, andin like mannerout ofthe(b)progression of coefficients form n lines in thefollow­ 109manner: 1.(a)Attach(m-1) zeros all totherightofthetermsinthe (a)progression; nextattach(m-2) zerostotherightandcarryover tothe left;nextattach(m - 3) zeros to therightandcarryover 2tothe left. Proceed in like manner untilallthe(m - 1) zeros are carriedover tothe left and none remainontheright. The m lines thusformed are tobewrittenunderoneanother. 1.(b)Proceedin likemannerto form n lines outofthe(b)progression byscattering (n-1)zerosbetween therightand left. 2.Ifwewritethesenlinesunderthemlineslastobtained, weshall have a solid square(m+n)termsdeepand (m+n)termsbroad. 3.Denotethelines ofthissquarebyarbitrary characters, which write down in verticalorderandpermute in every possible way, butseparate the permutations thatcanbederivedfrom one another byanevennumber of interchanges (effected between contiguCFus terms)from the restjtherewill thusbehalfof onekindandhalfofanother. 9JOnElimination andDerivation bymereInspection. 55 +.Nowarrangethe(m+n)linesaccordingly, soastoobtain i{(m+n)(m+ n-I)...2.I} squares of one kindwhich shall be called positive squares, and an equal numberoftheopposite kindwhich shall be called negative. Drawdiagonals inthesamedirection in allthesquares; multiply the coefficients thatstandin anydiagonal linetogether: takethe sum of the diagonal products ofthepositivesquares, andthesum of thediagonal products of thenegative squares; thedifference between thesetwosums istheprimederivative ofthezerodegree,thatis, is the resultofelimination betweenthetwogivenequations reducedtoitsultimate stateofsimplicity, there will benoirrelevant factors to reject,and notermswhichmutually destroy. Example. Toeliminate between ar+b»+c=0, l:rfJ+rna;+n=0, Iwrite down a,i.c,0, (1) 0,a,i,c, (2) l,m,n,0, (3) 0,t,m, n. (4) Ipermute thefourcharacters (1), (2), (3), (4),distinguishing theminto positiveandnegative jthusIwritetogether Positive Permutations. 3I12312321 4 4 4 2314 4 41 3 2213I 3 1 2 2 31444132 44 4 312321321I I andagain .LVegativePermutations. I123I4 4 421 3 213 231I123444132 I4 4 4!231 1 3 2321 312 I3 1 2321444 I , 56 OnElimination andDerivation [9 Irejectfromthepermutations of each species all those where 1 or 3 appearinthefourth place, and also those where 2 or 4 appearin the first place, for these will be presently seen to give rise todiagonal products which are zero. Thepermutations remaining are Positive effectual permutations. 1I33I1 21 4 3 321I4 4 4 2!2 i Negutive effectual permutations. 31 1 3 1432 4 3 2 1 224 4 I now accordingly form four positive squares, which are a,b,e,0, 0,a,b,c, l,m, n,0, 0,t,m, n,l,m, n,0, a,b.0,0, 0,a,b,c. 0,t,m, n,l,Tn,n,0, 0, l,m, n, a,b,c,0, 0,a,b.c,a,b,e,O. l,m,n,0, 0,t,m,n, 0,a,b,e. Drawing diagonal lines from left toright,andtakingthe sum of the diagonal products, I obtainatnt+lbtn+ltet+amte.Again,thefournegative squares t,m, n,0, a,b.e,0, 0,t,m, n, 0,a,b,e,a,b,c,0, 0,t,'Tn,n, l, m, n, 0, 0,a,b,c,a,b,c,0, l,m,n,O, 0,a,b,c, O.t,m, n,t,m, n,0, 0,a,b,e, a,b,e,0, 0,t,m, n, giveasthesum ofthediagonal products lbmc+alnc+ambn+lacn, thatis, lbmc+ambn+2acln. Thustheresultofeliminating between a:r;I+be+e=0, f,a;2+m:x+n=0, oughttobe, and is atnt+Vet-2acln+lbtn+amte-lbmc-ambn=O. 9J bya Process ofmere Inspection. Ruleforfindingtheprimederivative ofthefirstdegree, which is oftheformAx-B.57 Beginasbefore, only attachone zero less to each progression; we shallthusobtainnotasquare,butan oblong broaderthanitis deep, con­ taining(m+n-2) rows, and (m+n-1)termsin each row:in a word, (m+n-2) rows, and (m+n-1)columns. To findArejectthecolumnattheextreme right,wethusrecover a square a.rrangement (m+n-2)termsbroad and deep. Proceed with thisnewsquareaswiththeformerone;thedifference between the sumsofthepositiveandnegative diagonal products will giveA. To findB,dojustthesamething,withtheexception ofstrikingoffnot thelastcolumn,butthelastbutone. Ruleforfindingtheprimederivative ofanydegree, say the rth,namely, Arxr-Ar_1xr-1+......±Ao· Beginwithaddingzerosasbefore,butthenumber to beaddedtothe (a)progression is(m-1')and tothe(b)progression (n-1'). Therewillthusbe formed an oblong containing (m+n-21')rows, and (m+n-1')termsin each row, and therefore thesamenumberofcolumns. To findanycoefficient asA"strikeoffallthelast(1'+1) columns except thatwhichis (s) places distantfromtheextreme right,andproceedwiththe resulting squaresasbefore. Through the well-known ingenuity andkindlyproferred help ofadis­ tinguished friend, I trustto be able to getamachine made for working Sturm'stheorem, andindeedallproblems ofderivation, afterthemethod hereexpounded; on which subjectI haveagreatdeal more yettosay,than canbeinferred fromthisor mypreceding papers. 10. NOTEONELIMINATION. [Philosophical Magazine, XVII.(1840), pp. 379, 380.] THEobjectofthisbriefnoteis togeneralise Theorem 2in mypaperon Elimination - whichappeared inthelast Decembel' numberofthisMagazine. Thetheorem sogeneralised presents asymmetry which before was wanting. Here,as in so many otherinstances, the whole occupies in thememory a. lessspacethanthepart. To avoid theill-looking andslippery negative symbols,Iwarn my reader thatInow use two rows of quantities written oueovertheother,todenote theproduct ofthetermsresulting fromtakingawayeachquantity inthe underfrom each in theupperrow. Lethr,hi'"hmbetheroots of one equation ofcoexistence, kl,k,...k..oftheother, andlettheprimederivative ofthedegreerberequired. Takeanytwo integers pandq,suchthatp+q=r.Thederivative inquestion may be written f (hrh"Jh,,)(hp+1h,,+2'"h",)} ~( _1.)(_1.)( _k)(_k)k1k2kq•kq+Jkq+t...k.. ~la;10)... a;"7'a;I'••a;q(hI~hp)(klk2kq) • .hp+1h,,+211m.kq+1kq+t k.. N.B.Whatever pandqbetaken,so long only as p+q=r,theabove expression changesnothingbutitssign;which,therefore, upontranscendental grounds,itis easy to see is of one nameoranother, according aspisodd or even. Intheoriginalpaper,Iasserted thistheorem only for thecaseofp=O. orq=O. [*p.~sabove.ED.] 11. O~THERELATION OFSTURM'S AUXILIARY FUNCTIO~S TO THEROOTS OF AN ALGEBRAIC EQUATION. (Plynwuth BritishAssociation Report1841,(PtII.),pp. 23, 24.] THEauthoravailed himself ofthepresent meeting oftheBritish Association tobringunderthemoregeneralnotice of mathematicians his discovery, madeintheyear1839, of the real natureandconstitution ofthe­ auxiliary functions (so-called) whichSturmmakes use of in locatingtheroots of anequation: these are obtained byproceeding withtheleft-hand side of theequation and its first differential coefficient asif it were our objectto obtaintheirgreatest common factor;thesuccessive remainders, withtheir signsalternately changed andpreserved, constitute thefunctions inquestion. Eachoftbesemay beputundertheform of a fraction, thedenominator of which is a perfectsquare,or in fact theproduct ofmany:likewise the numerator contains ahugeheapof factors of a similarform. Thesetherefore, aswellasthedenominator, sincetheycannotinfluence theseriesofsigns,may berejected; andfurthermore we may, if we please, againmakeeveryotherfunction, beginning fromthelastbutone,changeits sign,ifweconsentto usechanges wherever Sturmspeaksofcontinuations ofsign, and vice versd. Thefunctions ofSturm,thusmodified and purgedofirrelevancy, the author,by way of distinction, andstilltoattribute honourwhereitisreally mostdue,proposes to call"Sturm's Determinators"; and he proceeds to lay baretheinternal anatomy oftheseremarkable forms. He uses theGreekletter"~.. toindicatethatthesquaredproductofthe differences of thelettersbefore which itis prefixed is to betaken. Lettheroots oftheequation be called respectively a,b,c,e...1,thedeter­ minators takenintheinverseorderare as follows :- ~«(l,b,c,el). !~(b,c,e l)IX-!a~(b,c, e...1). !~(c,e...1)g;2-!(a+b).~(c,e...1)x+!ab.'(c,e..•1). ••••••• !Ink,l)(x-a)(x-b)(1X-c)(IX-e)'"(IX-h»). 60 Sturm'sAuxiliar!l Functions. [11 Itmay be here remarked, thatthework ofassigning thetotalnumber of real and of imaginary roots falls exclusively uponthecoefficients of the leadingterms,whichtheauthorproposes to call .. Sturm'sSuperiors ":these superiors are only partialsymmetric functions ofthesquareddifferences, but complete symmetric functions of theroots themselues, differing intheformer respectfromthoseother(atfirstsightsimilar-looking) functions ofthe squareddifferences of theroots, in which, from thetimeofWaringdownwards, theconditions ofreality.have beensoughtfor.Itseemsto have escaped observation, thattheseriesoftermsconstituting anyoneofthecoefficients in theequation of thesquaresof the differences (withtheexception ofthe first and last)eachadmitofbeingseparated and classified intovarious subordinate groupsin such a way, thatinsteadof being treatedas asingle symmetric function of theroots,theyoughtto be viewed as aggregates of many.Infact,Sturm's superior No.1isidentical withWaring's coefficient No.1;Sturm'ssuperior No.2is apartofWaring's coefficient No.3;Sturm's superiorNo.3is apartofWaring's coefficient No.6;and so forth till we come to Sturm's finalsuperior, which is again coextensive andidentical with thelastcoefficient in theequation ofthesquaresofthedifferences. The theoryofsymmetric functions of forms which are themselves symmetric functions of simple letters,or even of otherforms, the authorstateshisbelief is here for thefirsttimeshadowed forth, butwould be beside his present objecttoenterfurtherinto. He would conclude by callingattention tothe importance to thegeneralinterests ofalgebraical andarithmetical science thatasearching investigation should be instituted for showing,apriori,how, when a set of quantities is.known to be made up partlyof possible and partly ofpairsof impossible values, symmetrical functions ofthese,one less in numberthanthequantities themselves, may be formed, from thesigns ofthe ratiosof which to unityand to one anothertherespective amounts ofpossible andimpossible quantities mayatonce beinferred: inshort,weoughtnotto restsatisfied, until,from the very formofSturm's Determinators, without caringto know how theyhave been obtained, we are able to pronounce upon theuses to which theymay beapplied. 12. EXAMPLES OFTHEDIALYTIC METHOD OF ELIMINATION ASAPPLIED TOTERNARY SYSTEMS OF EQUATIONS. [Cambridge Mathematical Journal, II.(1841),pp.232-236.] THISmethodis ofuniversal application, andatonceenablesus toreduce anycaseofelimination totheform of a problem, wherethatoperation is to beeffected between quantities linearly involved intheequations which containthem. Asapplied to abinarysystem,fo:=0,epa;=0,themethod furnishes arulebywhichwemayunfailingly arriveatthedeterminant, free from every species of irrelevancy, whether of alinear,factorial,ornumerical kind. TherilleitselfisgiveninthePhilosophical Magazine (London and Edinburgh, Dec.1840).Theprinciple oftherulewill be found correctly statedbyProfessor Richelot, ofKonigsberg, in alatenumber ofCrelle's Journal, atthecommencement of amemoirinLatinbordering onthesame subject(UNotaadEliminationem pertinens "), Myobjectatpresent is tosupplya fewinstances ofitsapplication to ternarysystems ofequations. Ex.1.Toeliminate e,y,z,between thethreehomogeneous equations Ay'-20'xy+Bx"=0, Bz'-2A'yz+Oy'=0, Cw-2B'z.c+Az'=O.(1) (2) (3) Multiply theequations inorderby -z',x",y2,addtogether, anddivide outby2xy;weobtain C'Z2+Cxy-A'xz-B'yz=O. Bysimilarprocesses weobtain A'a;"+Ayz-B'yx-C'e»=0, B'y2+Bzx-O'zy-A'xy=o.(4) (5) (6) '62 Examples inDialytic Elimination. [12 Between thesesix,treated assimpleequations, thesixfunctions of .a;.'I,z,namely,w,y',z~,xy,ae,yz,treatedasindependent of each other,may beeliminated; theresultsmaybe seen, by mereinspection, to come out ABC(ABC-AB'2-BC't-CA't+2A'B'C') =0, orrejecting thespecial(N.B.notirrelevant) factorABC,weobtain ABC-AB't-BC'~-CA't+2A'B'C' =O. Imayremark,thattheequations (1), (2), (3), or (4), (5), (6), expressthe condition of Ax'+Byt+Cz'+2A'yz+2B'zx+2C'xy, bavingafactorXx+p.y+liZ;ageneralsymbolical formula ofwhichI am in possession fordetermining ingeneralthecondition of anypolynomial of .anydegreehavinga factor, furnishes meatoncewitheitherofthetwo .systems indifferently. Theaversion Ifelttorejecteither,led me to employ both,andthuswastheoccasion of theDialytic Principle ofSolution mani­ festingitself. Ex. 2. A.:t.2+ayz+bzx+cxy=0, (1) lIly2+lyz+mzx+nxy=0, (2) RZ2+pyz+qzx+rxy=o. (a) Multiply equation (1) byfly+ryz,equations (2)and(3) byvzandICy respectively, andaddtheproducts together, weobtaintermsofwhichy'z .andyztaretheouly two intowhich a:doesnotenter. Makenowthecoefficients of each of thesezero,andwehave ary+lll+RIC=0, af3+Mil+pIC=O. LetII=a,IC=a,thenry=-(l+R),fl=-(Jf+p). Hence,multiplying asdirected, andthendividing outbyx,weobtain (mll+bry)z~+(rIC+cfJ)~+(bfJ+cry+nll+qIC)yz+Aflxy+Aryxz=0, -orbysubstitution, [ra- c(M+p)}yt+{rna-b(l+R)lzt+[an+aq-b(M+p)-c(l+R)} yz - A(M+p):cy-A(M+p)xz=O. (4) Similarly, bypreparing theequations so as to admitinturnofyandz .as a divisor, we obtain [ma-l(R+b)}z2+[mr-n(A +q)}w+{mc+mp-n(R+b)-l(A+y)}xz - M(R+b)yz-A(A+q) xy=0, (5) .[rm-q(A+n»)x'+ira-p(M+c)}yt+[rl+rb-p(A+n)-q(M+c)}xy -R(A+n)xz-R(M+c)yz=O. (6) 12J Examples inDialytic Elimination. 63 Between thesixequations (1), (2), (3), (4), (5), (6), {Loi,yO,Z2,xy,aiz,yz,may beeliminated; theresultwill be a function ofnineletters[threeoutof each equation (1), (2), (3f)equated to zero. Perhaps thedeterminant may be foundtocontainaspecialfactorofthreeletters jandifso, may be replaced bya.simpler function of sixlettersonly. Ex.3. Toeliminate between thethreegeneralequations Axs+By"+Czi+2Dyz+2Ezx+2Fxy=0, Lz2+My"+Nz"+2Pyz+2Qzx+2Rxy=0, fx+gy+hz=0', Byvirtueofoneofthetwocanonswhichlimittheforms in whichthe letterscanappearcombined inthedeterminant of ageneral systemof equations, we know thatthedeterminant inthiscase(freedofirrelevant factors)oughtto bemadeupineverytermofeightletters(powersbeing counted asrepetitions), namely,(A,B,C,D, E,F)mustenterinbinarycom­ binations, (L,M,N,P,Q,R)thesame,whereasf,g, hmustenterinquaternary combinations. Toobtainthedeterminant, write Axs+By2+Cz"+Dyz+Ezx+Fxy=0, Lx'+ltfy"+Nz"+Pyz +Qzx+Rxy=0, fr+gyx+he» =0, fxy+gy2+hzy =0, fxz+gyz+hz' =O.(1) (2) (3) (4) (5) Wewantoneequation more of threelettersbetweenr,yO,zi,xy,xz,yz. Toobtainthis,write (Ax+Ez+Fy)Xl+(By+Fa+Dz)Yl+(Cz+Dy+Ex)Zl=0, (Lx+Qz+Ry)Xl+(My+Rx+PZ)Yl+(Nz+Py+QX)Zl=0, f~+ gYl+ hs,=0. ForgetthatXl=X,Yl=y,Zl=s,andeliminate Xl>Yl'Zl>weobtain h{(AX+Ez+Fy)(My+Rx+PZ)} -(By+F«+Dz)(L»+Qz+Ry) {(Cz+Dy+Ex)(Lx+Qz+Ry)} +g-(Nz+Py+Qx)(Ax+Ez+Fy) +f{(Nz+Py+Qx)(By+Fx+Dz)}=o. -(Cz+Dy +Ex)(My+Rx+pz) 64 Examples inDialytic Elirninati()n. [12 Thismaybeputundertheform ax2+f3y'+ryzl+a'yz+fJ'zx+ry'xy=0, (6) wherethecoefficients areofthefirstorderinrespecttof,g,h, L, M, N, P,Q,R, A, B, 0, D, E, F;in all of thethirdorder. Between theequations marked from (1) to (6), theprocess oflinear elimination beinggonethrough, weobtainasequated to zero a function of 5+3, or of eightletters,twobelonging tothefirstequation, two tothe second,andfour tothethird;sothatthedeterminant isclearof allfactorial irrelevancy. Ex.4.Toeliminate e,y,zbetween thethreeequations Aw+Byll+OZIl+2A'yz+2B'zx+20'xy= 0, L:r;2+My"+Nz'+2L'yz+2M'z.'r+2N'xy=0, PxlI+Qyll+RZI+2P'yz+2Q'zx+2R'xy=0. Callthesethreeequations U=0,V=0,W=0,respectively. Write xU=O, (1) yU=0,(2) zU=O, (3) xV=O, (4) yV=O, (5) zV=O, (6) a:TV=0, (7) YW=0, (8) zW=0. (9) Wehaveherenineunilateral equations: onemoreiswantedtoenableus toeliminate linearlythetenquantities w,'!I,:fI,xlIy,arz,xyll,tezl,xyz,yllz,yzl. Thistenthmaybe found by eliminating e,y,zbetween thethreeequations x(Ax+B'z+O'y)+y(By+O'x+A.'z)+z(Oz+A'y+B'x)=0, x(Lx+M'z+N'y)+y(My+N'x +L'z)+z(Nz+L'y+M'x)=0, x(Px+Q'z+R'y)+Y (Qy+R'x+P'z)+z(Rz+P'y+Q'x)=0; for, byforgetting therelations between thebracketed andunbracketed letters, weobtain (AB'0')J(My+s»+L'z)(Rz+P'y+Q'x)} x+z+Y1.-(Qy+R'x+P'z)(Nz+ L'y+M'x) +&C.+&c.=0, which may be putundertheform a;t.:l+fJJI+ryz3+oary+=0·. (10) *Wemightdispense witha10thequation, usingthenineabovegiven,todetermine the ratiosof the ten quantities involved to oneanother; andthenbymeansofanysuchrelations as MJxxy3=X2ylXxlyl,orx3xy3=XlyXxyl,&c. obtainadeterminant. Butit is easy to see thatthis would be made up of terms,eachcontaining literalcombinations of the18thorder. Again, we mightuse five out of the nineequations toobtainanewequation freefrom y3, y2z,yzl, Z3;thatis,containing xin every term:whichbeingdivided by x,andmultiplied 12] Examples inDialytic Elimit'wtion. 65 Byeliminating linearlybetween theequations markedfrom(1)to (10), weobtainaszero aquantity ofthetwelfthorderin all,beingofthefourth order in respecttothecoefficients of each of thethreeequations, which is therefore thedeterminant initssimplest form. Ihavepurposely, inthisbriefpaper,avoided discussing anytheoretical question. I may takesomeotheropportunity ofenlarging uponseveral pointswhichhavehitherto beenlittleconsidered inthetheoryofelimination, suchastheCanonsofForm,-the Doctrine ofSpecialFactors,-the Method ofMultipliers asextended to asystemof anyorder,-the Connexion between themethodofMultipliers andtheDialyticProcess,-the IdeaofDerivations and ofPrimeDerivatives extended toultra-binary Systems. Forthepresent I conclude withtheexpression of mybestwishes for thecontinued success of thisvaluable Journal. byIf,orbyz,wouldfurnishalOthequation nolongerlinearlyinvolved in the 9alreadyfound. The~ant, however, found in thisway, would consistof14-arycombinations ofletters. Finally,wemight,insteadof aeystemof tenequations, employ a systemof 15,obtained by multiplying eachofthegiventhreebyany5outofthe6quantities .xi,y",Zl,zy,%z,yz;but the determinant, besidee being nottotallysymmetrical, wouldcontaincombinations of the 15th order. Imaytakethisopportunity ofjustadverting tothe fact, thatthemethodin thetextdoes in faclcontainasolution oftheequation XU+p.V+"W=x"y'z', wherer+'+t'"4,andX,,..," arefunotions oftheseconddegreein regard to%,y,ztobe determined. 8. 5 13. INTRODUCTION TO AN ESSAY ON THEAMOUNT AND DIS­ TRIBUTION OFTHEMULTIPLICITY OFTHEROOTS OF ANALGEBRAIC EQUATION. [Philosophical Magazine, XVIII.(1841), pp. 136-139.] IUSEthewordmultiplicity todenoteanumber, anddistinguish between thetotalandpartialmultiplicities oftheroots of an algebraic equation. Theremay berdifferent rootsrepeated respectively hI'It.....h;times. l'istheindexofdistribution. I~.~...h;arethepartialmultiplicities, andifh=h1+h'ij+...+h,. histhetotalmultiplicity. Thetotalmultiplicity itisclearmay be defined asthedifference between theindex of theequation and the number ofitsrootsdistinguishable from oneanother. InthisIntroduction, I propose merely to consider howexisting methods may beappliedtodetermine theamount anddistribution ofmultiplicity in agivenequation, and conversely, how equations ofcondition canbe formed which shall implyagivendistribution andamount. Letthegreatest common factor between fa;(theargument ofthepro­ posedequation) andelf:becalledj;a;. And in like manner,letthegreatest common factor of IIa;andd{tbe calledIta;and so on, till in theend we come to f;»,whichhasno common factorwithdia;• LetkI,kt•••k;denotethedegreesina;of/a;,J,.a; ...Ira;respectively. Itiseasyto seethat k,-~,partialmultiplicities, are lessthan2,thatis, are each units. k'ij-ks.partialmultiplicities, will be less than3,andtherefore either1 or 2 in value respectively, and so on tillwe come to kr_I-k;which will severally bebetween zero and r-I, and k;-0 of values intermediate between zero and r. 13]OntheMultiplicity ofanAlgelRaic Equation. 67 Hencetherewillbe lei-2lel+leamultiplicities each ofthevalue1, ~-2lea+k," "2, ler_1-2ler•••ofthevalue r- 1, and -ler ofthevalue r. InplaceofIxwith:wemightemployr:with~:fandso on for therest;thevaluesoflei'le,...k;willremainunaffected bythischange; buttheformermethodwould be more expeditious inpractice. Thetotalmultiplicity is, of course, =lei' Suppose nowthatwe propose to ourselves theconverse problem to determine theconditions thatanalgebraic equation may have agiven amountofmultiplicity distributed in agivenmanner. Ifh",~,h,...hrbeused to denotethegivennumber ofpartialmulti­ plicities whicharerespectively ofthevalues1, 2, 3...r,it iseasyto see thatthequantities derivedabove by leI'lei...k;arerespectively equalto h"+2h.+ +rhr, hi+2h,+ + rh..-I' ha+2h.+ + rhr-'l' hr. Now from1fihavinga factor of thedegreek,common withfo:weobtain k,conditions, from ~~havinga factor of thedegreeleicommon withIIxwe obtaink,more,andso on. So thataltogether weobtaininthisway leI+Ie..,+ + lerconditions. Butit mayeasilybe seenthatthetotalmultiplicity beingleI'thenumber ofconditions needneverto exceed leIinnumber, nomatterwhatitsdistri­ butionmaybe.Hence,besidestheenormous labouroftheprocess,andthe extremecomplexity oftheresults,weobtainbythismethodmoreequations byfarthanarenecessary, anditrequires somecautionto know which to reject. In myforthcoming paper(toappearinPhilosophical Magazine ofnext month) I shallshow,bya most simplemeans, how withoutthe use of derivedor othersubsidiary functions, to obtainthesimplest equations ofcondition which correspond to a givendistribution of agivenamountofmultiplicity. Thetotalmultiplicity, saym,beinggiveninasmanywaysasthat number can bebrokenintoparts,somanydifferent systems ofmequations canbe formed differing each from theotherinthedimensions oftheterms. 5-2 68OntheMultiplicity ofan Alqebraic Equation. [13 Thesesystems may bearranged inordersothateach intheseriesshall implyall those thatfollow it, andbeimplied in allthosethatgo before, withouttheconverse beingsatisfied. Thesubjectoftheunreciprocal implication ofsystems ofequations is a verycuriousone, upon which thelimitsassigned to mepreventme from enlarging atpresent. Itis closely connected withapartofthetheoryof elimination, which, as far as I am aware, has eitherbeen overlooked, or has notmetwiththeattention which it deserves; I meanthetheoryofSpecial Factors. Anexample maymakewhat I mean by theseclear. Letabe afunction (ifmyreaderplease)void of ai,whichequivalent to zeroimpliestwogivenequations inxhavinga common root. Letabe rid of all irrelevant factors,thatis,letabethesimplest form ofthedeterminant, whenthecoefficients of thetwoequations areperfectly independent qualities. Now suppose, as is quitepossiblein avarietyofways, thatsuchrelations areinstituted between thecoefficients alludedto asmake asplitupintofactors, so thata=LxMxN=O. Only one of thefactorsL, M, N willsatisfythecondition oftheco­ existence ofthetwogivenequations: theothersare clearly, however, not to be confounded with factors of solution, orirrelevant factors, as theyare termed,butare ofquiteadifferent nature,andenjoyremarkable properties, whichpointto anenlarged theoryofelimination, andconstitute whatI call special or singular factors. Ishallfeel much obligedto any of thereadersofyourwidelycirculated Journal, interested inthesubjectofthispaper, who would do me thehonour ofcommunicating withme upon it, and especially iftheywould(between now and thenextcomingoutoftheMagazine) inform me whether any­ thing,and if so how much, different fromwhatis herestatedhas been done inthematterofdetermining therelations between thecoefficients of an equation corresponding to agivenamountanddistribution ofmultiplicity in its roots. Ioughtto add,thatmymethod enablesme notmerelytodetermine theconditions ofmultiplicity, butalso to decompose theequations con­ taining multiple rootsintoothersfree of multiplicity, thatis, to find, apriori,thevalues of theseveralquantities fxIgx11Xf,x Ir-lxf, (j;X)2'U2X)2'......,(frx)" rX' Moreover, otherdecompositions, notnecessary to beenlarged upon10this place, may be obtained withequalfacility. 14. A NEW AND MORE GENERAL THEORY OFMULTIPLE ROOTS. [Philosophical Magazine, XVIII.(1841), pp. 249-254.] ISHALLbeginwithdeveloping thetheoryof polynomials containing perfectsquarefactors,one or more. First, let us proceed to determine therelations whichmustexistbetween the coefficients of such polynomials, and afterwards show how they may be brokenup intoothersof aninferiordegree. Aparallelogram filled with lettersstanding inonerow isintended to expresstheproduct ofthesquared difference of thequantities contained. Thus(ab)indicates (a-sr.(~~c)isused toindicate (a-b)S(a-C)2(b-c)2. and so forth. Suppose now thattwo oftherootsel>el•••enbelonging totheequation Iz=0 areequalto oneanother, itisclearthat(elles•••en)=0;and more­ overisasymmetric function, and can be calculated intermsofthecoefficients ofjz. Nextlet us suppose thatwe have two couples of equals (as for instance aandb,two oftheroots equal, asalsocanddtwoothers),it is clear, thaton leavinganyone oftheroots out, the(n-1)thatare left will stillcontain oneequality, andtherefore we have (~I~"en)=0,(flt,ea...en)=0...(~,~...en-I)=O. Noneoftheparallelogrammatic functions abovetakensingly,aresymmetric functions ofthecoefficients, buttheirsumis;so also is thesum ofthe productof each intothequantity left out. Nowingeneral,supposethatthepolynomial fzcontainsrperfectsquare factors,80 thatwe have rcouples of equal roots belonging totheequation fiO· . I h ( ) d 11hhn(n-1) (n-r+2)z=,It18 Cear tateT•eTH..·enanateot er 1.2(r_1) functions of which itisthetype are severally zero. Moreover, thesum of 70 Onanewandmore general [14 theseorthesumoftheproducts of each by anysymmetrical function ofthe (r-1)lettersleftoutwill be a symmetrical function ofthecoefficients of thepowers of xinfa:Toexpress nowtheaffirmative- conditions corre­ sponding tothecase oftherebeingrpairsofequalroots,wemightemploy therequations, (~e.:...-en)=0, ~(~~l-" .~n)=0, s(e,'"en)=0, ~(e..,e~~)=O. Butthese,exceptthelast,arenotthesimplestthatcanbeemployed jthat is to say, we canwritedownrothers,thetermsofwhichshallbe oflower dimensions inrespecttotheroots. LetJ,.denotethatanyrational symmetrical function oftheJoIothdegree is to betakenofthequantities whichitprecedes. Thentherequations inquestion areallcontained inthegeneralequation ~{J,.(el'es...e"_I)x(e..~e..+!...en)}=O; JoIobeingtakenfrom 0 up to (r-1) weobtainrequations, which in respect totherootsarerespectively of alldegreesbetween 11(n-:l)-,-,-.(n-r+2)and~(n-1).;,.(~-r+2)+(r-1) 1.2...(r-1) 1.2...(r-1) reckoned inclusively. Nowatthisstageitisimportant toremarkthattheaboverequations, although necessary, are not sufficient; andindeed,nomereaffirmations of equality can besufficient toensuretherebeingrpairsofequalroots. Tomakethismanifest, suppose ,.=2.Theninorderthatanequation 11ULyhave two pairsofequalroots, we musthave bytheaboveformula s(e"es-.:,-~n)=0,~{el(e.z,es-.~e~)}=o. Butifinsteadoftherebeingtwoperfectsquarefactorstherebe one perfectcubefactoruif»,it may be shown by thesamereasoning as above, that the very sametwoequations apply.Infact, it may be showningeneral thatnosuchequations asthosegivenabove can be affirmed inconsequence oftherebeinganamount rofmultiplicity consisting ofunitpartswhich may not be affirmed withequaltruthasnecessary consequences ofthesame •Theimportance of therestriction hintedatbythe use of the word affirmative willappear hereafter. 14] TheoryofMultiple Roots. 71 amonntdistributed inanyothermannerwhatever. How toobtainaffirma­ tiveequations sufficient as well as necessary (undercertainlimitations) will appearattheclose ofthispresentpaper. Itisworthyofbeingremarked, thatifwe make II'denotethe sum of theproducts ofthequantities to which itis prefixed, taken JloandJlotogether, theequations ofaffirmation becomeidentical withthoseobtained byelimin- atingbetweenfa;andt-. Itcanscarcelybedoubted thattheillustrious Lagrange, had he chosen toperfecttheincomplete theoryof equal roots givenintheResolution Num&ique, byapplying to it his own favourite engineofsymmetric func­ tions,could scarcely have failed of stumbling by a back passageuponSturm's memorable theorem. Letusnow proceed to show how a polynomial known to containone or moreperfectsquarefactors may be decomposed. Letusbeginwithsupposing thatitcontains butone such factor;so thatfa;=4Ja;(a;-a)2. I shall show how to obtaintheequations C(a;-a)=0,D4Ja;(a;-a)=0,E(x-a)a=0,F(4Jx)=0, eachin its lowest terms. 1.To form theequation Le+M=0,wherea;=a,itis easy to see that if wewritedown in general theexpression (x-e1)(~e~e~) thiswill becomezerowhenever theroote1leftoutis not one of the equal roots (et): sothatin fact(callingthetwoequalrootse1,e.respectively) ~{(a;-e1)x(e.,.~...en)}=(x-e1)X(~.--e-;;) +(x-e2)X(e1,-ea...en), orsimply =2(a;-a)(ea,ea'"en). Hence by making a;!(~,e3"-'e~)-Ihx(~,ea•••en)}=0, wehave anequation for finding theequal roots elJes- Again, it is easily seen upon thesamehypothesis, that I{(x-e.)(x-ea)(x-e.)...(x-en)X(e.,es...en)} =2(x-e,)(x-es)...(a;-en)X(e"ea...en). •SeemynoteonSturm'sTheorem, Phil. Mag., December. 1839 [po45above.ED.]. 72 On anewandmoregeneral [14 Hence,toformtheequation havingthesamerootsas(x-a)cf>x,wehave onlytomake xn-II (~,es•••en)-a;n-tI {(e2+ea+en)X(es,ea...en)}...... ±I{(~ea...en)X(e2,eaen)}=O. Suppose now ingeneralthatwe have rperfectsquarefactors, so that Ix=cf>x(x-a,')2(x-~)s(x-ar)2. To form theequation C(x-~)(x-Clt)(x-ar)=0, we have only to make I{(x-el)(x-es)...(x-er)X(er+1Jer+2'"en)}=O. And toobtain wemustmake I{(x-er+l)(x-er+2)...(x-en)X(er+l,er+2...en)}=O. Thetheoryofperfectsquarefactors is not yetcomplete untilithasbeen shown how to obtainconstructively cf>x,and,asanalogy suggests, thecom­ plementary partJY(x-~)'(x-a2)'•.•(x-a,,)2,each in itslowestterms. To effect thelatteritmightbe saidthatit is only necessary totakethe squareofC(x-~)(w-~)...(x-ar).Itistruethepolynomial soformed wouldcontaineverypairofequalfactors,butnotinthelowesttermsas regardsthecoefficients (asweshallpresently show). To solve thislastpartoftheproblem, letitbeagreedthattwo rows of lettersinclosed in aparenthesis shallindicate theproduct ofthesquares ofthedifferences got bysubtracting eachintherow from eachintheother, sothat (~)=(a-b)l,(bac)=(a-b)2(a-c'f,(~~)=(a-c)'(a-d'f(b-c'f(b-d)2. Letusbeginwithsupposing thatIxhasonepaironly ofequalroots; to formthesimplest quadratic equation containing thispair,writedown (x-el)(x-es)X(ea,e4...en)Xcea).eSte4...en Now if elandelarethetwoequalrootsinquestion neitherofthe multipliers of(w-el)(x-ea)vanishes. If~and~areneitherofthemequalroots(ea,e4...en)=O. Ifone ofthetwo only belongtothepairofequalrootscea) -0el,e4...en- . 14] TheoryofMultiple Roots. 73 Henceitis clearthat istheequation desired. In likemannerifthereber pairs of equal roots theequation of the (2r)th degree which contains themall maybewritten I{(a;-tlt)(a;-lit)...($-e,.)X(6tr+1...en)X(euelellT ) }=O. 6tr+1en The coefficientof afWinthisequation is clearly of (n-2r)(n-2r-1)+4r(n-2r), that is, of (11+2r-1)(n-2r)dimensions. The coefficient of:J!"intheequa­ tionwhich contains therequal roots unyoked together is of(n-r)(n-r-1) dimensions, and consequently thecoefficient of a;trinthesquare of this equation would beof 2(n-r)(n-r-1)dimensions, thatis, would be n2+6rt-(4r+1)ndimensions higherthanneedful. Finally, to obtainanequation clear ofsimpleas wellasdoubleappear­ ancesof the equal roots, we have only to write thecomplementary form I{(a;-etr+1)(a;-6tr-t\l)...(a;-en)X(6.:r+1+en)x(e1,eo•..e...)}=O.etr+l...en Letus, nowthatwe are more familiarized with thenotation essential to thismethod,reverttothequestion with which we setout, and endeavour to obtainrsuchequations asshall imply wnambiguously theexistence ofrpairs ofequal roots. Theexistence ofrsuch pairs enables us to assertthefollowing disjunc­ tiveproposition, whichcannotbeassertedwhenthesameamountofmulti­ plicityis distributed inanyotherway. To wit, on selecting anyrroots out of theentirenumber, eitherthese r will all be found againin those thatare left, orthosethatare left will containinterse,onerepetition atleast;sothatexceptonthelattersupposi­ tionany(r-1) may be absolutely sunkoutof those thatare left, and there willstill be oneroot common to the (n-2r+1)remaining, and tother originallyselected to be left out. Wherefore calling the roots e1,el•••en,andgiving IJ.anyvaluewhatever, wehave 74 TheoryofMultiple Roots. [14 Hencethesimplest distinctive equations indicative oftheexistence ofr pairsofequalroots are to be found by putting IJ.equal in succession toall values from 0 up to (r-1). Forinstance, if werequirethatanequation oftheseventhdegreeshall havethreepairsofequalroots, we need only to call theseven roots respec­ tively CL,b,C,d,e,j,g,andthenour type equation becomes Fromthis itappearsthattherdistinctive equations forrpairs of equal roots are of different dimensions fromthergeneraloroverlying ones corre­ sponding tothemultiples r,anyhow distributed; thelowest of thelatter beingof(n-r+1)(n-r),thelowest of the former of (n-r)(n-r-1)+2r(n-2r+ I), thatis, ofn (n-1)-3r (n-1)dimensions. Ingeneralwe shall find that themoreunequally distributed themultiplicity may bethelower are the dimensions ofthedistinctive equations, andareaccordingly lowest when the multiplicity isabsolutely undistributed ". •Itmustnot, however, beoverlooked, thattheequations above given, although decisive as tothe existence of r pairsofequal roots whenthemultiplicity is known tobenotgreaterthanr, do notenableus toaffirmwithcertainty theirexistence when this limitation isabsent: for shouldthemultiplicity exceedr,theninevitably (nomatterhow it may be distributed) (e~0T~;~") is always zero. andconsequently nullifies eachterm of every one of theequa­ tions in question. Infact(repugnant as it may appeartobetotheordinary assumptions of analytical reasoning). it is not possible toexpresswithabBolute unambiguity theconditions of therebeingamultiplicity (r)distributed in anyassigned manner bymeansofraffirmative equations alone. 15. ON ALINEAR METHOD OF ELIMINATING BETWEEN DOUBLE, TREBLE, ANDOTHER SYSTEMS OF ALGEBRAIC EQUATIONS. [Philosophical Magazine. XVIII.(1841),pp.425-435.] PARTI.BINARY SYSTEMS. LETUandVbe twointegercomplete homog-eneous functions oftxand y,one ofthemth,theotherofthenthdegree jandletitberequired to expressthecondition ofthecoexistence ofthetwoequations U=0,V=0 bymeansoftheequation C=0,whereCis free from all appearances of zory. Thisequation, according tothesystemofnotation developed in apre­ cedingpaper,andwhich has beensinceadopted andsanctioned bythehigh authority of M.Cauchy, I callthefinalderivative: thequantity Cisdesig­ natedthefinalderivee: anditisourpresentobjectto show how thismay beobtained in aprimeform,thatit>tosay,divested ofirrelevant factors: in thisstateitmustconsistofterms,eachcontaining m+nletters,ofwhich nbelongtothecoefficients ofU,andmtothoseofV. Ofcourseinapplying thisruleitistobeunderstood thateverycombina­ tion ofpowersinUorVhas asingleletterprefixed forits.coefficient, andthatinthefinalderivee powers are represented byrepetitions ofthe samecharacter. EveryterminUorVbeingoftheformCtxPyq,xPyqiscalledanargu­ ment,Citsprefix. Assume twointeger positive numbers randr',andalso two others sand s'isuchthatr+r'=n-l,s+s'=m-l, andform from U=O,V=O twonewequations, x"y"U=0,xly"V=O. Suchequations aretermedtheaugmentatives ofthetwogivenonesrespec­ tively; also x'y"Uanditsfellow are termedtheaugmentees ofUandV. 76 OnalinearMethodofEliminatmg between [15 randr'aretermedtheindicesofaugmentation belonging toU,8and8' thesamebelonging toV. Finally,it will be useful hereafter tocallthegivenpolynomials UandV themselves theproposees, and thegivenequations whichasserttheirnullity, thepropositive equations, or, briefly, thepropositives. Now as many augmentees ofeitherproposee canbe formed asthereare ways of stowing awaybetween two lockers (vacancies admissible) anumber ofthingsequaltotheindexoftheother- jhence we shallhavenaug­ mentees of U,andmofV:thustherewill bem+naugmentatives each of thedegreem+n-I, andthenumber ofarguments isclearlym+nalso, sothattheycan beeliminated linearly, and thefinalderiveethusfound, containing m+nletters(properly aggregated) in each term,will be in itsprimeform,thatis,incapable offurtherreduction, and void of irrelevant factors. Itisworthyofremark,thatthefinalderiveeobtained byarranging in squarebattalion theprefixes of theaugmentees, permuting therows or columns, and readingoffdiagonal products, affected each with theproper sign(according tothewell known rule of Duality), will not only be free from factorial irrelevancy, butalso oflinearredundancy, whichlatterterm I use to signify thereappearance ofthesamecombination of prefixes, some­ timeswithpositive andsometimes withnegative signs:furthermore, it followsobviously from thenatureoftheprocessthatnonumerical quantity inthefinalderiveewill begreaterthanthehigheroftheindicesofthetwo given polynomials. PARTII.TERNARY SYSTEMS. CASEA.Indicesall equal. Method1. Lettherebe now threeproposees, U, V,W,integercomplete homo­ geneous functions ofe,y,s,each ofthedegreen:let r+r'+r'=n-I,8+8'+8"=n-I,t+t'+t"=n-I, a!'yr'Z'.,U,a;'y-'zi'V,a;t!t'zt"W, will, as above, be called theaugmentees ofU, V, W, and every otherpartof thenotation previously described isto bepreserved. • ..TotAugment. utriusvis ell:lIlquationibus propositis formaripossuntquot modi sintinter duoreceptacula (utriviBvel ambobus omnino vacareHeet)rerum,quarum numerus indioem alteriusmquat,distributionem faciendi." 15]double,treble,andotherSystemsofAlgebraw Equations. 77 Suppose now u=o,v=o,w=o, weshall haveasmanyaugmentative equations formed from each proposee asthereareways of stowing awaynthingsbetween threelockers (vacancies admissible).", thatis.nn;1 of each kind;ina.ll,therefore, 3n(1_~2+1),and everyone ofthesewill be of thedegree2n-1, sothatthenumber of arguments to beeliminated isequaltothenumber of ways of stowing away2n-1thingsbetween threelockers(emptyonescounting), thatis 2n(2n+l) 2 Asyet,then,we have notenoughequations foreliminating theselinearly. Make,however, a+,8+'Y=n+1, andwriteu=a:-F+ylJF'+zyF"=0, V=a:-G+'!IG'+zYG"=0, W=a:-H+'!IH'+zyH"= 0, it willalwaysbepossible tomakethemultipliers ofa:-,'!I.zyinteger functions: forifwe look to anyargument inU, V,orW,it is oftheform ~!I'r,andone ofthelettersa,b,cmustbenotlessthanitscorrespondent a,{1,"I,forotherwise a+b+c wouldbenotgreaterthana+fJ+"I-3, thatis,nwouldbenotgreaterthan(n+1)- 3, or n-2,whichisabsurd: if nowanyone, asa,beequalto orgreaterthan«,itmay be madeto supplyanintegerparttothemultiplier ofa:-. Hereitmaybeaskedwhatis to be done withsuchtermsasK3fl'!lz", whentwolettersa,bareeachnotlessthantheircorrespondents «,,8:the answeris,suchtermsmay be madetoenterunderthemultiplier ofa:-, orof31,or tosupplyaparttobothinanyproportion atpleasurej. Fromtheequations aboveweget,bylinearelimination, FG'H"+GH'F"+HFG"-GFH"-HG'F'-FHG"=O. Thismaybedenotedthus:II(a,,8,"I)=0, which equation I call asecondary derivative, andtheleftsideofitasecondary derivee;«,,8,"Imaylikewise betermedtheindicesofderivation (asr, s,t,&c.areofaugmentation). Nowsincea+,8+"I=n+1,itisclearthattheindexofII(a,,8,"I) isalways n+n+n-(n+1);thatis,2n-1. •BeeforLatintranslation thepreceding note. tTheprefullll ofanysuchterms(sayK)maybeconceived as made up of twoparte,an arbitraryconstant, aseand(K-e); ewilldisappear spontaneously from the final derivee. 78Ona linear Metlwd ojEliminating between [15 Ist,Letany two of the indicesofderivation betakenzero,thenitis easily seen thatallthetermsin IT(a,13,"I)vanish, and consequently the secondary derivative equations obtained uponthishypothesis becomemere identities, and are of no use. or or2nd.Letanyoneofthembecome zero. Itismanifest, fromthedoctrine of simple equations, thatIT(a,13,"I)may be made equalto {xu+j'V+vw}~, {X'u+j"V+v'W}~, {x"U+j'"V+vllW}~ xy' upontheunderstanding that X=G'H"-G"H'.j'=H'F"-H"F',v=F'G"-F"G', X'=G"H-GH", j"=H"F-HF", v'=F"G-FG", X"=GH'-G'H, j'"=HF'-H'F,v"=FG'-F'G. Thethreerows of coefficients will be respectively ofthedegrees 0-~+~-~ 0-~+0-~ 0-~+0-~ Thusifanyone oftheindicesa,13,"Ibe zero, IT (a,13,"I)becomes identical withx'U+j"V+v'W,wherethemultipliers ofU, V,Wareof 2n-(a+13+"I)dimensions, thatis of(n-1)dimensions, andmayaccord­ inglybeputundertheform ~Ax"yr'zr"U+~Bw!l ~"V+'i.Cafyt'zt"W, thatis to say, becomes a linearfunction oftheaugmentatives, andtherefore if combined with themintheprocess of linearelimination wouldgiverise totheidentity 0=O. Hencewemustrejectall such secondary derivatives as have zero for one oftheindices of derivation. Butallothers,itmay be shown, will be linearly independent of oneanother, and oftheaugmentees previously found. Hence, besides 3 n (nt1)equations ofaugment ofthedegree2n-1,we shall have ofthesamedegreeso many equations ofderivation asthereare ways of stowingawaybetween threelockers(n+1)things,underthecondition that n (n-1)no locker shall ever be left empty.thatis 2 •. . n-1n(n+1) 2n(2n+l)Thus,then,IIIall we have n-2-+3 2 =2equations, which is exactlyequal to thenumberofarguments to beeliminated. Hence •Videpage76 for the Latinversion. 15Jdouble,treble,andotherSystemsojAlqebraie Equations. 79 thefinalderiveecan beobtained bytheusualexplicitrule ofpermutation, andmoreover will be itslowestform,foritwillcontaiuin each term n(n+1) .-~ prefixesbelongmg totheaugmentatives ofU,and a like number pertaining tothose of Vand ofW,as wellasnn;1belonging tothe secondary derivatives, eachprefix in anyoneof which is triliteral, containing a prefixdrawnoutof thosebelonging to each of theproposees. Th bercontaini n+1n-1hatifh..I us every mem er containing n2-+n2-'t atISn'0t eongma prefixes belonging toU, V, W, singly and respectively, thefinalderivee evolvedbythisprocess will be in its lowest terms;aswasto be proved. CASEA.Indicesall equal. Method 2. Itisremarkable thatwe may vary themethodjustgiven by making r+r'+r"=n-2,s+s'+s"=n-2,t+t'+t"=n-2. Theaugmentatives willthusbe of the degree2n-2. Furthermore, wemustmakea+f3+ry=n+2.Itwillstillbe possible tosatisfybyintegermultipliers theequations U=:If"F+yfJF'+zYF", V=:If"G+'!IG'+zYG", W=:If"H+'!IH'+zYH", [theseitwill be useful in futuretotermtheequations, :If",'!I,zYbeingthe arguments, andF, G, H, &c.thefactorsof decomposition] for otherwise callingtheindices of ta,y, zinanyoriginalargument a,b,e,theirsum01"n wouldbenotgreaterthan(n+2) - 3,thatis(n-1),which is absurd. Forthesame reasons asinthelastcaseno index of augmentation must bemadezero:thedegree of each will be(on-a)+(n-f3)+(n-ry),thatis (2n-2),andtheirnumber(n+21)n;thenumberofaugmentatives will be 3(11;l)nlinearly uninvolved, each of thedegree2n-2,andtherefore ..(2n-1)2ncontammg 2 -arguments. Now(n+1)n 3(n-1)n (2n-1)2n ~+2=2 . 80On alinearMethodofEliminating between [15 Hencethefinalderiveemaybe found, anditwill beinitslowestterms, foreverymember willcontain 3(n;1)nlettersduetotheaugmentative, d3(n+1)nd h . I derivati.. II h h "II an2ue to t e partia erivative equations; 10at en t ere WI be3n'lettersin eachterm. Thissecondmethod beingapplied tothreequadratic equations ofthe mostgeneral form,leadstotheproblem ofeliminating between sixsimple equations whichlieswithinthelimitsofpractical feasibility, anditis my intention toregister thefinalderivee uponthepagesof some one of our scientific Transactions asastanding monument fortheguidance ofhereafter comingexplorers -. SCHOLIUM TOCASEA. or elseIfweattempt tocarryforward theseprocesses toquaternary systems, it becomes necessary tomake IX+,8+'Y+cS=(r-2)n+1 IX+fJ+'Y+cS=(r-2)n+2, whereristhenumberofproposees. Now ifthefactorsintheequations ofdecomposition areallinteger, one oftheindicesofderivation mustbenotgreaterthanthecorresponding indexinanyoftheoriginal arguments, whichmayeasilybeshownto be alwaysimpossible for asystemofequations, complete inalltheirterms, whenever theirnumberrisgreaterthanthree,ifcz+(3+'Y+cS=(r-2)n+2; butifcz+,8+"I+cS=(r-2)n+1onlypossible forthecaseofn=2. PARTICULAR METHOD APPLICABLE TOFOURQUADRATICS. xW=0,xZ=0, yW=O, yZ=O, zW=O, zZ=O, tW=O, tZ=O.zV=O, tV=O,ZU=O, tU=O,LetU=0,V=0,W=0,Z=0, be four quadratic equations existing between e,y,z,t. Make xU=O, xV=O, yU=O, yV=O, •Elimination between tlDOquadratics leadstoa final derivee made up of ,eventermsonly; the final derivee of threequadratics ismadeup ofatleastseveralthouaand; nay,I believe I may safely say, severalmyriadl ofterms! 615Jdouble,treble,andotherSysternJ8ofAlgem-aic Equations. 81 Alsowrite U=wF+yF'+sE"+tF'"=0, V=:cIG+yG'+zG"+tGIII=0, W=:cIH+yH'+zH"+tHIII=0, Z=:cIK+yK'+zK"+tK'"=O. Byeliminating linearlyweget I{FIG'(H"Kill-HiliK")}=0, which will beofthethirddegree,sincethefactorsrepresented bythe unmarked lettersF, G, H, K areof zero,andalltherestofunitdimensions. Similarly we may obtainotherequations, sothatbesidesthesixteen augmentatives already written down, we have four secondary derivatives, namely, ll(2111)=O, ll(1211)=0, ll(1121)=0, ll(1112)=0. Thus we havet'Wlentyequations andasmanyarguments toeliminate, since aperfectcubicfunction of fourletterscontains twentyterms. The final deriveewillcontain16+4.4 letters,thatis 32, 8 or 28belonging toeachsystemoforiginal prefixesineachmember, andwilltherefore be in its lowest tenus:for one of thecanonsof form teaches us,dpriori,that everymember ofthederiveededuced fromanynumber ofassumed equations mustcontainin eachmember asmanyprefixes belonging to oneequation of the system asthereareunitsintheproduct oftheindicesof alltherest takentogether. COROLLARY TO CASE A. Eitherofthetwomethods givenasapplicable tothiscaseenablesus to determine integervaluesofX, Y, Z, whichshallsatisfytheequation XU+YV+ZW=FxPy'lz", whereFisthefinalderiveeandp+q+r=3n-2.Forbythedoctrine of simpleequations we know how to expressFintermsofthelinearfunctions, outof which itisobtained bypermutation, thatis weareabletoassign valuesof.A,B, C,andtheirantitypes,asalso ofLanditsantitype, which shallsatisfytheequation I(..dx"yr'zr"U)+I(Bx',!!z·"V)+I(C:Jfyfzt"W) +I[zn(a,13,'Y)}=Fx1yDzh, (1) where.A,B, C,aswell asLandallthequantities formedafterthem,are madeupofintegercombinations oftheoriginal prefixes. Nowthefunctions II(a,13,'Y)maybeexpressed inthreeways intenusof U,V, W,ashasbeenalreadyshown. s, 82OnalinearMethodofEliminating between [15 We may therefore suppose thesefunctions to bedivided intothree groups,andmake IL11(a,8'Y)=~QU+Q'V+Q"W+I~U+R'Y±R"l!' x" :rfJ SU+S'V+S"H'+~ .(2)x., Anditisevidentthattheequations (1) and (2)leadimmediately tothe equation xu+YV+ZW=FafH!yb+gz"+", if we call a,b,cthegreatest values attributed respectively to a,,8,"/. Now if we supposethefirstmethodto be followed, f+g+h=211-1. Andit will always be possible to make a,b,c ofwhatvaluesweplease subjecttothecondition ofa+b+ c=11-1;for01leatleastoftheindices ofderivation in 11(a,,8,"I)mustbenotgreater thanitscorrespondent amonga.b,c;otherwise a+,8+,,/ would be not less than(a+b+c)+3; but a+,B+'Y=1I+l a+b+c=n-l, which is absurd. Hencewe cansatisfyXU+YV+ZW=FxPyqz',p, q,rbeingsubjectto thecondition ofp+q+r=3n-2,butotherwise arbitrary. Moreover, we cannotdo soifP+q+rbelessthan3n-2, for,that wouldrequirea+b+c tobelessthann-1.Nowiftwo oftheindices ofderivation, asaand,8,bemadeequaltoa+ 1,b+ 1respectively, the third'Y=(n+l)-(a+b+2)=(n-l)-(a+b), and is therefore greater thanc:sothata+,8+"Iforthiscase becomes greaterthana+b+c,and themethodfalls totheground. In fact, I have discovered atheorem whichletsme know this,apriori, a law which serves as astafftoguidemy feet from fallingintoerrorin devising linearmethods ofsolution, andtheimportance of which all candid judgeswho have studiedthegeneral theoryofelimination cannotfail to recognize. To wit, if XhX~,Xa...X"benintegercomplete polynomial functions ofnletters XI'Xa...X",andseverally ofthedegreebl•bl,bl'"b,,; thenitis always possible to satisfytheidentity 15Jdouble,treble,andotherSystemsofAlgebraic Equations. 83 ifIXI+llt+~+...+ex"beequalto orgreaterthanb,+b2+b,+...+b"-n+1, butotherwise not-. Thisagainis founded immediately uponasimpleproposition, ofwhich Ihaveobtained averyinteresting andinstructive demonstration. shortlyto appear,andwhichmay be enumerated thus:"Thenumberofaugmentees ofthesame degree tludcanbeformed, linearly independent ofoneanother, outofanynumberofpolynomial functions ofasmany»ariables, maybe eitherequaltoor lessthan.thenumberofdistinctarguments contained insuch augmentees, butnevergreater. Thelatterwillbe the case when the index oftheaugmentees diminished byunityis less than the sumoftheindicesof theoriginal unaugmented polynomials each sodiminished; theformer, when theaforesaid indexisequaltoorgreaterthantheaforesaid sum." Toreturntotheparticular case offindingX. Y, Z tosatisfy XU+YV+ZW=F:CPy'lff. Thishasbeenalreadydoneaccording tothefirstmethod; ifweemploy thesecondmethodofelimination weshallhave f+9+h=2n-2. But, now sinceIX+fJ+"I=n+2,weshalleasilysee bythesamemethod asabove.thattheleastvalueofa+b+c{wherea,b,cdenoterespectively thegreatest valuesofIX,fJ,"I,appearing inthedenominator ofthefractional farmsusedtoexpressII(a,ta,"I»),willbe onegreaterthanbefore, or n;so thatf+9+h+a+b+cwillstillbeequalto3n- 2,aswemight,dpriori, byvirtueofourrule.have been assured. TERNARY SYSTEMS. CASEB.Twooftheindicesequal;thethirdlessby aunit. LetU=0,V=0.W=0, bethethreegivenequations severally ofthe degreen.n,(n-l). •Henceitisapparent, thatinapplying themethodofmultipliers, 110curiousandimportant distinction existsbetween the CIlo8eSoftherebeing two equations, andtherebeingagreater nnmbertoeliminate from:for inthefirst case the element ofarbitrarinesa needsnevertoappear; inthelatterHcannotpossibly beexcluded fromappearing inthemultipliers. Thi.willexplainhow it comes topassthatthemethodofthetextmaybeemployed togive NrWusolutions of theXU+YV+ZW=FxPyqZT; thusnotonlycanp, qandrbevariously IIIIdeup of(f+a),(g+b). (h+c), but also II(11,(J,'Y)when two of theindices (11,(Jsuppose) are eechnotgreaterthantheassigned greatest valuesa.bmaybemadetofigureindifferently either lIDdertheform 6-2 84On alinearHetlwdofEliminating between [15 Maker+r'+r"=n- 2, 8+8'+s"=n- 2,t+t'+t"=n- 1, bymultiplying Uintox''y''f',Vintoa:'y"z",Wintow'!f'zt",weobtain augmentees each ofthesame,namely,the(2n-2)thdegree. Thenumberoftheseis Again,make(n-I)n (n-l)n n(n+I)-2+--2-+2 . a+.8+'Y=n+l. Itwillstillbe possible, as before, to form equations ofdecomposition in whichx",'!t,zYarethearguments, andaffectedwithintegerfactors. Forif we look to Weven,allitsarguments are oftheformaf"ybzC,where a+b+e=(n-1),andeach ofthesecannotbe lessthanitscorrespondent, forthatwould be to saythat(n-1)is notgreater(n+1) - 3,dfortiori, UandVcan bedecomposed inthemanner described. Thus.then,we shallobtainasmanysecondary derivees asinthelastcase(Method 1), thatis,n (n2- I ){sincea+.8+'Yisstillequalto(n+1)1,asbefore. More­ over, each of thesewill be of (n-a)+(n-.8)+(n-1 -,,/),thatis of2n-2 dimensions. Altogether, therefore, we have {(n-I)n(n-I)nn(n+I)}(n-I)n 2+2+2+2 linearindependent equations ofthedegree2n-2,andthenumberof 1· · . (2n-1)2nNh' be Iarguments to ermmate IS2 . ow t ese two numrsareequa. Thusweobtainafinalderiveecontaining oftr«coefficients (n-;1)n+(n-2I)n , ,n(n+I) (n-I)nanequalnumber ofV'llbutofWs-----.+---' nown(n-I), 22' , n(11-1)andn2exactlyexpressthenumber thatoughttoappearofeach oftheserespectively: hencethefinalderiveeis clearofirrelevant factors. TERNARYSYSTEMS. CASEC.Twooftheindicesequal; the third one greaterbya unit. Here,callingnthehighestindex,theaugmentees musteach be made ofthedegree(2n- 3),theirnumberwillevidently be (n-2)(n-I) (n-I)n (n-1)n 2+2+--2--' andwhere15Jdouble,treble,andother Systems ofAlgebraic Equations. 85 makingthesumoftheindicesofderivation now,asbefore,equalto(n+1); itwillbestillpossible to formintegerequations ofdecomposition, which will giverise toaugmentatives ofthedegree(n-cz)+(n-1)-fJ+(n-1)-"/' thatis,of(2n-3)dimensions, Thetotalnumber ofequations, whatwith augmentatives andsecondary derivatives, willbe {(n - 2)(n- I) +(n-l)n+(n-=-!>_n}+n(n-I) =4n2-4n+2=(2n-2)(2n-I) 2 2 2 2 2 2' thatis,isequaltotheexactnumberofdistinctarguments contained between them. Alsothefinalderivative willc~ntainin eachmember (n-2)(n-1)n(n-I) 2+2 ' thatis,(n-1)(n-1),lettersbelonging tothefirstequation, and (n-l)n n(n-I)-2-+ 2 ' thatis,n(n-1)belonging to those of thesecondandofthethird,andwill therefore beinitslowestterms. COROLLARY TOCASESBAND C. Itisnotnecessary, afterallthathas been alreadysaid, to do more than justpointoutthattheprocesses applicable tothesecasesenableus todeter­ mineX,Y,Z,whichsatisfytheequation XU+YV+ZW=FwyuzA, j+g+h=3n-3 forCaseB, j+9+h=3n-4for CaseC. 10. MEMOIR ON THEDIALYTIC METHOD OF ELIMINATION. PART L [Philosophical Magazine, XXI.(1842),pp.534-539-.] THEauthorconfineshimselfinthisparttothetreatment of twoequations, thefinalandotherderivees ofwhichformthesubjectofinvestigation. Theauthorwasled toreconsider hisformerlaboursinthisdepartment ofthegeneral theorybyfindingcertainresultsannounced by M.Cauchyin L'Institut, MarchNumber ofthepresentyear,whichflowasobvious and immediate consequences from Mr Sylvester's ownpreviously published principles andmethod. Lettherebe twoequations in:e, U=ate"+bxn-I+c:e"-'J+ea;"-S+&c.=0, V=ax'"+{3:em-1+ry:em-2+&c.=0, andletn=m+"where,is zero or anypositive value(asmay be). Letanysuchquantities as:erU,.x'V,betermedaugmentatives ofUorV. Toobtainthederiveeofa.degreesunitslowerthanV,wemustjoin8 augmentatives ofUwiths+,ofV.Thenoutof2s+,equations a;OU=O, a;1U=O, wU=O, x'-IU=O, xoV=0,XlV=0,wV=0, a;<+1-1V=0, we mayeliminate linearly 2s+L-1quantities. Nowtheseequations contain nopowerofa:higherthanm+L+S- ] ; accordingly, all powers of e,superior to?It-s,may be eliminated, andthe deriveeofthedegree(m-s)obtained in itsprimeform. Thustoobtainthefinalderivee(whichisthederiveeofthedegreezero), wetake 111augmentatives ofUwithnofV,andeliminate (m+n-1) quantities, namely, ai,w,x',•.....upto:em+n-I. *Reprinted fromProc.Roy.Irisli Acad., Vol.II.(1840-1844), p. 130. 16] OntheDialytw MetJwdofElimination. 87 This process. founded uponthedialyticprinciple, admitsofR.verysimple modification. Letusbeginwiththecasewhere t=0, orm='11.Letthe augmentatives ofUbetermedtt;UI,U2,Us,...andofV,Vo,VI'V2,Vs...., theequations themselves beingwritten U=a:.c"+ba;n-I+ca;n-2+&c. V=a'x"+b'a;n-I+c'x"-'J+&c. Itwillreadilybeseenthat a'Uo-aV o, (b'U;-bVo)+(a'UI-aVI), (c'Uo-cyo)+(b'UI-bVI)+(a,'Us-aVs),&c. willhe eachlinearlyindependent functions ofx,:rfJ,...a;m-\nohigherpower ofzremaining. Whence itfollows,thattoobtainaderiveeofthedegree (m-s)initsprimeform, we haveonly toemploythe8ofthosewhichoccur first inorder,andamongst themeliminate xm-\~,...X..-H I.Thus, toobtainthefinalderivee, wemustmakeuse ofn,thatis,theentirenumber ofthem. ~ow,letussupposethattis not zero, butm=n-t.TheequationV maybeconceived tobeofninsteadofmdimensions, if wewriteitunder the form Ox"+OX"-I+oa;n-s+...+OXm+1+ax'"+{3a;m-1+&c.=0, and we are abletoapplythesamemethodasabovejbutasthefirsttofthe coefficientsin theequation abovewrittenare zero, thefirsttofthequantities (a'Uo-aVo),(b'Uo-byo)+(a'UI-aVI),&c. maybereadsimply -aVo,-bVo-(tVI'-CVo--bVJ-aV2,&c. andevidently theiroffice can be supplied bythesimpleaugmentatives themselves, Vo=O.VI=0,Vs=0...V._I=°; andthusLletters,whichotherwise would be irrelevant, falloutoftheseveral derivees. Theauthorthenproceeds withremarks uponthegeneraltheoryofsimple equations, andshows how by virtueofthattheoryhismethod contains a solution of theidentity XrU+YrV=Dr whereDris aderiveeoftherthdegreeofUandV,andaccordingly, X,of the form x+f£X+v.~+...+8xm-f'-t, andF,oftheform l+m:x+...+tx"-f"-I, 88 .OntheDialytic MethodofElimination. [16 andaccountsaprioriforthefact of not more than(n-r)simpleequations beingrequired forthedetermination ofthe(m+n-2r)quantities )..,f£,u,&C. l,m,n,&c.,byexhibiting theselatterasknownlinearfunctions of no more than(n- ,')unknown quantities left to be determined, Uponthisremarkable relationmay beconstructed amethodwelladapted fortheexpeditious computation ofnumerical values of thedifferent derivees. Henext,asapointofcuriosity, exhibits thevalues of thesecondary functions,«ti.:aVo, b'U,-bVo+a'UI-aVI' «ti,-cVo+b'UI-bVI+a'U 2-aV 2,&c. undertheform ofsymmetric functions oftheroots oftheequations U=0, V=0, by aid of thetheorems developed in theLondon and Edinburgh Philosophical Magazine, December 1839-,andafterwards proceeds to a more closeexamination ofthefinalderiveeresulting from two equations eachof the same (any given)degree. He conceives a number of cubic blocks each of which has two numbers, termeditscharacteristics, inscribed upon one of its faces,upon which the value of such ablock(itselfcalled an element) depends. Forinstance, thevalue oftheelement,whosecharacteristics arer,s,isthe difference between twoproducts: theone ofthecoefficient rthinorder occurring inthepolynomial U,bythatwhich comes sthinorderinV;the otherproduct isthatofthecoefficient sthinorderofthepolynomial U,by thatrthinorderofV;sothatifthedegreeof eachequation ben,therewill bealtogether -In(n+1) suchelements. The blocks are formed intosquares orflats(plafcm.ds) of which the b.n n+1 di . odd The fh . numerIS2or~, accormgasnISeven or . e first of t ese contains nblanksinaside,thenext(n-2), thenext(n-4),till finally we reach a squareof four blocks or of one, according asnis even or odd. Theseflats are laid upon one another 80as to form a regularly ascending pyramid, of whichthetwodiagonal planesaretermedtheplanesofseparation and symmetry respectively. The former divides thepyramid intotwo halves, suchthatnoelement ontheone side of itisthesameasthatof any block in the other. Theplaneofsymmetry, asthenamedenotes, dividesthe pyramid intotwoexactlysimilarpartsjitbeingarule,thatallelements lying in any givenlineofa square (plafond) paralleltothe plane ofseparation areidentical jmoreover, thesum ofthecharacteristics isthesame, for aU elements lyinganywhere inaplaneparalleltothatofseparation. [*p.40above.ED.] 16J OntheDialytic MethodofElimination. 89 All thetermsinthefinal derivee are made up by multiplying nelements ofthepiletogether, underthesolerestriction, thatno two or more termsof thesaidproduct shall lie in anyone planeoutofthetwosetsofplanes perpendicular tothesides of thesquares. Thesignof any such productis determined bytheplaces of eitherset of planes parallel to a side of the squares andtooneanother, in which theelements composing it may be conceived to lie. Theauthorthenentersintoadisquisition relatingtothenumberofterms which will appearinthefinal derivee, and concludes this first partwiththe statement of twogeneralcanons, each of which affords as many testsfor determining whether aprepared combination of coefficients can enterinto thefinalderiveeofanynumber ofequations asthereareunitsinthat number, but80connected astogether only to afford double thatnumber,less one,ofindependent conditions. Thefirst of these canons refers simply to thenumberoflettersdrawnout ofeachofthegivenequations (supposed homogeneous); thesecondto whathe proposes to calltheweightof every terminthederiveeinrespecttoeachof tJ~variables whichareto beeliminated. Theauthorsubjoins, forthepurpose of conveying amoreaccurate conception ofhisPyramid ofderivation, examples ofthemode in which itis constructed. When 71=1thereis one fiat, viz. 1~1 Letn=3,therewill be two fiats: ~I i1, 2 1, 3 1, 4 I1---1 1, 3 1, 4 2, 4 1, 4 2, 4 3, 4Whenn=2thereis one fiat, viz. I I~~ I~~ Letn=4,therewillstillbe two fiatsonly: -~--- ~~I 2, 4 3, 4 I 1, 2 1, 3 1, 4 1,5------I- 1, 3 1, 4 1,52,5-------- I, 41,52,53,5------- I,52,53,54.5 90 OntheDialytic MethodojElimination. Letfl=5,therewill bethreeflats:[16 13,41I2,32,41 2,5---1- 2, 4 2, 53,I) 2,I)3,I),4,I) I11, 2 1, 3 I'1, 4 1, I)!1, 6--------,-- 1, 3 1, 4 1, I)1, 6 2, 6 1, 4 1, I)1, 6 2, 6 3, 6 1,51,6 2, 6 3, 6 4,6-1- 1- - - 1, 6 2, 6 , 3,6 ,4,65,6 Letn=6,therewillbethreeflats: 3, 4 3, I) 3,I)4,I)2,32, 42,52, 6-------- 2, 42,52, 6 3, 6-------- 2,52,63, 64,6-------- 2, 63, 6 4, 6 5,6 1, 2[I,3I 1, 4 1, 51, 61, 7 __1_--I- I, 3 1, 4 1, 51, 6I1, 72, 7--- --'-- 1, 41,51, 6 1, 7 2, 73, 7------------- I,51, 61, 7 2, 73, 74,7--------1-- ---- I, 6 1, 7 2, 7 3, 7 I4, 75, 7----------1---- I, 7 2, 7 3, 7 4, tI5, 7 6, 7 Thusthework of computation reduces itselfmerelytocalculating nn;1elements, orthen(n+1)cross-products out of which theyarecon­ stituted, andcombining themfactorially afterthatlaw ofthepyramid, to whichallusionhasbeenalreadymade. 17. ELEMENTARY RESEARCHES INTHEANALYSIS OF COMBINATORIAL AGGREGATION. [Philosophical Magazine, XXIV.(18440),pp.2R5-296.] THEensuing inquiries will be found to relatetocombination-systems, thatis, tocombinations viewed in an aggregative capacity, whose species beinggiven, we shall have to discover rulesforrangingor evolving themin classesamenable tocertainprescribed conditions. Thequestion ofnumerical amountwill only appearincidentally, andneverbe made theprimary object ofinvestigation -. Thenumberofthingscombined will betermedthemodulus ofthesystem towhichtheybelong. Theelements takensingly,orcombined in twos, threes,&c.,will bedenominated accordingly themonadic, duadic. triadic elements, orsimplythemonads, duads, or triadsofthesystem. Letusagreetodenotebythewordsynthemet anyaggregate of com­ binations in which all the monads of agivensystemappear ~lIce,and onceonly. It ismanifest thatmany such synthemes totallydiverse in every term maybeobtained for ugivensystemto anymodulus, andforanyorderof combination. Let us begin with considering thecase ofduadsynthemes. Takethe modulus 40andcalltheelements a,b,c,d. (ab,cd),(ac, bd), (ad,cb)constitute threeperfectly independent synthemes, andthesethreesynthemes includebetween themalltheduad elements, 80thatno more independent synthemes canbeobtained fromthem. • Thepresentthrorymaybeconsidered asbelonging toapartofmathematics whiohbears 10\hecombinatorial analysis muchthesamerelation asthegeometry ofposition totha'of measure,orthetheoryofnumbers tocomputative arithmetic; number, place,andcombination (asit_mBtotheauthorofthispaper)beingthethreeintersecting butdistinct spheresof thoughttowhichallmathematical ideasadmitofbeingreferred. tFromIf;'"andTiIJT/p.•• 92 Elementary Researches intheAnalY8i~ of [17 Again, let a,b,c,d,e,fbethemonads; we canwritedown five independent synthemes, to wit, ab,cd, ef) ad,cf,ebl ac, de,fbj. af,bd, ce ae,df,be Wecanwriteno more thanthesewithout repeating duadswhichhave alreedyappeared'" We propose to ourselves thisproblem :--..1systemtoanyeventmodulus beinggiven,toarrange the whole ofitsduadstintheformofsynthemes; orin otherwords,toevolve a Totalofduadsynthemes toanygiven even modulus§. Whenthemodulusisodd, as before remarked, theformation of aduad syntheme is of course impossible, foranynumberofduadsmustnecessarily contain an even number of monadic elements; butthereisnothing to prevent us from forming inallcaseswhatmaybetermedabisyntheme or diplotheme, thatis, anaggregate ofcombinations, where each element occurs twiceandno more. Forinstance, iftheelements be called afterthelettersofthealphabet, h(ab,be, cd,de,ea)h bi h . I d I ~d'we ave b bdda,teisyntematictotato mo u us 0;an1Ilec,ee, eo, , *Suchanaggregate ofsynthemes may betherefore termedaTotal. tThemodulus mustbeeven,asotherwise i~ismanifest no single syntheme can beformed. Weshallbefore long extendthe scope of our inquirysoastotakein thecaseof oddmoduli. :Triadicsystems willbetreatedofhereafter. §Itisscarcely necessary toadvertheretothefadoftheproblem being in generalindeter· minateandadmitting ofagreatvarietyofsolutions. Whenthemodulus is fourthereisonlyouesynthematic arrangement possible, andthereis noindeterminateness ofanykind;fromthiswecaninfer,itpriori,thereducibility ofabiquad­ raticeqnation; fornsing<p,j,Ftodenoterational symmetrical forms of function, itfollowsthat {f{q>(a,b),q>(c,d)}} Fj{<p(a, c), <p(b,dl)isitselfarational symmetric function ofa,b,e,d. f{<p(a,ti),q>(b,el) Whence it follows thatifa,b,e,dbetherootsofabiquadratic equation, j{<p(a,b),<p(e,ti)}oan be found by the solution ofacubic:forinstance, (a+b)x(c+ti)can bethusdetermined, whence immediately the Bumof anytwo of the rootscomesoutfromaquadratic equation. To themodulus 6thereare fifteen different synthemes capableof being construoted ;atfirst sightitmightbesnpposed thatthese could be classedinnaturalfamiliesofthreeor of five each, on which supposition theequation of thesixthdegree could be depressed; butoninquirythis hope will prove tobefutile,not butwhatnaturalaffinities do exist between the totals;butin ordertoseparate themintofamilies eachwillhaveto be taken twice over, or in otherwords, thefifteensynthemes tomodulus 6 beingreduplicated subdivide intosixnaturalfamilies of five each.Again, it is truethatthetriadstomodulus 6(justlike the duadstomodulus 4)admitof beingthrownintobut onesynthematic total,butthenthiswillcoutaintensynthemes, anumber greaterthanthemodulus itself. 17J likemannerCombinatorial Aggregation. 93 ab,be,cd, de, ef, fg,gal ac, ce, eg, gb, bd, dj, fathetotaltomodulus 7. ad,dg, gc, cf, fb,be, ea Ingeneml,ifnbethemodulus, thenumberofduadsisnn;1;nbeing even,~duadsgo to each syntheme, andtherefore thetotalcontains (n-1) ofthese.Ifnbe odd,then,since always nduadsgo to abisyntheme, the numberof such in thetotalisn~1. Beforeproceeding tothesolution oftheproblem first proposed, let us investigate thetheory of diplothematic arrangement. Herewe shall find anothertermconvenient to employ. By a cyclotheme, Idesignate a fixed arrangement oftheelements in one or more circles, in which, although for typographical purposes theyarewrittenoutin astraight line,thelastterm istobe viewed ascontiguous andantecedent tothefirst;therecurrence maybedenoted bylayinga dot upon thetwo opened ends of thecircle; ri.b.c.d.ewillthusdenoteacyclotheme tomodulus 5;a.b.c.d.e.f.g.h.ic the same to modulus 9;so also isa.b.c,d.e.j,g.h.icacyclotheme of anotherspecies to thesamemodulus. Ingeneralthenumber oftermswill bealike in each division of a cyclotheme. Nowitisevidentthateverycyclotheme, ontakingtogether theelements that lie in conjunction, may be developed intoadiplotheme. Thus i.2.a=12, 23, 31, i.2.3.4 =12, 23,3+,41, (12, 23,31)(i.2.:3;4.5.(3;'7.8.9) =45, 56, 64 . 78, 89, 97 Hencewe shall derivearuleforthrowing theduadsof anysysteminto bisynthemes. Letm=3,wehavesimplyirhC, m=5, wewrite t.i.b•c.d.e, a.c.e.b.d, the second beingderivedfromthefirst byomitting everyalternate term; similarly below,thelines are derivedeach from its antecedent. m=7, we have a.b.c.d.e.f.g, a.c.e.g.b.d.j, a.e.b.f.c.g.d. 94 Elementary Researches intheAnalysisof [17 A verylittleconsideration will serve to prove thatinthisway,11lbeing 11l-1aprimenumber,-2-cyclothemes may be formed, such thatnoelement will ever be found more thanonce incontactoneitherside with any other; whencetherule for obtaining thediplothematic totalto anyprime-number modulus isapparent. Forexample, tomodulus 7thetotalreadsthus:­ Ist,ab,bc,cd,de, ef,fg,gal 2nd.ac, ce, eg, gb, bd,df,fa, 3rd.ae, eb,bf,fc,cg, gd, da andno more remainstobesaidonthisspecialcase. Letus nowreturntothetheoryof even moduli, and show how to apply whathasbeenjustdone to constructing asynthematic totalto amodulus which is thedouble or a primenumber. Supposethemodulus to besix, thenumberofsynthemes is five.Letthe sixelements, a,b,c,d,e,f,betakeninthreeparts,sothateachpartcontains two ofthem;letthesepartsbecalledA,B, C,where.Adenotesab, B, cd, andC,ef. Nowtheduadswillevidently admitof adistinction intotwoclasses, thosethatlie in one part,and those thatliebetween two;thusab, cd,ef will be each unipartite duads,therestwill bebipartite. Theunipartite duadsmay be conveniently formedintoasyntheme by themselves; itonlyremains to form thefourremaining bipartite duad synthemes. Writethepartsincyclothematic order,asbelow: ABC. It will be observed thateachpartmay bewrittenin twopositions; thus ab.Amay beexpressed bybor bya B....c dd c' feC" f"e Now we may form a cyclic tableofpositions asbelow: ABC 111 122 212 221 17J Combinatorial Aggregation. 95 Herethenumbers ineachhorizontal linedenotethesynchronic positions oftheparts. Oninspection it will be discovered thatAwill be found in each of its twopositions, withBineachofitstwo;similarly BwithC,andCwithA. Infactthe four permutations, 11,12, 21, 22, occur,thoughindifferent orders. in any two assigned vertical columns. Nowdevelope thepreceding table,andwehave aeead!be!bde, bdfbeeadeaef; andthesebeingread off (thesuperior ofeachantecedent withtheinferior of eachcousequent ")mustmanifestly givethefourindependent bipartite synthemes which we were in questof,videlicet (ad, ef, eb), (ae, de, fb),(bd,ce,fa),(be,df,ea); these four, together withthesyntheme firstdescribed (ab,ed,ej),constitute a duadsynthematic totaltomodulus 6. Beforeproceeding furtherlet~stakeoccasion toremarkthatthefore­ goingtableofpositions mayevidently beextended to any odd number oftermsbyrepetition ofthesecond and thirdplaces,asseen intheannexed tables of position. i.l.l.l.i i.2.2.2.2 2.1.2.1.2 2.2.1.2.ii .1.1.1.1.1. i, i.2.2.2.2.2.2t, 2.1 .2•1 •2 . 1.2, 2.2.1.2.1.2.i. Nowlet10 bethemodulus. Asbeforedividetheelements intofiveparts,which call A,B, C,D, E. Theunipartite duadsfallintoasinglesyntheme; theeightremaining bipartite synthemes may be found asfollows:- Arrange incyclothemes(n;1 innumber) theoddmodulus system .A,B,C,D, E. We have thus ABODE, ACEBD• •Anyo~r jiz~dorderofsuocessive conjunction wouldanswerequallywell. tItwillnotfailtobeborneinmindthatinoperating withthesetablesonlycontif/UOlU elements aretakeninconjunotion: thefirstwiththesecond,thesecondwiththethird,the thinlwiththefourth,&0.,andthelastwiththefirst;no twotermsbutsuchas lietogether are in&Dymannerconjugated with one another. 96 Elementary Researches intheAnalysisof [17 Leteachcyclotheme betakeninthefourpositions given 10thetable above, we have thus2 x 4,thatis,eightarguments. dbcd~.dB73i.~b7di.~Bc8~ a.fJ'Y3E,a.bede, afJc3e, ab7dE, deebcl.a7Ef3t.~C Ebt.a'Yef3d, a'YEfJ3,a.eebd,a7efJd,aCEbo. And each of thesesrgumente willfurnishonebipartite syntheme, byreading off, as before, thesuperior of each antecedent withtheinferior of each consequent; andtheleastreflection will serve to show thatthesameduad canneverappearin twodistinctarguments. Inlikemanner, ifthemodulus be 14andsevenpartsbetaken,the bipartite synthemes, twelve in number, may beexpressed symbolically thus: fi.1.1.1.1.1.i f· . +i.2.2.2.2.2.2 {1.B.C.D.E.F.~} 1.2 . x+A.C.E.G.B.D.F .+2.1..1.2.1.2 1 .. .+.LJ..E.B.F.C.G.D+2.2.1.2.1.2.1 Nay more,from theabove table, if we agreeto name theelements ~:~:'&c., we canatonce proceed to calculate each ofthetwelvesynthemes inquestion by an easy algorithm. Forinstance, (i.2.2.2.2.2.2) x(A.C.E.G.B.D.F) Andagain (2.1.2.1.2.1.2) x(1.E.B.F.C. G.D) =AsE lI,EIBI,B,IFs,FIClJCliGS'GIDl,DiAl; each figure occurring onceunchanged as anantecedent and once changed as aconsequent. Ifitwerethought worth while itwould not be difficult,by usingnumbers insteadofletters,toobtainageneral analytical formula, from which all similarly constituted synthemes to anymodulus mightbe evolved. Buttherule ofproceeding mustbe nowsufficiently obvious; themodulus being2p,wedividetheelements intopclasses; thesemaybearranged into p;1distinct forms of cyclothematic arrangement, and each of thecyclo- themestakenin four positions, thusgiving4xP;!,thatis,2p- 2bipartite synthemes, thewholenumberthatcan be formed to thegivenmodulus 2p. 17J Combinatorial Aggregation. 97 I shall now proceed to the theoryofbipartite synthemes to themodulus 2mxp,by which itis to beunderstood thatwe haveppartseachcontaining 2m terms, and pisatpresentsupposed to be a primenumber; thetotal number of synthemes to themodulus 2mpbeing2mp-1, and2m-1 of theseevidently beingcapable ofbeingmadeunipartite; theremainder, 2mp-2m,thatis,(p-1)2m,will bethenumber ofbipartites to be obtained P: p-l2m(p-1)=-2-x4m; p-l-2-denotes thetotalnumberofcyclothemes tomodulusp;4m,as will be presently shown, thenumberof lines or syzygies in theTableofposition. To fix our ideasletthemodulus be 4 x 3, andletA,B, 0bethreeparts: ~~~a41 bIi,b.i,theirconstituents respectively. CtC,C.C4 Give afixedordertotheconstituents of eachpart,theneach ofthemmay betaken in four positions; thusAmay bewritten itt~~a4' ataaa4~' Uaa4~ ~, a4UtUa~· Assumesome particular positionfor each, as, for instance, ~b,Cll atb;Ci, ~b.c.. a4b4c4, and read off by coupling thefirst and thirdverticalplaces of each ante­ cedentwiththesecond and fourthrespectively of eachconsequent; we have accordingly, ~b'JJbIc"CIUa, a.b4,b.c4,c,a4· Itisapparent thatthe same combinations willrecurif any two contiguous parts revolve simultaneously through twosteps;or inotherwords,that A~B,=Ar+2B'+2' wherep.is anynumber, odd or even. • Ingeneral,iftherebe11"partsofJ.'termseaoh,andJ.'1I"be even, the numberofbipartite IYDtbemet! is(11"-1)J.',asis easily shown from dividing the whole number ofbipartite duads bythesemi-modulus. L 7 98 Elementary Researches intheAnalysis of Symbolically speaking, therefore, asregardsourtableofposition, r:8=r+2 :8+2,[17 or more generally, =r+2±4i:s+2±4i. Sothat 1:1=3:3. 1:2=3:4, 1:3=3.1, 1:4=3.2,2:1=4:3, ~.2=4:4, 2.3=4:1, 2.4=4:2. Therearetherefore no more thaneightindependent unequivalent permuta­ tionstoeverypairofparts.Nowinspectthefollowing tableofposition:- i.1.i, 1.2.3, 1.3.2, 1.4.4,2.1.2, 2.2.4, 2.3.1, 2.4.3. Itwillbeseenthatinthefirst and second, secondandthird,thirdand first places, all theeightindependent permutations occurunderdifferent names;thelaw offormation ofsuchandsimilartableswill beexplained induetime;enough forourpresent objectto see how, by meansofthis table,weareabletoobtainthebipartite synthemes tothegivenmodulus 4 x:3;thenumber according toourformula is 2 x 4 x :3;1=8,andthey may bedenoted symbolically asfollows:- (1.B.0)(1.1 . 1 + 1 . 2 .3 + 1 . ~.2+1 .4.4). +2.1.2+2.2.4+2.3.1+2.4.3 Eachoftheeighttermsconnected bythesign of+givesadistinctsyntheme ; forexample, letusoperateon A.B.0x(2.3.1). 2.3.1denotes2.3,3.1,1.2. 2.3givesriseto2(3 + 1) + (2 +2).(3 + 3)=2 .4+ 4. 2. 3.1givesrise to 3 (1 +1) +(3+ 2) .(1 + 3) =3.2+1.4. 1.2givesriseto 1(2+1) + (1 + 2).(2 + 3)=1.3 + 3. 1. Thesyntheme inquestion istherefore AtB"A,B2,B,C2,BIC"CIA"C,AI, andso on for all therest,therulebeingthat r:8=r(8+ 1)+(r+2)(8+ 3). 17] Combinatorial Aggregation. 99 ~ow,asbefore,itisevidentthatif we look only to contiguous terms,the abovetableofpositionmay beextended to anynumberof odd terms, simply byrepetition of the second and thirdfigures in each syzygy; and hence the rule forobtaining thebipartite synthemes tothemodulus 4xpisapparent. Forinstance, letp=7,therewill be 8 x 7 ;1,thatis, 8 x 3of themdenoted asfollows:- • , Al'1.1.1.1.1.1.1 + 2.1.2.1.2.1.21 {A.B.C.D.E.F.(1)+.d.C.E.G.B.D.F x+1.2.3.2.3.2.3+2.2.4.2.4.2.4". +.A..E.B.F.C.G.iJ +1.s.2.3.2.3.2 +2.3.1.3.1.3.1 1{ +1.4.4.4.4.4.4+2.4.3.4.3.4.3 Asan example of the mode of development, let ustakethe term 1.E.B.F.C.G.Dx2.4.3.4.3.4.3, 2.4.3.4.3.4.3= (2:4,4:3,3:4,4:3.3:4,4:3,3:2) (2.I}4.4}3.I}4.4}3.I}4.4}3.3}) =+4.3+2.2+1.3+2.2+1.3 +2.2 +1.1' A.E.B.F.C.G.D=A. .E,E.B,B.F,F.C,C. G,G.D,D.A., audtheproduct =(AiEl'E4B4,B,FI,F4C4,C,GI ,G4D4 ,DIAl). A4E"E.B.,BIF"F.C"CIG"G.Di•DIAl Letthemodulus be 6 x 3;as before, give a fixedcyclic order to the constituents of each part,and each will admitofbeingexhibited in six positions. Writesimilarly asbefore, ~blCI, a.i;CI, a,b,C "a4b,C4, aaboCa, Ueb.Ca, andtakethe odd places of each antecedent with the even places of each consequent; itwill now be seen that r:8=r+ 2:8+ 2=r+ 4:8+4, andthenumber ofindependent permutations is?36=2.6;and soin general, if therebe2mconstituents in apart,thenumberofindependent . .2m.2m4permutations 18 =m.m "2 7-2 100 Elementary Researches in theAnalysis oj [17 Therulefortheformation ofthetablewill beapparent oninspection. I suppose only threeparts,astherulemay always be extended toany number byreiteration ofthesecond and thirdterms.Thetablewillbe found to resolve itselfnaturally intofourparts,eachcontaining 'Inlines. Letm=1, we have 1.1.12.1.2 1.2.22.2.1 m.=2, wehave 1.1.12.1.2 1.2.32.2.4 1.3.22.3.1 1.4.42.4.3 m=3, we have 1.1.12.1.2 1.2.32.2.4 1.3.52.3.6 1.4.22.4.1 1.5.42.5.3 1.6.62.6.5 m=4,we have 1.1.12.1.2 1.2.32.2.4 1.3.52.3.6 1.4.72.4.8 1.5.22.5.1 1.6.42.6.3 1.7.62.7.5 1.8.82.8.7 Sothate,goingthrough allitsvalues from 1 tom,thegeneralexpression forthefourpartsis I{1.X(2a:-l J+l(1n+a:)2a: ) +2.e .2a:+2(m+x)(2x-1).1. To show theuse ofthisformula, let us supposethatwe have seven parts, eachcontaining tenterms,thegeneral expression forthebipartite duad synthemes is 1 "{1.a:(2X-l)a:(2a:-l)a:(iX_l) 1 {.l1.B.e.D.E.F.G} 2 2 2 2+.oi•X.a:.e:»:a: +.A.e.E.G.B.D.F xI+1(5+x)2a:(5+x)2a:(5+x)2x J' +.A.E.B.F.e. G.D+2(5+a:)(2a:_1)(5+a:)(2a:-l)(5+a:)(2a:-l) 17J Combinatorial Aggregation. 101 Make,forexample, a:=3, one of thesynthemes inquestion outofthe twelvecorresponding tothisvaluewill be A.C.E.G.B. D. F x 2.3. 6 . 3.6 . 3,.6. Here .A..C.E.G.B.D.F=AC, CE, EG, GB, BD, DF, FA, 2.3.6.3.6.3.6= =2.4)3.76.4}3.7)6.4\3.7 f6.3) +4.6+5.9+8.6+5.9+8.6+5.9+8.5 +6.8r+7 .1+10.8+ 7.1\+ 10.8)+7.1+10.7) +8.10I+ 9 .3 + 2.10 + 9.3 +2.10+9.3 +2.9 +1002+1.5 + 402 . + 1.5+4.2+1..5+4.1 andtheproduct =AliGhG,E7,s.o;G,B7,B.Dt,D,F7,F.A, AtG.,G,Eg,EsG.,G,Be,B,D.,o.s;FaA" &c. &c. &co Toprovetherilleforthetableofformation, itwill besufficient toshow thatnotwocontiguous duadaevercontainthesameorequivalent permutations ; theequation ofequivalence itwillberemembered is r:s=r+2i±21n:8+2i±2m. Now,asregardsthefirstandsecondterms,itismanifest that1:xcannotbe equivalent, eitherto1 :x'norto2:x,nor to2 :x',wherex'isanynumber differing froma: Similarly, asregardsthelastandfirstterms,x:1cannotbeequivalent to :1/:1,nor tox:2,nor tox':2;therefore thereis nodangeras farasthefirst tennisconcerned, eitherasantecedent orconsequent. Again,itisclearthatx:(2x-1)cannotinterfere withx':2x',nor (m+x):2xwith(m+x'):(2x'-1);neithercan(2x-1):xwith2x':ai,nor 2.x:(m+x)with(2x'-1):(m+x'). . Again,ifpossible, let x:(2x-1)=(m+aI):(2a1-1); then m+x'-x=2i, and 2x'-2x=2i, therefore 2m=2i, or m=i, whichisimpossible, since+iisthedifference between twoindices, e90Ch lessthanm. 102Researches inAnalysisojCombinatorial Aggregation. [17 Similarly, m+a;:2a;cannot=a;':2a;', andviceversawith the termschanged 2a;:(m+a;)cannot=2a;':a;', and (2a;-1) :a;cannot=(2a;'-I):(m+a;'), which proves therule forthetableof formation. So much for thebipartite dUMsynthemes. Asregardstheunipartite synthemes littleneed be said, for every partmay betreatedas aseparate system, and as each will producean equal numberofsynthemes, thesebeing takenone with another, willfurnishjustasmanyunipartite synthemes of thewholesystemastherearesynthemes due toeachpart.Thusthenthe synthematic resolution of the modulus 2mxpmay be made to dependon thesynthematization of2mand thecyclothematization ofp.Thishasbeen alreadyshown(whatever mmay be) for thecase ofpbeingaprimenumber; butI proceed now to extendthe rule to themoregeneralcase ofpbeing anynumberwhatever. 18. O~THEEXISTENCE OF ABSOLUTE CRITERIA FOR DETER­ MINI~G THE ROOTS OF NUMERICAL EQUATIONS. [Philosophical Magazine, xxv.(1844),pp.442-445.] IWISHtoindicate inthisbriefnoticeafact which Ibelievehasescaped observation hitherto, thatthereexist,certainly in some cases,andprobably inall,infallible criteriafordetermining whether Il.givenequation hasallits rootsrationalor not. Intheequation oftheseconddegreeitisenough,inorderthatthismay bethecase,thattheexpression forthesquareofthedifference of theroots shallbeaperfectsquare; inotherwords, ifw-px+q=0 haveitsroots rational,pi-4qmustbe not only apositivenumber (thecondition ofthe rootsbeingreal),butthatnumbermustalso be a complete square. Inthis caseitisfurtherevidentthatpmustbeeitherprimetoq,or if not, the greatest common measure ofpiandqmustbe aperfectsquare;butthis condition is contained intheformer, which is a sufficient criterion in itself. Ifwe nowconsider theequation ofthethirddegree, :r:'-par+qa;-r=0, onecondition is,thattheproductofthesquareddifferences shall be a perfect square;inotherwords,theequation cannothave all itsrootsrational unless plq2_4q3-18pqr- 41h-271.2 beapositivesquarenumber. Thisremarkismadeattheend ofthesecondsupplement ofLegendre's TheoryofNumbers, and is indeed self-evident; andinlikemanner one condition may be obtained for anequation of anydegreewhichistohave allitsrootsrational; butthisis far from beingthesolecondition required. 104 OnAbsolute Criteriafordetermining [18 Intheequation ofthethirddegree, however, one othercondition, conjoined withthatabove expressed. will serve to determine infallibly whether allthe roote are rationalor not. Toobtainthiscondition, letus suppose thatbymaking3x=Y+Pwe obtaintheequation ya-Qa:-R=O. Calling the threeroots of thisnewequation a,fJ,'Y(all of which it isevidentmustberationalifthoseofthefirstequation are so), we have a+fJ+'Y=0, Q=-(a{3+a'Y+fJ'Y)=rr+a,fJ+tp, R= af3'Y' From the lasttwoequations itis easily seen thatiflcbeanyprimefactor common to QandR,letwillbecontained inQ,andletinR;or, inother words,kwill be a common measure ofa,fJ,'Y. We have therefore asecond condition, that9q-3rshallbeanegative quantity, which is eitherprimeto2pJ-9qp+27r,or else sorelatedtoit, thatthegreatest common measure of thecube ofthefirst and thesquare ofthesecond is a perfect sixthpower. I now proceed to show theconverse. thatif these two conditions be both satisfied (and itwillappearinthecourse of the inquirythatthefirst does notinvolvethesecond),therootscannothelp being all rational. Itisevidentthatthetwoconditions inquestion aretantamount to supposing thattheroots of theproposed equation arelinearlyconnected with those of anotherz'-Qz-R=°(byvirtueoftheassumption 3a:=kz+p), whereQmay be considered asprimetoR;and where 4Q'- 27JlIis a perfectsquare. Letnow4Q'- 27Rt=t»,thenDt+27JlI=4<2',ort»+3(3Rt=4Q'. Here.asQisprimetoR, Dcanhave no common measure but3, with3R. Firstly,letQbeprimeto3R. ThenputtingIt+3ft=CI.thecomplete solution of theequation Im­ mediately preceding iscontained inthetwo systems : 1st.D=2f,3R=2g. 2nd.D=(f± 3g),3R=Ii-g, and for both systems, I±g",!(-3)={h±3k",!(-3)}'. 18J theRootsofNumerical Equations. 105 The second system musttherefore be rejected, for9evidently contains 3, and therefore f=3R±9willcontain3, andthereforeDandtherefore Qwill dothe same, contrary tosupposition. HenceJ[~±J{-(~-~)}] =vi{~±~J(-;7)} =J{~+/J(-217)} =+3.y'(~3)~{j±9.y'(-3)} h =-K±3.y'(-3)A±/0'.y'(-3); and thethreeroots oftheequation being {{A+/o'.y'(- 3)}+{A-/0'';(-3)}, 1±~(-3){A+/0'';(-3)}+1+~(-3){A-/0'';(- 3)}, willevidently be allrational, which of course includes the necessity oftheir beingalso integer. Again, secondly, if we suppose thatQdoescontain3,J)twillcontain27, andconsequently Dwill contain 9;andweshall have HereRbeingprimeto~,it may be shown, as in thelast case, thatthe complete solutionis R D 2"±lS';(-3)={h±k';(-3)}', consequentlyJ{~±J(~-~)}=h±k.y'(-3)j andthethreeroots of the equation are 2h,h-3k,h+3k respectively,and are therefore all rational. Hereitmay be observed thatthecondition ofRbeing an even number, whichwe know,apriori,isthecase when all the roots are rational, is 106On Absolute Criteria ofNU'moical Equation». [18 involved inthetwo more generalconditions alreadyexpressed. Itwill now beevidentthatthefirstcondition by nomeansinvolves thesecond,asitis perfectly easy tosatisfytheequationp+3g'=Q'withoutsupposing anything relative tok,thecommon measure ofJ,g,Q,exceptthatitbeitselfofthe formx'+3~2,whichwill give anequation which can be solved in rationaltermsfor allvaluesofA.,~,r,8; andconsequently theproduct ofthesquaresofthedifferences oftheroots may be a square,andatthesametimetherootsthemselves may be irrational", Ibelieveitwill be found on inquirythattheequation tJf'-qx+r=0 will always have two rational rootsif (n-l)n-l.qn_nn.rn-1 be acomplete square,provided thatqbeprimeto r. Furthermore, viewing thestriking analogy ofthegeneralnatureofthe conditions ofrationality alreadyobta.ined, to thosewhichservetodetermine therealityoftheroots ofequations, I amstrongly ofopinionthatatheorem remains to bediscovered, which will enableus topronounce ontheexistence ofinteger,asSturm's theoremonthatofpossible roots ofacomplete equation ofanydegree:theanalogy ofthetwo cases fails however inthisrespect, thatwhileimaginary rootsenteranequation inpairs,irrational rootsare limitedtoentering ingroups,eachcontaining twoorMORE. ..Thusthenitappearsthatthetotalrationality of therootsof theequationx'-qz-r=O maybedetermined by adirectmethodwithouthavingrecourse tothemethodofdivisorsto determine the roots themselves; the twoconditions beingthat4qJ-27r' shallbeaperfect_quare, and thegreatestcommon measure ofqlandr'aperfectnzthpower. 19. AN ACCOUNT OF A DISCOVERY INTHETHEORY. OF ~UMBERS RELATIVE TOTHEEQUATION Az8+Bys+Ozl=Dxyz. [Philosophical Magazine, XXXI.(1847),pp.189-191.] FIRST GENERAL THEOREM OFTRANSFORMATION. IFintheequation Az8+By'+Ozl=Dxyz, (1) AandBareequal,or intheratioof twocubenumbers to oneanother, and if27ABO-])3(whichIshallcalltheDeterminant) is free from all singleor squareprimepositive factors of theform6n+1,butwithout exclusion of cubicfactors of such form, andifAandBare each odd, and0thedoubleor quadruple of an odd number, or ifAandBareeach even and0odd,then, Isay,thegivenequation may be made to dependuponanotheroftheform A'u'+Hv'+O'uJ=D'uou:j where A'HO'=ABO, D'==D, uvw=some factor of s. The following aresome of theconsequences which I deducefromthe abovetheorem. Instatingthemit will be convenient to usethetermPure Factorialtodesignate anynumberintothecomposition of which no singleor squareprimepositivefactor of theform6n+1enters. Theequations z8+ys+2z3==Dxyz, z8+y'+4z3==Dxyz, 23,.1+2YS+z3==Dxy~, areinsoluble in integernumbers, provided thattheDeterminant in eachcase isaPureFactorial. 108 On aDiscovery IntheTheoryofNumbers. [19 Theequation re'+ya+Az3=9B.'C'yz isinsoluble inintegernumbers, provided thattheDeterminant, for which in thiscase we may substitute A-27JJ3,is apurefactorial whenever Aisof theform9n±1,andequalto2p:riZlor4p3iZI,pbeinganyprimenumber whatever. I wishhowever tolimitmyassertion astotheinsolubility ofthe equations above given. Thetheorem fromwhichthisconclusion is deduced doesnotpreclude thepossibility of two of thethreequantities e,y,zbeingtakenpositive ornegative units,eitherinthegivenequation itselfor in one or theotherofthoseintowhich it may admitofbeing transformed. Shouldsuch values of two of thevariables afford aparticular solution, theninsteadofaffirming thattheequations areinsoluble, Ishould affirmthatthegeneral 8olution can be obtained byequations infinite differences-. SECOND GENERAL THEOREM OFTRANSFORMATION. Theequation pre'+g''!t+h-z3=Ka;yz may always be made todependupon an equation oftheform Aus+Bva+Cur=Duvw, where ABC=Rs-8', D=3Rj anduvw=some factor of fa;+gy+lis. Rrepresenting K+6fgh,(2) 8"K-3fgh. •Takeforinstance theequation r+ys+2z s=9xyz. TheDeterminant 27.25isaPure Factorial; consequently if thesolution bepossible, since in thiscasethetransformed must beidentical withthegivenequation, thislattermustbecapableof being satisfied bymakingx andypositive ornegative units.Upontrialwe find thatx=I,y=I,z=2willsatisfythe equation. I believe, but have not fully gone through the work of verification, thatthesearethe onlypossible values(primetooneanother) whichwillsatisfytheequation. Shouldthey not be so, mymethodwillinfallibly enableme to disoover andto give the law for the formation ofall theothers. Here,then,underanycircumstances, isanexample, thefirst on record,of thecomplete resolution ofanumerical equation ofthethirddegree between threevariables. 19)OnaDZBcovery in the Theory ofNumbers. 109 I have not leisureto show theconsequences ofthistheorem oftrans­ formation in connexion with the one first given, butshallcontentmyself with a single numerical example ofitsapplications: afl+!tI-r'= -6xyz maybemade to dependontheequation u3+v3+ur=0, andistherefore insoluble. Itismoreover apparent thattheDeterminant ofequation (2)trans­ formedisingeneral- 27R3,and istherefore always a PureFactorial, and consequently theequation f3afl+g3y3+hSza=Kxyz willbeitselfinsoluble, beingconvertible intoaninsoluble form,provided that K+6fghisdivisible by 9.and provided furtherthat(K+6fgh'f-(K-3fgh)S belongstotheformmsQ,where Q is of the form 9n±1,and also of one or the otherofthetwo forms 2.r""',4p8''"."Pbeinganyprimenumberwhatever. Pressing avocations prevent me from entering intofurtherdevelopments orsimplifications atthispresenttime. Itremains for me tostatemyreasonsforputtingforwardthesedis­ coveriesin soimperfect ashape.Theyoccurred to me in thecourse of arapidtouronthecontinent, andtheresultswerecommunicated by me to myillustrious friend M. Sturmin Paris, who kindlyundertook tomakethem knownon myparttotheInstitute. Unfortunately, intheheatofinvention I got confused aboutthelaw of oddnessandevenness, to which thecoefficients of thegivenequation are in the firsttheorem geTleraUy (in order for thesuccessful application of my methodasfarasitisyetdeveloped) required to besubject. Istatedthis lawerroneously, andconsequently drewerroneous conclusions from my Theorems ofTransformation, which I am very anxious to seize theearliest opportunity of correcting. Iventure toflattermyselfthatasopening out a new field in connexion withFermat's renowned LastTheorem, andas breaking groundinthesolution ofequations of thethirddegree,these resultswill begenerally allowed to constitute animportant andsubstantial accession to ourknowledge oftheTheoryofNumbers. 20. ONTHEEQUATION INNUMBERS AxI+ByJ+Cr=D:xy~,AND ITSASSOCIATE SYSTEM OF EQUATIONS. [Philosophical Magazine, XXXL(1847), pp. 293-296.] INthelastNumber ofthisMagazine Igaveanaccountof aremarkable transformation to which theequation AxI+By'+Cr=Dxyz issubjectwhencertainconditions between thecoefficients A, B, 0, D are satisfied; whichconditions Ishallbeginbyexpressing with more generality andprecision thanI wasenabledto do in my former communication. 1.Two ofthequantities A,B, Care tobeto oneanotherintheratioof two cubes. 2. 27ABC-j)Jmustcontainnopositive primefactorwhatever ofthe form611+1.I erredinmy former communication innotexcluding cubic factors of thisform. 3.If2'" isthehighestpower of 2 which entersintoABC,and2ftthe highestpower of 2whichentersintoD,theneithermmustbe oftheform 371±1, orifnot,thenmmustbegreaterthan371. Thesethreeconditions beingsatisfied, thegivenequation can always be transformed intoanother, A.'u'+B't,ri+C'w=D'uvw, where A'B'C'=ABC,D'=D,uvw=a factor of s, Theconsequence ofthisis, asstatedin my former paper,thatwherever A, B,C,D,besidessatisfying theconditions abovestated,aretaken 80as likewise to satisfythecondition,-firstly, ofABCbeingequalto21m'"'!,or secondly, of ABCbeingequal to 21m'"'l.pm,",!,provided inthesecondcase thatABOis oftheform9m±1, andthatDis divisible by 9, Pbeingin 20]OntheEquation inNumlJers .AX-+By+Ow=Dxyz. 111 bothcasesa prime, thenthegivenequation will begenerally insoluble. And Iam nowenabledto addthattheonlysolution of which itwill in any case admit,isthesolitaryone found by makingtwo ofthetenus.Aw,Byl,Or equalto oneanother jsothat,forinstance, ifthegivenequation shouldbe of theform rei+ya+.ABOr=Dxyz, thentheaboveconditions beingsatisfied, theonesolitarysolution of which theequation can possibly admit,isx=1,Y=1. .Ar-De+2=0, whichmayormay not havepossible roots. Icallthisasolita?'yorsingular solution,becauseitexistsaloneandnoothersolution can bededuced from it; whereas in generalIshallshowthatanyonesolution oftheequation .AreI+By'+ce=Dxyz canbe made to furnishaninfinityofothersolutions independent oftheone supposed given,thatis,notreducible theretobyexpelling a common factor fromthe newsystemof values of e,y,zdeduced from the givensystem. The following is theTheorem ofDerivation inquestion: Let .Act+BfJI+Cr=D7.fJry. Thenif wewrite andmake x=F'G+G'H+HtF-3FGH, Y=FGt+GH~+BF'-3FGH, 1z=15(FI+GI+HI-3FGH}. or =afJ'Y{F'+G'+H2_FG-FH -GHI. weshallhave :xl+y'+.ABCzI=Dxyz. Iam hence enabled to show thatwhenever :xl+y'+.AzI=Dxyzis insoluble, therewill beawhole family of alliedequations equallyinsoluble. Forinstance. because :xl+ya+r=0 isinsoluble inintegernumbers, I know likewisethat :r;B+'!/+z8=wy'+relza+ylzl :r;B+'!/+z8=wya+wzl-2y'r areeachequallyinsoluble. 112 InfactOntheEquationm .Numbers [20 (.x3+y'+zI)x(~+ys+~-.x3!1-.x3z1 -y'zI) x(~+ys+~-.x3y'-.x3z1+2y'zI) x(~+ys+~-!/zI-!/w +2.x'zI) x(a!+YS+~-rr;SzI-zl!/+ 2!/rr;S) =ul+1f+ur, whereu, v,warerationalintegral functions ofe,y,z. Henceeach ofthefactorsmustbeincapable ofbecoming zero-. As aparticular instance of mygeneral theoryoftransformation and elevation, taketheequation rr;S+yl+2Z"=Mxyz. Then,withtheexception ofthesingular orsolitarysolution a:=1,Y=I,of which Itakenoaccount, I amableto affirm thatfor all values of JIbetween 7and- 6,bothinclusive, withtheexception ofM= -2,theequation is insoluble inintegernumbers. Takenowtheequation whereM= -2,namely trfl+'!f+2z1+2xyz=O. Oneparticular solutionofthisis x=l,y=-I, z=l. Another, which I shall call thesecondj, is o:=1,Y=3, Z=-2. Fromthefirstsolution I candeducein succession thefollowing: x=11, Y=5. z=-7, x=-793269121, Y=1179490001, Z=-1189735855, &~ &~ &~ Fromthesecond, x= -10085, x=&c.y=8921, y=&c.Z= -8442, z=&c. Asanotherexample, taketheequation rr;S+!I+6z3=6xys. •Itis however sufficiently evident fromtheirintrinsic form, whioh maybereduced to i(M'+3N·).thatthisimpossibility existsforallthefactorsexcept the first. tSeePostscript. 20J Ax'+By+Gz'=Dxyz. 113 Onesolutionofthetransformed equation ul+2tt+3'1.0"=6uvw isevidently u=1,v=1,W=1. HenceIcandeduceaninfiniteseriesofsolutions ofthegivenequation, of whichthefirstinorderofascentwill be II:=5,Y=7,z=3. Again,thelowestpossible solution inintegers oftheequation w+yI+6z1= 0 willbe 11:=17, y=37, z=-21. Theequation admits of thesolutions II:=1, Y=2,z=-1, II:= -271,Y=919,z=-438. Itrustthatmyreaderswill do me thejusticetobelievethatI am inpossession ofastrictdemonstration of allthathasbeenhereadvanced without proof. Certainofthewriter'sfriendsonthecontinent have, in their comments uponone of his formerpaperswhichappeared inthisMagazine, complimented his powers of divination attheexpense of hisjudgment, in rathergratuitously assuming thattheauthoroftheTheoryofElimination wasunprovided withthedemonstrations, which he wastooinertor toobeset withworldlycaresanddistractions topresent tothepublicinasufficiently digested form. Theproofofwhatever hasbeenhereadvanced existsnot merelyas aconception oftheauthor's mind,butfairlydrawnoutinwriting, andinaform fit for publication. P.S.Itmustnotbesupposed thatthetwoprimary orbasicsolutions abovegivenoftheequation w+yI+2z1+2.xyz=0, namely, II:=1,Y= -1,z=1, II:=1,Y=3,z=-2, areindependent of oneanother. Thesecondmaybederivedfromthefirst, as I shall show in afuturecommunication. Infactthereexistthreeinde­ pendentprocesses, bycombining whichtogether, oneparticular solution may bemadetogiverise to an infiniteseriesofinfiniteseriesofinfiniteseriesof correlated solutions, which it maypossibly bediscovered contain between themthegeneral complete solution oftheequation :xl+yI+AzI=DII:yz. 8. 8 21., ONTHEGENERAL SOLUTION (INCERTAIN CASES) OF THEEQUATION a;8+y'+Ai'=Mwyz,&0. [Philosophical Magazine, XXXI.(1847), pp. 467-471.] ISHALLrestrict theenunciation oftheproposition I amaboutto advance to much narrower limitsthanI believe are necessary tothe truth,with a view to avoid makinganystatement which I may hereafter haveoccasion to modify. Letusthensuppose in theequation a;8+y'+Ai'=Mwyz thatAis aprimenumber, andthat27A-M'ispositive,butexemptfrom positiveprimefactors of theform 6i+1.ThenI say, and have succeeded indemonstrating, thatallthepossible solutions inintegernumbers ofthe givenequation may be obtained byexplicitprocesses from one particular solution orsystemof values of e,!I,z,which may be calledthePrimitive system. Thissystemof roots or of values of x,y,zisthatsystemin which the value of thegreatest ofthethreetermsx, y,Ai.z(which may be called the Dominant) istheleastpossible of allsuchdominants. I believe thatin generalthesystemoftheleastDominant isidentical withthesystemofthe leastContent,meaning bythelattertermtheproductof thethreetermsout of which theDominant is elected. I proceed to show thelaw ofderivation. Toexpressthissimply, I mustpremisethatI shall have to employ such anexpression as8'=ep(8) toindicate, notthatacertainquantity, 8',is afunction of8,butthatacertainsystemofquantities disconnected from oneanother, denotedby8',areseverally functions of acertainothersystem ofquantities denoted by8jand,asusual, I shall denoteepep8byep18, epepiSbyep'8,and so forth. LetnowPbethePrimitive system of solution of theequation a;8+'!I+Ai'=Mwyz, Pdenoting acertainsystemof values of and writtenintheorderofthe 21] OntheEquatwn r+y+.AzI=Mxyz, &c. 115 letters 3:,y,s,which may always befound by a limitednumber oftrials (provided thattheequation admitsof anysolution). Thatthisisthecaseis obvious,since we haveonly to give theDominant every possible value from theintegernextgreatest toAiupwards; and combine thevalues ofw,y', A~sothatnone shall ever exceed ateachstepthe cube of such dominant, and wemustatlast,ifthereexistanysolution, arriveattheSystemofthe LeastDominant. ' . Now, every system of solution is of one or theotherof twocharacters. Either 3:andymustbe odd and zeven, or 3:andymustbe one odd and the other even and zodd.Thatallthreeshould be odd is inconsistent with the given conditions astoAbeing odd and Meven;and if all threewere even,bydrivingoutthecommon factor we shouldrevertto one or theother ofthe foregoing cases. Thesystemsofsolutionwherezis even may be termedReducible, those wherezis oddIrreducible. Letcf>denoteacertainsymbol of transformation hereafter tobeexplained. ThentheReducible systemsofthefirstordermay beexpressed by cf>P,cf>sp,cf>sp,adinfinitum j or ingeneralbycf>"'P,'ntbeingabsolutely arbitrary. I willanticipate by statingthatthefunction cf>involves no variable constants; thatis tosay, ~(S)may be found explicitly fromSwithoutanyreference totheparticular equation to which Sbelongs. Letnow"denoteanothersymbol of trans­ formation,also hereafter tobe defined, and differing fromcf>insofarasit does involveasconstants thethreevalues of 3:,y,zcontained inP:thenthe generalrepresentations ofIrreducible systems ofthefirstorderwill be denoted by "CP'"P. Itispropertostateherethatthesymbol" isambiguous; andtcf>n.p, whenPand1ltare given, will have two values, according totheway in whichthetermsrepresented byParecompared with3:,y,zinthegiven equation w+y'+A~=M:xyzj foritisobviousthatif3:=a,y=b,z=csatisfies theequation, so likewise .".ill 3:=b,y=a, z=c. Each however of thesevalues of"cp",pgivesasolution ofthekindabove designated. Proceeding in likemannerasbefore,theReducible system of thesecond ordermaybedesignated bycf>"'.tcf>'".P,theIrreducible bytcf>"'."cf>n,.Pj andingeneraleverypossiblesystemof values of 3:,y,zsatisfying the proposed equation,in which ziseven,iscomprised undertheform cf>n,.."cf>n,.-1."...¢"'."cf>'".P;8-2 116 On theGeneral Solutionof [21 and every possible systemof such values, in which Z18odd. iscomprised underthe form thequantities 'nt.~...nrbeingof course all independent of oneanother, andunlimited innumber and value. Thusthenwe may be saidto have thegeneralsolution ofthegiven equation inthesame sense asanarbitrary sum ofterms.each of a certain form, is in certaincasesaccepted asthecomplete solution of apartial differential equation. Asregardsthevalue of thesymbols" and¢.¢indicates theprocess by whicha.b.e becomes transformed intoa,fl,ry,therelations between thetwo sets ofelements beingcontained inthefollowing equations: fl=a'b"+b'e/2+e'a"-3alb'e', ry=abc[a'2+b'2+c't-a'b'-a'e'-b'e/}. Next,astotheeffect of theDuplexsymbolt.Lete,g.,bethe elements ofthePrimitive systemP:tbeingthevalue of zande, 9of xandytakenineithermode of combination, each with each, which satisfy the proposed equation 0;3+~+Az3=Mxyz. Leti,m. nrepresent anysystemS. A.flo,vrepresent anysystem,,(8), tShastwo values, which we may denotebyt'S,'tSrespectively, and accentuating theelements A.#.vaccordingly tocorrespond, we shall have A'=3gm(gl-em)+3Am(tl-en)-M(g,l'-e'lm), #'=3Am (m~-gl)+3el(em-gl)-M(e,m'-g'lm). v'=3el(en-,l)+3gm (gn-m~)-M(eg1l2-,2lm): we havethen 'ir'S==A'.flo',v', and in like manner '.r~s-''''' I Ior=A,#'u, I"Sbeingderivedfrom'ir'Sbythemereinterchange ofeand9onewiththe other. 21J theEquation :C+'!I+Ai'=Mxyz,&c. 117 I havestatedthateverypossiblesolution oftheproposed equation comes underone ortheotheroftheorders,infiniteinnumber andinfinitetothe power of infinityinvarietyofdegree,abovegiven:thisis notstrictlytrue, unless we understand thatallsystems ofsolution areconsidered to be equivalentwhich differ only in a multiplier common to all threeterms ofeach;thatis to say, which may be rendered identical bytheexpulsion ofacommon factor. So thatma,mf3,m'Yas asystemistreatedasidentical witha,f3,'Y,which of coursesubstantially itis;anditshouldberemarked • thatthere is nothing toprevent theoperations denoted byepand"intro­ ducingacommon factor intothesystemswhichtheyserve to generate, and thelatterinparticular will have a strongtendency soto do. I believe thatthistheorem may beextended withscarcely anymodifica­ tion tothecasewhereA,insteadofbeingaprime,isanypowerofthesame, andtosuppositions stillmoregeneral. Ibelievealsothat,subjecttocertain verylimitedrestrictions, thetheorem mayprove to applytothecase where thedeterminant 27A-M3becomes negative. Thepeculiarity ofthiscasewhichdistinguishes itfromtheformer, IS that itadmitsof allthethreevariables x,y.zintheequation ar+ys+AzS=Mtxyz havingthesamesign, which is impossible whenthedeterminant ispositive; orinotherwords,thecurveofthethirddegreerepresented bytheequation r-+X3+1=~XY(in which I callthecoefficient of XYthecharacter­ istic),which, as long as thequantity lastnamedis lessthan3, is asingle continuous curveextending on both sides to infinity, assoon as the characteristic becomes equalto 3assumes toitselfanisolated point,the germofanoval or closed branch,whichcontinues to swell out(alwayslying apart from theinfinitebranch) asthecharacteristic continues indefinitely to increase. Ioughtnottoomitto callattention tothefactthatthetheorem above detailedis always applicable tothecaseoftheequation 3,3+ys+AzS=0, whenAisanypower of a primenumbernotoftheform6i+1;inother words,theabovealwaysbelongs to theclass ofequations havingMonogenous solutions, which for thesake ofbrevitymaybetermedthemselves Mono­ genousEquations" . •ThUlltheequation x3+11'+9z"=0alludedto byLegendre isMonogenous, and thePrimitive IJ.~ofsolution isx=l,y=2,z=-1,from which every otherpossible solution inIntegers maybededuced. 118 OntheEquation X-+Y+Az'=Mxyz,&c. [21 Ontheprobable existence of such aclass of equations Ihazarded aconjecture attheconclusion of my lastcommunication tothisMagazine. As I hope shortlytobringoutapaperonthissubjectina morecomplete form, Ishallcontent myselfatthistimewithmerelystatingatheorem of much importance tothecompletion ofthetheoryofinsoluble andof Monogenous equations ofthethirddegree; towit,thattheequation 10 integers a(.1:8+'!/+zll)+e(x2y+y2z+Z2X+X'!/+yz2+zw)+exyz=0 may always be transformed soastodependupon the equation ju3+g";+hw=(00-e)uvw, wherein jgh=ae2-(e2+:3a2)e+9a2-3a&-2cJ. By means of theabovetheorem, amongotherandmoreremarkable consequences, we are enabled to give atheoryof theirresoluble and monogellous cases of theequation w+'!/+m3zll=Mxyz, whenmis some power of 2, or of certainothernumbers. 22. O~THEINTERSECTIONS, CONTACTS, AND OTHER CORRE­ LATIONS OF TWO CONICS EXPRESSED BYINDETER­ MINATE COORDINATES. [Cambridge andDublinMathematical, Jou'MULl, v.(1850),pp.262-282.] LETU=0,V=0betwohomogeneous equations oftheseconddegree withrealcoefficients, between thesamethreevariables E,'T/,~. Thedirectandmostgeneralmode of determining theintersections of theconics expressed bytheseequations would be to make aE+b."+c~=t, a'E+b''T/+c'~=u: eliminating E,'T/.,between thefourequations in which theyappear,there resultsabiquadratic equation betweentandu.Thenatureoftheinter­ sectionswilldependupon the natureoftheroots of thisbiquadratic; and thus the conditions may be expressed analytically, which will represent the several casesof alltheintersections beingrealor allimaginary, or one pairrealandtheotherimaginary. Theseanalytical conditions willdepend uponthesignsofcertainfunctions ofthecoefficients of thegivenandthe assumedequations beingof anassigned character; myendeavour has been toobtainconditions of acharacter perfectly symmetrical and free from the coefficients arbitrarily introduced. Inthisresearch I have only partially succeeded, butthemethod employed, and some of the collateral results,will, Ithink,be found of sufficient interesttojustifytheirappearance inthepages of thisJournal. Adopting Mr Cayley's excellent designation, letthefourpointsofinter­ sectionof the two conics be calledaquadrangle. Thisquadrangle willhave threepairsofsides;theintersections of each pair,fromprinciples of analogy,I calltheverticesofthequadrangle. Then,inasmuch asthe four 120 Onthe Correlations oftwoOonic« [22 sets ofratiost .1]:t,corresponding withthefour sets of theratiot:u, mustbe sorelatedthatwe may always make ~=a+b.v(- 1),~=c+d,/(-1), ~=a-b.v(-1), ~:=c-d.v(-1), ~=ex+.B.v(-l), f:='Y+S.v(-l), ~=ex-,8.v(-1),f:='Y-S.v(-1), wemay easily draw thefollowing conclusions. Ifallthefourpointsofthequadrangle ofintersection arereal,thethree verticesandthethreepairsof sides are all real. Ifonly two pointsofthe quadrangle are real, one vertexandone ofthethreepairsof sides will be real;theothertwovertices and two pairsof sides beingimaginary. Ifall fourpointsofthequadrangle areunreal,onepairof sides will be real andtheothertwo pairs imaginary, as inthelastcasejbutallthethree verticeswillremainreal, as in thefirst case. Hencewe have a directand simplecriterion fordistinguishing thecase ofmixedintersection frominter­ sectionwholly real or wholly imaginary; namely,thatthecubicequation oftheroots of which thecoordinates oftheverticesare reallinearfunctions shall have apairofimaginary roots.Thisisthesole and unequivocal condition required. Theequation inquestion is, oroughtto be, well known to be thedeter­ minantinrespectto~,1],tofxU+p.V.Infact, if we write U=a~2+b1]2+cr+2a'1]t+2b"~+'2c'~, V=ar+,81]2+'Yr+211'1]t+2,8't~+2'Y'~, xU+P.V=(aX+ap.)~2+&c.=A~2+B1]2+Or+2..1'1]'+2B't~+20'~. theratiosofthecoordinates ~,1],tofthevertexofxU+p.Vmay easily be shown to be identical with AB-02:0'..1'-B'B:B'O'-..1'..1, andwill be real or imaginary asX:p.is one or theother. Ifthenthecubicequation inA.:p.,namely, ~(XU+Il.V)=0, has apairof imaginary roots,thatis,if00(x.U+Il.V)isapositive quantity, theinter­AjA,E'" sections ofUandVare of a mixedkind,thatis,thetwo conics have two realpointsin common. 2"2] expressed byIndeterminate Coordinates. 121 I mayremarkhere,enpassant,thatifwe form thebiquadratic equation intandu,ep(t,u)=0 fromtheequations U=O, V=O, aE+In]+cs=t, a'E+b'."+c'~=u, andif any reducing cubicofthisequation beP(8,ro)=0,thedeterminant ofP(e,ro)must,fromwhathasbeenshown above, be identical with ;~(XU+p.V)multiplied by some squared function oftheextraneous coefficients a,b,c; a',b',c', If00(XU+p.V)is anegative quantity, itremains todistinguish betweenthecases oftheconicsintersecting reallyin fourpointsornotatall. Themostobviousmode of proceeding todistinguish between purelyreal andpurelyimaginary intersections would be asfollows. LetXl'JI-I;">..t,Jl~; AI>11-1,bethethreesetsofvaluesofX,p.whichsatisfytheequation o(XU+P.V)=° and make Al=aA1+(J,IJ-J.,A.=a">..t+aP-2, Aa=aXa+aJ.'a, C1=c~+'YJI-I, Cs=c~+"11-'., O,=ex,+'Yll-a, B/=b'~+fl'JI-I,B.'=b'">..t+fJ'11,Ba'=b'Xa+fJJ.'a, A1C1-B/'=e1,AIC._11;'=e..AaCa-Ba'l=ea' ~owiftheequation AEI+B.,,'+Cr+2A'.,,~+2B'sE+2C'~=° represent a pairofstraight lines,itmay bethrownintotheform AC-B»Au'+--A.- v"=0, whereuandvarelinearfunctions ofE,.",S,andthestraight lineswill be realorimaginary, according asB'I-ACispositive ornegative; hence one or elseall ofthequantities e1,e2,ea,will benecessarily negative, andtheinter­ sectionswill beallrealor allimaginary, according asallthreearenegative oronly one isso. Acubicequation illemay be formed containing e1,e.,ea 88its roots by eliminating between theequations e=AC-B'l jo(XU+p.V)=O, and theconditions fortherealityoftheintersections will bethatall four coefficients ofthiscubicshallbe ofthesamesign,whichinrealityamount onlyto two,sincethefirstandlastmustin all cases have thesamesign. 122 OntheCorrelations oftwoOonie« [22 thatisThesameobjection however ofwantofsymmetry andconsequent irrelevancy andcomplexity attaches tothisasmuchastothemethod originally proposed. Thefollowing treatment ofthequestion relievesthe objection ofwantofsymmetry asfar asthecoefficients ofthesameequation areconcerned, butinitspractical application necessitates anarbitrary and therefore unsymmetrical election to bemadebetween thetwosetsof coeffi­ cientsappertaining tothetwoequations. Itishowever, Ithink,toocurious andsuggestive to besuppressed. Iobservethatifthefourintersections areall real, an imaginary conic cannotbedrawnthroughthem;fortheequation toanimaginary conicmay alwaysbereduced totheformA.t2+BlI+Oz2=0,whereA,B,0areall positive andcantherefore haveatutmostonerealpoint.Consequently thecase oftotalnon-intersection isdistinguishable fromthatofcomplete intersection bythepeculiarity thatintheone case I-'may be so takenthat U+p.V=0shallrepresent animaginary conic,thatis,U+p.Vwill be a. function whosesignneverchanges forrealvaluesoff,'TJ,~.whereas inthe lattercase novalueofp.willmakeU+p.V=0theequation to animaginary conic,andtherefore U+I-'Vwill have values on bothsidesof zero. On the otherhand,itisobviousthataninfinitenumber ofrealaswellasunreal conics may be drawnthrough fourimaginary pointsofintersection. Con­ sequently ifwemakeU+p.V=0(supposing theintersections ofUandV to beimaginary), therewill be a rangeorrangesof values of p.consistent, andanother rangeorrangesofvaluesofp.inconsistent withrealvaluesof f,1},~;inotherwords,U±p.V=0treatedasanequation between thefour variables E.1},~,p.,willgiveone ormoremaxima orminima valuesofIJ. inthecasesupposed, butnosuchva.lueswhentheintersections aretwoor all ofthemreal. T.)determine thesevaluesofp.,letdp.=0;thenwe have fE(U-p.V)=0, d d1}(U-p.V)=0, d d~(U-p.V)=0, E~(U-p.V)=o. Inorderthatanyvalueofp.found from thisequation maybeamaximum orminimum, Lagrange's condition requires that (d d d)!l hdE+kd1}+ld~p. maybe afunction ofunchangeable sign. 22] Nowexpressed byIndeterminate Coo-rdinates. dUdVdp. d~=p.d~+VdE'123 therefore sincedp.=0, Hence similarly ddId d dE.d7J=VdE.d7J{U-p.V}, Makingnowasbefore&c. &c. &c. U=aE2+b7J2+&c., V=aE2+{37J2+&c., a-ua=A.b-p.{3=B,&c., thecondition forp.,aroot of 0{U-p.V}=0,givingp.amaximum or mini­ mum, may be expressed bysayingthat Ah2+B/c'J.+Cl2+2A'H+2B'hl+2C'hle shall beunchangeable in sign for all real values of h,k, l. Theabovequantity, byvirtueoftheequation 0=0, is always the product of two linearfunctions. Hencewe see, as above indicated, thatif allthesepairsare real, thatis,if allthepointsofintersection ofUandV arereal,thereisnomaximum orminimum value of p.jbutif only one pair bereal and the othertwopairsbeimaginary, thatis,if allthefourinter­ sections are imaginary, thentwo ofthevalues of p.,namelythosecorrespond­ ingtotheimaginary pairs,are real maxima orminimavalues of p.,butthe third is illusory. Now I shall show thatifV=0 is arealconic,buttheintersections of UandVareallunreal,thevalue of p.which makes U+P.Vtheproductof reallinearfunctions of~,"7,~,is always one or the otherextremeofthethree valuesof p.whichsatisfytheequation o (U-p.V)=o. Assume asthethreeaxes of coordinates thethreelinesjoiningthe vertices of thequadrangle each with each, thetwonon-intersecting conics mayevidently bewrittenundertheform U=c(a;2+y2)-e(y2+Z2)=0, V= -'Y(a;2+y2)+E(y2+z2)=0j 124 OntheOorrelations ojtwoOonics [22 theseequations beingonlyothermodesofwriting U=Aa;'+By2+OZ2, V=A'a;'+B'J/+0'#, inwhichA, B,OJA',B',0'will bereal,because byhypothesis D(U+p.V)=O hasallitsrootsreal. Hencee,y,Zarelinearfunctions ofE,'T},~,andconsequently, by asimple inference from a theorem of Prof. Boole", therootsof~{U+p.Vjare identical withthoseof D!U+p.Vj=O. "1/0 . c ec-eTheselatterareevidently-,-,--;thethirdof which is theone''''1ery-€ whichmakesU+p.Vtheproduct of tworeallinears,for we have 'YU+cV=(C€-"'Ie)(y2+Z2), €U+eV=(€C-e<y)(a;'+y2), ('Y-€)U+(c-e) V=(C€-ery)(Z2_ X2)t . Nowc c -ee'Y-C€ ;Y-~-€='Y(ry-~)' ec -eery-C€ ;-ry-€=€(ry-€); ande,"'Iaresupposed tohavethesamesign,asotherwise Vwould be an unrealconic;hencetheascending ordescending orderofmagnitudes of c ec -ethethreevaluesofXfollowsthescale-,- ,---,as was to be shown. ryery-€ Imagine nowlengths reckoned on a line corresponding to allvaluesof p.from - 00to+00,andmarkoffuponthisline bythelettersA, B, 0, thelengths corresponding withthethreerootsofD (U+p.V)=O.Then observing thatwhenp.=±ce,U+I-'Vis ofthesamenatureasV,andis therefore apossible conic by hypothesis, andagreeing tounderstand by a possible andimpossible regionofp.,arangeofvaluesforwhichU+p.V corresponds toapossible andimpossible conicrespectively, one ortheotherof theannexed schemes willrepresent thecircumstances ofthecasesupposed: -00~POSll.Reg. -00~Pass. Reg.AImposs, Reg.BPass. Heg. CPas-OJ.Reg.~+oo APoss. Reg. BImpos.'l.Reg.CPoss. Reg. ~+oo Butineitherschemeitisessential toobserve thatthemiddlerootof D(U+p.V)=0dividesapossible from an impossible region;audtherefure *SeePostscript. tZl -x2=0 of course represents arealpairoflines. ~2] expressed byIndeterminate Coordinates. 125 ifwecan find n, u,any two values lying between thefirst and second and second and thirdroots of theaboveequation arranged inorderoftheir magnitude, one ofthetwoequations U+vV=0,U+nV=0, willrepresent apossible and theotheran impossible conic:one such couple of values mayalways be found by takingtheroots ofthequadratic equation ddp.0{U+p.Vj=o. Hencecallingthetwo roots thereofmandM,we see (which is in itself atheorem) thatoneatleastoftheconicsU+1nV=0,U+MV=0,must beapossible conic, provided onlythatV=0 beapossible conic:ifboth U+mVandU+MVare possible conics, theintersections ofUandVare all real, and if not, not ",Thecriteria fordistinguishing possible from impossible conics beingwell known need not be statedinthisplace. We may of course proceed analogously by forming the two conicslU+V, LU+V,wherelandLare roots of ~0{XU+Vj=0 upon the supposition ofU=0beinga possible conic. Ifeitherof the two UandVbenotpossible, theirintersections are of courseimpossible, andthequestion isalreadydecided. Itwill be seen aspre-indicated thatthismethod only fails in symmetry because of the choice between thecouples 1n,M,andl, L.Butmoreover a perfect methodfor thediscrimination ofthetwo cases of unmixed inter­ sectionone from the othershould(perhaps ?)requiretheapplication of only a singletest(in lieu of thetwoconditions whichtheabovemethodsupposes), over and above the condition which expresses thefact oftheintersections being so unmixed. SuchmoreperfectmethodI have not yetbeen able toachieve. Another interesting question ofintersections remains tobe discussed, namely,supposing the two conics are known tobenon-intersecting, howarewe toascertain iftheyareexternal to oneanother, or if one contains theother? In order to settlethispointwemustfirstestablish acriterion fordetermin­ ingwhethera givenpointisinternal orexternal to agivenconic;thepoint being in generalsaidto beexternal when two real tangents can bedrawn fromittothecurve, and internal whenthiscannotbe done. •Itmustbewell observed however thatthepossibility of the conics U+mVandofU+MV doesnot imply the realityof theintersections unless the conic Visknown to be possible. ForifVbeimpoesible £and'Yhateopposite signa,andtherefore c_-eisintermediate beween e'Y "1-£eandf'andthescheme for p.willbeashereannexed : -..,'"QImJlOMible. .AP088ible. BPOlsible. CImp<l88ible.~+"" BOthatU+mVandU+MVwill both represent possible conics. 126 OntheCorrelations oftwoOonics [22 Letnow ¢(x,y, z)=a:rJl+by!+czt+2a'yz+2b'zx+2c':xy=0, betheequation toanyconic:l,m,nthecoordinates of anypoint.Let A=be-a'l,B=ca-b'l,C=ab-C'2, A'=aa'-b'c',B'=bb'-c'a', C'=ee'-a'b'. Thenthereciprocal equation totheconic is Ar+B.,,2+Cr+2A'7]'+2B'~+2C'~=0, andinmaking l~+m7]+n'=0.theratiosof~.7],,mustberealifthe tangents drawnfroml,m. narereal:thiswill be found to implythatthe determinant A,C'B',, C'B,A'm ,, B',A'C,n, t.'Tn,n,°shallbenegative-. Thisdeterminant may be shownt to beequaltothe product ofthedeterminant a, c',b' c',b,a' b',a',ci bythequantity all+bml+cn2-2a'mn-2b'ln-2c'lm, thatis,equalto¢(l,m, n)x o. Hencel,m,nisinternal orexternal to¢(x,y,z)according as<p(l,m,n) and0¢havethesameorcontrary sign. If¢(l,m, n)=0,thepointlies ontheconic,andthepointisneither internal norexternal; if0¢=0,theconic becomes a pairofstraight lines, andnopointcanbesaideitherto bewithinorwithout suchasystem. Henceourcriterion fails, asitought to do, justinthevery two cases where thedistinction vanishes. Ibelievethatthiscriterion isheregivenfor thefirsttime. • Seetheorem ofthe"Diminished Determinant" inPostscript tothispaper. tAs we know Iiprioribyvirtueofatheorem given by M. Cauchy, andwhich is included &8 aparticular casein atheorem of my own, relating toCompound Determinants, thatis, Deter­ minants ofDeterminants, which will takeits place &Banimmediate consequence of myfund". mentalTheorem givenina Memoir abouttoappear. Thewell-known rule for the multiplication ofDeterminants is also a directandsimpleconsequence from my theorem onCompound Determinants, which indeed comprises, I believe, in one glance, all the heretofore existing Doctrine ofDeterminants. 22J expressed byIndeterminate Coordinates. 127 Toreturntothetwonon-intersecting conics. Letusagainthrowthem undertheform U=(xl+y2)-e"(Zl+y2), V=k(xI+y')-hl(ZI+y'), eandebeingreal,thatis,UandVbeingbothfunctions corresponding to possible conics. Suppose Uexternal toV;thenanypointinUis an external pointtoV. Take inUeitherofthetwopointsrepresented bytheequations y=0, r=e"z2;substituting thesevaluesofyand 11:,Vbecomes k(e"-e')z', and0Vbecomes -k'e'(1 -e');therefore (1 -e')(If-e')mustbepositive, that is,e'mustbe one of theextremes of thethreevalues1,e",e'.Inlike manner, if Visexternal toU,ewill be also one of theextremes ofthesame threequantities; and hence, ifthetwo conics are mutually external, unity willbethemiddlemagnitude ofthegroupIf,1,£'. Nowthethreeroots of 0(V+XU)=0, are e' l-e'h.=-kX=-k- h.=-k--,If' I-e'" HenceifUandVbewithout oneanother, or,asitmaybetermed,are extra-spatial, thethirdvalueofXwill be of a different signfromthefirst two;butifthetwo conics be co-spatial, thatis, if one includes theother,all the three valuesofXwillhavethesamesign.Hencewehavethefollowing elegantcriterion ofco-spatiality of twopossible conicsexpressed bythe equations U=0,V=0,between indeterminate coordinates ~,"I,~;the coefficients ofthecubicfunction 0(XU+p.V)mustgiveonlychanges ort",onlycontinuations ofsign. Ifthistestbenotsatisfied, it willremaintodetermine which of thetwo COlliescontains, andwhich is contained bytheother.LetUcontainV, then the orderofmagnitudes will be1,e',e';therefore k11-e'isgreater-6' thank,andtherefore k~=~,which isthatroot oftheequation o(V+xU)=O whichisalwaysone ortheutheroftheextremes, isthegreatestofthethree. Hencetheschemefortheimpossible andpossible regionsofXwill be as below: -<Xl-gPOI!8. AImp088. BP088. CPOI!8.r:::r+eo Henceifthetwo roots of d~{V+XU}=0belandL,and of the two coniesV+lU=0,V+L U=0,theformer be thepossible, and thelatterthe impossible one,UcontainsVor iscontained in itaccording aslisgreater orlessthanL. 128 Onthe Oorrelations oftwoOonie« [22 Observe thatifUandVbenon-cospatial, sothatthethreevaluesof p.in0(U+p.V)=0 have not all thesamesign and consequently zerolies between thegreatest andleastofthem,itwill not be necessary tomake trialofthecharacters ofthetwocurvesU+1nV=0,andU+MV=0, inordertoascertain whetherUandVintersect ornot;foritwill be sufficient to find which of thetwoquantities 1nandMsubstituted forp. in0(U+p.V)causesitto have theopposite sign to 0(U+0V),thatis, oU,andthisone ofthetwoitis, ifeither,which will makeU+p.Van impossible conic. and will thusaloneservetodetermine whether theinter­ sectionsofUandVareunreal,orthecontrary. Itmightbe acuriousquestion toconsider whether, in acertainsense, conicsnotbothpossible may not be said to lie one withinorwithoutthe other.Upongenerallogicalgrounds, Ithinkitnotimprobable thattwo impossible conicsmightbediscovered eachto contain the other; hutthisis aninquirywhich I have not had leisuretoenterupon. I havethusfarsupposed therootsof0(AU+V)=0 to be all distinct from one another. I nowapproach thediscussion ofthecontactof two conics, in which eventtwo ormoreoftheroots will be equal. Thecondition forsimplecontactisevidently 00(xU+p.V)=o. Ap.t'l' Theunpaired value of Ain0(XU+V)makesxU+Vanimpossible pairof lines, and therefore, intheschemeforXdrawn as above, will separate thepossible from theimpossible region. Whether theconicsintersect in two real or two unrealpoints,besidesthe pointofcontact, will be known atonce by ascertaining whetherU+p.V=0 represents two real or two imaginary lines.Ifthelatter,thetwocurveslie doe-a-dos or onewithintheother,according asthe successions of sign in o(XU+V)are all of thesamekindornot;iftheybe all of thesamekind, one will includetheother,namely,UwillincludeViftheequalrootsare greater,andbeincluded initiftheybelessthantheunequal one.Thislast conclusion however, itshouldbe observed, is inferred upontheprinciple of continuity, bymaking twovaluesofXapproach indefinitely nearto one another, butcannotbestrictlydeduced fromtheequations givenforUand Vapplicable tothegeneralcase, in which theaxes ofcoordinates arethe threeaxesjoiningthevertices; sincetheselatter,inthecasesupposed, reduceto two only, andconsequently suchrepresentation ofUandV becomes illusory. Ifallthreevalues of Aare equal, thethreevertices cometogether, and hence thetwo conics will have threeconsecutive pointsin common, thatis, will have thesame circle of curvature. Onthissupposition thetwo curvescutatthepointofcontact, andall fourpointsofintersection are of course real. 22] expressed byIndeterminate Coordinates. 129 Theclassification ofcontacts between two conics may be statedas follows: Simplecontact=one case. Seconddegreecontact=two cases, namely,common curvature ordouble contact. Thirddegreecontact=one case, namely,contactin fourconsecutive points. These four cases of coursecorrespond totheseveralsuppositions ofthere being-twoequalroots,threeequalroots, two pairsofequalroots, or four equalrootsinthebiquadratic equation obtained between twovariables by elimination performed inanymanner between thegivenequations inthe twoconics. The first speciesandthefirstcaseofthesecond species have beenalready disposed of. Iproceed toassigntheconditions appertaining tothesecond caseofthesecond species, when UandVhave adoublecontact. LetA,A',B,B'bethetwopairsofcoincident pointsinwhichthe conicsaresupposed tomeet;eitherpairof linesAB,A'B',andAB',A'B, becomes acoincident pair.Hencesuch a value of p.can be found as will makeU+p.Vthesquareofa linear function of~,1],~.Iftherefore we makeU+p.V=W,andformthedeterminant d2Wd2Wd2W -dr' d~d~' d~d~'P d"Wd2Wd2W d1]dE'd1]2' d1]d~'q d2Wd2Wd2W d~d~' d~d1]' d~2'r p,q,r,° =Ap2+Bq2+01'2+2Fqr+2Grp+2Hpq, whereallthecoefficients are quadratic functions ofp.,and make A=0,B=0,0=0,F=0,G=0,H=0, eachofthesesixequations inp.willhaveoneandthesameroot in common. Itis, however, enough toselectanythreeiifthesevanishtogether forany value of p.,theremaining threemustalso vanish. Thisisasimple application of a generallaw·which will appearin aforthcoming memoiron "Determinants andQuadratic Forms,"of which thispaperis to beconsidered asanaccidental episode. • Forstatement ofthislaw caned the Homaloidal Law, see Philo.ophical Magazine ofthis month.. On CertainAdditions, &c."[po150 below. ED.] ~ 9 130 On the Correlations oj twoConice [22 Takenowanythreeofthesixequations whichforthesakeofgenerality callP=0,Q=0,R=0.Thehypothesis ofdoublecontact requiresthat PandQ,QandR,RandPshallhave a factor in common; butthese conditions arenotsufficiently explicit forourpresentobject,sinceP,Q.R mightbe oftheform ,,(A.-a)(A.-b),,,'(A.-b)(>..-c),tc"(>..-c)(A.-a), andwouldthussatisfytheconditions abovestated,withoutP,Q,Rhaving acommon factor. A sufficient criterion isthat.fQ+gRandPshallhave acommon factor for all valuesoffandg. Letthentheresultant of.fQ+gRandPbe Lfi+Mfg+Ngi, wemusthave L=O,M=O,N=O, where Listheresultant ofPandQ, N" ""RandQj andMis a new function, which if we call Q=ep(>..),R='t(A.),andsuppose aandbto bethetworootsofP=0, is easily seen to be equalto epa.'tb+epb.'tao ThisI calltheconnective ofP.QandP.R. L, M, N mayconveniently bedenoted bytheforms P.Q,P.R,Q.P.R. We may now takemoregenerally aP+bQ+cR, «P+fJQ+'YR, which will haveafactorincommon for allvaluesofa,b,c,rx,fJ,'Y. I amindebted to MrCayleyfortheremarkthattheresultant ofthese twofunctions is a new quadratic function, which,according tomynotation justgiven,maybeputundertheform PQ(afJ-ba.)t+QR(b-y-cfJ)i+RP(crx_a-y)t +PRQ(b-y-cfJ)(crx-a'Y)+QPR(ce -a-y)(afJ-brx)+RQP(afJ-brx)(b-y-cf3). Ternary systems ofthesixcoefficients formeduponthetypeof(PQ, PQR,QR),I callcomplete systems, becausethethreefunctions included in suchasystemequated severally to zero, implythattheremaining three coefficients areall zero. Suchasystemas(PQ, QR, RP)Itermanincom­ pleteternarysystemas notdrawing withitthelikeimplication. Probably (?) weshouldfind on investigation thatPRQ, QPR, RQP,would also be an expressed byIndeterminate Coordinates. 131 incomplete system,butthatsystemsformedafterthetypeofPRQ, RQ, RQP arecomplete. Thishowever is only matterofconjecture, as J have been too much occupied with otherthingstoenterupontheinquiry. Thedistinct types of ternarysystems are altogether six innumber, namely, four of a symmetrical species, PQ, PRQ, PQ, PRQ.QR, QPR, PQR, RQ,RP. RQP. QR, RQP; and two of an unsymmetrical species, namely, PQ. PRQ,PQR, RQ,PR, QPR.· Ifinsteadof confining ourselves tothreeoutof the six originalquantities, .A,B, C; F, G, H, wetakethemallintoaccount, andwritedownthe resultant of aA+bE+cC+fF+gG+hH, a.A+fJB+'YC+epF+XG+'TJH; weshallobtainaquadratic function of 15variables (nothowever all indepen­ dent)having120 coefficients, all of which mustbe zero.Itwould be extremely interesting todetermine how many complete ternarygroupscan be formed outofthese120terms. Itwill berecollected thatwe have assigned asthecondition ofcontact inthreeconsecutive points,thatacertaincubicequation shall have all its roots real. Now, aswellremarked byMr Cayley, we cannotexpressthis factbylessthanthreeequations inintegraltermsofthecoefficients. Thus ifthecubicbewritten aX'+3bx'+3cX+d=0, wehaveasone of such ternarysystems, u=ac-b"=0,V=bd-c'=0,W=be-ad=O. Thesignificant partsoftheseequations are of course, however, capable of beingconnected byintegralmultipliers U', V', W', suchthat U'U+V'V+W'W=O. • PQ, QR, RP, maybecompared inageneralway with the angles, andPRQ. QPR, RQP, withthesideaofatriangle. 9-2 132 OntheCorrelations oftwoConics [:?2 Anynumber offunctions U, V, W sorelated, I callsyzygetic functions, andU',V', W' Itermthesyzygetic multipliere". Theseinthecase supposed are c,a,b,respectively. Inlikemanneritisevidentthatthemembers of anygroupoffunctions, morethantwo innumber, whosenullityisimplied intherelation ofdouble contact, whether suchgroupform acomplete systemor not,mustbe in syzygy. ThusPQ, PQR, QR, mustform asyzygy; nor isthereanydifficulty inassigning asystem ofmultipliers toexhibit suchsyzygy. Calling P=q,(X),R='ir(X),aandbthetwo roots of Q=0, Ihavefoundthat l('ira)2+('irb)2]PQ-(q,a.'ira+q,b.'irb)PQR+{(q,a)2+(q,b)2)QR=O. Again,ifwetaketheincomplete system (PQ), (QR), (RP), itwill be found that L (QR)+M(RP)+N (PQ) =0, provided that,callinga,b;c,d;e,j:therootsofP=0,Q=0,R=O. respectively, wemake , (a-c)(a-d)(a-e)(a-f)L=(ko+kla+kla2+ksaS+k.a~),--'b-------a- +(ko+k lb+k.bl+k3b"+k.b~)(b-..c)_(bbd)(b=e)jb_-f),-a (c-a)(c -b)(c -d)(c -e)(c-f)1lf=(ko+ k1c+/..'lcl+ k.c'+ k,c') ---' ---d------c- +(ko+kId+kad2+ksds+k,d')(d..=..a) (c!-..-=-b~Sdd_= c)(d-e)(d_-.n,-c (e-a) (e-b)(e-c)(e-d)N=(ko+k1e+k2el+kse"+/..'.e')--------f----e- +(ko+kJ+kdl+/..·afs+k.j')(L=a2<L- J~{-=-Cl(f -d); i;kl,kl,s;k,beingquitearbitrary, andL, M, N, although presented ill afractional form,beingessentially integral. Thisfact ofL, M, N constituting asystemofmultipliers tothesyzygy QR, RP, PQ, iseasilydemonstrated; for QR=(c-e)(c-f)(d-e)(d-f), RP=(e-a)(e-b)(f-u)(f-b), PQ=(a-c)(a-d) (b-c)(b-d). *Therewillbe ingeneralvarious such systemsofmultipliers. 2"2] expressed byIndeterminate Coordinates. 133 Hence L(QR)+M(RP)+N(PQ) =(a-c)(a-d)(a-e)(a-f)(b-c)(b-d)(b-e)(b-j)(c-e)(c-f)(d-e)(d-f) x~.__ko±kla+k2a2+k.al+k.a&_° (a-b)(a-c)(a-d)(a-e)(a-j)- . Mytheoryofelimination enables me toexplain exactlythenatureof L, M, N, andthereasonoftheirappearance assyzygetic factors. LetLT)u..s.signifywhatL, M, N become, when all thek'sexceptkr aretakenzero.Thenthetheorygivenby me in thePhilosophical Magazine furtheyear1838, or thereabouts j',showsthatLoX+LIistheprimederivee ofthefirstdegreebetween thetwoequations PandQxR,or, inother words,will betheremainder integralized of9{: Inlikemanner Mox+MI,NoX+N1aretheintegralized remainders of Rtandof~Qrespectively. Ifnowtheresultant ofP,QandofQ,Rareeach zero, buttheresultant ofPandRisnotzero, it will be evidentthatP,Q,Rmustbe oftheform f(X+a)(X+c), g(X+c)(X+d),h(X+d)(X+b); andthereforePxRwillcontainQ,andconsequently wemusthave Mo=O, M1=0. Moregenerally, if wewrite Q=o, XQ=O, X2Q=0, PxR=O, andeliminate dialytically, thatis,treating X·,XI,XI,Xasdistinct quantities, weshallobtain- X·:XI:X·:X:1 ::M.:M.:MOl:M1:Mo; andtherefore whenPxRcontainsQ, Mo=O, MI=O, MOl=0,M.=O, M.=O. •Thiscannotbeobtained directlyfromwhatisstatedinthepaperreferredto,although contained inthe general theoryofderivation theregiven. The arbitrary functions whichenter intotheexpreasion forthegeneralderivees have been inthatpaperevaluated onlyfortheprime deril'eeB, which however are only particular phenomena, withreference tothegeneralresultsof Dialytic Elimination. Hereafter I may give amoregeneral exposition ofthisremarkable, although ignoredorneglected theory. TheprimederiveesoffxandI'xareSturm'sFunctions, clesredofquadratic factors,andareexpre88ed byvirtueofthegeneraltheorems therelaid down ufunctions ofxandofsymmetrical functions oftherootsoff», [tp.40above.ED.] 134 OntheCorrelations oftwoConics [~2 Inlikemanner, whenQxPcontainsR, No=0,N1=0, N2=O,N.=O, N4=0; andwhenRxQcontainsP, ~=~ ~=~ ~=~ ~=~ ~=Q Accordingly, we see from theequation L(QR)+M(RP) +N(PQ)=O, thatifQR=0,RP=0;butPQnot=0,thenN=0;andtherefore No=0, N1= 0,.I..Vs=0,N,=0,lV.=0, andso in like mannerfortheremaining corresponding two supposit.ions ", Beforeproceeding toconsider theremaining case ofthehighestspecies ofcontact,Imustobservethatbesidestheequations involved in thecondi­ tionthatA,B, C: F, G, H,or, which is thesamething,thatanythree ofthemshallall have a factorin common, we musthave0(U+AV)con­ tainingthesquareofsuchcommon factor. In thememoirbeforeadverted to ageneraltheorem will begivenand proved, which shows thatthislatter condition isinvolved intheformer one jin fact, more generally (butstill only as a particular case)thatwhenUandVarequadratic functions ofn letters,butU+£Vadmitsofbeingrepresented as acomplete function of (n-2)quantities only, which are themselves linearfunctions of thenletters, then0(U+AV),which is of courseafunction ofAofthenthdegree,will containthefactor(A_£)2, Whenthetwo conics have four consecutive pointsin common, the characters ofdouble-point contactand ofcontactinthreeconsecutive points mustexistsimultaneously jandconsequently thefactor common to A, B,C'; F, G, H, willenternot as a binarybutas aternaryfactorinto0(U+AV). Thisgivestheextracondition required. Asanexample takethetwo conics, y2 U=1 _k+,xi-z?=0, V=y2+W-2kxz+('2k-1)Z2=0, U+AV=(1~k+A)y'+(1+A)W- {I+A(I -2A:»)Z2-2kXxz. • Since we areabletoassign the values of the syzygetic multipliers in theequations L(PQ)+JI (QR)+N(RP)=O, L'(PQ)+M' (PQR)+N' (QR)=O, L"(QR)+JI" (QRP)+N" (RP)=O, L'"(RP)+.lI'" (RPQ)+N"'(PQ)=O, itfollowsthatwe may eliminate between thesefourequations anythreeof the six quantities (PQ),(PRQ),&c.,andthusexpressanyone ofthemintermsofanytwoothers:thismethod, however, is not practically convenient. I may probably hereafter returntothissubject. 22J expressed byIndeterminate Ooordinates. 135 Thecomplete determinant ofU+AVisthen -I 1I-k{I+(I-lc) A){(I+A)2-2kA(I+A)+k2A2)=-I-k {I+(1-k)A}s. .A,B. Carethedeterminants ofU+>..V,whenx=0,y=0.z=0.respectively. Thus A= (1~k+A)(1 +>..). B=(1~-k+A){I+x(I -2k»), C=k2>,,'-(I +>..)(I+A(I-2k»)=A2(I-k)2-2>..(I-k)-I; I I;\=-1..::IemakesA=0,B=0,C=0,andthefactorA+f _kenterscubed into0(U+AV). Hencethetwo conics haveacontactoftbethirdorder. Thisiseasilyverified; forifwepassfromgeneral toCartesian and rectangular coordinates. andmakezunity;U=0 willrepresent anellipse withcentreattheorigin.eccentricity vk,andmeanfocaldistance I,and V=0thecircleofcurvature attheextremity oftheaxismajor", Ihadintended tohaveaddedsomeotherremarks connected withthe presentdiscussion, andalso tohaveappended anaposteriori proofofthe propositions relative totherealityandotherwise oftheverticesandchordal pairs of intersection whichIhave,atthecommencement ofthispaper, deduced quitelegitimately, butin amanner notatfirstsightperhaps easily intelligible, fromthegeneral principles ofconjugate forms;butthisdis­ cussion hasrunonalready toa.lengthsomuchgreater thanIhad anticipated andthantheimportance oftheinquiry mayseemtojustify, thatImustreservefor afuturenumber oftheJournal whatfurthermatter I mayhavetocommunicate concerning it. POSTSCRIPT.-As I have alluded toProfessor Boole's theorem relativetoLinear Transformations, it may bepropertomention my theorem on the subject, which isofamuch more general character, andincludes Mr Boole's (so far asitrefersto Qnadratic Functions) asacorollary toaparticular case.Thedemonstration will begiven intheforthcoming memoir above alluded to. LetUbeaquadratic function of any numberofletters31,:l;...x"'and let anynumber roflinearequations of the general form Jarx1+~rx.J+ + "arx"=0, • Wehavethusdiscussed allthe four casesofbiconieal contact: for anexactlyparallel diAcusion ofthetheoryofcontactofaplanewith the curve of double curvature in which two mrlllCEllofthesecondorderintersect, see thepaperin thePhilosophical Maga.tine fortRismonth, beforereferredto.[p, 148 below. En.] 136 OntheCorrelations oftwo Conics [22 beinstituted between them:andbymeansoftheseequations letUbeexpressed asafunction ofany(n-r)ofthegivenletters,sayofXr+l,XrH......x..,andlet U,soexpressed, becalledM.Let be called Lr•Thenthedeterminant ofMinrespecttothe(n-r)lettersabove givenisequaltothedeterminant of U+LIXn+l+L~XnH+..,...+Lrxn+TI considered asafunction ofthe(n+r)letters dividedbythesquareofthedeterminant I::::::::::::: I I•••••••••••••••••• I Ian-P-r...... ~r This I call thetheorem ofDiminished Determinants. Ifnow we have Uafunction ofrletters,andVofrotherletters,andVis derivedfromUbylineartransformations, thatis,byrequations connecting the 2rletters; then,sinceTJmay beconsidered asafunction ofaUtlte2rletterswith abortive coefficients for all thetermswhereanyofthesecondsetofrlettersenter, we may applyourtheorem ofdiminished determinants tothequestion so con­ sidered,andtheresultmay be found to represent MrBoole'stheorem inaform rathermoregeneralandsymmetrical, hutsubstantially identical withthatgiven byMrBoole. Thussuppose!ar+bx!/+!cy2sayP,and!au~+p?w+!yVsayQ,aremutually transformable byvirtueofthelinearequations lx+my=AU+PO", l'x+m'y=A'u+po'v, Pmay beconsidered asafunction ofx,y,U,v,andQasthevalueofP,whenwe eliminate xandyhyvirtueofthetwolinearequations LI=la:+my-AU-pov=0, L~=rx+m'y-A'u-po'v=o; we have therefore byourtheorem thedeterminant ofQequaltothesquared reciprocal ofthedeterminantI~:,:'Imultiplied bythedeterminant a,b,0, 0, l, l' b,c,0, 0, m,m' 0, 0, 0, 0, - A,-A' 0, 0, 0, 0, -po,-po' l,m,-A,-po,0,° I', 111',-A',-po',0,° 22] expressed byIndeterminate Coordinates. "137 whichlastdeterminant isevidently equaltothedeterminant ofPmultiplied by thesquareofthedeterminant I~:"",!.Whence we seethatthedeterminant ofQ,,.,., divided bythesquareofI~".I,isequaltothedeterminant ofPdivided by,1\.,"" thesquareofI~,:'I.Thereis alsoanother way more simple, butlessdirect,by meansofwhichthetheorem ofdiminished detenninants may be madetoyield MrBoole's theorem oftransformation *.Someunavowed usehasbeenmade intheforegoing pagesofthisformertheorem, one ofthehighestimportance in theanalytical andgeometrical theoryofquadratic functions. Ithasbeennearly ayearinmypossession, andItrustandbelievethatI amcommitting noactof involuntary misappropriation inannouncing itasa.resultof my own researches . •Namely, byconsidering PandQastachderived from some common function ofe,y.u,V. It,bymeansoftheequations L1=0.L2=0;the law of Diminished Determinants willthenindicate thedeterminants ofPandQ.eachunderthe form of fractions havingtheBamenumerator, but whoeedenominators willbel~;1J.,12 andIll;171,Irespectively. 1\.1J.I .711 23. ANINSTANTANEOUS DEMONSTRATION OF PASCAL'S THEOREM BYTHEMETHOD OF INDETERMINATE COORDINATES. [Philosophical Magazine, XXXVII. (1850),p.212.] THE new analytical geometry consists essentially of twoparts-the one determinate, theotherindeterminate. Thedeterminate analysis comprehends thatclass of questions inwhich itisnecessary toassume independent linearcoordinates, orelsetotake cognizance oftheequations by which theyareconnected iftheyarenot independent. Theindeterminate analysis assumes atwill any numberof coordinates, andleavestherelations whichconnect themmore or less indefinite, and reasons chiefly through themedium ofthegeneral pro­ pertiesofalgebraic forms,andtheircorrespondencies withtheobjectsof geometrical speculation. Pascal's theorem ofthemystichexagon, andthe annexed demonstration ofitsfundamental property, belong to thisbranchof thesubject,and afford an instructive andstrikingexample oftheapplication ofthepuremethodofindeterminate coordinates. Letx,y,Z,t,u,vbethesides of a hexagon inscribed intheconicU.Let thehexagon bedividedby a new line epinanymannerintotwoquadri­ laterals, sayxyzep,tuvep. Then ayep+bxz=U=aUI/>+f3tv; therefore (ay-au)ep=f3tv-bxe; therefore ay-auandeparethediagonals of thequadrilateral txvz. Byconstruction, episthediagonal joininge, v(thatis,theintersection of xandv)withz, t;andthuswe seethatay-auisthelinejoiningt,xwith v,z;butthisline passes throughy,u.Therefore x,t;y,t£;Z,vlie in one andthesamerightline.Q.E.D. 24. ON A NEW CLASS OF THEOREMS INELIMINATION BETWEEN QUADRATIC FUNCTIONS. [Philosophical Magazine, XXXVII. (1850),pp.213-218.] Isaforthcoming memoirondeterminants andquadratic functions, Ihave demonstrated thefollowing remarkable theorem as aparticular caseofone much more general, alsotheregivenanddemonstrated. LetUand.yberespectively quadratic functions ofthesame2nletters, andletit besupposed possible to institute nsuchlinearequations between theselettersasshallmakeUandVbothsimultaneously becomeidentically zero".Thenthedeterminant ofxU+p.V,which is of course a function of Xandp.ofthe2nthdegree,will become thesquareof afunction ofXandp. ofthenthdegree; andconversely, if thisdeterminant be aperfectsquare,U andVmay be made to vanishsimultaneously bytheinstitution ofnlinear equations between the2nletters'[. LetnowPandQberespectively quadratic functions ofthreelettersonly, sayx,y,z;andlet U=P+(lx+my+llz)t, V=Q+k(lx+my+nz)t. Thedeterminant ofxU+p.Vinrespecttox,y,z,tiseasily seen to be (X+kp.rxthedeterminant of xP+p.Q+(lx+my+nz)t inrespecttox,!I,z,t.Henceif we call xP+p.Q+(lx+my+nz)t=W, and make CWasquaredfunction ofX,p.or which is thesamething,ifryzt DO[lV)=O, A,.xgzt • Inthemoregeneraltheorem abovealludedto,thenumberoflettersisanynumberm,the numberof linearequations beinganynumbernot exceeding ~. tWhen n=I.....eobtainatheorem ofelimination between two quadratics, which has been llreadygivenbyProfessor Boole. 140 On anewClassofTheorems [24 UandVwillvanishsimultaneously whentwolinearrelations areinstituted between thequantities (all orsomeofthem)e,y,e,t. Inorderthatthismaybethecase,itwill beseento besufficient that P=0,Q=0,(lx+my+nz) =0, shallcoexist jforthentwoequations betweenx,y, zofwhichlx+my+nz =0 willbeone,will suffice to makeUandVeachidentically zero.Hencewe havethefollowing theorem: DO{XU+fLV+(lx+my+nz)t) ~,.~lIzt isafactoroftheresultant of P=0,Q=0,lx+my+nz=O. Acomparison oftheordersoftheresultant andthedeterminant shows thattheymustbeidentical, a-ci-pree, ofanumerical factor,which,ifthe resultant betakeninitsgeneral lowestterms,maynodoubtbeeasilyshown to beunity. As anillustration ofourtheorem, let P=xy+yz+zx, Q=cxy+ayz+be». Then l 01n,X+bu; X+afL, 0, m,x+CfL, 0, X+afL,I0, o{XP+fLQ+(lx+my+1tz)t) =X+CfL, :tIIzt X+bp.. It. =nl(X+CfL)1+mO(X+bfL)!+l2(X+afL)1 - 2lm(X+bfL)(X+afL)-2mn(X+CfL)(X+bfL)-2nl(X,+afL)(X+CfL) =XI{n!+m!+l!-2lm-2mn-2nl] +2XfL{enl+bm»+al'-hn(a+b)-mn(b+c)-nl(c+a») +fL2{&nl+b!rlt!+a2l2-2ablm-2bcmn-2canl). Andwethusobtain,finally, o0{AP+fLQ+(lx+my+nz)t] IIjo"'lIzt =(nl+ml+l2_2lm-2mn-2nl) x(c'nl+b~n2+a2l2-2ablm-2bcmn-2canl) - {(enl+bm2+all-lm (a+b)-mn(b+c)-nl (c+aW = -4lmn{(a-b)(a-c)l+(b-a)(b-c)m+(c-a)(c-b)n]. :?4JinElimination betweenQuadratic Functions. 141 Now toobtaintheresultant of xy+yz+sa:=0, cxy+azy+bxz=0. lo:+my+11Z=0, we ueed only findthefoursystems intheirlowesttermsofx:y:z,which satisfythefirsttwoequations, andmultiply thefourlinearfunctions obtained bysubstituting thesevaluesofx,y, zinthefourth:theproduct willcontain theresultant ofthesystemaffected withsomenumerical factor.Inthe presentcase,thefoursystems ofe,y,zare a:=0,y=0,z=1, y=O, z=O.x=1. z=0,a:=0,y=1, x=(lL-b)(a-c), y=(b-a)(b-c), z=(c-a)(c-b), andaccordingly theproductof lXI+mYl+nzl, lx,+my~+nz~, lXa+mys+nzs, lx;+my.+nz., becomes lmn((1-b)(a-c)l+(b-a)(b-c)m+(c -a)(c-b)nJ, agreeing withtheresultobtained by mytheorem,-a specialnumerical factor4,arisingfromthepeculiar form of theequations, havingdisappeared fromtheresultaut. Ageometrical demonstration maybegivenofthetheorem which IS instructive initself,andwillsuggest aremarkable extension ofitto functions containing morethanthreeletters; theequation o\xU+/-,V+(lx+my+nz)t)=0,ZI/ttt which is aquadratic equation inA:/-"mayeasilybe shown to implythatthe conicxU+/-'Vistouched bythestraight line le+my+nz=O. And we thusseethatingeneraltwo conics, xU+/-,V=O, passingthrough theintersections of twogivenconics, U=O, V=O, 142 On anewClassofTheorems [24 may be drawntotouchagivenline. If, however, thegivenlinepasses through any of the four pointsofintersection, in such caseonlyone conic can be drawntotouchit;accordingly o0txu+f£V+(lx+my+nz)t) mustbe zero when l,m,71aresotakenas tosatisfythiscondition, thatis,if le,+my)+llZ)=0, or lx,+my,+nz,=0, or lx,+my,+nz,=0, or le,+my.+nz.=0, whencethetheorem. NowsupposeUandVto be each functions of fourletters, ai,y,z,t; when o{XU+f£V+(lx+my+nz+pt)u) =0, zrzltc theconoidxU+f£Vtouchestheplane lx+my+nz+pt=0 ; and0=0beinga cubicequation, ingeneralthreesuch conoids can be drawn. Considerations ofanalogy makeitobvious to theintuition, thatinthe particular case of two of thesebecoming coincident, thegivenplane lx+my+nz+pt mustbe atangentplanetothosetwocoincident conoidsatone ofthepoints whereitmeetstheintersections ofU=0,V=0;thatis lx+my+nz+pt=0 will pass through atangent lineto,or inotherwords,maybetermed .atangent planetotheintersections. Hencethefollowing analytical theorem, derived fromsupposing q,r,S,tto beproportional totheareas ofthetriangular faces of thepyramid cutout of space by thefour -coordinate planesto which e, y. z, t refer. As theseplanesareleft indefinite, q,7',S,tareperfectly arbitrary. Theorem. 1. 2. 3. 4.Theresultant of U=O}V=0,whereUandVarefunctions ofa;y,z,t; l»+my+nz+pt=0 ; dUdUdU dU I dx'dy'dz'dtI dVdV dV dV dx'dy,de 'dt=0; l, g,m,n, 8,P t 24] ~nElimination betweenQuadratic Functions. 143 whichsystem,itwillbeobserved, consists ofthreequadratic functions, and onelinear function ofe, y, z, t, contains thefactor DO{XU+p.V+(lx+my+nz+pt)u}. ).p.%Vzt Thislastquantity is ofthe4 x3th,thatis,the12thorderinrespectof the coefficients in UandVcombined; ofthe4x 2th,thatis,the8thorder inrespect of l,?n,n,p;and ofthezeroorderinrespectofq,r,8,t. Theresultant whichcontains it is ofthe(4+4+2.4)th,thatis,16th orderinrespecttothecoefficients in UandV;ofthe(4+8)th,thatis,the 12th,inrespectofl,m,n,p;and ofthe4thinrespectofq,1",S,t.Hence the special (and, asfarasthegeometry ofthequestion is concerned, the unnecessary,I may not say extraneous orirrelevant) factor which entersinto theresultant is ofthe4thorderinrespecttothecombined coefficients of U andV·;andofthesame order in respecttol,?n,n, p,and inrespectto q,r,8,t. I havenotyetsucceeded indivining itsgeneralvalue. In theveryparticular example, ofthesystem, a:JfJ+f3yi=0, cz2+dt'=0, .orla:+my+nz+pt=0, ax,f3y,0,°0,0,ce,dt=0, l, V~,n,p q,0,0,° Ifindthatthedoubledeterminant is c2d2a2f32(cpt+dni)t(m2a+l2f3)2, andtheresultant is f/c'd'a2{3f(cp2+dn2)4(m2a+l2f3)2, givingasthespecialfactor f/f32(cpt+dn2)i. I'believethatthetheorem which I have here givenfordetermining the conditionthatle+my+nz+ptshall be a tangent planetotheintersection oftwoconoids UandV,namely,thatthedeterminant of AU+p.V+(la:+my+nz+pt)u shallhave two equal roots, is altogether novel. •Andoollllequently of the second inrespect to the separate coefficients ofeach, 144OnanewGlassofTheorems inElimination. [24 Whatisthemeaning of allthreeroots of thisdeterminant becoming equal,thatis, of only one conoid beingcapableofbeingdrawnthroughthe intersection ofUandVtotouchtheplane lx+my+llZ+pt? Evidently (exvi(walogi(e) thatthisplaneshallpassthrough threecon­ secutive pointsofthecurveofintersection, thatis,thatitshallbethe osculating planetothecurve. Ifwereturntotheintersection of twoco-planar conics,andifwesuppose a line to be drawnthrough two ofthepointsofintersection, theconics capableofbeingdrawnthrough thefourpointsofintersection totouchthe fine, besides becoming coincident, evidently degenerate eachintoapair ofrightlines.Itwould seem, therefore, by analogy, thatif aplanebe drawnincluding any two tangent lines to thecurveofintersection of two surfaces of theseconddegree, thisshouldbetouched by twocoincident conesdrawnthrough thecurveofintersection, andconsequently everysuch doubletangent plane to theintersection of two conoids (anditisevident thatone or more of thesecan betakenateverypointofthecurve)must passthrough one oftheverticesofthefour cones in which theintersection may also be considered tolie;andit would appearfrom this, thatingeneral fourdoubletangent planesadmitofbeingdrawntothecurve, which is the intersection of two conoids, ateachpointthereof. Atparticular points atangent planemay be drawnpassing through morethanone ofthe vertices, andthenof course thenumber ofdoubletangent planesthatcan bedrawnwill be lessened. Theseresults,indicated byanalogy, become immediately apparent onconsidering thecurveinquestion astracedupon anyone ofthefourcontaining cones.Fortheplanedrawnthrough a tangent atanypoint,andthevertexoftheconebeingatangent plane tothecone,mnstevidently tonchthecurveagainwhereitmeetsit. We thushave an additional confirmation oftheanalogy between apointof intersection of twocurvesandthetangentatanypointoftheintersection of two surfaces. Imightextendtheanalytical theorems which have been established for functions ofthreeand four to functions ofagreaternumber ofvariables; butenoughhas been done to pointoutthepathtoanewandinteresting classoftheorems atonce inelimination andingeometry, which is all that I haveatpresentleisureorthedisposition toundertake. 25. ADDITIONS TO THE ARTICLES·, "ONA NEW CLASS OF THEOREMS," AND"ON PASCAL'S THEOREM." [Philosophical Magazine, XXXVII. (1850),pp.363-370.] FIRSTaddition.-I havealludedinthesecond of theabovearticlesto a moregeneraltheorem, comprising, asaparticular case,thetheorem there given for thesimultaneous evanescence of two quadratic functions of 2nletters, on nlinearequations becoming instituted between theletters. Inordertomakethisgeneralization intelligible, Imustpremise a few wordsontheTheoryof Orders, a termwhich I have invented withparticular reference toquadratic functions, although obviously admitting of a more extended application. Alinearfunction of allthelettersentering intoa function or systemoffunctions underconsideration I call an orderofthe letters, or simplyan order. Now it is clearthatwe may always consider a function of any numberoflettersas a function of asmanyordersasthere areletters;butincertaincases afunction may be expressed intermsof a fewernumberofordersthanithasletters,as when thegeneralcharacter­ isticfunction of a conic becomes thatof apairof crossing lines or a pairof coincident lines, in which eventitlosesrespectively oneandtwo orders, and 80forthecharacteristic of a conoid becoming thatof a cone, a pairofplanes or twocoincident planes,in which severalevents, a function of four letters becomesthatof onlythreeorders, or two orders, or one order, respectively. Whenafunction may beexpressed by means of rorders less thanitcontains letters, I call itafunction minusrorders. I now proceed to statemy theorem. LetUandVbefunctions each ofthesamemletters,andsupposethat thedeterminant 10respectofthoselettersofU+J.I.Vcontainsipairsof [*pp.138, 139 above. ED.] 8. 10 146 On a new Class ofTheorems. [25 equallinearfactors of JJ.;thenitispossible, by means of ilinearequations instituted between theletters,to make UandVeachbecomefunctions of the same m - 2iorders;and conversely, if by i equations between theletters UandVmaybe made functions ofthe same m - 2i orders, thedeterminant ofU+JJ.Vconsidered as a function of JJ.willcontainisquarefactors. Thuswhen 'In=2nandi=n, UandVwill each become functions of zero orders, thatis, willbothdisappear, provided thatontheinstitution of acertainsystemofnlinearequations, amongthelettersof which UandV are functions, thedeterminant of(U+JJ.V)is aperfectsquare,-which is thetheorem ~ven inthearticlereferredto. So forexample ifUandVbequadratic functions of four letters,and therefore thecharacteristics of two conoids, 0(U+JJ.V)beingaperfectsquare, expresses thatthese conoids have a straightline in common lyingupon each oftheirsurfaces. IfUandVbequadratic functions ofthreelettersonly, and admitthere­ fore ofbeingconsidered asthecharacteristics of two conics, 0 (U+JJ.V) containing asquarefactor, is indicative oftheseconicshavinga common tangentata common point,thatis, oftheirtouching eachotherat some point;foritis easily shown thatthedisappearance of two orders from any quadratic function byvirtueof onelinearfunction of itslettersbeingzero, indicates thattheline, plane, &c.of which thelinearfunction isthe characteristic is atangenttothecurve, surface, &c.of which the quadratic function isthecharacteristic .. I pass now to a generalization ofthetheorem which shows how to express,undertheform of a double determinant, theresultant of onelinear and two quadratic homogeneous functions ofthreeletters(which I should have given in theoriginal paper,had I not therebeen more intentupon developing anascending scalethanofexpatiating upon asuperficial ramifi­ cationofanalogies), andwhichconstitutes mySecondaddition tothatpaper. towit- HUandVbe homogeneous quadratic, andL,. L,...Lnhomogeneous linearfunctions of(n+2)letters Xl'Xt...Xn+2,thedeterminant oftheentire systemofn+2functions isequalto U I ===oJ {AU+JJ.V+LI~+L2t..a+ ..•+Lntn); A,ItzIJXt···:1;a+til'I •...t.. thedemonstration isprecisely similartotheanalytical one given 10the September Number- fortheparticular case ofn=1. Whenn=0,werevertto Mr Boole's theorem ofelimination between U andValreadyadverted to.Theproof,itwill be easily recognized, does not requiretheapplication of the more generaltheorem relativetothesimul- I"p.140above.] 25] On a new Glass ofTheorems. 147 taneousdepression of orders of two quadratic functions, butonlythelimited onebeforegiven, which supplies theconditions oftheirsimultaneous disparition, I now proceed to develope more particularly certainanalogies between thetheoryofthemutualcontacts of two conics, and thatofthe tangencies totheintersection of two conoids. ButhereagainImustanticipate some of theresultswhich will be given in myforthcoming memoir onDeterminants andQuadratic Functions. byexplaining whatis to be understood by minor determinants, andthe relationin which theystandto thecomplete determinant in which theyare included. Thispreliminary explanation, andthestatement oftheanalogies abovealludedto, willconstitute myThirdandlastaddition. Imagine aoydeterminant setoutundertheform of a squarearrayof terms.Thissquaremay beconsidered asdivisible intolines and columns. Nowconceive anyone line and a.nyone columnto bestruckout, weget inthiswayasquare,onetermless inbreadthanddepththantheoriginal squareiand byvaryingin every possible mannertheselection oftheline andcolumnexcluded, weobtain,supposing theoriginalsquaretoconsistof nlinesandncolumns, nSsuchminorsquares, each of which will represent what ItermaFirstMinorDeterminant relative totheprincipal orcomplete determinant. Now suppose two lines and two columns struckout from the original square, ~eshallobtainasystemofr(\-=--!2r squares,each two termslowerthantheprincipal square,andrepresenting adeterminant of onelowerorderthanthose above referred to.Theseconstitute whatIterm asystemof Second Minor Determinant!'! iand so in generalwe can form asystem ofrthminordeterminants bytheexclusion of rlines and r columns, andsuchsystemin general willcontain {n (n-1)...(n-r+1)}S 1.2...r distinctdeterminants. Isay"ingeneral" ibecauseiftheprincipal determinant betotallyor partially symmetrical inrespecttoeitheror each of its diagonals, the numberofdistinctdeterminants appertaining to eachsystemof minors will undergo amaterial diminution, which is easily calculable. ~ow I have established thefollowinglaw:- Thewholeof asystemofrthminorsbeingzero,impliesonly(r+1)S equations, thatis, bymaking(r+1)1oftheseminorszero, all will become zeroiandthisistrue,nomatterwhatmay bethedimensions or form of the complete determinant. Butfurthermore, ifthecomplete determinant be formedfromaquadratic function, 80as to be symmetrical aboutone of its diagonals, thenl(r+1)(r+2) only of therthminorsbeingzero, will serve 10-2 148 OnanewGlassofTheorems. [25 toimplythatalltheseminorsarezero. Of course, in applying these theorems, caremustbetakenthattoe(r+IforHr+l)(r+2)selected equations mustbemutually non-implicative, andshallconstitute indepen­ dentconditions. Intheapplication I amabouttomakeoftheseprinciples, we shall have only to deal withasystemoffirstminorsandofasymmetrical determinant. Ifthreeoftheseproperly selected be zero, from theforegoing it appears thatallmustbe zero. NowletUandVbecharacteristics of two conics, thatis,let eachbe afunction of onlythreeletters,it may be shown (seemypaper-inthe Cambridge andDublinMathematical Journal forNovember, 1850)thatthe different species of contacts between thesetwo conics will correspond to peculiar properties ofthecompound characteristic U+11.V. Ifthedeterminant ofthisfunction have two equal roots, theconics simplytouch;ifit havethreeequalroots,theconics have a singlecontact of ahigherorder,thatis,thesamecurvature; ifitssix first minors become zero simultaneously forthesame value of 11.,theconics have a double contact.Ifthesamevalue of p"whichmakesallthesefirst minors zero, beatthesametimenotmerelyadoubleroot (as of analytical necessity italwaysmustbe)butatrebleroot of o (U+11.V)=0, thentheconics have a singlecontactofthehighestpossible ordershortof absolute coincidence, thatis,theymeetin fourconsecutive points. Theparallelism between thistheoryandthatof twoquadratic functions P,Q,andonelinearfunctionLtof fourletters,sayx,y,z, t,isexactj. ForletP+Lu+p.Qbe nowtakenas ourcompound characteristic (a func­ tion,itwill be observed, of five letters,a,y,z,t,u);ifitsdeterminant have twoequalroots,Lhas two consecutive pointsin common withtheinter­ sectionofPandQ,thatis, passes through atangent tothatintersection; if it have threeequalroots,Lhasthreeconsecutive pointsin common with thesaidintersection, thatis, is an osculating planethereto; ifitsfifteen firstminorsadmitof allbeingmadesimultaneously zero,Lhas adouble contactwiththeintersection ofPandQ,thatis, it is a tangent planeto some one of thefour cones of thesecondordercontaining thisintersection; [*p. 119above.] tObserve thatP=O,Q=0,L=0nowexpress theequations to twoconoids andaplane respectively. :::Thisparallelism maybeeasilyshownanalytically toimply,andbeimplied, inthe geometrical fact,thattheoontactoftheplaneL withtheintersection of the two surfaces Pand Q,is ofexaotlythesamekindasthecontact(whichmustexist)between the two oonics which aretheintersections ofPandQrespectively withtheplaneL. 25] OnanewClassofTheorems. 149 ifthesamelinearfunction ofp.whichentersintoallthesefirst minors be contained cubically inthecomplete determinant, thentheplaneLpasses through fourconsecutive pointsoftheintersection ofPandQ,andthe pointswhereitmeetsthecurvewill bepointsofcontrary planeflexure j and,asitseems to me, atsuchpointsthetangential direction ofthecurve mustpointtothesummit of one or otherofthefour cones above alluded toe.Inassigning theconditions forLbeinga double tangent planetotheintersection ofPandQ,we may takeanythreeindependent minorsatpleasure equal to zero. One of thesemay be selected so as to be clearofthecoefficients of Ljin fact,thedeterminant ofP+p.Qwill be a first minor of P+p.Q+Lu;p.maythusbedetermined by abiquadratic equation; andthen,byproperly selecting thetwootherminors, we may obtaintwoequations in which onlythe first powers of thecoefficients of e,y,Z,tinLappear,and may consequently obtainLundertheform of (ae+a)x+(be+(3)y+(ce+"I)Z+(de+0)t, \\'herea,crjb,13;e,'Y;d,0will be known functions ofanyone ofthefour valuesofp..Thepointofcontactbeinggivenwillthenserve todetermine e,andweshallthushavetheequation to each of thefourdoubletangent planesatanygivenpointfullydetermined. Inthe foregoing discussions I have freely employed thewordcharacter­ Uticwithoutpreviously defining itsmeaning, trusting tothatbeingapparent fromthemode of its use. Itis atermofexceeding value for itssignificance andbrevity. Thecharacteristic of ageometrical figuretisthefunction which,equated to zero,constitutes theequation to such figure. Plucker, Ithink,somewhere calls ittheline orsurfacefunction, as thecase may be.Geometry, analytically considered, resolves itselfintoasystemof rules fortheconstruction andinterpretation ofcharacteristics. One more remark, andI have done. A verycomprehensive theorem has been givenatthe commencement ofthiscommentary, forinterpreting theeffect of a complete determinant of alinearfunction of two quadratic functions(U+P.V),having •Ifthisbe80,thenwehavethefollowing geometrical theorem:-"The.ummitofOIUofthe four Clm~.oftheseconddegreewhichcontain theintersections oftlCOsurface«ofthesectnuiorder dralt'ninanymannerreB]Jeclit'ely through tirOgivenconic.lyinginthe.ameplane,andhaving .nthoneanothera contact ofthethirddegree,urilialu:ay.befoundinthe.alllerightline,fUJnlely iftthetangentlinetothetwogivt71conic.atthepointofcontact:' tMoregenerally, thecharacteristic ofanyfact orexistence is thefunction which,equated to zero.expresses thecondition of theactuality of suchfactorexistence. Perhaps themostimportant pervading principle ofmodern analyeis,butwhichhasnever hitherto beenarticulately expressed, is that,according towhich we infer,thatwhen one fact of whatever kindisimpliedinanother, thecharacteriatie of the first mustcontain lUIa factor the c:haracteristic ofthesecond; andthatwhen two facts aremutually involved, theircharacteristics willbepo....ers of the sameintegral functi~n. Thedoctrine ofcharacteristics, appliedtodependent .y.tem.of facts, admitsofawide development, logicalandanalytical. 150 Ona newGlassofTheorems. [25 one or more pall"Sofequalfactors(e+t:p,).Buthereafarwidertheory presents itself, of which theaim should be to determine theeffectand meaning ofthisdeterminant, havinganyamountanddistribution ofmulti­ plicitywhatsoever among its roots. Nor mustourinvestigations endat thatpointjbutwemustbeable todetermine themeaning and effect of common factors, one or more entering intothesuccessive systems of minor determinants derived from thecomplete determinant ofU+p-V. Nor are we necessarily confined to two, butmaytakeseveralquadratic functions simultaneously intoaccount. Aspiring tothesewidegeneralizations, theanalysisofquadratic functions soars to apitchfrom whence itmay look proudlydown on thefeebleand vainattempts ofgeometry propertorise to its level or to emulate it inits flights. The law which I have stiltedforassigning thenumberofindependent, or tospeakmoreaccurately, non-coevanescent determinants belonging to agivensystemof minors, I call theHomaloidal law, because itis a corollary to aproposition whichrepresents analytically theindefinite extension of aproperty common to lines and surfaces to all loci (whether inordinary ortranscendental space)of the first order, all of which loci may, by an abstraction derivedfromtheidea of levelness common to straight linesand planes, be called Homaloids. The property inquestion is,thatneithertwo straightlines nor two planescan have a common segment jinotherwords, if nindependent relations ofrectilinearity or of co planarity, as the case may be,existbetween triadicgroups of a series of 7!+2, orbetween tetradic groupsof a series of n+3pointsrespectively, theneverytriadortetrad of the series, according totherespective suppositions made, will be in rectilinear or inplaneorder. So, too, ifnindependent relations ofcoincidence existbetween theduadsformed out of n+1 points, every duadwill con­ stitutea coincidence. This homaloidal law hasnot been statedintheabovecommentary in its form of greatest generality. Forthispurpose wemustcommence, not withasquare,butwith an oblong arrangement oftermsconsisting, suppose, ofmlines and ncolumns. Thiswillnotinitselfrepresent adeterminant, butis,Il.Sitwere, aMatrixoutof which we may form various systems of determinants by fixing upon a numberp,andselecting at willpliues and p columns, thesquarescorresponding to which may be termeddeterminants of thepthorder. We have, then,thefollowing proposition. Thenumberof uncoevanescent determinants constituting asystemofthepthorderderived fromagivenmatrix,ntermsbroad and mtermsdeep, may equal, butcan neverexceedthenumber (n-p+l)(m-p+l). 25] OnPascaT:s Theorem. Remark onPASCAL'S andBRIANCHON'STheorems.151 Iomittedtostate,intheSeptember Number of theJournal.", thatthe demonstration theregivenby me for Pascal's, appliedequallytoBrianchon's theorem. Thisremarkis ofthemoreimportance, because thefaultofthe analytical demonstrations hitherto given of thesetheorems hasbeen,that theymakeBrianchon's consequence ofPascal's, insteadofcausingthetwo toflowsimultaneously fromtheapplication ofthesame principles. No demonstration can be held valid in method,orastouching theessence of the subject-matter, in which theindifference oftheduadiclawisdeparted from. Untiltheserecenttimes,theanalytic method ofgeometry, asgivenby Descartes, had been suffered to go on haltingasitwere on one foot. To Plucker wasreserved thehonourofsettingitfirmly on its two equal supports bysupplying thecomplementary systemofcoordinates. This invention, however, had become inevitable, aftertheprofound views pro­ mulgated bySteiner,intheintroduction to hisGeometry, had once taken hold of theminds of mathematicians. To make thedemonstration inthe articlereferred toapply,totidem literis, toBrianchon's theorem (recourse beinghad to the correlative systemofcoordinates), itis onlyneedfulto consider Uas thecharacteristic ofthetangential envelope of theconic, x,y,z,t,U,vasthecharacteristics ofthesixpointsofthecircumscribed hexagon, 4>thecharacteristic ofthepointin which thelinee, vmeetsthe lines,t;ay-auwillthenbe shown to characterize thepointin which t,xmeetsv,z;andthuswe seethaty,u;i,x;v, z,thethreepairsof opposite sides of thehexagon, will meetin one and thesamepoint,which isBrianchon's theorem. [*p. 138above.] 26. ONTHESOLUTION OF A SYSTEM OF EQUATIONS INWHICH THREEHOMOGENEOUS QUADRATIC FUNCTIONS OFTHREE UNKNOWN QUANTITIES ARERESPECTIVELY EQUATED TONUMERICAL MULTIPLES OF AFOURTH NON-HOMO­ GENEOUS FUNCTION OF THESAME. [Philosophical Magazine, XXXVII. (1850),pp.370-373.] LETU, V, W bethreehomogeneous quadratic functions ofe,y,z,and let",beanyfunction ofa,y,Zofthenthdegree,andsuppose thatthere isgivenforsolution thesystemofequations U=..d"" V=B"" lV=0",. Theorem. Theabovesystemcan be solved by thesolution of acubic equation, andanequation ofthenthdegree. ForletDbethedeterminant inrespecttoe,y,Eof IU+gV+hW, thenDis acubicfunctionoff,g,h.Nowmake D=0,AI+Bg+ Oh=0; theratiosofI:g:hwhichsatisfythelasttwoequations canbedetermined bythesolution of acubicequation, andtherewillaccordingly bethree systems off,g,hwhichsatisfythesame, as ~,s.s; 12,92'h2 , Is,g..i; NowD=°impliesthatIU+9V+hWbreaksupintotwolinearfactors j accordingly weshallfind (llx+7n1Y+n1z)(:\IX+/l-IY+VIZ)=0, (l2x+ll'-2Y+~z)(~x+P-2Y+v2z)=0, (l,x+nI,y+n,z)(AIX+P-sy+VIZ)=0, 26J On thesolution.ofa System ofEquation». 153 in which theseveralsets ofl,m,n;A.,fL,vcan beexpressed without diffi­ culty intermsof the several values of -Jf,";g,";h. Let the above equations bewrittenundertheform pP'=O, QQ'=O, RR·=O. Sincethegivenequations areperfectly general,itisreadilyseenthat theequations (P=O,P'=O), (Q=O,Q'=O), (R=O,R'=O), willseverally represent pairs ofopposite sides of a quadrangle expressed by generalcoordinates x,y,z;sothatone of the two functions R, R'will be a linearfunction ofPandQand also of P'andQ',andtheotherwill be a linearfunction ofPandQ:and also of P'andQ•. Inorderto solve theequations, we need only consider two such pairs asPP'=0,QQ'=0;wethenmake P=o,Q=O, or P=O,Q'=O, or P'=O, Q=O, or P'=O, Q'=O. Anyone ofthesefoursystems will give theratiosof$:y:z;andthen, bysubstitution inanyone of thegivenequations, weobtainthevalues of e,y,zbythesolution of anordinary equation ofthenthdegree. The number of systems $,y,zistherefore always4n. Theequations connected withthesolution ofMalfatti's celebrated problem, ofInagiventriangle toinscribe threecircles such thateach circle touchestheremaining two circles and also two sides of the triangle," given byMr Cayley in the Novemher Number for1849of theCambridge and DublinMathematical Journal, to wit, by'+cz'+2fyz=lJ2a(bc-f2)=A, czl+axl+2gzx=8'b(ca-g')=B, ax'+by'+2/w:y=IJ2c(ab-h')=C, comeunderthegeneralform which has justbeen solved. It80happens, however,thatinthisparticular case r:s..h1) r;g2'l~r' j.,g"ha) • Were it notforthisbeing the case, the number ofsolutions would be n timesthenumber ofweJsofobtaining duadsoutofthreesets of two thinga,excluding theduadsformingthesets, thatis, thenumberofsolutions would be 12/1in place of 4n,thetruenumber. 154 OnthesolutionojaSystemojEquations. [26 becomerespectively 0,1 1 Jj'-0 10,1 -B'0 1 10-0'Jj' andthecubicequation isresolved without extraction of roots. Itfollows from my theorem thattheeightintersections ofthreecon­ centricsurfaces ofthesecondordercan be found by thesolution of onecubic andonequadratic equatiun; andingeneral, if we have 1/>,V'(Janythree quadrntic functions ofe,y,e,andI/>=0,V=0,e=°bethesystemof equations to be solved, provided thatwe can by lineartransformations express 1/>,V,eundertheform of U-aw, V-bw, W-cw, V,V,Wbeinghomogeneous functions, andwanon-homogeneous function ofthreenewvariables, a',y',s',wecanfindtheeightpointsofintersection ofthethreesurfaces, ofwhichU, V,Warethecharacteristics, bythe solution of onecubicandonequadratic. But(asI amindebted to MrCayley forremarking to me)thatthismay be possible, impliesthecoincidence oftheverticesof one cone of eachofthesystems of four cones in which the intersections ofthethreesurfaces takentwoandtwoarecontained. I mayperhapsenterfurtherhereafter intothediscussion ofthiselegant littletheory. Atpresent Ishallonlyremark,thatasomewhat analogous mode of solution isapplicable to twoequations, U=aP'Z, V=bP'Z, inwhichU, Varehomogeneous quadratic functions, andPsomenon-homo­ geneous function ofe,y. We have only tomakethedeterminant offU+gVequal to zero,andwe shallobtaintwosystems ofvaluesoff,g,wherefrom wederive llX+m..y=±.,I(afl+bg1)P, l2X+'Tnty=±.,I(af2+bgs)P, fromwhicheandymay bedetermined. 27. ON APORISMATIC PROPERTY OF 'l'WO CONICS HAVING WITH O~EANOTHER A CONTACT OF THETHIRD ORDER. [Philosaphical J.lfagazine, XXXVII. (18.50), pp. 438, 439.] IFtwo conics have withoneanotheracontactofthethirdorder,thatis, if theyintersect in fourconsecutive points,itwill easily be seen thattheir characteristics referredtocoordinate axesin theplanecontaining themmust beof therelative formsa:'+yz,k(y2+a;2+yz)respectively, ycharacterizing their common tangentatthepointofcontaot", Henceifwetakeplanesof reference in space, and call tthecharacteristic oftheplaneoftheconics, the equations to any two conoids drawnthrough themrespectively will be of therelativeforms U=a;2+yz+tu=0, V=y2+a;2+yz+tv=O. UsingWtodenoteV-U,and(W)todenotewhatWbecomes when ey issubstituted fort,we seethatWand(W)are oftherespective forms !f+twandyO;showingthattheformer is the characteristic of a cone which willbecutby anyplanet-eydrawnthrough theline(t,y)in apair ofrightlines;or, inotherwords,thatone oftheconescontaining theinter­ sectionofthetwovariable conoids(VandU)will have its vertexin the intariable linewhich is thecommon tangent tothetwo fixed conics:this provesthetheorem statedby mehypothetically in a foot-note in one of my papersin thelastnumberoftheMagazinet. Thestepsofthegeometrical prooftherehintedatareasfollows. •Theserelativeorconjugate forms are takenfrom Cotablewhich Ishallpublishin afu'are Dumberof'hisMagtUi~, exhibiting theconjugate characteristios in'heirsimplest forms, eorrespondent \0all'hevariousspecies of contacts possible between lines andsurfaces of'he IelXlnddegree. Thistableisasimportaut tothegeometer asthefundamen'al trigonometrical formul.\0theaualys', or themultiplication table'0thearithmetician; audit issurprising &batDOODehashitherto thoughtofconstructing such. [tP.U9above.] 156 Ona Poriematie Property oftwoOonice. [27 The four consecutive pointsin which thetwo conics intersect willbe consecutive pointsinthecurve of intersection ofthetwovariable conoids. Thiscurvelies in each of four cones of thesecond degree. Everydouble tangent planetoitpassesthrough thevertexof oneamongst these.The planecontaining four,thatis, two(consecutive) pairsofconsecutive points, is adoubletangentplane, and will therefore passthroughavertex;butfour consecutive pointsof a curve of thefourthorderdescribed upon a cone, andlyingin onetangent planethereto, can only be conceived generally as disposed in theform of an I,of which thebellypartwillpointtothe vertex;or, inotherwords,atanypointwhere two consecutive osculating planescoincide so thatthespherical curvature vanishes, thelinearcurvature will also vanish, thatis,therewill be a pointof inflexion at which, of course, thetangent linemustpassthrough thevertexofthecone. This isthe assumption felt to be true,butstatedby mehypothetically in thepaper referred to,because areadydemonstration did not at themoment occur to me. The legitimacy ofthisinference is nowvindicated bytheabove analytical demonstration. Themethods ofgeneralandcorrelative coordinates and ofdeterminants combined possess a perfectly irresistible force (to which I can only compare thatofthesteam-hammer inthephysical world) for bringing underthe graspofiutuitive perception themostcomplicated andrefractory formsof geometrical truth. 28. O~THEROTATION OFARIGID BODYABOUT A FIXED POINT. [Philosophical Ma.gazine, XXXVII. (1850),pp.440-444.] INtheCambridge andDublinll/athematical Journal for March 184,8, anarticlebyProfessor Stokes,oftheUniversity ofCambridge, isusheredin withthewordsfollowing:- UThe most generalinstantaneous- motionof arigidbody moveable in alldirections abouta fixedpointconsistsin amotionofrotation aboutan axispassingthrough thatpoint.Thiselementary proposition issometimes assumedasself-evident, andsometimes deduced astheresultof ananalytical process.Itoughthardlyperhaps to beassumed, butitdoes not seem desirable to referto a long algebraical process for thedemonstration of atheorem so simple, YetI am not aware of a geometrical proofanywhere published which mightbereferredto." Thelearnedandingenious professor is indubitably right,andmighthave trustedhimselftoassertlesshesitatingly thenecessity ofdemonstrating thisproposition, which possesses none of thecharacters ofaself-evident truth;butitis to beregretted thatheshouldhavestateditin such a form asnaturally toleadtheincautious readertomistake thenatureandgrounds ofitsexistence, whichconsistinthisfact-that anykind ofdisplacement ofabody moveable abouta fixed axis, whether instantaneous andinfini­ tesimal,orsecularandfinite, is capableofbeingeffected by a singlerotation aboutasingleaxis. Theannexed simpleproof of thiscapitallaw hastheadvantage of afford­ ingaruleforcompounding intooneanytwo(andtherefore anynumberof) rotationsgivenindirection, magnitude andorderofsuccession. • The italics do not existin theoriginal. 158 OntheRotation oja [28 QItwillsomewhat conduce to simplicity if we fix our attention upon a spherical surfacerigidlyconnected wishtherotating body, and havingits centreatthefixedpointthereof. Whenthe positions of two pointsin thisare given, thepositionofthebody iscompletely determined. Nowevidently twopointsA,Bmay be brought respectively toA'B' (ifAB=A'B')by tworotations; thefirsttakingplaceabouta polesituated anywhere inthegreatcirclebisecting ..1...1.'atrightangles,thesecondabout A',theposition intowhichitisbrought bythefirstrotation. Thisview leads us to consider theeffect of two rotations takingplacesuccessively abouttwoaxesfixed intherotating body.Or again, we may make theplane A'B'revolveintothepositionABrounda poletakenatthenode in which thetwoplanesintersect, andthenthepointsA, Bswingintotheirnew positions A',B'bymeansof arotation aboutthepole ofthegreatcircle, of which A'B'forms a part. Thismode of effecting thedisplacement naturally suggests theconsideration oftheeffect of rotations takingplace successively abouttwo axes fixed in space. First,then,let usstudytheeffect of thecombination of arotation (a)havingPfor its pole, followed by another(f3),of which Qisthepole, PandQbeingpointsinthesurface of the revolving sphere. Indrawing theannexed figure, I have supposed thatthetworotations r are ofthesame kind, each tending, whena spectator isstanding with his head to the respective poles and his feet to thecentre,to makeapointat hisright-hand passinfrontof hisfacetowards hisleft-hand. LetnowPQ R'revolvethrough ~positively intothepositionof PR,andthrough ~negatively intothatofQR. ThenI saythatthetwoimpressed rotations a and{3aboutPandQwill beequivalent to asinglerotationaboutR,equal to twice theacuteanglebetweenQR,RP. LetthefirstrotationaboutPbringQtoQ'andRtoR'jitisclearthat QPR, Q'PR, (lPR'are allequaltriangles. Therefore R'(lR=2PQR={3. Consequently thepositive rotation {3aboutQ'(thenewposition ofQ)will carryR'back again to R,itsoriginal position. Hencetheactualmotion whichresultsfromthesuccessive rotations combined beingconsistent with Rremaining at rest,mustbeequivalent toasinglerotationaboutR. To finditsmagnitude, letthesecondrotation carryPtop'.;thenthe angular displacement PRP'(which is therequired rotation ofthewhole •The reader is requested tofillinthepointpiand joinP'R. 28] RigidBodyabout a FixedPoint. 159 body) is equal to twice theacuteanglebetweenQ'R,RP,which is thesame asthatbetween QR,RP,as was to be shown. Thuswe seethatthesemi­ rotations aboutthreepoles(considered astheangularpointsof aspherical triangle), which,takenin order, would bringthesphereback to itsfirst undisturbed position, areequaltotheincluded anglesatsuchpoles respec­ tively. Ifinourfiguretheorderoftherotations hadbeenreversed, PQr, QPr would have beentakenrespectively equaltoPQR,QPR,butontheopposite side ofPQ,andr would have been theresultant pole,theresultant rotation remaining inamountthesameas before. Ifeitheroftherotations had been negative, theresultant pole would be found in QRproduced, namely,attheintersection ofrQorrPwithPQ. Callingtheresultant rotation 'Y,we have always .a..{3.'Y.QR.RP.PQsin2:SID2':sm'2::sin : SID :sin. Whenthecomponent rotations areinfinitesimal inamount,Randr will cometogether inQP;theorderofsuccession oftherotations willbe indifferent, andweshallhave a:f3:'Y::sin;: sin~: sin~:: sinQR:sinRP:sinPQ, which gives therulefortheparallelogrammatic composition of twosimul­ taneously impressed rotationa'", If,next, we consider theeffect of rotations abouttwo poles, PandQ, fixedin space (supposing, as above, thattheytakeplace first aboutPand thenaboutQ),wemusttakeQPrequaltohalfthecontrary oftherotation aboutP,andPQrtohalfthedirectrotation aboutQ(theanglebeingnow takenpositive which was on thefirstsupposition negative, andviceversa); sothat,retaining theoriginal figure,thefirstrotation willbringr toR, andthesecondcarryRback tor;showing thatristheresultant pole, andthattP'1'P,theresultant rotation, will be double theacuteangle betweenQr,rP,as intheformercase. Topopular apprehension theimportant doctrine ofuniaxial rotation maybemadeintelligible bythefollowing mode of statement. Takea pocket-globe, open thecase and roll aboutthespherewithinit in any mannerwhatever; thenclosingthecase,therewillunavoidably remaintwo pointson theterrestrial surfacetouching thesametwopointsonthecelestial surfaceastheywere inapposition withbeforethespherewas soturnedabout inits case. • Compare MrAiry'sTracts,Art...OnPrecession andNutation." t1"isnotexpressed in the figure given. 160 On theRotation ofa [28 Itisrighttobearinmindthatthewhole of thisdoctrine iscomprised in, andconvertible with,thefollowing easygeometrical proposition relative to arcs of greatcircles on anyspherical surface,including theplaneasan extreme case. ..Thearcsjoiningtheextremities (eachwitheachineitherorder)of two otherequalarcs,subtend equalanglesateitherofthepointsofintersection of twogreatcirclesbisecting atrightanglesthefirst-named connecting arcs·... Thespherico-triangular mode of compounding rotations giveninthe abovesimpledisquisition may easily be madetheparentof a whole brood of geometrical consequences, which, however, I mustleave to theingenuity and care of thosewho have aturnforthiskindofinvention. ButIoughtnot toomittoinviteattention to aremarkable form, which may beimparted tothetheorems abovestatedforthecomposition offinite rotations, orrathertoatheorem which may be derivedfromthembyan obvious process of inference. LetP,Q,R...X, Zbeanynumber ofpointsonaspherecapable of movingaboutitscentre,joinedtogether by arcs of greatcircles so as to form aspherical polygon. Imagine anynumber ofrotations totakeplaceabout thesepointsin succession aspoles.Itmattersnot which is considered the first pole of rotation,buttheorderofthecirculation mustbesupposed given, as,forinstance, PQR...XZ,orQR...XZP,orR...XZPQ,&c.This will be one order;thereverseorderwould be PZX... RQ,orQPZX...R,&.c. I shall suppose thecirculation to be of thekindfirstabovewritten. N'ow we may make two hypotheses:- 1.Thatthepolesar~fixed in space. 2.Thattheyare fixed in therotating body. Inthefirstcase,lettherotations aboutthegiven poles P,Q,R, S...X, Z be double theamounts which would serve to transport PQtoQR, QR to RS...XZtoZPrespectively. Inthesecond case, lettherotations bedoubletheamounts which would carryPZtoZX...SRtoRQ, RQ toQPrespectively. Then,oneither supposition, thesum ofthecombined rotations iszero;or, to use a more convenient andsuggestive form of expression, ifthepoles of rotation form a closedspherical polygon whose anglesarerespectively equaltothesemi­ rotations aboutthepoles,theresultant rotatiou is zero. •Thisproposition will be seen to be immediately demonstrable, by thecomparison ofequal triangles, when viewed &8theconverse ofthisother...Thearcs(orfightlines)joiningthe correspondent extremities of thebasesof twosimilarisosceles spherical (orplane)triangles havingacommon vertex,areequaltoeachother:' 28J RigidBodyalJoutaFixedPoint. 161 Thisproposition isimmediately derivable fromthefundamental one relativetothreepoles,givenabove, by dividing thepolygon intotriangles byarcs,joininganyoneofthepoleswithalltherest, or(aspointedout to meby myeminent friend Prof. W. Thomson) itbecomes apparent asa particular case ofamoregeneralproposition, onrepresenting themotion aboutthesuccessive axes aseffected by two equal pyramids havinga common vertexatthecentreof motion, of which theone is fixed in space,andtheotheris fixed in therevolving body and rolls over the first,80thatthecorresponding equalfaces are successively brought into coincident apposition. P.S. To find thepole ofrotation whereby PQmay bebrought intothe positionP'l,we may use thefollowing simpleconstruction. Measure off from 0 thenode of thegreatcircles (or rightlines) con­ tainingPQandpl'l,twodistances intheproperdirection upon each (four distinct assumptions maybemade), say ORandOSequalto oneanother andtothedifference betweenOPandOP,thenthepole ofrotationrequired, sayE,isthecentreof the circle described aboutROS,andtheamountof rotation istheanglesubtended byORorOSatE.Thewriterofthispaper suggests thataxisofdisplacement would be a convenient termfordesignating thelinewhereby any finite changeintheposition ofabody moveable about a fixedcentremay bebroughtabout;ageometrical theoryofrotation leading totheinvestigation of a very curiousspecies of correlation, now opens tlpon theview,thegeneralobjectof which may be statedasfollows: ..Givenuponasphereorplaneany curve considered asthelocus of successive poles of instantaneous rotation, andtheratiooftherotation abouteach pole to itsdistance fromtheonethatfollows.", toconstruct thecurveofthepoles of displacement, and todetermine theamountof rotation corresponding to each such pole." Thediscussion ofthisquestion offersafine field for theexercise of geometrical tasteand skill. • Which by analogyma.ybetermedthe ..densityofrotation." s. 11 29. ONTHEINTERSECTIONS OF TWO CONICS. [Cambridge andDu1JlinMathematical Journal, VI.(1851), pp. 18-20.] LETthetwo conics be written U=ag;I+by'+czS+2a'yz+2b'z,x+2c'xy=0, V=a.W+{3'!f+ryzl+2a.'yz+2f3'z,x+2ry',xy=0, and make U+}..V=Ax'+B'!f+Czt+2A'yz+2B'z,x+2C',xy. In mypaperinthelastnumberoftheJournal:", I showed thatthecase of intersection of the two conics in two pointswasdistinguishable fromallother cases by theequation 0 (U+)..V)=0havingtwoimaginary roots.When alltheroots are real, thecurveseitherintersect in fourpointsor notatall. Ontheformersupposition, -C',+AB,-A"+BC,-B"+CA, which are quadratic functions of}..,will benegative for allthreevalues of }.... Onthecontrary supposition, one value of }..will make all thesethree quantities negative, buttheothertwo values with eachmakethemall threepositive. Hence we obtainasymmetrical criterion (which I strangely omittedto statein my former paper)by forming thequantity A'lI+B"+C"-AB-AC-BC. A cubicequation Ly'+M'!f+ Ny+P=0 may bethenconstructed, of which thethreevalues of theabovefunction corresponding tothreevalues of }..will betheroots. Thecondition forrealintersection isthatL, M, N, P should be all of the same sign. The conics being supposed real, LandParenecessarily inboth cases ofthesame sign. The condition istherefore satisfied ifeitherL, M, [*p.119 above.] 29] OntheIntersections ofTwoOoniee. 163 N,orM, N, P be ofthesame sign, and is consequently equivalent tothe condition that~andfshall be both positive, or:and~both positive. Itdoesnotappeartobepossible in thenatureofthequestion to finda criterion fordistinguishing between thetwocases,dependent onthesign ofonesinglefunction ofthecoefficients. Thecaseofdouble contact,abstraction beingmade of binary intersection, isasortofintermediary statebetween intersection in fourpointsand non­ intersection jand accordingly, asshown in my former paperforthiscase,the twoequalvalues of Xwill make thethreequantities AB-C", BC -A",CA-B" allrealj80thattwo ofthevalues of ycorresponding totheequal values of X arezero, and thecriterion becomes nugatory as itoughtto do. Again, when thetwo conics do not intersect, Idistinguished twocases according astheylieeachwithout, or onewithintheother,thatis,according astheyhave four common tangents or none. But,asMr Cayley haswellremarked to me,asimilardistinction exists whentheconicsintersect in fourpoints jinthatcasealso they may have four common tangents or notany:whentheyintersect in twopoints-they havenecessarily two and only two common tangents. Thereisno difficulty inseparating these four cases. Lettheconics be written (U)=E'+7]1-r, (V)=AE'+B7]I-cr, (U)and(V)beingwhatUandVbecome when thecoordinates are changed frome,y,ztoE,'1,~. A,B, Carethethreevalues of Xintheequation o(V-XU)=O. Ifthecurvesintersect A-C,B-Cmusthavedifferent signs,thatis,C mustbeanintermediary quantity betweenAandB. Again,thetangential equations totheconics expressed by thecorrelative systemofcoordinates will be El'+'11'-~l'=0, Ell+'11'_~ll_O.ABC -I andthatthesemay have four real systemsof roots, 1 1 1 1:A-c'c:» musthavethesame sign jandconsequently, asA-CandC-Bare 11-2 164 OntheIntersections ofTwoConics. [29 supposed to havethesamesign,AandB,andtherefore allthreeA, B, 0, havethesamesign. We have therefore thefollowing rule: Lettheequation inA.,namely, 0(U+A.V)=0, be called (J=0,andthe equation iny,above given, Cl)=0. By an equation beingcongruent or incongruent, understand thatits roots have all thesamesign ornotall thesamesign. Then Cl)congruent, (Jcongruent, impliesthattheintersections and common tangents arebothreal; Cl)congruent, (Jincongruent, impliesthat theintersections arereal,butthecommon tangents imaginary; Cl)incon­ gruent, (Jcongruent, impliesthattheintersections andcommon tangents arebothimaginary; Cl)incongruent, (Jincongruent, impliesthattheinter­ sections areimaginary, butthecommon tangents real. Inlikemanner,asthecasesofcontactof linesarelimiting casestothose whichrelatetotherelativeconfigurations oftheirpointsofintersection, 80 thecases of contactof surfaces are limiting casesin which thecharacters whichusuallyseparate thedifferent forms of theircurve of intersection exist blendedandindistinguishable. Thefirststeptherefore tothestudyofthe particular speciesofthecurveofthefourthdegree.",inwhich two surfaces oftheseconddegreeintersect, is toobtaintheanalytical andgeometrical characters oftheirvariousspecies of contact. Accordingly I have made an enumeration of these different species, no less than12 innumber, many of themhighlycurious and I believe unsuspected, whichthereadermayconsult inthePhilosophical Magazine forFebruary, 1851t. Bytheaidof theselandmarks, Ihavelittledoubt, should time andleisure permit,ofmapping outanaturalarrangement oftheprincipal distinctions of formbetween thatclassat least of lines in space of the fourthorderwhich admitof being considered thecomplete intersection of two surfaces. *I have found thatthe 16pointsofspherical flexure in thiscurvearethe foursetsof four pointsin which it meetsthe four facesof thepyramid whosesummits arethe vertices of the four cones of theseoond degree in which the curve is completely contained, whioh 16 pointsreduceto 4whenthetwosurfa.cee haveanordinary contact,andto1 whentheyhaveacuspidal contact: ofcoursein the case of contactthepyramid abovedescribed inamannerfoldsupandvanishes, asthereare nolonger4distinotvertices. I havefoundalsothatwhenthefactorsof 0(U+"Y). (UandVbeing the cha.racteriBtios of the two surfa.ces) areallunreal,thepointsof flexure areall unreal, Whentwofactorsare realandtwoi.ma.gina.ry, two ofthefacesof thepyramid (namely, itstworealfaces)willeachcontainone (and only one) pairofrealpointsof flexure, andtheother twoplanesnoue;andlastly,when the factorsof 0(U+"Y)are all real, theneitherall thepoints of flexure areimaginary, or elsealltheeightcontained inacertaintwo of the pyramidal faces arereal:andthese two casesadmitof being distinguished byamethodanalogous in itsgeneral features tothatwhereby I haveshown in the textabove how to distinguish between the casesof 4realand4imaginary pointsofintersection of two conics. Where the two surfaces havean ordinary contact, the curve of intersection, it is well known, hasadoublepoint;andwherethe surfaces haveahighercontact,the curve hasacusp.Thusin thefactof the 16 flexures reduoing to4andto 1intheserespective cases, we see abeautiful analogytowhatWokesplacewith the 9 flexures ofaplanecurve of thethirddegree, which contract to3and1,aooording astheourve hasadoublepointoracusp, [tp,219 below.] 30. ONCERTAIN GENERAL PROPERTIES OF HOMOGENEOUS FUNCTIONS. [Oambridge andDublinMathematical Journal, VI.(1851), pp. 1-17.] LETXdenotetheoperation d d d XId-+X,.1_+...+Xn-d'al~ an and.A.theoperation d d d ~da:,+asllA;+ ... +anda:n: and now suppose that Q),ahomogeneous function oftdimensions of lit,as...€tn,andnotof any of thequantities ~,X,...Xn,issubjected to thesuccessive operations indicated by.A.'Xr• We have (d d d)-: d dJr.A.x'"Q)=litdxl+asdx,+...+andXnXId~+X2~+...+Xnaa Q) (d d d),=rlit~+asda,+ ...+andanx:Q) =r(t-r+l)xr-IQ). forXr-IQ)is of(r-1)dimensions, lowerthanQ)(which is of tdimensions) in lit.a,...a,.. Hence A'x'Q)=r(t-r+1)AI-IXr-lQ) =&c.=[r(1'-I)...(r-s+I)} [(t-1"+l)(t-1'+2)...(t-r+s)[X-r-'Q). (1) 166 OncertainGeneralProperties [30 Now intheexpression supposethatwewrite (3)ICs=~+~e, we have, by Taylor'stheorem, ""OJ=U"OJ+.AU"OJe+.A'U"OJI~2+ ...+A"U"OJ1.2.~...r' whereU"OJdenoteswhat')(OJbecomes, on substituting u'sforx's,andAnow represents Thisexpansion stopsspontaneously atthe(r+l)thterm,because ')(OJis only ofrdimensions in1Ct,1Cs...xn• Applying nowtheorem (1), weobtain ""OJ=U"OJ+r(£-r+1)Uf'-1OJE+ir(r-1){(£-r+1)(£-r+2)}Ur-IOJe' +...+{(£-r+l)(£-r+2)...£}we".(2) Inusingthistheorem inthecourseoftheensuingpages,itwill be found convenient toassigntoeaspecific value, .and I shall suppose itequalto Xnhi .-;tISgivesan CLt 1£:1=Xl- -Xn,an £l.,ru,=x,--Xn,an anUn=Xn--X"an =0. Andinasmuch astheUsymbol now contains CLt,a"...an,80thatUU"no longerequalsUf'+\IshallwriteU..forU".Theorem (2)willthusassumethe form ""OJ=U..w+r(£-r+1)U..-IOJ::+1r(r-1)(£ -r+1)(£-r+2)Ur-IOJ(~' (~\" +...+{(£-r+1)...£}OJa:.J' 30] ojHomogeneous FuncliO'JUI. 167 whereUpfor all values of f"denoteswhat (d d d)P 0;clat+x,~+ ... + IX,,-lclis.-I Ct) becomes, on substituting Ut,~,...Un-Ifor0;,IC"•••Xn-I>afterthe processes ofderivation have been completed: thisitisessential to observe, because Ut,~,...u..-Inow involve ai,~,...a..-I,an.ThetermXn~isomittedfrom thesymbol of linearderivation, becauseinthesubstitutions IX"will be replaced by zero. Asan example of thislasttheorem, take Ct)=as+1JI+c'+kobo; then XCt)3a'x+31JIy+3c'z+kbc:r:+kcay+kab», ')(Ct)6az1+6by'+6cza+2kexy+2kayz+2kbzx, ,)(Ct)=6z1+6y'+6z1+6kxyz. UICt)=3aa(x-:z)+3ba(y-b;)+kbc(x-:Z)+kca(y-~), (az)'(bz)'(Z)(bz) U,Ct)=6ax-C+6bY-C+2kcx-acy-c ' (az)S(bz)'U.Ct)=6x-c+6Y-C' andit willbefoundthattheequations given by theorem (3) are satisfied, namely zXCt)=UCt)+3-Ct),c z z' ,)(Ct)=U,Ct)+2.2cUCt)+2.3CiCt), z ~ zI ')(Ct)=U~+3cUsO'+3.1.2CIUCt)+1.2.aCiCt). Probably, asthis theorem is of rathera novelcharacter, theannexed sketchof asomewhat different course of demonstration may be not un­ acceptable tomy readers. We have (d d d )XCt)... Xlclat+.1;d~+ ...+x..da~ Ct)j and bythewell-known law for homogeneous functions, (d d d)lCt)=~da l+~~+ ...+andan Ct). 168 Hence .HenceOncertainGeneral Properties (11:,,)v: d d )X-t~ro=U,da;+~cia,+...+Un-Ida..-Iro =Uro. Xro=(U+I11:,,)ro, a"[30 X:ro={U+(t-1)~}(U+t::)ro, x'ro= {U+(t-2)~}{U+(t-1)::}(U+t~ro, &c.=&c. Butinperforming theprocessindicated bytheseveralfactorsitmustbe carefully borne in mind thatUUris not=Ur+I;thiswouldbethecasewere itnot fortheterms-~11:",-~11:",&c.,whichenterintou"~.•.Un-I'Butana" onaccountoftheseterms,we have for Henced d d II:ndalVI=~~=...=dan-I Un_1= -~. LetII:nbecalled e; we findan X=U+If, X'=lU+(t-l)f}(U +se) =UU+(21 -1)fU+(t-1)tf' =Us+2(I-l)fU +(t-l)tf'; x'=IU+(I -2)f)x' = UU,+2(t-1)fUU+(t-1)telU +(t-2)fU,+2(I-2)(t-1)f'U+(t-2)(t-1)t€' =Us+3(I-2)fU,+3(t-i)(t-1)f'U+(t-2)(1-1)t€'. SO] ojHomoqeneous Fumctions. 169 Thesameprocessbeingcontinued willleadtoresultsidentical withthose previously obtained andexpressed intheorem (3). Theexpansion ofX..,treatedaccording tothissecondmethod, appearsto requirethesolution ofthepartialequation indifferences a..+1.1+1=ar,1+1+(L-2r) a..," 4e,Ibeinggivenasunityfor8=1andaszero for all othervaluesof8. Itisprobable howeverthatthesolution ofthisequation mightbeevaded bysomeartificepeculiar totheparticular caseto bedealtwith. I do not propose to dwell upon thisinquiry, which would be foreign to theobject ofmypresentresearch. Itmay however notbeoutof place to make the passing remark,thattheequations expressing X"intermsof powers of U admiteasily of beingreverted, asindeedmaythemoregeneralform 1X..=ur+e..Ur-1+ 1. 2 £"£f'-1u..-t+ &c. whichbecomes theequation offormula(3), onmaking er=r(L+l_r)x.. ,X..=X"Q),andur=U..Q);a.. for let then whence andthereforeX..=£1£"" £ry.., Vf'-2 Vf'-'&y..=vr+V..-1+1.2+1.2.3+c.; d Vr=e-dryr _ y..-,y..-,&.- y..-Y..-1+1.2-Lf:3+coo Thusweobtain,fromequation (3), U..Q)=XQ)-r(L-r+1)Xr-1Q)X..+&c.an Asafirstapplication oftheorem (3), I shall proceed to show how Joachimsthal's equation tothesurfacedrawnfrom a given point(a,13,'Y,S) through theintersection of two surfaces ep(x,y,e,t)=O.8(e,y,z,t)=0, may beexpressed undertheexplicitform oftheequation toacone. Theequation inquestion isobtained byeliminating xbetween 1ep).,m+Xep).,m-1+1.2X'ep).,m-I+&c.=0, 1 18).,'"+-XtrAm-1+ -X,'trAm-2+---X'8Xm-,+&c. =0,1.2 1.2.3 170 whereOncertainGeneralProperties [30 Bytheorem (3)Jthesetwoequations, onwriting IXn=e,becomean q,).'"+{Uq,+mq,e})."'-1 ).m-t +{U'q,+ 2(m-1)Uq,e+(m-1)mq,E'} 1.2+&c.=0_ ).-, (JAn+{U()+n()e}).n-l +(Ut(J+ &c.)1.2+ {U'8+ 3(n-2)UI(Je ).n-I+3(n-2)(n-1)U8~+(n -2)(n-1)neI}i.~.3+&c.. Now onwriting).=p.-e,theseequations taketheforms ()p.n+U(Jp."-1+U2(J~~;+&c.=0, asis easily seen by substituting back'"+ein place of p..Consequently enolongerappearsinthecoefficients of thetermsoftheequations between whichtheelimination is to be performed, andtheresultant willaccordingly comeoutasafunction only of¢'.Uq"U'¢,&c.,thatis,ofCl,f1J'YJB; andof Cl IX- -t~,fJy--St,z-~t, showing thattheequation inIX.y.s,t,is oftheform ofthattoacone,aswe knowaprioriitoughtto be.Precisely asimilarmethodmay beappliedto theelucidation ofthecorresponding theorem forasystemofraysdrawnfrom agivenpointthrough thelocus of the intersection of two curves. Beforeentering upon some furtherandmoreinteresting applications of theorem (3)_itwill beconvenient toexplainanomenclature whichhasbeen employed by me on anotheroccasion, andwhich is almostindispensable in inquiries ofthenaturewe are now engaged upon.Homogeneous functions may becharacterized bytheirdegree,bythenumber ofletterswhichenter intothem,andlastly,bythelowestnumberoflinearfunctions oftheletters which may be introduced in place of theletterstorepresent suchfunctions. Anysuchlinearfunction Idesignate as an order, and am now ableto dis­ criminate between thenumber oflettersandthenumber oforderswhich enterintoagivenfunction. Thelatternumber, generally speaking, isthe sameastheformer;itcanneverexceed it, butmaybeanynumber ofunits .essthanit. 30J ojHomoqeneous Functions. 171 I needscarcelyobservethatapairofpointsbecoming coincident, aconic becoming apairof lines, aconoid becoming acone, and 80forth, for the higherrealms of space,will be expressed by thehomogeneous function of the secondorder which characterizes such loci.",losing one order, thatis,having anorderlessthanthenumber oflettersentering therein. Callingsuch characteristic tP(0;,y,Z...t),it is well known thatthe condition of such loss of an order is thevanishing ofthedeterminant d'epd'epd'ep d:i:i'da;dy...do;dt d'epd'e/>d'ep dydo;'d;l/'...dydt d'epd'e/>d'ep dtdo;'dtdy...dii A conoid becoming a pairof planes, acone becoming a pairofcoincident lines,apairofpointsbecoming indeterminate, will, in like manner, be denoted bytheircharacteristic losing two orders, and 80forth, for the higherdegrees of degradation. In like manner, in general, ahomogeneous function ofthreelettersofanydegree losing anorder, typifies thatthe locus to which it is the characteristic willbreakupintoa system of rightlines. Nowlet Cdbea.homogeneous function of a,{1,'Y•••8.and suppose thatwe havetheequations Cd=0,Xcd=0,JCCd=0, where Xasabove d d d d =0;da+yilfJ+6d-y+...+td8' Ieaythatoneliminating anyofthevariables 0;,y,Z•••tbetween the second andthirdoftheaboveequations, theresulting equation will be of one order lessthanthenumberofletters,thatis,theexpulsion of one letterwill be attended bytheexpulsion of twoorders. Forwe have, by theorem (:3), Xcd=UCd+20;"Cd=0,an x'cd=U,cd+2::Ucd+2(71j'Cd=0, andbyhypothesis Cd=O. Hence we have also Ucd=0, U,cd=OJ and since UCd,U,cdcontain one order less thanthenumber oflettersin •IfU=Oistheequation toanylocus,Umaybesaidtocharacterize thesame,ortobeits cbaraderiatic. 172 OncertainGeneralProperties [30 0),theresultant oftheelimination between themwillcontaintwoorders lessthanthenumber oflettersinCI)jandconsequently, whichever ofthe letters IX,'!I,z...tweeliminate between XCI)=0andX2(1)=0,provided that 0)=0,theresultant equation willcontainoneorderlessthanthenumberof lettersremaining. Thuswe see how itisthatthetangent line toaconicmeetsitin two coincident points,thetangent planetoaconoid in two intersecting lines, andso forth, for thehigherregionsofspace". Forinstance, ifwetake CI)(x,'!I,z,t)=0,theequation toII;conoid, and a,fJ,"I,8,thecoordinates to anypointtherein, weshallhave CcJ(cr,fJ,'Y,8)=0, (d d d s-.. IXda+'!IdfJ+Zd"f+td8)CI),thatIS,XCcJ=0, and CI)(z,'!I,Z,t),thatis,~CI)=0, e,'!I,s,trepresenting thecoordinates of anypointintheintersection ofthe conoid by thetangentplane. Consequently, bywhathasbeen shown above, on eliminating anyone of thefourletterse,'!I,z, t,theresultant function ofthreeletterswillcontain only two orders, andwillthusrepresent apairof lines, real or imaginary, intersecting oneanotherattt,fJ,"I,8. Thefact which hasjustbeendemonstrated (thattheresultant ofXCcJ=0, X'Cl)=0,loses an orderifCI)=0),indicates thatonexpressing one ofthe quantities w,'!I,Z•••tintermsoftheothers,bymeansofthefirstequation, andthensubstituting thisvalueinthesecond,thedeterminant ofthe equation soobtained mustbe zero. Now by virtueofatheorem whichwasgivenby me in anotetto my paperinthepreceding numberofthisJournal, thisdeterminant will beequal tothesquared reciprocal ofthecoefficient in theequation XCcJ=0ofthe lettereliminated multiplied bythedeterminant inrespecttox,'!I,Z...t,Xof ~CI)+XCI)X. Thislatterdeterminant istherefore zero;butthisdeterminant isthe resultant oftheequations d(d d)'d(d d)dxIXda+'!Idb+&c.CI)+d:c,xda+'!Idb+ ... CcJ=0, d(d d 2d(d d \dyxda+'!Idb+&c.) CcJ+d'!lxda+'!Idb+ ...) CI)=0, &c. &c. &c. &c. XCI)=0,thatIS,(x:fa+'!I;b+...) CI)=0, •Thusatangential seotion of ahyperloous of the second degree at any pointouts it in two cones. [tp.185above.] 30] ofHomoqeneous Functions. 173 Thusweobtainthesingular law,thatthesymmetrical determinant dddd dd d ciadaCIl,dO,dbCIl,•••ciadlCIl,dO,CIl dddd dd d dbciaCIl,db db CIl,•••db dl CIl,dbCIl dddd dd d dedaCIl,dedbCIl,•••dedlCIl,deCIl d d dl da CIl, d dO,CIl, iszerowhen CIlis zero.d d d d dl db CIl,•••dld1CIl, d ddbCIl,•••dlCll,o Thisiseasilyshownindependently by means of a remarkable and Ibelievenoveltheorem, relativeto homogeneous functions. If0)beanyhomogeneous function of tdimensions ofa,b,c...l,we have(byEuler'stheorem alreadyrepeatedly applied), remembering that dO)dO)dOJ da'db...dlare all homogeneous, -LCIl+(afa+b~+ + l~)CIl=0, dO)(d d d d d d)-(t-I)da+ adada+bciadb+ ..·+ldadl CIl=O, -(,-1):+ (a;bCi~+ +l~:z)CIl=O, &c. &c. &C. -(t-I)a;+(a~fa+ +l~~) CIl=O. Between theseequations we mayeliminate all the letters, a, b, C •••l,andwe obtaintheequation dddd dd d do'daC1I,da db CIl,...dad1CIl,daCIl d d dd d d d db da CIl,db db CIl,...dbdiCIl,dbCIl dd d d d d d de da CIl,de db CIl,...deCilCIl,-CIl=0.de ................................................. d d d d d d d dldCtCIl,dl db CIl,...dld1CIl,dlCIl d d d t do'CIl,dbCll,...d1CIl,-CIlt-I 174 OncertainGeneralProperties [30 Asacorollary to thistheorem, weseethatitQ.I=0thedeterminant obtained intheprevious investigation becomes zero, agreeing withwhat has been alreadyshown;infactthelast-named determinant isalways equalto £-1--Q.lX £d d dada Q.I, d dilldaQ.I,d d d d dadbQ.I,'.'dO,dlQ.I Thisremarkable theorem, whichIhavecommunicated to friends nearly atwelvemonth back, is bere,Ibelieve,published for the first time", Suppose nextthatQ.I(x,y,z)isthecharacteristic ofaline of any degree, to which atangentisdrawnatthepointa,{3,"1,usingUinamannercorre­ spondent toitsprevious signification todenote (x-~z)~+(y-@z)!!-, "1da "1d{3 andunderstanding Q.I(a,{3,"1)byQ.I,we have for determining thepointof intersection, Q.I=0,XQ.l=0,XRQ.I=0;andconsequently, by aid of our theorem (3), we shall obtain Q.I=o, UQ.I=o, By means of thetwolatterequations, weobtain •Thusletzbeahomogeneous function inzandyof ,dimensions, andlet d.zdztPll£liz£liz dZ'dll'liz"dzdll' dys' be called p, q, r, II,t;weshallfind r,I,p I,t,q=0,, p,q,,-=-1w thatis,w-~rq'-2pql+tp'-, rl-I' ' 30] ofHomogeneous Functions. 175 whereFandGareeachof only(n-2)dimensions, and serve to determine theintersections ofthetangent withthecurve,extraneous tothetwo coincident ones atthepointofcontact. Again,suppose that'"is afunction of anydegreeof anynumber of letters(1,13,"/'&c.,andthatwe havegiven",=0,x'"=0,x'"=0,...Xm",=0j it isevident from our fundamental theorem thattheseequations may be replaced by andconsequently thattheexpulsion of(m-1)letters,by aid of the lastm ofthegivenequations, will beattended bythedisappearance ofmorders, or, inotherwords, the resultant will beminusan order, thatis,will have one orderlessthanthenumberoflettersremaining in it. Inapplying tospaceconceptions thepreceding theorem, itwill be con­ venienttouseageneralnomenclature forgeometrical species of various dimensions. Thuswe may call a line amonotheme, a surface a ditheme, thespecies beyondatritheme, andso on,adinfinitum. Asystemofpointsaccording tothesamesystemofnomenclature would becalledakenotheme. Ann-theme hasforitscharacteristic ahomogeneous function of (n+2) letters. Again,itwill beconvenient to giveageneralnametoallthemesex­ pressedbyequations ofthefirst degree. Rightlines and planes agree in conveyingan idea of levelness and uniformity jtheymay both be saidtobe homalous. Ishalltherefore employtheword homaloid tosignify in general anythemeofthefirst degree. Nowlet'"(x,y,z...t)bethecharacteristic toann-theme ofthenth degree. Thenumberoflettersx,y,z...tis(n+2). Asusual,let",represent", (a,13,"/...0),andsuppose '"=0,x'"=0,X''''=0...X"'"=0, andconsequently U1",=0,U,,,,=0...U"",=O. Oneliminating (n-1)lettersbetween thenlastequations, theresulting functionwillbeofthreelettersbutof only two orders, and of the 1.2.3...n degree. Hencewe seethatateverypointofann-theme ofthenthdegree, 176 OncertainGeneralProperties [30 andlyinginthetangent homaloid thereto,1.2....nrightlines may be drawncoinciding throughout withthen-theme. Thusonerightline can bedrawnateachpointof alineofthefirst orderlyingontheline;tworightlinesateachpointof asurfaceofthesecond orderlyingonthesurface;sixrightlinesateachpointof ahyperlocus ofthe thirddegree, and so forth. Itis obvious thata surface may be treatedasthehomaloidal sectionof atritheme, justasaplanecurvemayberegarded asasectionof asurface. Ishallproceed to show upon thisview, how we may obtainatheorem given by MrSalmonfor surfaces of thethirddegreeof aparticular character from thelawjustlaid down, according to which a tritheme ofthethirddegree admitsof sixrightlinesbeingdrawnuponitateverypoint". Let Q)(:c,'!I,z,t,u)bethecharacteristic ofanytritheme of thethird degree; a,p,'Y,S,e,coordinates toanypointinthesame.Then Q)(a,P,'Y,S,f)=0,andtheequation tothetangent homaloid willbe XQ)(a,P,'Y,S,f)=0, andtheequation tothepolaroftheseconddegree tothegiventritheme inrelation totheassumed pointasorigin,(thatis, theinfinitesystemofhomaloids thatmay be drawn from thepointto touchthetritheme), will be ~Q)(a,P,'Y,S,f)=O. Butthesectionofanypolarthrough itsoriginisthepolarofthesection tothesameorigin;hencethepolartotheintersection ofQ)(:c,'!I,z,t,It)=O. withXQ)(a,/3,'Y,S,f)=0, istheintersection ofXQ)=0 with ~Q)=O. Theprojections oftheseintersections uponthespacea,'!I,z,twill be found by eliminating u,andgettingthecorrespondent twoequations between :c,'!I,z,t.Hencewe seethattheprojection ofthelatterinter­ section uponanyspace:c,'!I,s,tisacone;or, inotherwords,this intersection itself,thatis,thepolartotheintersection ofthetritheme withitstangent homaloid, is acone;thatis to say, thesurfaceofthe thirddegreeformed by cutting atritheme ofthe,thirddegreebyany tangent homsloid hasa conical pointatthepointofcontact; sothat everysurfaceofthethirddegreewith a conical pointmay be considered astheintersection ofatritheme ofthethirddegreewithanytangent homaloid thereto'[. • Thereduction ofanyequation oftheBixthdegreetodepend upon one of the firthmay be shownby MrJerrard's methodtobeequivalent todrawingastraightline upon a tritheme of the thirddegree,justas thereduction of theequation of the fifth degree toatrinomial form may be showntobedependent upon our being able todrawastraightline upon aditheme of thesecond degree. Now at every pointofatritheme straightlinesmaybedrawn,butastheykeeptogether ingroupsof sixes theycannotbe found in generalatagivenpointwithoutsolving an equation ofthesixthdegree. tSoin likemannerasurfaceof thethirddegree with more thanoneconicalpointmay be genera.ted by theintersection of thetritheme withaplurl-tangent plane;and80too wemayget otherva.rieties by takinghomaloidal sectionsoftrithemes whosecharacteristics areminusone or moreorders. 30] ofHomoqeneous Functions. 177 Ifnow we makeHencethenwesee,asaninstantaneous deduction from our general theorem, thatatany conical point(when one exists) of asurface of the thirddegreesixrightlines may be drawn lyingcompletely upon it. This theorem is thusbrought into animmediate andnaturalconnexion with the well-known one, thatateperypointin a surface of the second degree, two right lines canbedrawnlying wholly upon thesurface.", Thelastgeometrical application ofthetheorem (3) which I shall make, referstotheequations employed by Mr Salmon in No. XXI.(NewSeries)of thisJournal, toobtainthe locus of thepoints on any surface atwhich tangentlinescanbedrawnpassingthrough four consecutive points. I may remarkinpassingthattheseequations may beobtained byrathersimpler considerations thanMr Salmon hasemployed so to do, and without any reference toJoa.chimstha.l's theorem; for if we takeE,'TJ,~,8,astheco­ ordinates of anypointin one of thetangentlines above described, and if we takethefirstpolartothesurface with thispointasorigin,threeout ofthe fouroriginalpointswill be found in such polar consecutive butdistinct; and consequently in the second polar, referredtothesameorigin, two will con­ tinueconsecutive butdistinct,andconsequently one will remainover in the third polar. Hencewritingtheequation to the surface 0)(:e,y,6,t)=0, and using Dto d d d d .denoteEd:e+'TJdy+~dz+8dt'we shallevidently have fI)=O, (1) DO)=0, (2) J)JfI)= 0, (3) J}1f1)=0, (4) asobtained by Mr Salmon. And the samekind of reasoning precisely appliestothetheoryofpointsof inflexion in curves; threeconsecutive points in arightline inthiscasecorresponding to four such in thecase aboveconsidered :e'=--8=u ~t' .,,-!L8=v 'ft ' z"'--8=w ~t' •Ifwehaveanindeterminate systemofalgebraical equations consisting of onequadratic and anotherft'fonction ofthreevariables, thismaybecompletely resolved by considering thefirst&8 anequation toasurfaceofthesecond degree, finding atanypointthereofthetwolines whioh lieuponthesurface,anddetermining theirrespective intersections withthesurfacerepresented bytheIIeCOndequation. Thiswillrequiretherefore thesolution only ofaquadratio and an ftC eqaat.ion. Inlikemanneranindeterminate systemoftwoequations of fourvariablea, one of the thirdandtheotherofthenthdegree, may beoompletely resolved (with the aidofthetheoremin thetat)by means of twoequations, one of the sixthand theotherof theftthdegree. 8. 12 178 OncertainGeneralProperties [30 ~=ell+X{311 a.=~+Xt32'theequations (2), (3), (4), by our theorem, may beexpressed intermsof tt,v,w,whichbeingeliminated weobtainanequation between a,y,e,t, which will express thesurfacewhoseintersection withthegivensurface Q)=0 serves to determine thelocus of thepointsinquestion. Henceif we proceed in theordinary mannertoeliminate two ofthefour letters,as~and'f},between theequations (2), (3),(4),theresultant will be of theformMxI/>(t8), where Mdoesnotcontain ~,'f},~or(),and where by thegenerallaws ofelimination I/>(~,8) will be anintegral function ofthe sixthdegreeinrespectto~,(): anditismanifest thatMxI/>(~,8) will be identical withtheresultant of (2),(:3),(4)expressed intermsofu,tI,w, whenuandvareeliminated cy-presof anintegralizing factor,showing that I/>(t8)isWSintegralized, thatis. isequalto(t~-ZO)8. Consequently asMI/> isoftheorder(n-I)2.3+(n-2)1.3+(n-3)1.2, thatis,lln-18 in respecttoe,y,s,t,it follows thatM=0,theequation tothesecond surface spokenof above, will be of theorderlIn-24,agreeable toMrSalmon's showing. I shall conclude thispaperby showing theapplication of ourtheorem to thesubjectpropounded by MrJerrardandSirWilliam Hamilton, ofsystems ofequations containing a sufficient number ofvariable lettersforeffecting thesolution without elevation of degree. Ifwe have nhomogeneous equations containing asufficient numberof letters ~,a,z...amtoenableus to express thesolution of(11-1)ofthe equations underthe form an.=am+Xt3m, where ell'~•••elm,13"132'"13maresupposed known, and Xisindeterminate, itisevidentthatbysubstituting thesevalues in thenthequation, Xmay be found by solvinganequation ofthesamedegreeasthatequation contains dimensions of~,a....a".. Letusthenpropose thisquestion: how many lettersalla....aTare neededtoobtainalinearsolution of asystemofnequations t/>.=0,1/>2=0,...epn=0, oftheseveral degrees 'I'~...'n,without elevation ofdegree; by alinear solution beingunderstood asolution undertheform ~=ell+X{3" a.=~+X{32' whereXis leftindeterminate. 30J ojHomoqeneous Functions. 179 (8)LetussupposethattXt,~•••a,.,substituted respectively forall~...ar, satisfythegivensystemofequations. Thedetermination ofthesevalues without elevation ofdegreewill, from whathas been Raidbefore,depend uponthelinearsolution of asystemofequations differing fromthegiven systembytheomission of anyone ofthematpleasure. Nowmake d d dD=!X)da,.+~d~+ ...+a,.dar. andthenwrite Dcf>,.=0,P!f»=0'"D'.cp)=0I ~~.~.~'..~~~.~~::'.~:~,~.~ " D!f>n=0,Pcpn=0 ...D'"!f>n=0 Thevaluesof~,~...arderivedfromthissystem,say(a)1I(a),..,(a).., g1Ve ~=el)+X(a~,~=a,+X(a)"•..e,=a,.+X(a)r' asolution undertherequired form, where Xis leftindeterminate . .Thesolution ofthisnewsystemwithoutelevation ofdegreedepends on thelinearsolutionof allbutone ofthem;thisexcepted one may be taken theone whose dimensions trarethehighestor ashighasanyofthe quantities t),~•••£no Consequently, ifwe usethesymbol(k),ks...kr)todenotethenumberof lettersrequired forthelinearsolution (without elevation ofdegree)ofk, equations ofthefirstdegree,k,ofthesecond,ksofthethird,.."k;of therth,itwouldatfirstsightappearfromthepreceding reduction that wemusthave where(kl•k,... kr)=(KlIK,...Kr-llK;j, K,=k,+ks+ '" +kr-1+kr, K,=k,+ ... +kr-l+krJ Kr-l=kr-l+k" K;=kr-1. Butnowstepsinourtheorem (3), and shows thatthesystem(8)may be superseded byanother, in which thevariables, insteadofbeingaI,~...an, will be al ~ Cln-l ~--an,CZs--Un,...£tn-l--an;an ex,. ex,. consequently thenumberofreallyindependent variables is only(n-1);we musttherefore have 12-2 180GeneralProperties ofHomoqeneous Functions. [30 whereSincetheintroduction ofanewsimpleequation isequivalent tothe requirement of one more disposable letter,we may write theabove more symmetrically undertheform (klJk,...kr)=('KlJK,...Kf'-lJK/), 'K1=1+k,+k,+ ...+kr, Kr'= kr-l. BymEl&DSofthisformula of reduction (klJks...kr)may be finally brought down to theform(L),andthevalue of(L)beingthenumber ofletters required forthelinearsolutionofasystemofLlinearequations isevidently L+2. Thus, to determine thenumberoflettersrequired forthe'linearsolution ofasinglequadratic, we write (0, 1)=(2)=4. Fortwoquadratics, we write (0, 2)=(3, 1)=(5)=7; foraquadratic andacubic, (0,1,1)=(3,2)=(6,1)=(8)=10; for two cubics, (0,0,2)=(3,2,1)=(7,3)= (11, 2) = (14,1) = (16) =18. Theseresultscoincide withthoseobtained bySirWilliam Hamilton in hisReporton MrJerrard's Transformation oftheEquation oftheFifth Degree in theTransactWn8 oftheBritishAssociation. I have much more tosayonthesubjectof thelinearsolution ofasystem of indeterminate equations, andam,I believe, able topresentthesubjectinamoregeneral lightthanhashitherto beendone;butmyobservations onthismattermust. bedeferred untilasubsequent communication. 31. REPLY TOPROFESSOR BOOLE'S OBSERVATIONS ON A THEOREM CONTAINED INTHELAST NOVEMBER NUMBER OFTHEJOURNAL. [Cambridge and Dublin Mathematical Journal, VI.(1851),pp.171-174.] TOErestricted spacethatcanbesparedfor discussion in these pages, necessitates metocompress withinthenarrowest limittheremarks which I feelboundto make on Mr Boole's extraordinary observatious "inthe February number ofthisJournal, on mytheorem contained intheante­ cedentnumberthereoft, whichstatements Icannot,intheinterests oftruth andhonesty, suffer to passunchallenged. Theobjectofthattheorem was toshow how thedeterminant of thequadratic function resulting fromthe elimination ofanysetofthevariables between agivenquadratic function andanumberoflinearfunctions ofthesamevariables, could be represented . withoutperforming theactualelimination byafraction, of which thenume­ rator would be constant whichever set ofthevariables mightbe selected for elimination, andthedenominator thesquareofthedeterminant corresponding tothe coefficients of thevariables 80eliminated. Thenumerator itselfisa determinant. obtained by forming thesquarecorresponding to thedeterminant ofthegivenquadratic function, and bordering ithorizontally andvertically withthelinesandcolumnscorresponding tothecoefficientsof allthevariables inthegivenlinearequations. Animmediate coroUary fromthistheorem leads toMrBoole's, Conversely upon theprinciple that"toutestdanstout" Mr Boole devotes a page andahalfof close printmerely to indicate the stepsof a method by which from his theorem mine is capable of being deduced, endingwiththeannouncement, tha.tthenumerator inquestion isequaltothequantity (the symbols above employed beingMr Boole's own), andconcludes with essuring his readersthat..hehasascertained thatMrSylvester's resultis reducible totheaboveform."MrSylvester would be very sonytoputhis [.CambroandDubUnMath. JUUT.VI.(1851),pp.90,284.] [tp.135above.] 182 ReplytoProfessor Boole's Observations [31 resultunderanysuchform. Mr Boole couldscarcely havereflected uponthe effect of his words whenheindulged intheremarkwhich follows-i-" there cannotbea doubtthatforthediscovery oftheactualrelationinquestion, the abovetheorem is far more convenient thanMrSylvester's." Ofthevalueto beattached tothisassertion theannexed comparison ofresultsissubmitted asaspeCImen. Letthequadratic function be aafl+bJl+cz'+dt'+2exy+2ezt+2gxz+2ryyt+2hyz+2'1]xt, andthelinearfunctions (takentwo innumber) lx+my+nz+pt, l'x+m'y+n'z+p't: Mynumerator will bethedeterminant (hereinafter citedastheextended determinant ), ae9'T/r eb h m1n, ry 9hcenn' dp p, '1]rye lmnp0 0 rm,n'p'00 To find thenumerator of Mr Boole's fraction, wemustformthe symbolical operator fl'd,d,d,d )da+m db+n dc+Pdd I d d d d d dJ ~+2lmde+2npde+2lndg+2mpdry+2lp dh+2mnd'T/ {l" d "d"d rd }da+mdb+n dc+Pdd x+2l'm':e+2n'p':e+2l'n'~+2m'p':ry+2l'p':h+2m'n':~ andafterexpanding thedeterminant hereunder written e 9 '1] b h ry 'ghce ."ryed perform theoperations aboveindicated upontheresultsoobtained. Thesearetheoperations andprocesses which,onProfessor Boole's authority, we are to accept uaswithout doubtfarmoreconvenient" than theonesimpleprocess of forming, andwhennecessary, calculating the 31J on aTheorem ofMrSylvest('jf"s. 183 extended determinant a.bovegiven. Hereforthepresent I leave the casebetween Mr Boole and myselftothejudgment of thereadersof thisJournal. IntheAprilNumber ofthePhilosophical Magazine-, I have shown thattheextended determinant serves, not only to represent thefull and complete determinant ofthereduced quadratic function, butlikewise all theminordeterminants thereof; thelastsetof which will be evidently no otherthanthecoefficients themselves. Forinstance, in theexample above given,if we wish to find thecoefficient of x'afterzandthave been eliminated, we have only to strikeoutthelineandcolumnebh'Ymm'from theextended determinant ; if we wish to find thecoefficient ofy2,wemust strikeouttheline and column a e9"1l l';to findthecoefficient of xy,we muststrikeoutthelineae9"1ll'andthecolumnebh'Ym m',orviceversa. Ineach ofthesecasesthedeterminaut soobtained isthenumerator of theequivalent fraction; thedenominator remaining alwaysthesamefunction ofthecoeffici~ntB oftransformation asintheoriginaltheorem. Again,iftherebetakenonly one linearequation, andby aid of itxis supposed tobeeliminated; and ifthereducedquadratic function be called Ly'+MzI+Nt'+2Pzt+2Qyt+2Rzy, thesameextended determinant asbeforegivenwill serve, when stripped of itsouterborder,consisting ofthelineandcolumnl'm'n'p',to produce thevariousequivalent fractions: thusformthesquare L RQ R M P QP N. Thenumerator ofthefractionequivalent toI~~\'thatis, toLM-Jl2, maybe found by strikingout from theform oftheextended determinant the lineandcolumn "1'Yedp;thatcorresponding toI~~\'thatis,LP-RQ, will befound by striking outtheline9 hc€nandthecolumn "1'Yedp,orvice fJeTSa;andso forth for all thefirst minor determinants; andsimilarly the second minors, thatis,L, M,N, P, Q,R,may beobtained bystriking out in each case a correspondent pairof lines and pair of columns. Thus, to find thenumerator ofLthesamepairof lines and columns, namely, (ghc€n), (11'Yedp),mustbe elided. To find the numerator ofR,thepairoflines (ghcen),("1'Yedp),and thepairofcolumns (ebh'Ym),("1'Yedp),orvice t1e7"sa,will have to beelided;and so forth for theremaining second minors. I may.conclude withobserving, thatthetheorem contested by Mr Boole is animmediate corollary from thegeneralTheoryofRelative Determinants alludedt tointhe"Sketch" insertedin thepresentnumberoftheJournal. [*p.241below.] [tp, 188 below.] 32. SKETCH OF A MEMOIR ON ELIMINATION, TRANSFORMATION, ANDCANONICAL FORMS. [Oambridge and Dublin Mathematical Journal, VI.(1851),pp.186-200.] THEREexistsapeculiar systemofanalytical logic, founded upon the properties of zero,whereby, fromdependencies ofequations, transition may be made totherelations offunctional forms, and vice'Versd:thisI callthe logic of characteristics. Theresultant ofagivensystemofhomogeneous equations ofasmany variables, isthefunction whosenullityimpliesandisimpliedbythepossi­ bilityoftheircoexistence, thatis, isthecharacteristic of suchpossibility j butinasmuch asanynumerical product ofanypower of acharacteristic is itselfanequivalent characteristic. inorderto givedefiniteness tothenotion of aresultant, itmustfurtherberestricted to signify thecharacteristic taken inthelowestformof which itingeneraladmits. Thefollowing very generalandimportant proposition forthechangeof theindependent variables intheproce8.'lofelimination, is animmediate consequence ofthedoctrine ofcharacteristics. Lettherebe twosetsofhomogeneous forms of function j the1st, .pHcf>t•••!/>n, the2nd. VI'Vs'"V..· Lettheresultsofapplying theseforms to any setsofnvariables be called (cPt),(.p,)...(4)..). (VI),(Vs)···(V ..)j thenwilltheresultant (inrespecttothosevariables) of 4>1{(VI),('h)..·(V..»)• .p.{(VI).('Ir.)...(V..»). !/>n[('/rJ).('Ir.)...('Ir,,)l, 32JElimination, Transformation, andOasumieal Forms. 185 betheproduct of powers (assignable bythelaw ofhomogeneity) ofthe separate resultants ofthetwosystems, {(</>I),(cPi)••.(ep,.)I, [(Vi),(Vi)...(V..)}· Bymeansofthedoctrine ofcharacteristics thefollowing generalproblem mayberesolved. Givenanynumberoffunctions ofasmanyletters,and aninferiornumber offunctions ofthesameinferiornumber ofletters,obtained bycombining, inter86,in a known manner, thegivenfunctions, todetermine thefactor by which,theresultant ofthereducedsystembeingdivided,theresultant ofthe originalsystemmay beobtained. Ifinthetheorem forthechangeoftheindependent variables bothsets of forms of funotions betakenlinear,weobtainthecommon ruleforthe multiplication ofdeterminants: ifwetakeone setlinearandtheothernot, wededucetwo rules, namely,Thattheresultant of agivensetoffunctional forms of agivensetofvariables, entersasafactorintotheresultant, 1st,oflinearfunctions ofthegivenfunctions ofthegivenvariables; 2nd, ofthegivenfunctions oflinearfunctions ofthegivenvariables: theextraneous factor in each case beingapower of whatmay be con­ veniently termedthemodulusoftransformation, thatis,theresultant of theimported linearforms of functions. Fromthesecond of theserulesweobtainthelaw first statedIbelieve forfunctions beyondtheseconddegreeby Mr Boole, to wit, thatthedeter­ minantofanyhomogeneous algebraical function (meaning thereby the resultant ofitsfirstpartialdifferential coefficients) is unaltered byany lineartransformations ofthevariables, exceptso farasregardstheintro­ duction of apower of themodulus oftransformation. Thisis also abundantly apparent fromthefact,thatthenullityof such determinant impliesanimmutable, thatis,afixedandinherent, property of acertain corresponding geometrical locus. Thereexist(asis now well known) otherfunctions besidesthedeter­ minant, calledbytheirdiscoverer (Mr Cayley) hyperdeterminants, gifted withasimilarproperty ofimmutability. I havediscovered a process for findinghyperdeterminants offunctions of anydegreeof anynumber of letters, by meansofaprocess of Compound Permutation. All MrCayley's formsforfunctions of twolettersmay beobtained inthismannerbytheaid of one of thetwo processes (towit,thatone which will hereafter becalled thederivational process), forpassingfromimmutable constants toimmutable forms.Suchconstants andforms,derivedfromgivenforms, may be best 186 Sketchofa Memoir on Elimination, [32 termedadjunctive; atermslightlyvariedfromthatemployed by M.Hermite inamorerestricted sense. Thetwo processes alludedto may be termedrespectively appositional and derivational. Theappositional iiifounded upon theproperties ofthebinary functionx,+Y17+z~+...;in which, whether wesubstitute linearfunctions ofx,y,e,&c., orlinearfunctions of""1,~,&c., in place of a;y,z,&c., or e,"1,~,&c.,theresultisthesame. Consequently, if weapplythe form 4>toe,"1...~,andtakeanyconstant (inrespectto',"1'''~) adjunctive to 4>(""1...n+(xe+Y"1+...+ z~+kt)r:', callingthisquantity" (x,y...s,t),theformVisevidently adjunctive tothe form4>:andifweexpand 80astoobtain "(x,y...z,t)=VI(x,y...z)".+Vv(x,y...z)tfJ+&c., itisevidentVI'"V,&c. will be eachseparately adjunctive to4>.These forms, when Visobtained by finding thedeterminant inrespecttoe,"1...~ ofS,are, in fact, identical withHermite's" formesadjointes." Thederivational mode of generating forms from constants depends upon theproperty oftheoperative symbol d d d X.=~(&+"1dy+...+ ~dz' appliedto4>afunction ofe,y...Z;namely,thatif in4>,in place of these letters,wewritelinearfunctions thereof,to witx',y'...z',we maywrite 1:'d,d f"d X.=~dJ;'+"1dy'+...+ ~dz" wheref,"1'...rwillbethesamefunctions ofe,"1...,thatx',y'...z'are of e,y...z. Suppose now, inthefirstplace,thatinregardtoe,"1...~,V(x,y...z)is adjunctive tox.r4>(x,y...z);thenistheform"adjunctive totheform,p,for onchanging x,y ...ztox',y'...s', (I:d d rd)r'"(" ')\~dJ;+"1iy+"'+~dz 'f'x,y...z b (t 'd,d f"d)'"(" ') .ecomes ~dX'+"1dy'+...+ ~dz''f'x,y...z , andconsequently t(x,y...z)becomesV(x',y'...z'),multiplied by a power ofthemodulus oftransformation, themodulus ofthattransformation, beit well observed, whereby x',y'...z'would be replaced byx,y...z,andnotas intheappositional mode ofthatconverse transformation according to which 32J Transformation, andOanonical Forms. 187 :E,Y..•zwould be replaced byx',y'...z'.Itis onaccount ofthisconverse­ nessofthemodesoftransformation thattheappositional andderivational modesofgenerating formscannotexceptforacertainclass of restricted lineartransformations becombined in asingleprocess. More generally, if instead ofasinglefunction xT4>(x,y...z),wetakeasmanysuchwith different indicestoXastherearevariables, andformeithertheresultant inrespectto"'TJ••,"oranyotherimmutable constant inregardtothose variables, (presuming inextension ofthehyperdeterminant theoryandasno doubtistheease,thatsuchexist),everysuchresultant orotherconstant will give a form of function ofx,y...Zadjunctive tothegivenform4>. Itmay beshownthateverysuchresultant so formed will contain 4>as a factor. A~in.intheformermoreavailable determinant mode of generation, if wetakethedeterminant inrespectto"'TJ•••"itmaybeshownthatallthe adjunctive functions soobtained will bealgebraical derivees ofthepartial differential coefficients of4>inrespecttox,y..,z;thatistosay,ifthesebe respectively zero, all suchadjunctive functions soderived, aslastaforesaid, will be zero, or in otherwords, each suchadjunctive is asyzygetic function of thepartialdifferential coefficients oftheprimitivefunction. To:MrBoole is duethehighpraiseofdiscovering andannouncing, under asomewhat different andmorequalified form and mode of statement, this marvel-working process of derivational generation ofadjunctive forms.Iwas ledbackto it, in ignorance or'whatMr Boole haddone, by thenecessity whichIfelt toexistofcombining Hesse'sso-called functional determinant, underacommon pointof viewwiththecommon constant determinant of a function; underpressure ofwhichsense of necessity, itwasnotlongbefore Iperceived thattheyformedthetwo ends of a chainofwhichHesse'send exists fur all homogeneous functions, buttheotheronly when suchfunctions arealgebraical. Infact,ifwegivetoreveryvalue from 2upwards, thesuccessive determinants inrespectto','TJ•••'of (d d d)T,dx+'TJdy+...+'dz4>(x,y, z), willproduce thechaininquestion. which, when 4>isalgebraical andof ndimensions, comes to anatural termination whenr=11-1.Thelast member ofandthenumber oftermsinthischainareidentical withthe lastmember ofandthenumber oftermsinSturm's auxiliary functions, whenthevariables arereduced to two. Thereis some reasontoanticipate thatthischainoffunctions maybemadeavailable insuperseding Sturm's chain of auxiliaries; andif so,thenthefatalhindrance toprogress, arising fromtheunsymmetrical natureofthelatter,is overcome, andweshallbe 188 SketchofaMemoironElimination, [32 able topassfromSturm's theorem, whichrelatestothetheoryofKeno­ themes,orPoint-systems, tocertaincorresponding butmuchhighertheories for lines, surfaces, and n-themes generally. Therestriction of space allowed to me in thepresentnumber ofthe Journal willpermitme only to alludein thebriefesttermstothetheory ofRelative Determinants, which,asitwill be seen, plays an important part intheeffectuation ofthereductions ofthehigheralgebraicsl functious to theirsimplest forms.Norcantheeffect of theprocesses to beindicated be correctly appreciated withoutaknowledge ofthecircumstances underwhich theresultant ofagivensystemofequations can sink in degreebelowthe resultant ofthegeneraltypeof such system. Abstracting fromthecase whentheequations separately, or incombination, subdivide intofactors,this lowering of degree,asmay be shown by thedoctrine ofcharacteristics, can onlyhappenin one of two ways. Eithertheparticular resultant obtained is arationalroot ofthegeneralresultant, orthegeneralresultant becomes zero forthecase supposed, and theparticular resultant is ofadistinctcharacter fromthegeneralresultant, beingin factthecharacteristic ofthepossibility notofthegivensystemofequations beingmerely able to coexist (for thatis alreadysupposed), butoftheirbeingable to coexist for a certainsystemof valuesother than agivensystemorgivensystems. Sucharesultant may be termedaSub-resultant; thelowestresultant intheformercasemay be termedaReduced-resultant. ThetheoryofSub-resultants is onealto­ getherremaining tobeconstructed, and is well worthyequallyofthe attention ofgeometers and ofanalysts. As tothetheoryofRelative Determinants, theobjectofthistheory is toobtainthedeterminant resulting fromeliminating asmanyvariables ascan beeliminated, chosenatpleasure from asetofvariables greaterin number thantheequations containing them;andthemode of effecting thisobjectisthrough themethodof theindeterminate multiplier. To avoid thediscussion of the theoryofsub-resultants andotherparticularities, Ishall contentmyself with givingtheruleapplicable tothecase(theonly one of whichasyetapractical application hasoffereditselfto me in thecourse of mypresentinquiries) when all butone ofthefunctions are linear. IfU,LI,Lv...LmbethefirstannOandtheotherslinearfunctions ofn variables, anditbedesiredto findthedeterminant oftheresultant arising fromtheelimination of anymout ofthenvariables, the following is the rule: Findthedeterminant, thatis,theresultant ofthepartialdifferential coefficients in respecttothegiven variables, and of AI,Av...Amof U+LIAI+L'lAv+...+LmA,R' 32) Transformation, andOanonical Forms. 189 Thisresultant, initslowest form, will be always arational (n-1)throot of theresultant ofthehomogeneous systemofequations towhichthesystem abovegivencanbereferred as itstype;and this reduced resultant divided byapower(determinable bythelaw ofhomogeneity) oftheresultant of ~,Ls...L.,.,when all buttheselected variables are made zero, will be the resultant determinant required".Asregardswhathallbeensaidconcerning thereducibility ofthe.generaltypicalresultant inthecase before us, thisis aconsequence of, and may be brought intoconnexion with, thefollowing theorem, which is easily demonstrable bythetheoryofcharacteristics. If QIJQ2...Q.,.be mhomogeneous functions ofmvariables ofthesame degree, r of which enterineachequation only as simplepowers uncom­ binedwithany oftheothervariables, thenthedegreeofthereduced resultant isequaltothenumber oftheequations multiplied bythe (m-r-l)thpOlVerofthenumber ofunitsinthedegreeof each, subjecttotheobvious exception thatwhen r is m, (therebeing in fact butOMstepfrom r=m- 2 to r =m),insteadofr,(r-1)mustbe em­ ployed iu theabove formula. Asanexample ofasub-resultant as distinguished from a reduced-resultant, Iinstance thecaseofthree quadratics U, V,W,functions ofe,y,z,ineachof which no squared power of zissupposed toenter:itmayeasilybe shown by my dialytic methodthatinsteadof sixequations, between whichtoeliminate a:',ys,zI, :ry,es,yz,weshallhave only 5, thethreeoriginal ones and two insteadof threeauxiliaries between which to eliminatew,yJ,a:y,a:z,yz,theapparent resultant isaccordingly ofthe9thinsteadofthe12thdegree. Butthisis notthetruecharacteristic ofthepossibility ofthecoexistence ofthegiven systems, which in fact is zero, as isevidenced bythefactthattheyalwaysdo coexist, since theyarealwayssatisfiable by onlytworelations between the variables, towitx=0,y=O.Tbeapparent resultant isthensomething different, and whathas been termedbytheaboveaSub-resultant. Itakethisopportunity ofentering my simple protestagainsttheappro­ priationof mymethodof finding theresultant of anysetofthreeequations ofdegreesequal or differing only by a unit,one from those of theothertwo, byDrHesse, 80far asregardsquadratic functions, withoutacknowledgment, fouryearsafterthepublication of my memoir in thePhilosophical Magazine: thefundamental idea ofDrHesse'spartialIilethodisidentical withthatof mygeneralone.Stillmoreunjustifiable isthesubsequent use ofthedialytic principle, bythesameauthor,equallywithoutacknowledgment, and incases wherethereis nopeculiarity of form of procedure togiveevenaplausible ground for evadingsuchacknowledgment. Itiscapableof moral proofthat • The lIllIIU!methodappliesnot only totheFinalorConstant Determinant, but likewise to alltheFunctional De&erminants inthechainabovedescribed, extending upwards fromthisto theHeeBiau, or&8itoughttobetermed, thefirstBoolian Determinant. 190 SketchofaMemoir onElimination, [32 what I had writtenonthematterwassufficiently known in Berlinandat Konigsberg, ateach epoch of DrHesse's use of themethod. I now proceed to tbeconsideration of the more peculiar branchofmy inquiry,whichisastothemode of reducing Algebraical Functions totheir simplest and most symmetrical. orasmyadmirable friend M. Hermite well proposes to callthem,theirCanonical forms. Every quadratic function of anynumber ofvariables may always be linearlytransformed into any other quadratic functions ofthesame, and thattoo in an infinitevarietyofways; butin every otherinstance therewill be only a limitednumber of ways, whereby, when possible, one form will admitofbeingtransmuted intoany other:andwiththesoleexception of a cubic function of two letters,such transmutation willneverbe possible, unless acertaincondition, orcertain conditions, besatisfied between theconstants oftheforms proposed for transmutation. Thenumber of such conditions isthenumber ofpara­ metersentering intothe canonical form. andis of course equal to the number oftermsinthegeneral form of the function diminished bythe squareof thenumber ofletters. Thusthereis oneparameter inthe canonical form for thebiquadratic function of two and the cubic function ofthreeletters,and no parameter in the cubic function of twoletters. Hitherto no canonical forms have been studiedbeyondthecases above cited,butI have succeeded. aswillpresently be shown. in obtaining methods forreducing totheircanonical forms functions withtwoandfour parameters respectively. Owing to whathasbeenremarked above,the theoryofquadratic functions is atheoryapart.Simultaneous transforma­ tion gives definiteness tothattheory,buthasnoexistence for any useful purpose forfunctions ofthehigherdegrees. Wherethetheoryofsimul­ taneous transformation ends.thatof canonical forms properly begins;and inwhatfollows,thecaseofquadratic forms is to be understood asentirely excluded. Suchexclusion beingunderstood, thereis no difficulty in assigning thecanonical, thatis,thesimplest andmostsymmetrical general, form to which every function of twolettersadmitsofbeingreducedbylineartrans­ formations. Ifthedegreebe odd,say2m+1,thecanonical form will be ••1Im+1+'IJ,m+!++ulm+1•"Ig... m+l' ifthedegreebe even, say2m,thecanonical form will be ullim+ttg'nn+...+um,m+K(~Ut...Um'f, alltheu'sbeinglinearfunctions ofthetwo given variables. Itiseasyto extendananalogous mode of representation tofunctions of anynumber of letters. Fromtheabove we see thatfor cubic, biquadratic, andquintic functions of two letters,the canonical forms will be respectively u·+tI,u·+tr+KUS";,uB+if+we, withalinearrelationinthelast-named casebetween u,'II,W. 32) Transformation, andCanonico "Forms. 191 Firstastothereduction ofany4°function" u4+v4+K-//'toCayley's form Thismaybeeffectedin agreatvariety r '''IfJ. thesimplest asregardsthecalculation-f ways, of whichthefollowing is not themodulus oftransformation, whpdrequired, butthemostobvious. Let F(x,y),becomes transmuted int:febythegivenbiquadratic function, say determinant ofFbe called ~~itscanonical form, be calledM;letthe respecttoEand'7ofllJandthedeterminant ofthedeterminant in dY ~.4)(E!--+'7~rF(xy)whichlatter,for d»dy," instricterjustievity'ssake,maybetermedtheHessian ofF,(although calledDt•TketheBoolian wouldbethemoreproperdesignation), be weretheveaen,byexamining thecanonical formitself(whichisasit weshallobypalpitating heartofthefunction laidbaretoinspection), cainwithout difficulty thetwoequations (1-9m2)2=MI2D!14B' m'(I-9mt)2(mt_1)2=M'UD~Elimins t121i44• ,tingtheunknown quantityM,weobtain mt(mi-l)i m'-m h(192'"=c,or1~9-2=el,wer -m -.m ~eis aknownquantity. sh~'hiscubicequation forfindingmis ofapeculiar form;itbeingeasyto 11'Iiapriori,bygoingbacktothecanonical form,thatitsthreerootsare Fot,8(m),fP(m).where m-I (J(m)='J~-1''Jm+ 8beingaperiodical form offunction suchthat(J3(m)=m. Thisitis which accounts forthesimpleexpression form,thatmay be obtained by solvingthecubicabove given. A betterpractical mode is to take,insteadofthedeterminant ofthegivenfunction anditsHessian, the twohyperdeterminants andeliminate asbefore:acubicequation having precisely thesameproperties, andin factvirtually identical withtheformer, willresult. Whenmandconsequently Marefound,thereis nodifficulty whatever, callingthegivenfunctionFanditsHessianH(F),informing linearfunctions ofthetwo,as q,(m)F+ v(m)H(F)} l/>I(m)F+VI(m)H(F), whichshallbeequalto,thatis,identical with,(u'+v2'"andU2V2,whence IIandvarecompletely determined. 192\, "-Sketclt-,p! a Memoir onElimination, [32 where, Another andinteresting '~odeofsolution is totake,besidesthegiven functionFanditsHessian, ei erthesecondHessian orthepost-Hessian ofthegiven function, by thepost-essianunderstanding thedeterminant in respectofEand"lof (dd)'Eda;+"l F:., anythreeofthefour functions will be linear related,anditmaybeshown that,callingeitherthesecondHessian(thatis, heHessianoftheHessian) orthepost-Hessian H',we shall have H/(F)+aH(F)+bF=0, whereaandbwill berational andintegerfunctions of F,andnumerical multiples of twoquantities Rand ,suchthatthe determinant ofFwill be equal R'+82jandthis, beitob rved, without anyprevious knowledge oftheexistence ofthesehyperde rminants R and8. Ifnow we go to Hesse'sform for acubicfunction ofthree\letters, we shall find thatprecisely similarmodes of iuvestigation applysteirforstep. CallingthefunctionFanditsHessianH (F),andthe.post_Hessianl\r second HessianatchoiceH'(F),weshallfind • H'(F)+mSH(F)+nR'F=0, where m and narenumerical quantities andR'+&equalthedeterminant ofF.Itisinteresting tocontrast thisequation withtheonepreviously mentioned asapplicable tothe4"functions of twoletters,namely, H'(F)+mRH(F)+nSF=O. Inbothinstances thereis no difficulty in assigning therelations between theoriginalRand8, andtheRand8 of any adjunctive form. All Aronhold's resultsmaybethusobtained andfurtherextended without theslightest difficulty. Asregardstheequation forfindingtheparameter inHesse'scanonical form for thecubic of threeletters,thiswill be of the 4thdegreeinrespecttothecube of theparameter, andtheroots will be functionally representable as X;e(x)j ~(x)jv(x), (Jt(x)=~2(x)=~(x)=xj e~(x)= ~e(x)=V(x), 4>t(e)=V~(x)=e(e), Ve(x)=e",(x)=~(x); owing to which property theequation is soluble underthepeculiar form observed by Aronhold. 32J Transformation, andOanonical Forms. 193 and where whereIpasson now to abriefaccountofthemethod,orratherofamethod (forIdoubtnotof being able to discover othersmorepractical), ofreducing a function of the5th degree of two letters(sayofa:andy)to its canonical formu5+'I!'+'lif,subjecttothelinearrelationau+bv+cw=0, where the ratiosa:b:c, and the linearrelations between u,v,wandthetwogiven variables aretheobjectsofresearch. HereI have found greataid from the method of Relative Determinants; and I may notice thatthe successful application of more compendious methods to thequestion would be greatly facilitated werethereinexistence atheoryofRelative Hyperdeterminants, which is stillall to form, butwhich Ilittledoubt,withtheblessing of God, tobeableto accomplish. Itmay some little- facilitate thecomprehension of what follows, if cbeconsidered asrepresenting unity. Callingasbefore the givenquinticfunctionF,themodulus of transforma­ tionM,theHessian andpost-Hessian ofF, HandH',and itsordinary or constant determinant D,we shall find ai'l!'w+b'uJu8+&u'vI=M'H, PIP,P8P~ =MSH', PI=aivw+btwu+c.·uv, Pi=a1vw-btwu-ciuv, Ps= -aivw+biwu-ciuv, p~= -aivw-biwu+ciuv; alsoD=M'l/Jmultiplied bytheproductofthesixteenvalues of at+bt(l)!+ct(l)l. Fromtheaboveequations itmay be shown thatH'(a known function of the8thdegreeofthegivenvariables ai,y)mustbecapableofbeingthrown undertheform L{(x-UtY)(x-a,y)x(x-UaY)(x-a~y) x(x-a5y)(x-asY)x(x-a.,y)(x-a,y)}, (al-a,)iX(as-a~'tx (c, -as)iX(a.,-as'! D=L,=K, sothatKisaknownquantity ".Accordingly thesaidequation ofthe8th degree,considered asanalgebraical equation in::,may by known methods bey • Or inotherwords,thepost-Hessian determinant ofagivenfunction in twolettersof the seeonddegree.maybedividedintofourquadratic fllOtorain such awaythattheproductofthe cletennin&nts oftheeeI18veralfactorsshallbeequaltothedeterminant of the given function. 8. 13 194 Sketchofa Memoir on Elimination, [32 found by meansofequations notexceeding the4thor even the3rddegree: in fact, to do thisitis onlynecessary to form theequation tothesquaresof thedifferences oftherootsof::intheequationH'+'!l=0, which new equa-y tion will be of the28thdegree.Ifwethenform two otherequations ofthe 378thdegree,onehavingitsrootsequalto,.;Kmultiplied bythebinary products ofthetwenty-eight roots of theequation lastnamed,theotherto ,.;Kmultiplied bythereciprocal of such binaryproducts, theleft-hand members ofthesetwoequations expressed undertheusual form will have afactorin common, which may be found by theprocess of common measure andwill be of the6thdegree, whoserootsconsisting ofthreepairsof reciprocals may be found by thesolution of cubics only. Inthisway, bymeansofcubicsandquadratics, (al-~)2,(a,-a4)2,(aD-a~)', (~-as)S, can be found, which beingknown, Hencecan bedetermined inpairsbymeansofquadratics fromtheequation H'+'!I=O.Thisbeingsupposed tobe done, we have PI=IL I, PS=gL 2, Ps=hLs, P4=kL4, whereLI,L2,L;Lhareknownquadratic functions of$andy.To determine theratiosoff,g, h, k,we have threeequatious "obtained from theidentity ILl+gL2+hLa+kL4(=PI+P2+Pa+P4)=OJ I:9:h:kbeingknown,ILl:9Ls:hL,:kL4areknown ratios. But PI+P,=2atvw, PI+Pa=2biwu, PI+P4=2ciuv. aivw="A.P, biwu="A.Q, c'uv="A.R, whereP,Q,Rare known quadratic functions ofe,y. ofallthem zero.*For wemusthave the coefficients of x2,xyandyllin fLI+gL2+hLs+kL4• 32) Transformation, andOanouieal Forms. 195 Hencea:b:C!Daybe found by means of theidentical equation a"~tJvS+b"usw+c''IIuS=H(F), whereby theratiosa-!:b-t:C-tcan be obtained without anyfurther extraction of roots, showing thatthereisbutonesingletruesystemof ratiosas:bB:c3applicable totheproblem; a:b:c being thusfound,Xis easilydetermined, andthusfinallyu, v,warefound in termsofa:andy•. I havelittledoubtthatamoreexpeditious mode of solution thanthe foregoing-f will be afforded by an examination oftheproperties andrelations ofthequadratic and cubic forms,adjunctive tothegenera.lquinticfunctions, andindeedto every (4n+1)·function of twolettershereinbefore adverted to. Sufficient space does not remainfordetailing thestepswhereby the general cubicfunction offourlettersmay, by aid of equations nottrans­ cending thefifthdegree,bereduced toitscanonical form US+'II+'Ill+r+T, whereinu, v,w,p, qareconnected by alinearequation au+bv+cw+dp+eq=0; thefourratiosof whose coefficients a:b:c:d:egivethenecessary nnmber :::::-42parameters furnished bythegeneral rule. Suffice it for the present tosay,thattheanalytical mode of solution depends upon a cir­ cumstance capable ofthefollowing geometrical statement: Everysurface ofthe4thdegreerepresented by afunction which is theHessian to any givencubicfunction whatever of fourletters,haslyinguponittenstraight linesmeeting threeandthreeintenpoints,andthesetenpointsaretheonly pointswhichenjoythefollowing property inrespecttothesurface of the3rd degreedenoted byequating to zerothecuhicalfunction inquestion, towit, thattheconedrawnfromanyone of them asvertextoenvelop thesurface, willmeetitnot in a continuous doublecurveofthe6thdegree,butin two curveseach ofthe3rddegree,lying inplaneswhichintersect inthe tenlines respectively abovenamed;sothatto each of the tenpointscorresponds one ofthetenlines:thesetenpointsand lines are theintersections taken respectively threewiththree,andtwo with two, of asingleandunique tty8temof fiveprincipal planesappurtenant toeverysurface of the3rd degree,andtheseplanesare nootherthanthosedenoted by u=~ v=~ w=~ p=~q=Q • Theproblem thussolved may bestatedasconsisting inreduoing thegeneralfunction cazS+bz'y+cry2+dzifHZJt+fyS tothe form (/.x+my)s+ (l'z+m'y)1+(l"z+m"y)$. tThecoefficients inthereducing recurrent equation of the 6th degree in the processabove detailedmayriseSObeof541632dimensions inrespecttotheoriginalcoefficients in F. 13-2 196 Sketchofa Memoir on Elimination, [32 I have found also by thetheoryofSub-resultants, thattheanalogy between lines and surfaces of thethirddegree, in regardtotheexistence of double and conical points,ispreserved inthiswise:thatinthesame wayasa double pointon acurveofthe3rddegreecommands theexistence of a double pointonitsHessian, so doesaconicalpointin a surface of the 3rddegreecommand over and above the10necessary, and so to speak naturalconicalpoints,atleastoneextra,thatis tosayan11thconical pointonitsHessian. And here for thepresentImustquitmybriefand imperfect notice of this subject, composed amidsttheinterruptions and distractious ofanofficial and professional life. Observation. Itmay be somewhat interesting andinstructive tomy readers,to have a tableofthesuccessive scalar-determinants of aquintic function of two letterspresented tothemat asingleglance. Preserving the notation above [page 193], we have thefollowing expressions: The given function =u'+11'+vi, itsHessian =Mt(a'v"w'+b~us+~u;,a), itspost-Hessian =M'xtheproductofthefourforms of itsprreter-post-Hessian =MItxtheproductofthenineforms of atvDwD+b!(I)DwDuD+c!(I)DuDvD, andthefinaldeterminant =M"xtheproductofthesieteenforms of al+(l)lbt+(l)icl. The success of themethodapplieddepends (asabove shown) upon the fact of a certainfunction of theroots ofthepost-Hessian (which is an octavic function of the variables) beingknown, which fact !tingesuponthecircum­ stancethat P.S. I have much pleasure insubjoining thecubicalhyperdeterminant ofthe12thdegreefunction of two letters,worked out upon theprinciple of Compound Permutation hintedatintheforegoing pages, for which I am indebted tothekindness and skill of myfriendMrSpottiswoode. • BywhichImeanthedeterminants inrespectto~."of (dd)P ~dZ+"dV F(zy). 32] Transformation, andCanonical Forms. 197 Thefunction being called --"12b-'1 12.11_-'0'&clU......- + ;<;-Y+-2- r.;w-y+ ....+ y, thefollowing is·its cubical hyperdeterminant: agm-6ahl+15aik+lOaP-6bfm, -24bhlc+30bgl+20bij-240fl+l14cgk, -145ci'+50chj+15cem+20cgi+20ch',• -400dgj+280dhi+20del+50dfe+IOd'k, +385egi-135e'k-290eh'+705fgh, -330fli-50fl. MrSpottiswoode willI hope publish theworkitselfin the next Dumber of theJournal, in which I shall also show how thehyperdeterminants ofthe cubical function of threeletters,Aronhold's SandT,may be similarly obtained. [0Seebelow,p.202.) · 33. ONTHEGENERAL THEORY OFASSOCIATED ALGEBRAICAL FORMS. [Cambridge (tOODublinMathematical Journal, VI.(1851),pp.289-293.] THEfollowing briefexposition ofthegeneraltheoryof Associated Forms, asfarasithasbeenasyetdeveloped bythelaboursorgeniusofmathema­ ticians, is intended aselucidatory and, toacertainextent,emendative of some of thestatements in mypaper-onLinearTransformations, inthe preceding number oftheJauT'IWl. Inthefirst place, let a linearequivalent of any given homogeneous function beunderstood tomeanwhatthefunction becomes when linear functions ofthevariables aresubstituted in place of thevariables them­ selves,subjecttothecondition ofthemodulus oftransformation (thatis, thevalue of the determinant formed by the coefficients of transformation) beingunity. Secondly, lettwosquarearraysofterms(thedeterminants corresponding to each of which are unity)besaidto becomplementary when each termin theonesquareisequaltothevalue of whatthedeterminant represented by theothersquarebecomes when thecorresponding termitselfistakenunity, butalltheothertermsin thesameline and column withitaretakenzero. Thisrelationbetween thetwosquaresis well known to be reciprocal. Thus, forinstance, abc afJ'Y a'b'«anda'fl', 'Y a"b"e" a"fl"II 'Y will be said to be reciprocally complementary to oneanotherwhenthetwo determinants whichtheyrepresent are each unity,and when we have [.p.184,above.] 33JGeneral TheoryofAssociated AlgebraiMl Forms. 199 a=10 0 0f3'I 'Y 0f3"I' 'Y b=010 aI0I 'Y a"0'YII b'=a0'Y 010 a"0'YII &c.a= {J= f3'=10 0 0b'0' 0b"0" 010 a,00' a"00" a00 010 a"00" &c. Accordingly, twotransformations, sayofF(x,y,z)andG(u, v,w)respect- ively,maybesaidtobeconcurrent wheninFfora,y,Z,wewrite ax+by+oz, a'x+b'y+o'z, a"x+b"y+c"zj and inGforu, v,w,wewrite au+00 +cw, a'u +b'v +c'w, a"u+blv+o"Wj butcomplementary when for u, v,w,wewrite au+{Jv+tyW, a'u+{J'v+'Y'W, a"u+{J"V+'Y"Wj• a,b,0,&c.,a,{J,'Y,&c.beingrelatedinthemannerantecedently explained. Twoforms,eachofthesamenumber ofvariables, may be saidto be associate forms when thecoefficients oftheonearefunctions ofthoseofthe otherjandwhenithappens thatthecoefficients ofthefirstareallexplicit functions ofthoseofthesecond,thelattermaybetermedtheoriginant and theformerthederivant, IfDOWallthelinearequivalents of one or of two associated formsare similarly relatedtocorresponding- linearequivalents oftheother,sothat eachmaybederivedfrom each by thesamelaw,theforms so associated will besaidtobeconcomitant each totheother.Thisconcomitance may be of twokinds,andveryprobably, inthenatureofthings,onlyor'thetwokinds abouttobedescribed. 200 On theGeneral Theoryof [33 Thefirstspecies ofconcomitance isdefined bythecorresponding equivalents ofthetwoassociated formsbeingdeduced byprecisely similar, or,aswe have expressed it,concurrent transformations orsubstitutions, each fromitsgivenprimitive. Thesecondspeciesofconcomitance isdefinedby thecorresponding equivalents beingdeduced notbysimilarbutbycontrary, thatis,reciprocal orcomplementary substitutions. Concomitants ofthefirst kindmay be called covariants ;concomitants ofthesecondkindmay be calledcontravariants. Whenofthetwoassociated forms one is a constant, thedistinction between co-andcontra-variants disappears, andtheconstant may be termedaninvariant oftheform with whichitisaaaociated ".It followsreadilyfromthesedefinitions thatacovariant ofacovariant anda contravariant of acontravariant areeach ofthemcovariants jbutacovariant ofacontravariant andacontravariant of acovariant areeachofthem contravariants; and also thataninvariant, whether ofacovariant or of acontravariant, is aninvariant oftheoriginal functiouf. Itwill also readilybeseenthatasregards functions uf twoletters acontravariant becomes a covariant bythesimpleinterchange ofe,y with-y.ai,respectively. Covariants areMrCayley's hyperdeterminants; contravariants include, butarenotcoincident with,M.Hermite's formes­ adjoiutes, if weunderstand bythelast-named termsuchformsasmay be derivedbytheprocessdescribed by M.Hermite inthethirdof hislettersto M.Jacobi,..Surdifferents objetsde laTheorie desNombres," (whichprocess isanextension ofthatemployed fordetermining thepolarreciprocal of an algebraical locust). M.Hermite appeart', however, elsewhere tohaveused itAccordingly aninvariant toagiven form may bedefined to besuchafunction ofthe coefficients of the form, asremains absolately unaltered wheninsteadof the given form any linearequivalent theretoissubstituted. Of course if the determinant of the coefficients of the transformations correspondent to therespective equivalents be nottakenunityassupposed in thisdefinition, theeffect;,willbemerelytointroduce asamultiplier some power of the deter. minantformed by the coefficients of transformation. tItmay likewise be shown thatlinearequivalents ofcovariants andcontravariants are themselves relatedto oneanotherascovariants andoontravariants respectively, thetransforma­ tionsby which theequivalents areobtained beingtakenconcurrent intheoue case andcontrary orreciprocal in theother;andofcourseanyalgebraic function ofanynumber ofoovariants is acovariant andofcontravariants acontravariant. :::Thishasbeenfurthergeneralized by me in thetheorem§ given in thelastnumberofthis Journal, where I have shown in effect thatanyinvariant inrespectto~,'I...(Jof I(~.'I...(J)+(x~+y'l+ ...+t(J+p)p ..-l, (fbeingsupposed to be ofthedegreen)isacontravariant ofI(x,y...t).Whenthisinvariant is thedeterminant ofI,it maybeshownthatweobtainM.Hermite's theorem. Itissomewhat remarkable thatcontravariants shouldhave been in useamongmathematicians &8well in geometry asthetheoryofnumbers (although theircharacter &8suchW&8notrecognized) before oovariants hadevermadetheirappearance. Invariants of course first cameup with the theory of theequation to thesquaresof thedifferences of therootsofequations, thelastterminsuch equation beingan.invariant. I believe thatI amcorrectin s&ying th&toovariants firstmade theirappearance in one of MrBoole'spapers,in thisJournal; butHesse'sbrilliant application [§p. 186 above.] 33J Associated Algebraical Forms. 201 thetermforme-adjointe inasenseaswideasthatin which I employ eontravariants. Forinstance, hehasgiven a must remarkable theorem, whichadmitsof being statedas follows: Ifwe have afunction of anynumber ofletters,sayofx,y,Z,as m(m-l)a:r!"+mba;m-Iy+rnca;m-l Z+2d:rf"r-tyJ+&c., andifIbeanyinvariant ofthisfunction, thenwill (xm.!i+Xm-ly.!i+xm-1Z.!i+xm-2y2.!i &c)rIda db dedd' bea"forme-adjointe" ofthegivenfunction. Itisperfectly trueandadmits ofbeingveryeasilyproved, as I shall show in your nextnumber,thatthisis acontravariant ofthegivenfunction-jbutitis not (as far as I cansee)a forme-adjointe inthesense in which theuse ofthatword isrestricted inthe letteralluded to.If, however, we adoptasthedefinition offormes-culjoiniee generally, thatproperty inregardtotheirtransformees which M. Hermite hasdemonstrated oftheparticular classtreatedof by him in the letter alluded to,thenhisformes-adjoiutes become coincident with my contra­ variants. Itwillthusbe seenthatcovariants andcontravariants form two distinctandcoextensive species of associated forms, which dividebetween themthewide and fertileempireoflineartransformations so far as its provinces have been asyetlaid open by theresearches ofanalysts. In yournextnumber I propose to entermuch more largelyintothesubject generally. Moreparticularly I shalldescribe thenewmethodofPermutants, including thetheoryofIntermutants andCommutants (whichlatterare aspeciesof the former, butembrace Determinants as aparticular case), and theirapplication tothetheoryofInvariants. I shall also exhibitthecon­ nexionbetween thetheoryofInvariants andthatofSymmetrical Functions, andsomeremarkable theorems onRelative Invariantaj-. Someofyourreadersmay like to be informed thataSupplement tomy lastpaper,underthetitleof "An Essay on Canonical Forms,"hasbeen since publishedj ; andthatI havetheregiven a much simplermethodofsolution oftheproblem ofthereduction ofquinticfunctions totheircanonical form thanintheoriginal memoir, andextended themethodsuccessfully to the of onefromamongtheinfinitevarietyoftheseforms to the discovery of the pointsofInflexion inacarveof thethirdorder.inotherwords, to the Canonical Reduction of the Cubic Fuaction ofThreeLetters, appearstohave been the first occasion oftheirbeingtarnedtopractical lICCOant. •TbiBis alsotrueif1betakenanycovariant insteadof aninvariant ofthefunction. tItwillbereadilyapprehended thatthedefinitions andconceptions abovestated,respecting covarianta andoontrsvariants of twosinglefunctions, may be extended so as to comprehend systemsoffanctions covariantive orcontrsvariantive to oneanother. ::ByMrGeorgeBell,University Bookseller. FleetStreet. [po203 below.] 202General TheoryofAssociated Algehraical Forms. [33 reduction of allodd-degreed functions totheircanonical form. I may take thisoccasion to statethattheLemmagiven in Note(B) oftheSupplement, upon which thismethodofreduction is based, is an immediate deduction fromthewell-known theorem forthemultiplication ofDeterminants. Thereis anumerical errorin"TheCubical Hyperdeterminant ofthe TwelfthDegree," workedoutafterthemethodofcommutants by MrSpottis­ woode,givenattheend of my paperintheMayNumber. Thecorrectresult will bestatedinthenextnumber of theJournal, where I hope also to be ableto fixthenumber ofdistinct solutions of theproblem ofreducing a SexticFunction toitscanonical form US+v.e+111+mu211w2• Forodd-degreed functions thereis never more thanonesolution possible. as shown in theSupplement referred to. P.S. Since the above was sentto press, I have discovered an uniform mode of solution forthecanonical reduction offunctions, whether of odd or even degrees. Thecanonical form however, exceptfor thefourthandeighth degrees,requires tobevariedfromthatassumed in myprevious paper.Thus, for thesixthdegreethecanonical form will be auS+W+cui+muvw(v- w)(w-u)(u- v), whereu, v,waresupposed to heconnected bytheidentical equation u+v+w=o.Andtherewill be only twosolutions-a remarkable and mostunexpected discovery. Forfunctions oftheeighthdegreethereare fivedistinctsolutions, and in generalthereisthestrongest reason for be­ lieving(indeeditmay bepositively affirmed) thatwhenthecanonical form hasbeen1ightlyassumed forafunction of the even degreen,thenumberof solutions will bei(n+2) wheninis even,buti(n+2) wheninis odd.It turnsouttherefore thatthetheoryforfunctions ofthesixthdegreeis in somerespects simplerthanfor those of thefourth. Theinvestigation into canonical forms here referredtohasled me to thediscovery of amost unex­ pectedtheorem for finding all theinvariants of acertainclass,belonging to functions of twolettersof an even degree. 34. ANESSAY ON CANONICAL FORMS, SUPPLEMENT TO A SKETCH OF AMEMOIR- ONELIMINATION, TRANSFOR­ MATION AND CANONICAL FORMS. SINCEtheabovepaperwasinprintI havesucceeded inobtaining a canonical representation ofthequadratic and cubic functions adjunctive to thegeneralquintic(5thdegreed) functions of twoletters. LetFthequinticfunction ofX,y, =u5+v5+w, and au+bv+cw=0, Mbeingthemodulus ofthetransformation, whereby transition ismade from X,ytou,v.Thenthequadratic adjunctive is M'(j{a'vw+lrwu+c'uv}; andthecubicadjunctive issimply 1 CiM8(abc)luvwt. Hence we can, in accordance withwhatIventured topredictinthepreceding sketch,findu, v,w,by means of a simpleandpractical co-process. To wit, call F=lxa+5mx'y+lOn,xSyi+1<>pari'+5qWJt+ryS. [- p.184above. See p,201,nete:1:.] tThe knowledge of theexistence of these loweradjunctive forms is mainlyaconsequence of MrCayley's splendid discovery of hyperdeterminant constants. Infact, they are respectively thequadratic andcubichyperdeterminants inrespectto~and'Iof1.2.:.4.5(~1x+'I~)•F ; zandybeingtreatedasconstants. Thefortunate proolaimer ofanewoutlying planethas been justlyrewarded by the offer of abaronetcy andanational pension, whichthewriterofthiswishes him long life andhealth toenjoy. In themeanwhile, whathasbeen done in honourof thediscoverer of a new and inexhaustible region of exquisite analysis? 204 Form the determinantOnOamonieal Forms. [34 le+my,'InX+ny,me+P'!J I7I.'l:+'fly,na;+py,pa:+qy me+py,p:c+qy,qa;+ry Letthiscubic function, by solving it as acubicequation, bemade equal to thenL(a;+fy)(a;+gy)(a;+hy), o(F) uvw=.e;{!00(F)}'u=k(a;+fy),v=l(a;+gy),W=m(a;+hy~ Bymeansof theidentity, F=ul+vi+wi,l',m',f1.1,are known by the solution of linearequations, andthusu,v,w,aredetermined by solving a cubicequation insteadof one of theeighthdegree,asinthemethod first given,andtheprocess ofcanonising a quinticfunction is rendered practically possible. Forbrevitysakeleterepresent unity. The constant determinant ofthe cubicadjunctive willbefound to be 3Mao(abcyo. Calling, then, the cubic adjunctive ofF, 0(F),we have the remarkable equation Itmay also be shown thatif we call theHessianofF, H(F),we shall have the following equally remarkable equation: oH(F)=!oFx oO(F). Again,callingthequadratic adjunctive of'F,Q(F),we shall easilyfind r(a'+b'+el)} (a'+b'-e') oQ(F)=MIOi(a'_bl+e'), l(al-b'-e') or, if we please, {a10+b10+&0 }=MIO_2a'b'_2a'c"-2b'e'. Whenu, v,wareknown,a,b,c,which aretheresultants ofv,W;W,u;u,v respectively areknown. Buttheirratios, or, if we please to sayso,theratios ofa~:b~:c",may be found independently andveryelegantly asfollows:- Let Mloxproductofthe4 forms of at+lib'+ltel=A, M'JOxproductofthe 16forms of a~+lib!+liel=B, =0. 34] OnCarwnical Forms. 205 .d,B, 0are known quantities, beingrespectively whatwe have called a Q(F),0(F)-,i00(F). Itmay easily be shown that B-AI=128MlIOasbsc"(as+bs+d'). HenceMSas,JPbs,Msc"aretheroots ofpinthecubicequation t+B2-::c11 pi+ifB;,~~I-A}p+Cf=O. A,B,0,itwill beobserved, areindependent and, astheymay betermed, primeorradicaladjunctive constants. Hitherto muchmystery and un­ certainty haveattached tothetheoryofhyperdeterminants, fromitshaving beentacitlyassumed thatthey were always eitherof lower dimensions than theordinary determinant, or elsealgebraical functions of such, and of the determinant. Whereas we nowsee that,whilstthedeterminant of a function intwolettersofthefifthdegreeis ofeightdimensions, one of its radicalor primitive hyperdeterminants is of four, buttheotherof twelve dimensions. Thisis amostvaluable consequence, and would seemtoindicate thatthe number ofradicalhyperdeterminants toafunction, over and above the common determinant, is always equaltothenumber ofparameters entering intoitscanonical form. Theimportance ofthisascertainment of an un­ suspected thirdradicalconstant, adjunctive toaquinticfunction oftwo letters,inmaking to march the theoryofhyperdeterminants, canhardly beover-estimated. Fromtheequation lastgivenwe areenabled toassigntheconditions in orderthattwofunctions ofthefifthdegreemay becapableofbeinglinearly transformed eitherintotheother.Forif we call FandF'two such linearly equivalent quinticfunctions, theymustbecapableeach of being thrown underthesame form US+va+(lu+mv)S,whereland7nshall bethesame for each.Consequently wemusthave the roots of pinthesameratioforF andF',whichconditions may beexpressed by means of thetwoequations B-AIH-A" OJ-=-O'!-' • Morestrictlyspeaking (andthiscorrection shouldbesupplied throughout inthe"Sketch"), Bisthenegative determinant ofiF.After finding, by the methodofcharacteristics, or any apeeialanmces, thealgebraic partof the value of a resultant ordeterminant, a process frequently of80mecomplexity remainsover inassigning itsnumerical multiplier; thispartof theoperation beinganalogous tothatwhichoccursin theIntegral Calculus, ofdetermining theconstant to be addedafterthegeneralform of an integralhasbeendetermined. Inthe"Sketch," acorrection for thenumerical multiplier remains also to be appliedto theexpressions given for the successive HMsiandeierminanY. 206 OnOmwnUxd Forms. [34 .A',B',0',of course representing thesamefunctions ofthecoefficients of F'asA,B, 0,respectively ofF. Thetwoconditions required intbeirsimplest form are accordingly AA' Oi=O'i' BB' 0.=elf' or AI:J1t:0::A":B'2:0', thatis tosay,all quintic functions oftwo letters ofwhichthedeterminant istothesubduplicate poweroftheradicalhyperdeterminant ofthetwelfth order and to the sesquiduplicate poweroftheradicalhyperdeterminant ofthe fourthorder ingivenratios,aremutually convertible. So forthequartic(thatis,biquadratic) function of twoletters,callingR andStheradicaladjunctive constants ofthesecond and thirdorders,the condition ofconvertibility between different forms of thesameis,that Rs:~shallbeagivenratio.And, in general. we may inferthatthe condition ofconvertibility between different functions of anydegreeis, thattheseveralradicaladjunctive constants of each raisedrespectively to such powers as will make themof likedimensions, shallbe to one another in'givenratios. Of course all cubic functions of twoletters,according tothis rule,aremutually convertible without anycondition, theyhavingbutone radicaladjunctive constant jandin fact all such functions, beingrepresent­ ableasthesum of two cubes of new variables linearlyrelatedtothosegiven, arenecessarily convertible. I havefurthersucceeded in obtaining thecanonical form ofthequadratic adjunctive toanyodd degreed function of twoletters,whichpresents a wonderful analogy tothetheoryofrelative determinants. ofquadratic function» ofanynumberofletters,andconstitutes animportant steptowards theconstruction ofthetheoryofrelativebyperdeterminants. Letafunction of twolettersoftheodddegreem(=2n- 1) bethrown underitscanonical form, and letthereexistthen-2equations, ~Ut+~u,+'"+a,.u,.=0, bJUt+b2~+...+bnUn=0,(1) (2) (n-2) Then,ifMbethemodulus ofthetransformation whichconverts Ut,~into 34J OnCanonical Forms. 207 e,y.andif, onmaking 8118i•••8ndisjunctively equalto 1, 2...nwe use(8,,-1,8n)todenoteingeneralthedeterminant a'I'aBi'..a'._i, b'I'b,....b,,,-o t,'l"...l,,,_. thequadratic adjunctive of-~(11) 2Fwill be l1tm... Mm-l.~j8 8m-l( )_(1,2r-1c:l(n,) ur•u,) . N.B. By meansofthisformula, andofthetheorem forfindingrelative determinants ofquadratic functions, wecanobtainthegeneral canonical formfor onesetofthebiquadratic adjunctive constants (hyperdeterminants ofthefourthorderin MrCayley's language) ofanyodddegreed function oftwoletterst. Thus,forthefifthdegree.preserving thenotation ofthe"Sketch," we havethebiquadratic adjunctive constant =0,c4,b',a c4,0,a4bu», b4a',0,cx&0-', a,b,c,°Fortheseventh degree,if wesupposethefunction to beequalto u'+v7+w'+87, and au+bv+cw+d8=0, a'u+b'v+c'w+d'8=O j thebiquadratic adjunctive constant will beMl4 bythe (cd'_c'd)I'multiplied determinant 0, (ab'-a'b)S,(ac'-a'c)S,(ad'-a'd)S,a, a, (ba'-b'a)S, 0, (bc'-b'c)S,(bd'-b'd,!.b,b' (ca'-c'a)S,(ab'-e'b)S, 0, (cd'-e'd)S,c,e' (cia'-d'a,!.(db'-d'b,!,(dc'-d'c)S, 0, d,d' a, b, c, d, 0,°a', b',s: 0,°, e, • Thecondition m=2n-1is onlynecessary inorderthatT.,,'(14m)maybeacanonical. ~Wlea possible anddeterminate, form for any given function of themthdegree. But the­ theoreminthetext,80far&8itservestoobtainthequadratic adjunctive ofT.,,'(14m),istruefor alloddValUflIIof m.whethergreateror Ieasthan211-1. tBeeNote (A)of Appendix. 208 On Oanonical Forms. [34 ; ,./\Thedeterminants oftheHessian, thepost-Hessian, andtheprseter-post­ Hessian ofFwill be found (in thecase ofthequinticfunction) to be always multiples of powers of thedeterminant ofthegivenfunction, and of itscubic adjunctive; and I believe thatingeneralfor afunction of twolettersof any degreethedeterminants of allthederivedforms in theHessian scale ",will benecessarily algebraical functions ofanytwo of.them. I hope very shortlyto accomplish thereduction of functions, ashighas theseventhdegreeof twoletters,totheircanonical form, and also to present acomplete theoryofthefailing or singular casesof canonical forms. SincetheabovewasinprintI have discovered thefollowing GENERAL THEOREM forreducing afunction oftwo letters ofanyodd degree toitscanonical form. Letthedegreeof thefunction be(2n-1);thenits canonical form is UI2n-1+u,'m-l+'" + u,,2n-l, with(n-2)linearrelations between Ul>u..z,...Un. To find ~,u..z,.•.Un,proceedasfollows. Letthe given function of the (2n-1)thdegreebesupposed to be 2n-2 ~wn-l+(2n-1)~a;m-'Jy+(2n-1)-2 ~:rn-'ylJ.+ ...+a.m.y'ln-l. Formthedeterminant -,-,a2x+UsY,a,x+a4ya"x+a,.+IY asa;.J..a4y, a"+lx+UnHY a"x+~IY'a,.+JX+a,.+lJY,.••.....•aw_1x+awy Thisdeterminant is a function of xandyofthenthdegree, and by resol ving anequation ofthenthdegree, may be decomposed into11factors,say (llx+m1y)(l2x+1nrlY)••.(l"x+m,.y); • I usethetermHesaian (moreproperly speaking the Boolian) Scale,todenotethedeter­ minantlinrespectof~and'Iof (~:x+'I~+&c.)lJF. NeitherHesse,however, nor anyotherwriterup tothepresenttime,hadthought ofCOD­ structing, andstillIesaofturningtoaccount, thefunctions (the first only excepted) which figure inthisscale. 34] weshallthenhaveOnOarwnical Forms. Ut=PJ.(~x+~y), u,.=1J,J(~x+'1Tlgy),209 Un=P«(lnx+m"y), wherethel'sandm's are known, andthe(2n-1)thpowers of thep'smay befoundlinearly, by means of theidentical equation 'i,um-1=F(z,y).Thus forexample afunction oftheseventh degreeof twolettersmay bereduced toitscanonical form .(lx+my'f+(l'x+m'y)7+(l"x+m"y)7+(l'''x+m"'y)', bytheresolution of abiquadratic equation. My.demonstration ofthis extraordinary andunexpected conseqnence restsuponthefollowing lemma.", itselfa verybeautiful andstriking theorem (nodoubtcapableof much generalisation) inthetheoryofdeterminants, Formtherectangular matrix consisting ofnrowsand(n+1) columns T..T2,i;Tn+I, T2,t;T,Tn-t-I, i;T"T....TnH, where Thenallthen+1determinants thatcan be formed by rejecting anyone columnatpleasure outofthismatrixareidentically zero. Inorderthebettertorealisetheproof,suppose n=4,sothat2n-1=7. Let F(x,y)=~x'+7a'J~y+21aa~y2+35a,x'yI+35aawy' +21aswy·+7arX'lf+aay'· Suppose t'+u'+v7+W'=F(e,y)=G(u, v), at+bu=v, a't+b'u=W. Then,ifM18themodulus oftransition frome, ytou,vthehyper- 8.• SeeNote(B)of Appendix. 14 210 OnOanonical Forms. [34 determinant, or, toadoptmy new expression, thepermutant P,(meaning thereby) ~a;+a~y, ~a;+asy,a,a;+a.y,a.a;+a,y, ~a;+asy,asx+a.y,a.a;+a.y,a.a;+a.y a.a;+a.y,a.a;+a,y,a,a;+asy,aua;+a,y a.a;+a,y,a,a;+asy,asa;+a.,y,a.,a;+a'd'!l which is a constant adjunctive inrespectto,and'1/of('d:+'1/;yyF,will. according totheprinciples laid down in thepreceding "Sketch," bethe productof a power of Mmultiplied bythecorresponding adjunctive constant of(,tu+'1/:vyG(u,v),and istherefore amultiple of thedeterminant (1+AI)t+Asu,Ast+A,u,Ast+A.u,A.t+A.u I ' A,t+Asu,A,t+A.u,A4t+A.u,A.t+A.u A,t+A.u,A.t+A.u,A,t+A.u,A.t+Aru A.t+A.u,A.t+A.u,A.t+Aru,Art+(1+As)u where Al=a7+«,A,=a6b+a'·b',As=a'b'+a"b"...As=b7+b'r. Inthisdeterminant thecoefficient of u·is As,A••A.,A. A"A.,A,...10 A.,A.,A.,Ar A"A.,Ar,1+As which is numerically equal to A"A.,A, As,A.,A. A.,A.,A.,Ao- AsA.,A.,A. lA"Ao,Ar A.,A.,Ar AI"A"A. IA.,A.,A. +ArA.,A.,A.- (1+As)A.,A.,A, A.,A.,ArIA.,A"A. =0,because thesecond factors of theproducts are all zero by thelemma. Hencethepermutant P;vanishes whent=0, andconsequently itcontains tasa factor, and in like manneritmay be proved to containu, v, w. Hencet,u, v,warethealgebraical factors of p.,and precisely the same proofappliesto show in thecase of a function ina;andy.sayF2'Il-I,of any 34] OnCanonical Forms. 211 odddegree(2n-1)whatever, thatthecorresponding permutant p..will containthefactorstl:I,'Il2...u"linearfunctions of$,y,suchthat tl:IlIfl-l+~:In-I+ ... + u"lIfl-l=FlIfl-l aswastobeshown. Whenever P"hasequalroots,thiswilldenoteeither(which is themore generalcase)thattheusual canonical form fails and gives place to a singular form, (owing to some of thecoefficients of transformation becoming infinite), or, which is themore special supposition, thatthecanonical form becomes eatalectic by one or more of thelinearroots-disappearing. Thusinthe cubicfunction, ifP2hasequal roots, and consequently itsdeterminant (whichiscoincident withthatofthefunction itself)vanish,thenthecanoni­ cal form in generalfails; so that,forexample, a:r;3+barycannotingeneral beexhibited asthesum of two cubes:if, however, certainfurtherrelations obtainbetween thecoefficientsof F,thecanonical form reappears catalectically, thefunction becoming infactrepresentable asasinglecube. So, again, for thequinticfunction (referring backtothenotation above [page 205]), ifPIhave equal roots, thatis if0=0,thecanonical form fails, unless at the same timeB-A2=0, in which case thefunction becomes thesum of two fifth powers; butiffurthermore A=0,thenthiscatalectic formagain gives place to asingular form, which, on thesatisfaction of afurthercondition between thecoefficients, againin itsturngives way before a (bicatalectic, thatis)doublycatalectic form, namely, a singlefifth power. Itisremarkable, thatthe form to which Mr Jerrard's methodreducesthe function ofthefifth degree, expressed homogeneously asaa!+bflJ'!/+cy",isa singular form,beingincapable ofbeingexhibited asthesum ofthreecubes; such, however, is not thecasewiththeformax"+bwy2+cy".Itmayfurther beremarked, thatalthough the singly catalectic formofthequinticfunction is expressible by twoconditions only,namely, 0=0,B-A2=0,it will be indicated byPI(whichbeinga cubicfunction ofxandycontains fourterms)completely disappearing, sothatapparently fourconditions wouldappearto berequired orimplied. Butof course thesemustbecapableofbeingshown to be non-independent, and to be merelytantamount tothetwoindependent ones, 0=0,B-.A2=O.Thetheoryofthecatalectic forms of functions ofthe higherdegreesof twovariables presents manystrongpointsofresemblance and ofcontrast tothatofthecatalectic forms of quadratic functions of several variables. Oneimportant andimmediate corollary from theGeneral Theorem is, thattheconstants whichenterintothelinearfunctions appurtenant to the canonical form of any function of an odd degreeform a single and unique system;or, inotherwords,thecanonical forms for such functions are void of •tit,~...u.maybetermedthelinearroots of the form F210-1' 14-2 212 OnOasumical Forms. [34 multiplicity, aresultcontrary towhatmighthave been anticipated, and towhatwe know isthecaseforthecanonical forms of functions of aneven degree. Itmayfurtherbe shown thatif we have the(n-2)equations aju,.+~~++anu.a=0, b1u,.+bt~++bnUn=0, llu,.+ls~+...+lnUn=0, andcallMthemodulus oftransformation inrespecttoUll'Ut,and if we make Pn=Ku,.'Ut...Un, then ~,a•...a,.inln-l) ls,l•...In is equal to theproductofthein(n-1)factorsoftheform alit•..aB..., bllt•••bB. _s, . l6J.'lBt'"lB•.., 011OS",On-ibeingany(n-2)numbers out ofthennumbers 1, 2, 3...n. Itmay hence be shown that mbeinganumberwhichisafunction ofn,and which may be shown to be equal to 0(xn-1y+:r;yn-l)+productofthesquareddifferences of theroots <1:,11 ofIn-i=1,thatis {(n-1)11-rm= =(-n)n-i-(n-2) , andthus • 0meansiliedeterminant inrespectto11:and y. 11,.. 34) OnOasumieal Forms. 213 Asanexample ofthemode offinding u j,~...Un,let F=&cD+20wy'+1010/, then 3x,2y,2x P,=2y,2x,2y=4:cB-4y'x. 2x,2y,2x Hence u=fai,v=9(x+y),w=h(x-y). Tofind,f,g,h,we have u'+'If+wi=F,hence f'+9'+h'=3j9'+h'=2;9'-h'=1; whencewe have F=:c'+(x+y)'+(x-y'l. Again,we find o(4:c'-4y2X )= -4'x12, (_\~)0Pst=4, andaccordingly x(x+y)(x-y) ={(_3):~oPs)i' according tothegeneralformula above given. As a second example let F=3x1+42:c'If+70wy'+14:cy8+y1j then 3x,2y,2x,2y 2y,2:c,2y,2x P,= =4(:cSy-fLy)=4.xy(x-y)(x+y),2x,2y,2:c,2y 2y,2x,2y,2x+y and accordingly we shall find .x"+y7+(x-y)7+(x+y)7=F. Moreover and Thuso(4.x'y-4XYS)=tV, ~-2 =42. agreeable tothegeneralformula.. 214 OnCanonical Forms. [34 Asa corollary to ourgeneralproposition, itmay be remarked, thatif Fm-Ibe asymmetrical function ofe,yofthe(2n-l)thdegree,Pn(F'ZIl-I) will be also a symmetrical function ofxandy,and may therefore beresolved intoitsfactors by solvingarecurring equation ofthenthdegree, which may, by well-known methods, be made to dependonthesolution ofanequation ofthelnthort(n-l)thdegree,according asnis even or odd. Hencethereduction of afunction of twolettersofthedegree4m±1 to itscanonical form as thesum of powers lOay be made to dependonthe solution of anequation ofthemthdegree;sothat,forexample, asymmetrical function ofai,y,as high asthefifteenth orseventeenth degree, may be reducedbymeansof abiquadratic equation only. InashorttimeI hope to presenttothepublicacomplete solution ofthecanonical forms of functions of twolettersof evendegrees,andpossibly toexhibitsomeimportant applications oftheprinciples ofthemethodtothe theoryofnumbers. APPENDIX. NOTE(A). Thepermutants (meaning, in MrCayley's language, thehyperdeter­ minants) ofFm+l(x,y)ofthefourthdimension inrespecttothecoefficients ofF,may be all obtained bytakingthequadratic permutant inrespecttox andyofthequadratic permutant inrespectofEand."of (dd)2lEdz+."£yFm+1(x,y), lhavinganyintegervalue from 1 to n. Inextension of atheorem intheforegoing Supplement, whichapplies only to thecase ofl=n,Iamable tostatethefollowing more general theorem, in which thesamenotation ispreserved as above [page 207]. Thequadratic permutant inrespecttoEand."of 1(dd)1l (2n+1)2n...(2n-2l+2)Edx+."elilF2n+1(x,y), isequalto (1~2~1~{(Br ,er«;U,)in+I-tl}. Ifnow we proceed to form thequadratic permutant oftheabovesum inrespecttoxandy,we knowapriori,byreasonofMrCayley's invaluable researches, thatweshallnotgetradically distinctresultsfor all values, but only forcertainperiodically changing values of l. 34J OnOanonical Forms. 215 Ihavenotyethadleisureto seek for anexplicitdemonstration ofthis remarkable law,founded upontheabovegivencanonical representation. NOTE(B). Thelemma,upon which thegeneral method forreducing odddegreed functions totheircanonical form is founded, maybestatedrathermoresimply andmoregenerally as follows i-i- Thedeterminant TTl' TTI+lt' Trl+~'...Tr• ...Tr,.+lt ...Tr,,+l.t Trl+lro_1,Tf'J+lro-1•••Tr"+l,,..l whereT,denotesAIUt'+All~'+ '"+Amam'provided that m islessthann, isidentically zero.Inthetheorem, asthusstated,thereis nosubstantial loss ofgenerality arisingfrom.theomission of theb's. Thusstatedthetheorem anditsextensions evidently repose upon the sameorthelike basis as thetheoryofpartialfractions. NOTE(0),referring totheoriginal" Sketch." TheBoolo-Hessian scale of determinants furnishes a veryprettygeneral theorem ofgeometrical reciprocity inconnexion withthedoctrine of suc­ cessive polars. LetF(x,y,a), acubichomogeneous function ofe,y,s equated to zero, expressingeneral acurveofthethirddegree; then (a:f.x+bd~+c;z)Fwillexpressitsfirstpolarinrespecttothepointa,b,C, thatis,theconic which passesthrough thesixpointsin which thetangents drawnfroma,b,Ctotouchthegivencurvemeetthesame. Again,ifwetakel,m, nthecoordinates of any new point, (d d d)(d dd)ld:c+mdy+ndzad:c+bdy+Cdz F willexpress thepolar,thatisthechordofcontactoftheabove conic, in respecttothelastnamedpoint.Ifnow we eliminate l,m,nbetween the threeequations (d d d)(d d d)ldx+mdy+nd.zad:c+b dy+Cdz F=0, (ld d a-.(,db' d,d)F0d:c+mdy+nds)ad:c+dy+Cdz=, (d d d)("d b"d"d)F 0ld:c+mdy+ndz ad:c+dy+Cdz=, 216 OnCarwnical Forms. [34 itiseasily seen thattheresultant oftheelimination isthesquareofthe determinant a,b,c a',b',a' a",u,e" multiplied bytheHessianofthegivenfunction. And, moreover, thatifwe eliminate e,y,zwe shall obtainprecisely thesameresultwiththeletters l,m, nsubstituted fore,y,z.Henceitfollows,thatif wetakethedoubly infinitesystem of firstpolars to a givencurve of thethirddegree, in respect to allthepointslying in itsplane, and thenfrom any pointintheHessian tothegivencurve, draw pairs of tangents to each conic of thesystemso generated, thenallthechords of contactwillmeetin one and thesamepoint, which will itselfbe alsoapointsituated upontheHessian andconjugate to theformer. So, ingeneral, for afunction of anydegreeof anynumber ofletters, viewedwithrelation to thedoctrine of successive polars, thedeterminants oftheBcolo-Hessian scaletakeoneanotherup inpairs;namelythefirst takesupthelastbutone,thesecondthelastbuttwo, and so on jand consequently, ifthedegreeofthefunction be odd, thatfunction which (making abstraction oftheconstant determinant attheend) lies in the middle of thescale pairs with itself, and, in a sense analogous tothatabove exhibited for afunction ofthethirddegree, may be said to be always itsown reciprocal. P.S. I have justdiscovered themethod of reducing functions of two lettersofevendegrees totheircanonical form, which will shortlybepublished in a second Supplement. AtpresentI offertheannexed theorem (whichstrikingly contrasts with thelaw ofuniqueness demonstrated offunctions of an odd degree)asa foretaste oftheenchanting developments with which I hope shortlytopresent myreaders:- Ifa givenhomogeneous functionofa;and yofthe degree 2n besupposed to bethrownunderitscanonical form, Ut2n+u,2n+ ...+14.271+K(u1u,...Un,!, then will Knhaven2-1ingeneraldistinctvalues, to each ofwhichwill correspond a single distinct sy,ftemofthelinearfunctions ofxand y, 1 1 1 In~,l"u"...In14.• 35. EXPLANATION OFTHECOINCIDENCE OF ATHEOREM GIVEN BYMRSYLVESTER INTHEDECEMBER NUMBER OFTHIS JOURNAL,WITHONESTATED BYPROFESSOR nONKIN INTHEJUNENUMBER OFTHESAME. [Philosophical Magazine, (Fourth Series) I.(1851), pp. 44-46.] IWISHtostate,withoutloss of time, thatinthetheoremgivenby me- for thecomposition of two successive rotations aboutdifferent axes, I have been anticipated by Prof. DonkinintheJuneNumber ofyourJournal. To myshameImustconfess,that,although an occasional contributor to, Iamnotinvariably aconstant readerof yourvaluable miscellany, otherwise Ishouldnothaveintroduced thetheorem inquestion without due acknow­ ledgment of Professor Donkin's claims to whatever meritmayattachto the priorityofpublication. Thefactis,thatI made out thetheorem for myself nineyearsago, and had some communication onthesubjectwithProfessor DeMorgan, who was thenwritingtheseventeenth chapterof hisDifferential Calculus. Arecentconversation withthisgentleman hasbrought backto myminda vividrecollection ufthecourse of thatcommunication. Ibrought underProfessor De Morgan's noticetheanalytical memoirofSrGabrioPola onthesubjectintheMemoirs of theItalianSocietyof Modena, and satisfied myself of theexistence ofthesingleaxis ofdisplacement bycompounding the tworotations inthemannergivenin mypaper,which, for thecase of two axes fixed in space, is thesame as Professor Donkin's, and for two axes fixed intherotating body ismaterially, although notformally thesame. Itthenoccurred to me thata moresimpledemonstration oughtto be deducible from thepossibility of always findingthepointon asphere,by revolution aboutwhich, as apole, one equalarc could actually be shown to betransportable intotheplace of another. Butinproceeding to work out this idea I fell into aremarkable blunder,in which I have since been followed bymorethanone able friend to whom I have proposed thequestion. The [0 p.168ahove.] 218 Explanation, etc. [35 blunder wasofthiskind:-Twoarcs have to be drawn, bisecting atright anglesthearcsjoiningtheextremities of twoequalarcs;thepointofinter­ section of thetwobisecting arcsmustin allcasesfalloutsidethequadrilateral formed by theequalandjoiningarcs. I supposed it to fall inside. There appearsto beafataltendency todo so in all who takethesubjectinhand. Inconsequence ofthiserror,thecause of which I did not atthemoment perceive, I wasdrivento denyandadmitin onebreaththesameproposition. Mr De Morgan sentmethecorrectproofafterthismethod(thesameasthat given by him atpage 489 of his Calculus), I aminclined tothinkafterIhad myselfdetected myerrorjbutofthisIcannotfeelcertain. This isthemethodalludedto by me in thewords"itisrightto bear in mind, &c.," atthetimeofwritingwhich all recollection of thesamething havingbeenpublished by Mr De Morgan had vanished from my memory. The proof of thetriangle ofrotations is so simple, that,asProfessor Donkinstates(in aletterwhich he has done me thefavour of addressing me onthesubject) wasthecasewithhimself, I thought itincredible thatit shouldnothaveappeared in some elementary work, and I was therefore at no pains to publishit as my own jnor should I have writtenatall onthe subject,hadit not been for thesurprise occasioned to my mind by falling in with Professor Stokes's articleintheCambridge and DuhlinMathematical Journal. todemonstrate theexistence of aninstantaneous axis, which proceeds in apparent unconsciousness ofthesosimplydemonstrable law, thatanynumber ofrotations of anykind(andtherefore thosethattake place in an instantof time) are representable by a single rotation about asingleaxis. I shall feel obligedby the early insertion ofthisexplanation, more injusticetomyselfthantoProfessor Donkin, whosehighandworthily earnedreputation, not tospeakofthedisinterested love oftruthforitsown sake,apartfrompersonal considerations, whichanimates thelabours of the genuine votaryof science, mustmake him indifferent towhatever credit mightbesupposed toresultfromthefirstauthorship orpublication ofthe verysimple(however important) theorem inquestion. 36. ANENUMERATION OFTHECONTACTS OF LINESAND SURFACJi:S OFTHESECOND ORDER. [Philosophical Magazine, I.(1851), pp. 119-140.] IT is well known thatingeneral any two homogeneous quadratic functions of the same systemofvariables may be simultaneously trans­ formed,soasto beexpressed each ofthemaspurequadratic functions of anewsystemofvariables equalinnumber andlinearlyconnected withthe originalones;apurequadratic function meaning one in which only the squaresofthevariables areretained. Everyhomogeneous quadratic function may be treatedasthecharacter­ istic·of a locus of theseconddegree: ifthefunction be of two letters,the locus isabinarysystemofpointsin a line wherein thedistances of two fixedpointsfromeitherpointofthegivensystemor given multiples of such distances correspond tothevariables; if ofthreeletters,thelocus is a conic, thedistances or given multiples of thedistances of every pointinwhich from threegivenlines intheplaneoftheconic are represented bythevariables; if of four letters,thelocus is a surface of thesecond order, thecoordinates beingthedistances ormultiples ofthedistances of anypointthereinfrom fourplanesdrawnin the space in which thesurface is contained, andso on for loci of four and higherdimensions. I propose, however, in thepresentpapertorestrictmyself to thetheory of thecontacts of loci not transcending thelimitsofvulgarspace, by which I meanthespace cognizable through thesenses[,and shall accordingly be •According tothedefinition statedby me in a previous paper,thecharacteriatic of a locus is U1efunction which,equatedtozero,constitutes theequation thereto. tIftheimpressions ofoutward objectscameonlythrough thesight,andtherewere nosense oftouchorresistance, would not spaceof three dimensions have been physically inconceivable? Thegeomp.try oftbreedimensions inordinary parlance wouldthenhavebeen called trans­ cendental Butin verytruththedistinction is vainandfutile. Geometry, to beproperly understood, mustbestudiedunderauniversal pointof view; every (even themostelementary) proposition mustberegarded as a fact, andbutas asinglespecimen oflIDinfiniteseriesof bomologous facts. Inthiswayonly(discarding asbutthetransient outward form of a limitedportionofan infinitesystemof ideas, allnotionofextension asessential totheconception ofgeometry, however nsefnlas a suggestive element) we may bope toseeaccomplished anorganicandvital development of thescience. 220 .AnEnumeration oftheContactsofLines [36 almostexclusively concerned indetermining thesingular cases of conjugate systemsofquadratic forms of two, three.andfourlettersrespectively. Inorderthatthereduction of any such system.sayUandV,to apure quadratic form may be possible (as itgenerally is),itisnecessary thatnone oftheroots ofthecomplete determinant ofU+>..Vshallbeequal;ifany relation ofequality existbetween theseroots,the,general reduction is generally nolongerpossible; underpeculiar conditions, however, aswill hereafter appear, inspiteoftheequality ofcertainoftheroots,the irreducibility initsturnwillcease,andtheordinary reduction becapable ofbeingeffected.Itiseasily seen, thattoeveryrelationofequality between theroots ofthedeterminant ofU+>..Vmustcorrespond aparticular species ofcontactbetween theloci which UandVcharacterize. Butweshould makeagreatmistakewere we to supposethateverysuchrelationofequality corresponded withbutonespeciesofcontact; forinstance, thecharacteristics ofUandVoftwo conics are functions ofthreeletters,and0(U+>..V)will be a cubic function ofA..Suchafunction mayhavetwo roots, or all itsroots equal:thiswould seem to give buttwo species of contact,whereas we well knowthatthereare no less thanfour species of contactpossible between two conics. Accordingly we shall find, that,inordertodetermine thedistinctive characters of each species of contact, wemustlook beyond thecomplete determinant, andexamine intotherelations (inthemselves andto one another) oftheseveralsystems ofminordeterminants thatcan be formed fromU+>..V, Bypursuing thismethod. we may assign aprioriallthepossiblespecies ofcontactbetween any two loci of thesecond degree. How important this methodis will be apparent fromthefact,thatnotonly have thedistinctive characters ofthevariouscontacts possible between surfaces ofthesecond orderneverbeendetermined, buttheirnumberandthenatureofcertainof themhaveremained untilthishourunknown andunsuspected. Themethodwhich we shallpursueis anexhaustive one,andwillconduct us by anaturalorderto asystematic arrangement of allthedifferent modes andgradations of suchcontacts. In apaper-inthisMagazine forNovember 1850, Iexplained thedecline ofminordeterminants, andstateda law, called thehomaloidal law, con­ cerning them. IfUandVbecharacteristics ofthetwo loci whose contacts areto be considered, U+>..Vwill bethefunction, theproperties of whose complete determinant, and oftheminorsystems ofdeterminants belonging to it, will serve to specify thenatureofthecontact. Itwill beremembered, that,whatever bethenumber ofvariable letters in anyquadratic function U,threeofitsfirstminordeterminants beiugzero, [*p.160above.) 36J andSurfacesoftheSecondOrder. 221 makes all thefirst minors zero jsix of its second minors beingzero,makesall thesecondminorszerojand so on for thethird,fourth, &c. minor systems according totheprogression of thetriangular numbers. Itis well known thatwhatever lineartransformations beapplied to aquadratic functionW,thecomplete determinant thereofwillremainun­ altered,exceptbyamultiplier depending uponthecoefficients introduced intotheequations oftransformation; consequently theroots of A.inthe equation obtained bymaking thedeterminant ofU+;\Vzeroremain unaffected bysuchtransformation jand any relationorrelations ofequality amongtheroots oftheequation 0(U+;\V)=0 is animmutable property ofthesystemU, V,which is unaffected bylineartransformations. Another and more gent'ralkindofimmutable property (comprehending theabove as aparticular case), to which I shall have occasion to refer, is thefollowing. Suppose all the minors of any order of U+;\Vhaveafactor A.+Ein common; thisfactor will continue common to thesamesystemof minors whenUandVaresimultaneously transformed. Thisis a very important proposition, andeasilydemonstrated; for if;\+Ebe a common factor to all therthminorsofU+;\V,(U-EV)will have itsrthminors zero, andthere­ fore,asexplained by me in thepaperabovereferred to,U-EVwill be degraded rordersbelowUorV.This isclearlyaproperty independent oflineartransformation, consequently ;\+Ewillremaina factor of the transformed rthminors. Inlikemanneritisdemonstrable thatanynumber ofdistinctfactors A+£1';\+£2•••common to therthminors of one form of U+;\V,will remain common factors of any otherlinearly derived form of thesame. It isconsequently necessary thateachrthminorof one form of any quadratic function Wshall be aeyzygetic "function ofalltherthminors of anyotherform ofthesame;andconsequently afunction of;\of any degree,whether all its factors be or benot distinct, which is common to therthminorsof one form of U+;\V,willremainso totherthminors of anyotherform ofthesame. The law exhibiting theconnexion of each rthminorof one form of W (anyhomogeneous quadratic function) with all therthminorsof anyother formofW,will form thesubjectof adistinct communication. Finally, to fully comprehend theannexed discussion, the following principle mustbeapprehended. •If.d=pL+qM+rN+&c., wherep, q, roo. do not any of them become infinitewhen L, H, N...or any of thembecome zero, Amaybetermedasyzygetic function ofL,H,N.... Inthetheorem above alluded to, it will be shown (as mightbeexpected) thatthe syzygy inthe easeooncerned is of the simplest kind,thatis,thateachrthminorofaquadratic function of any Dumberoflettersisahomogeneous linearfunction ofalltherthminoraof the same qua.dra.tio. functionlinearly transformed. 222 AnEnumeration ofthe Contacts ofLines [36 Ifany factor X"enterintoalltherthminorsofW,and ifXibethe highestpower of Xcommon to all the(r+l)thminors,thenX,.......will be a common factor to all the(r-1)thminors. Letrbetakenunity;itiseasilyproved-thatthecomplete determinant ofanysquarematrixmay beexpressed bythedifference between twopro­ ductef-,each of two first minordeterminants divided by a certainsecond minordeterminant. Theproposition istherefore demonstrated forthiscase, andtherebyin factimplicitly for every case,inasmuch asthefirst minors of ~lVeryrthminorare(r+l)thminors of theoriginal matrix. Henceit follows,thatif anysystemofrthminordeterminants have a common factor E',thecomplete determinant mustcontainat lowest thefactor Elr+1)' ,andany systemof(r-s)thminordeterminants thereunto willcontainat lowest the factor E(H11'. I now proceed to applytheseprinciples tothedetermination ofthe relative forms of conjugate quadratic functions representing geometrical loci of the second order. I shall beginwith two binarysystems ofpoints IIIarightline. Thegeneralcharacteristics UandVof two such systemsmay bethrown undertheform U=x2+y9 } V=aa;2+b~. When 0(V+AU)=0hasits two roots equal,thesesystemshaveapoint in common. Theabove forms ceaseto beapplicable, andconvertinto U=xy } V=ax2+bxy wherea:=0represents thecommon point. *Thiswillappearin mypromised paperonDeterminants andQuadratic Functions. tWhenthematrixissymmetricalnbout one of its diagonals (as it is in the casewhich we are concerned with), one of these products becomes a square. I maytakethisoccasion ofhinting, thatthetheoryofquadratic functions mergesinalargertheory of binaryfunctions, consisting of the sum of the multiples ofbinaryproducts formed by combining each of one set of quantities, x,y,z ...witheach of the same number ofquantities ofanother set, asx',y',z' ...•For instance, axx'+bxy'+cxz' +a'yx'+b'yy'+c'yz' +a"zx'+b"zy'+c"zz' would be abinaryfunction, anditsdeterminant (nolonger,as inaquadratic function, symmetrical abouteitherdiagonal) wouldcorrespond to thesquarematrix abc a'b'c' Almostall theproperties ofquadratic apply,withslightmodifications, tobinaryfunctions. 36] andSurfacesoftheSecondOrder. 223 LetUandVnowrepresent two conics. Whenthereis nocontact,we have asthetypesoftheircharacteristics U=rr+y'+z', V=ar+by'+cz'. Thethreeroots of 0(V+>..U)=0are >..=-a, >..=-b, >..=-c, showingthattherearethreedistinctpairsof lines in which theintersections ofUandVarecontained, theequations tothreepairsbeingrespectively (b-a)y'+(c-a)z'= 0, (c-b)z'+(a-b).x'=0, (a-c)rr+ (b-c)y'= 0; the four pointsoftheintersection beingdefined by theequations corre­ sponding to theproportions a;:y:z::-/(b-c) :-/(0-a):-/(a-b). Nowlet0(U+>..V)have two equal roots;thecharacteristics assumethe form U=rr+y'+es, V=a.:rf+by'+cez-. Twoofthepairsof lines become identical, thatis, two of thefourpointsof intersection coincide. • Wemay ifwepleasemakea::b;for itmaybe shown thattheequations, intheirpresent Corms,containanarbitrariness of 10degrees; namely, 9 onaccount ofx,y,zbeingarbitrary IinearsoCr.TJ,8;2 onaccount of theratiosa:b:c;together 11 reduced by one degree on aecountofx,y,s,changed intoLx,Iy, ls,leavingU=O, V=O uneJIeoted. Now the degreesof arbitl'arine88 in twoconics,subjecttoMtiStyonly one condition, is 2 x5- 1 or 9. Hence there iaone degree of arbitrariness tospare.Infact,onmakinga=b,theaxiszbecomes theline joiningthe two pointsofintersection distinctfrom the pointofcontact; xremaining thetangent atthepoint ofcontact, andy,strangetosay,stillarbitrary, subjectonlytopassingthrough the point ofcontact; if, however, ybemadetopassthroughthepointofcontact, andeitherone of thedisUnctintersections, thisform, U=:r:'+y'+X!I, V=az'+ay'+cxz, bellomllllnolongertenable, butgiVll8placeto U=y'+yx+xz, V=ay'+ayx +cxz, wberezisthetangentatthepointofcontact, zthe line joiningthe twointersections with one another,and z,x+yrespectively thelinesjoiningeitherofthemwith the pointofcontact; ifthe multiplierof yxinVin the above be made binsteadofa.xremainsthetangentasbefore,y bewmesanylinethrongh thepointofcontact,andzanylinethrough one of the distinctinter­ sections. Asystematio view of similarmodulations of form and the studyof the laws of arbilrarinll88 connected withthem,asapplicable tothegeneral subject-matter ofthispaper, mtIAbedeferred toasubsequent occasion. 224 AnEnumeration ofthe Oontacts ofLines [36 (1) (2) (3) (4)Thismaybetermed"Simple Contact." Thetangentatthepointof contactisx=0;thisequation makingUandVeachbecome of only one order. Theintersections are x=0.y=0, x=0,y=0, "f(a-c)x+"f(b-c)Y=0.z=0. "f(a-c)x-"f(b-c)y=O, z=O. Theseareobtained bymakingV-aU=0, which gives a:=0 orz=o. x=0 givesy2=0.thatis,y=0 twice over, andz=0gives (a-c)w+(b-C)yl=O. Thenumberofconditions to besatisfied inthiscaseis one only. Nextlet0(U+}..V)have all itsrootsequal.Thiscondition willbe satisfied (stillleavingUandVasgeneralastheycanremainconsistent withtheseconditions) bymaking U=w+yz+ yx, V=aW+ayz+byx. Hereonly one distinctpairof linescanbedrawntocontaintheinter­ sections, showing thatthreeoutofthefourpointscometogether. Thismay betermed" Proximal Contact." Thenumber ofaffirmative conditions to besatisfied is two.andthecontactistherefore entitled ofthe second degree. Thetangentatthepointofcontactisy=0.andthefourintersections become x=0, x=0. x=0, x=0,y=o, y=O. y=O, z=O. andoneThesemay beobtained fromtheequation V-aU=0, which gives y=° orz=0;theformerimplying concurrently withitselfa;2=0, andthelatter yz=O. Thusweobtainthreesystems, x=0.s>0, x=0,Z=0, corresponding tothreeconsecutive pointsandthesingledistinctone. 36] andSurfacesofthe Second Order. 225 Thedeterminant ofU+XVbeingonly of thethirddegreeinX,we haveexhausted thesingularities ofthesystemU, Vdependent ontheform of thecomplete determinant ofU+xV. LetnowthefirstminorsofU+XVhave a factor in common; thiswill indicatethatU+XVmay be made to lose twoordersbyrightly assignin~ A, inotherwords,thattheintersections ofUandVarecontained upon apair ofcoincident lines.Hereitisremarkable thattheoriginal forms of UandV reappear, butwith a special relationofequality between thecoefficients: we shallhave, in fact, U=a?ry'+Zl, V=aafJ+ayl+bz'. Thisgivesthelaw fordouble,or,asIpreferto call it, diploidal contact.". B)"virtueoftheHomaloidal law, we know thatifthreefirstminorsof U+AVbe zero, all are zero;we have therefore toexpressthatthree quadratic functions ofXhave a root in common. Thisimpliestheexist­ enceof twoaffirmative conditions; thecontactofthetwo conics taken collectively may therefore bestillentitled ofthesecond degree, although thecontactateach of thetwopointswhereittakesplace is simple, or of thefirstdegree. Thesepointsareevidently defined by theequation {x+v(-l)y=O, z=OJ, {x-v(-l)y=O, z=O}, andtheordinary algebraical solution oftheequations U=0,V=°would naturally leadtothefoursystems x+v(-1)Y=0,z=0, x+v(-l)y=O, z=O, x-v(-l)y=O, z=O, x-v(-1)Y=0,z=0; the twotangents atthepointofcontactarex+v(-1)Y=0,a:-v(-I)Y=0, andthecoincident pairof linescontaining theintersections isZl=O. •Seemyremarkst ontheconditions whichexpressdoublecontactintheCambridge Journal, SOY.1850.Ifnfunctions, beingallzero, bethecondition ofafact,butrindependent syzygetic equations admitof being formedbetween thesefunctions, thenumber ofaffirmative conditions requiredisnotn,but(n-r);becausethefactmaybeexpressed byaffirming (n-r)equations and denyingcertainothers. ThusifP=O,Q=O,R=O,8=0expressafact,and PP'+QQ'+RR'+8S'=0, PP"+QQ"+RR"+88"=0, lhefactisexpressible byaffirming P=0,Q=O,anddenying R'S"-R"8'=0, forthenP=O,Q=O ..illimplyR=0, 8=0;or, in like manner, byaffirming anyothertwooutofthefournecessary eqaations, anddenying theotherequations. Observe, however, thatalltherequired equations ~coexistintheabsenceofsuchrightofdenial. [tp.129above.] ~ 15 226 .AnEnumeration ofthe Contacts ofLines [36 Itmayatfirst view appearstrange,thatwhilstnocondition isrequired inorderthatUandVmay besimultaneously metamorphosed intotheforms ofw+y2+zi,a:;c2+by2+cz2,ct,bandebeingallunequal, forthismetamor­ phosis to be possible when any two become equal, not one buttwoconditions mustbe satisfied. The reasonofthisis,thatthecoefficients of transform­ ation,which,aswellasa,b,c,arefunctions ofthecoefficients of theg-iven quadratic functions, become infinite Oilconstituting between thesaid coefficients such relations as arenecessary forsatisfying theequation a=b, ora=c,orb=o,exceptupontheassumption of some furtherparticular relations between themover and above thatimplied in such equality. Intheordinary caseofdiploidal contact,thefirstminorshavinga factor in common, thisfactor will entertwiceintothecomplete determinant of U+A.V,butitmayenterthreetimes:thiswillindicate, thatnotonly dothefourintersections lie on a coincident pairof lines,butfurthermore, thatthereisbutonepairof lines of anykindon which theylie. Intheordinary case ofdiploidal contact,itwill beobserved thatthis lattercondition doesnotobtain; thefourintersections lie onacoincident pairoflines;buttheylie also on a crossing pair,namely,inthetwotangents atthepointsofcontact. Inthishigherspecies of diploidal contact, it is clearthatthetwopointsofcontact, which are ordinarily distinct, come together, andthatall four intersections coincide. This Icallconfluent contact; theforms ofUandVcorresponding thereto willbe U=W+y2+ XZ, V=ay2+axz; thecommon tangent atthepointofcontact beingx=0, andthefour coincident points r,e2=0,y2=0. Thenumber ofaffirmative conditions to besatisfied beingthree,the contactis to be entitled ofthethirddegree. Observe, thatitis of no use to descend belowthefirst minors in this case;becausethesecond minors, beinglinearfunctions ofA.,couldnothave a factor in common, unless V:Ubecomes a numerical ratio,which would implythattheconicscoincided -. Fortified bythesuccessful application ofourgeneralprinciples tothe preceding morefamiliarcases of contact, we are now in a condition toapply withgreaterconfidence thesameapriorimethod totheexhaustion and characterization of allthevariedspecies of contactpossible between surfaces •No-contact andcomplete coincidence maybeconceived as the two extremecasesin thescale oCrelativeconjugate Corms. 36] andSurfacesoftheSecondOrder. ofthe second order;aportionofthesubjectcomparatively unexplored, and neverbeforethoughtsusceptible ofreduction toasystematic arrangement. Whenthereis nocontact,we may write U=x'+yi+Zi+ti, V=ax'+byi+OZI+dtl, andtheintersection ofthesurfaces will lie in each of thefour cones, ~-~x'+0-~~+0-~~=~ (a-b)x'+(0-b)Zl+(d-b)tl=0, (a-o)x'+(b-0)y2+(d-o)ti=0, (b-a)y2+(0-a)Zi+(d-a)tl=O. Whenever thesurfaces are in contact,certainofthesecones will coincide withcertainothers,sothattheirnumberwill be always less thanfour. Also, 88weshallfind in such event,theymaydegenerate intopairsofintersecting orcoincident planes. Letusbeginwithconsidering thecasesofcontactfor which thefirst minors(andconsequentlyafortiori the minors inferiortothefirst) have nofactor in common. Here0(V+XU)is abiquadratic function. IfXhaveall its roots unequal, we haveUandVasabove given. Iftwo roots are equal, thecharacteristics assumethe form U=x'+yi+zI+a:t } V=ax'+by2+OZi+da:t. Thetouching planeisa:=0jthepointofcontactisx=0,y=0,z=0jthe curveofintersection is one of thefourthdegree, with a double pointatthe pointofcontact. There isbutonecondition to be satisfied, andthecontactmay beentitled "simple"andof the first degree. NextletA.havethreeequalvalues, the equations become U=x'+yz+ti+xy, V=x'+yz+at2+bX!J. Thetangentplaneatthepointofcontacty=0,andthepointitselfx=O, y=0,t=O.Thecurve of intersection is a curve of thefourthorder, with a cuspatthepointofcontact. Thenumber ofaffirmative conditions to be satisfiedistwojthecontactis ofthesecond degree, andmay be termed "proximal" or cuspidal, 15-2 228 AnEnumeration ofthe Contacts ofLines [36 Nextlet0(U+XV)have two pairsofequalroots, we shallfind U=a:J+.xy+zt, V=ayz+bxy+czt. Thelinex=0,Z=0 will be common tobothsurfaces. Thecurveof intersection willtherefore breakupintoarightlineanda line of the thirdorder. Theformer will meetthelatterin twopoints,which will be each of them pointsofcontact. Thecontactistherefore diploidal; butasthereisanother speciesofdiploidal contactto which we shallpresently come,itwill be expedient tocharacterize each of thembythenatureoftheintersections ofthetwosurfaces; accordingly thismaybetermedunilinear-intersection contact, or more briefly, unilinear contact. Thenumber ofaffirmative conditions to besatisfied beingtwo,itmay besaidtobecollectively oftheseconddegree,but(obviously 7)thecontact ateach ofthetwopointsis ofthenatureofsimplecontact. Lastly,letussupposethatall four rootsofU+XVareequal;weshall find, asthemostsimpleexpressions ofthemostgeneralforms of thetwo surfaces, U=a:J+.xy+yz+zt, V=axy+bz2+azt. Inthiscasethetwopointsofintersection ofthecurveofthethird degree,andtherightline on which thesurfaces intersect, cometogether, so thattherightline becomes a tangenttothecurve.Thenumberofconditions to besatisfied isthree:thereisbutonepointofcontactwhichmaybe con­ sideredastheunionof two which havecoalesced, and thespeciesmaybe definedasconfluent-unilinear contact. Ifwethrowtheequations totheconoidshavinganunilinear contactinto theform weobtainx(x+y)+zt=0, xy+Z(y+ct)=0, (x+y)(y+ct)-yt=0, whichlastequation is nolongersatisfied byx=0,Z=0,thesesystems of rootshavingbeenmadetodisappear bytheprocess of elimination. Thecurveofthethirddegree,in which thetwogivenconoidsintersect, maythusbedefined astheircommon intersection withthenew conical surfacedefinedbythethirdoftheaboveequations. 36] andSurfacesoftheSecondOrder. 229 andMoregenerally, itisapparent thatthethreeconoids, a:u-yt=Oj yv-zu=O , zt-::r:v=O in which e,y,s,t,u,'Vmay any of thembeconsidered asahomogeneous linearfunction of fourothers,intersect inthesamelineofthethirddegree. Besideswhich,thefirst and second intersect intherightliney,u;thesecond andthirdine,'V;thethirdandfirst ine,t;each of which lines itisevident isachordofthecommon curveofintersection. Forinstance, y=0,U=0 maybesatisfied concurrently withalltheabovethreeequations bysatisfying theequation zt-::r:v=0, which, astwolinearrelations existoriginally be­ tweenthesixletters,and two more have been thrownin, becomes aquadratic equation between anytwo oftheletters. The only case of exception tothisreasoning is, when y=0,U=0 can be satisfiedconcurrently withz=0,v=0, andwitha:=0,t=0;butinthiscase the surfaces all become cones;andasthereis nolongeracurveofthethird degree,.. Caditqusestio," Evenhere, however, theintersection of any two of thesurfacesbecomes a conic, and two coincident generating lines on thetwo cones; so thatif wetakeone oftheseandtheconic torepresent adegenerate formof a line of thethirddegree,theremaining straightlinepassesthrough adoublepointofsuchdegenerate form,andthecasepassesintothatof confluent-unilinear contact. The two double pointsintheintersection ofthetwo conoids U=x(x+y)+zt=0, V=xy+z(y+ct)=;0, bywhich I mean thepointsofintersection oftheconicwiththerightline common to them,are found by making x=0,z=0, andsubstituting inthe derivedequation (x+y)(y+ct)-ty=0, whichgives y=0, ory+(c-1)t=O; sothatthetwopointsrequired are x=0,y=0, z=0, x=0,y=(1 -c)i,z=o. Itappears also thattheentireintersection iscontained in each of thetwo cones, tj:V;thatis,xI+z{(l-c)t-y} cU-V;thatis,cxS+y{(c-1)x-z}, therespective verticesof which are at-thepointsabovedetermined. 230 AnEnumeration oftheContactsofLines [36 Theequations forconfluent-unilinear contact, a;(a;+y)+z(y+t)=0, xy+z(cz+t)=0, gIve (a;+y)(cz+t)-(y+t)Y=0; which, on making a;=0,Z=0, issatisfied byyl=0;showing thatthe confluence takesplaceatthepoint a;=0,y=0,Z=o. Thenumberoftermsinthetwoequations forordinary unilinear contact beingsix, and in thosegivenforconfluent unilinears seven,andtheempirical rulein allothercasesbeingthatthetermstendtodiminish andnever increase innumber asthedegreeofthecontact(expressed bythenumber ofconditions to besatisfied) rises,Iamled tosuspectthattheconjugate systemforthelatterspeciesofcontactmayadmitofbeingreduced tosome moresimpleform. Imuststatehereonce for all, thatallthedistinctsystems of(atleast consecutive) conjugate formsthathavebeen,andwill begiven,aremutually untransformable, Thisitiswhichdistinguishes singular fromparticular forms. Aparticular form isincluded initsprimitive; butasingular form is one, which,whileitresponds tothesameconditions assomeothermoregeneral form, is incapable ofbeingexpressed asaparticular case ofthelatter,on account oftheadditional condition orconditions whichattachtoit. Ipassnow tothesingularities whicharisefromthefirstminordeter­ minants ofU+AVhavingafactorin common, thesecondminorsbeing supposed to bestillwithout acommon factor. Whenthiscommon factorislinearinrespecttoA.,letitbesupposed toenternotmorethantwice(twice,we know, by thegeneral principle enunciated atthecommencement ofthispaper,itmustenter)intothe complete determinant. Two oftheconescontaiuing theintersection ofUandVthenbecome coincident, anddegenerate eachintothesamepairofcrossing planes. This may betermedbiplanar-oontact, Thecharacteristics ofsuchcontactare U=w+yl+Zl+tJ, V=au:"+ay'+bzl+ct'; thepointsofcontactaretwo innumber, beingattheintersection ofthetwo planeconicsintowhichthecurveofintersection breaksup.Thetwoplanes 36J andSurfacesofthe Second Order. x+v(-l)y =0, x-v(-I)Y=0,t=O, t=0,z=O, z=O,inwhichtheselie aregivenbytheequation (b-a)z'.l+(c -a)t'=0;these intersect intherightlinez=0,t=0,whichmeetsbothsurfaces inthesame twopoints, thetwocommon tangentplanesatthesepointsbeing x+v(-l)y=O, x-v(-I)y=O respectively. This,then,isanotherspeciesofdoublecontactbetween two conoids, and, as farasIknow,theonlykindhitherto recognized assuch.Thenumberof conditions to besatisfied remains two, as in theformerspecies. Nextsuppose thatthecommon factor of thefirstminorentersthree timesintothecomplete determinant insteadoftwiceonly, as in thelast case. Thecorresponding characteristics will be found to be U=a;I+et+y'+z', V=ax'+azt+by'+cz'. Theintersection ofU, Vstilllies in two planes, (b-a)y'+(c-a)z'=O; buttheintersection ofthesetwoplanes, y=O,z=O, meetsthesurfaces inthetwocoincident points, y=0,z=0,a;t=o. This,therefore, IcallconBuent-biplanar contact; thetwo conics con­ stituting thecomplete intersection, insteadofcutting, touchandattheir pointofcontactthetwoconoids haveacontactofasuperior order.The conditions to besatisfied forthiscase are threeinnumber. Nextsuppose thatthecommon factorofthefirstminorsentersonly twiceintothecomplete determinant, butthattheremaining twofactors become equal. Heretheanalytical characters ofunilinear andbiplanar contact are blended; in fact, theintersection consists of a conic andapairofright linesmeeting oneanother andtheconic.Thecharacteristics are U=a;I+y'+z'+zt, V=rt:rP+ay'+bzl+est. 232 AnEnumeration oftheOontactsofLines [36 Theintersection iscontained inthetwo planes z=O,(b-a)z+(c-a)t=O, and consists of thetwo lines z=0,xl+yI=0,lyinginthecommon tangent planez=0, andtheconic (b-a)z+(c-a)t=O ) (a-c)xl+(a-c)y2+(b-c)Z2=o]. Therearethreepointsofcontact, namely, the pointx=0,y=0,z=0, where the two rightlines cut, and xl+y2=0,t=0,Z=0, where theselines meettheconic. This, then,is acaseoftriplecontact. I distinguish itby thename of bilinear-contact. Thenumberofconditions is stillthree. Now all else remaining asbefore, let thetwopairsof equal roots in the complete determinant becomeidentical, or, inotherwords,letthecommon factor of .the firstminors be contained fourtimesinthecomplete deter­ minant. Thecharacteristics become U=xz+xt+yl+Z2, V=axz+bxt+by2+bz'. Theintersection becomes thetworightlines a:=0,y2+Z2=0, and the conic z=o, xI+y2=O. Allthesemeetinthesamepoint, x= 0,y= 0,z=°j 80thatinsteadofcontactinthreepoints,thecontacttakesplaceaboutone only, in which the threemay be conceived asmerging. This I call confluent­ bilinearcontact. Itrequires thesatisfaction of four conditions. Nextletus suppose thatthetwodistinctfactors are common to each of thefirstminors. This will imply theexistence of four affirmative conditions. Thecomplete determinant will ofnecessity containeach ofthesefactors twice, so thatnoadditional singularity canenterthrough thisdeterminant. Thecharacteristics assumetheform U=xl+y2+Z2+tl, V=aW+ay2+bz'+bt'. Thetwo surfaces will meet in four straightlines, forming a wry quadrilateral, whoseequations are x±.v(-l)y=O, z±v'(-l)t =0. 36J andSurfacesoftheSecondOrder. 233 Theseintersect eachotherinthefourpoints a:=0,y=0, z~+tl=0, z=0,t=0,W+yl=0, eachof which will be adistinctpoint.ThisItermquadrilinear contact. Nowletthetwo factors common to each of thefirstminorsbecome identical; sothatasquared function, instead ofanordinary quadratic function ofX,is nowtheircommon measure. Thefactor which enterstwiceintoeach ofthefirstminorswillenter fourtimesintothecomplete determinant; thenumber ofconditions to be satisfied is one more thaninthepreceding case,namelyfive, and the characteristics becomeu=W+y2+a:z+yt, V=aa:2+by'+cez+cyt. Hereartsesasingularity of form in theintersections utterlyunlike anything whichhasbeenremarked inthepreceding cases.Foritwill notfail to have been observed, thattheintersection intheninepreceding caseswas always a line or systemof lines of thefourthdegree,soas to becut byanyplanein four points. Butinthiscase,thefact ofthefirstminorshavinga factor in common, showsthattheintersection iscontained in twoplanes(which is of course to be viewed asadegenerate species of cone);andthefact ofthecomplete determinant havingallitsroots equal, shows thatthereisbutonesystemof apairof planes in which theintersection iscontained, and no more. Sothatthetwopairsof planes, intowhichthewryquadrilateral was divisibleinthecaseimmediately preceding, now become a singlepair.This canonly be explained by two of theopposite sides of thequadrilateral becoming indefinitely nearto oneanother, butstillnotcoinciding inthe sameplanes; sothattheactualvisibleorquasi-visible- intersection will be inthreerightlines, of which themiddleone meets each of the twoothers. Thiswillfurtherappearbyproceeding regularly to solvetheequations U=O, V=O. (z+Ii=0,Y-lea:=0), (z-kt=0,y+leo:=0),(a:=0,y=0) ; ($=0,y=O);V-cU=0 givesy=±ke,wherek=J(~=~),andtherefore a:z+ka:t=0, ora:z-k:r:t=0;whence we see thatthecomplete intersection isrepresented bythelines • Iusethetermqnaei-visible, because theintersection maybecome in partor whole imaginary. 234 AnEnumeration oftheOontaeteofLines [36 showing thattherearebutthreephysically distinct lines,asalready premised. This,then,may be considered asderived fromthepreceding caseof awryquadrilateral intersection, byconceiving twoopposite sides of the quadrilateral to come indefinitely near,butwithout coinciding. Letthesetwo lines be calledPandP";takeanypointinPand any two pointsinP'indefinitely nearto oneanother andthepointfirsttaken,then thisindefinitely smallplanewill be common to both surfaces, and consequently theyoughtto touch along fmerypointin the line P. Thisisagainconfirmed by the forms giventoUandv:Forat anypointwherethecoordinates are 0, 0,s,0theequations to thetangentplanestothetwo surfaces respectively are soX+Oy=0, csoX+cOy=0, thatisto say, are identical. Whilst, therefore, certaingrounds ofgeometrical, andstillstronger grounds ofanalytical analogy, mightseem to justifythisspecies of contact takingthename of confluent quadrilinear, yetas,in fact,theintersection is trilinear, andas, moreover, thetwoindefinitely proximate linesmustbecon­ sidered,notascoincident, butasturnedaway from one another throughan indefinitely small angle and outofthesame plane, I prefertotakeadvantage ofthisstriking property ofcontactat every pointalongaline (aproperty entirelydistinctfrom any thatwe have yetconsidered), and confer upon the species of contactwe have been considering thedesignation ofunilinear­ indefinite contact. Wheretheline ofindefinite contactmeetsthetwootherlines of the intersection, thecontactis of course of a higherorderjthusoffering a parallel towhattakesplace in ordinary unilinear contact, in which there isnocontact,exceptonlyat two points oftherightline forming partofthe complete intersection. I believe thatthiskind ofcontact,which forms anaturalfamily with two othersaboutto bedescribed, andwhich will close thelist, has neverbefore beenimagined, and would at first sighthave been rejectedasimpossible. Havingnowexhausted thecasesofthefirst class, in which theminors have no factor in common, and thetwosectionsof the second class, in which thesecond minors have no common factor, butthefirst minors of U+AVa linearorquadratic function ofXin common, I descend to thethirdclass,in whichthesecond minors, which are quadratic functions ofX,aresupposed to haveacommon factor. Thiscommon factor mustentertwiceintoeachofthefirst minors by virtueofthelawpreviously indicated, andcannotentermorethantwice,as 36] andSurfacesofthe Second Order. otherwise thefirstminorsofU-+XVcould only differ from one anotherbya. numerical multiplier, which is obviously impossible, exceptwhenU+XVis oftheform(k+X)U,thatis, when thetwosurfaces coincide. Again,thecommon factor of thefirstminormustenterthreetimesinto thecomplete determinant; butthereis noreasonwhyitmaynotenterfour times,andthustwocasesarise.Inthefirst,thecharacteristics takethe formu=a;2+y'+Z2+t2, V=a:r;2+ay~+az2+bt2• Theseconddeterminant having a.factor in common, shows thattheinter­ sectionU, Viscontained in apairofcoincident planes;butthecomplete determinant, havingtwodistinct factors,evidences thattheseplaneinter­ sections, viewedasindefinitely nearbutstilldistinct, lie inthesame cone, whichwill be a cone enveloping boththesurfacesUandVallalongtheir mutualintersections. Thisis also seen easily from theforms of UandV; for wehaveV-aU=(b-a)t2,whichprovesthattheintersection lies in thecoincident, or, tospeakmorestrictly,consecutive planest2=0;andat anypointa:=~,y="I,z=~,thetangentplaneto each surface becomes ~x+'TJY+SZ=O. Asthereare sixindependent, thatis,non-necessarily co-evanescent second minors,thatthesecondminorsystemsshall all have a common factor, implies thesatisfaction of five conditions. Thisspecies of contactIcallcnrvilineo­ indefinite; itis, I believe, theonly kind of indefinite contactbetween two surfaces ofthesecondorderhitherto takenaccount of. Thereisstill,however, a higherspecies of contact,videlicet, when all the fourroots of thecomplete determinant ofU+AVareidentical withtheroot common to each of itssecond minors. Inthiscasethecommon enveloping cone becomes identical withtheplane(considered as acoincident pairof planes)in which thesurfaces intersect. Thecharacteristics taketheform U=x2+xy+zt, V=xy+zt. Theintersection iscontained completely inthecommon tangent plane II:=0, andconsists ofthetworightlines, (x=0,z=0), (x=0,t=0). This,thehighestandcrowning species of contact, I callbilineo-indefinite. Itisdefinedbysixconditions. Ateachpointof the two lines of intersection ofUandVthereiscontact, andaverypeculiar species of contactattheintersection ofthesetwo lines themselves. 236 AnEnumeration oftheContact»ofLines [36 To form a distinctidea.ofthis,letthephysical visible or quasi-visible intersection ofU. Vtakeplacealongthetwo lines L,M;theratiOflal inter­ sectionmustbe conceived as made up of thewryquadrilateral, L, MjL',M', in whichLisindefinitely neartoL',andMtoM'.Itfollows,therefore, that thereiscontactatthefouranglesofthequadrilateral; butasthereis nothingto fixtherelativedirections ofthediagonal joiningtheintersection ofLandMtothatofL'andJI',because thereisnothing to·restrictthe position ofthelatterpoint,exceptthatitshalllie upon eithersurface.", it appears thatnot only is therecontactatthejunction ofthetwolines constituting thecomplete intersection ofthetwosurfaces, butthatthese surfaces continue totouchatconsecutive pointstakenallroundthisfirst, andindefinitely near to it in any direction'], Bilineo-indefiuite (thehighest) contact for two conoids is strictly analogous to confluence, thehighest species of contact between conics. Forthislattermay be conceived asanintersection made up of two co­ incident pairsofcoincident points;andtheformer. as an intersection made up of two coincident pairsof crossing rightlines;andapairofcrossing lines is to a planelocus of theseconddegreewhatacoincident pairofpoints is to arectilinear locus ofthesame degree. Inthesubjoined tableI havebrought underonepointof view the characters andalgebraic formswhich I call thecondensed formscorre­ sponding to each species of contactabovedetailed. A.Quadmtic loci in a rightline. Simplecontact. }xy Onecondition. x~+xy B.Quadratic loci in a plane. 1stClass. Simplecontact. Onecondition. Proximal contact. Twoconditions. 2nd Class. Diploidal contact. Twoconditions. Confluent contact. Threeconditions.} } } }~+yS+ xz ~+by2+CXZ x2+yx+yz ax2+byx+ayz :C+y2+ZS ~+ay2+bz2 ~+i+xz y2+xz •Thiswillbebetterseen by reference to theanalogy presented by the case when thetwo conoidstouchallalongacurve.Therationalintersection ismadeup ofthiscurveandanother indefinitely nearit. The two curves.whatever be theposition oftheirnode,will lie in the same enveloping oone, so thattheposition of the node is indeterminate. tAs the two surfaces jutone close intotheotheratthispoint,it would perhaps benot improper todesignate thecontactatsuchpointasumbilical. 36J andSurfacesoftheSecondOrder. 237 C.Quadratic lociinspace. IstClass. Simplecontact. Onecondition. Proximal contact. Two conditions.}:rP+y2+Z2+ret} a:rP+by2+cz2+dxt }W+y2+xt+zt } aa;2+by2+ext+azt Unilinear contact. 1st species of diploidal. Two conditions. Confluent-unilinear, or triplecontact. Threeconditions.}W+xy+ztl ayz+bxy+czti }:rP+yz+xy+zt} az2+bxy+bzt 2ndClass,1stSection. Biplanar contact. 2nd species of diploidal. Two conditions. XZ+ret+y2+Z2 } axz+bret+by2+bz2}:rP+~l+Z2+zt}or{XZ+yt a:rP+ay2+bz2+czt axt+byz }Confluent-bilinear con­ tact.Fourconditions.Confluent-biplanar con-}:rP+zt+~+Z2} tact.Threeconditions. a:rP+azt+by2+cz2 Bilinearcontact. Threeconditions. 2nd Class, 2nd Section. Quadrilinear, orquad­ ruplecontact. Fourconditions. Unilineo-indefinite con­ tact.Fiveconditions.}:rP+y2+Z2+t2 }{xy+zt aa;2+ay2+bz2+btloraxy+bzt }:rP+y2+XZ+yt } aW+by2+cxz+cyt 3rdClass. Curvilineo-indefinite contact. Five conditions. Bilineo-indefinite con­ tact. Six conditions.}X2+y2+z2+t2 } aa;2+a~+az2+btl }:rP+xy+zt} xy+zt 238 AnEnumeration ofthe Contacts ofLines [36 Another (and, in a physical sense, more) naturalmode of grouping the twelvespecies of conoidal contact, which.without observing the same lines ofdemarcation, leavesintactthe sequence of thespecies, is intothethree families. The first, or definite-continuous. for which thesurfaces touchin asinglepoint,andintersect in anunbroken curve, comprises simple and cuspidal contact. Theseconddefinite-discontinuous, for which the surfaces touch in one, two,threeor four points, butintersect inacurve more or less brokenupinto distinctparts,comprises all the species from thethirdtotheninthinclusive. Thethirdnaturalfamily is thatofindefinite contact,andcomprises the three lastspecies.Itwill of course be observed thattherearefive species of single contact,thatis,contactat onepoint,namely, simple, cuspidal, and thethree confluent species, two of double, one of treble,one ofquadruple, andthreeof indefinite contact; thelastbeingdistinguishable interse-lineo-indefinite as beingspecial at two points, curvilineo-indefinite ashavingnospeciality, and bilineo-indefinite asbeingspecialatonepointonly. Imightnow proceed to discuss more particularly thenatureofthe 'Contacttaken,not collectively, butwithreference to each single pointwhere itexists. This, however, must.bereserved for afuturecommunication; as also, among otherimportant andcuriousmatter,theascertainment ofthe singular forms of quadratic conjugate functions of five or more letters. At presentI shallcontentmyselfwithstatingthefollowing generalproposition, whichnaturally suggests itselffrom aconsideration ofthecasesalready 'Considered. In aconjugate quadratic systemof anynumberofletters.thelowestand alsothehighestdegreeofsingularity will be always unique; theconditions to be satisfied in theformer case beingonly one in number, and inthelatter kr(1'-1), whererdenotesthenumberoftheletters. Thefirstpartofthis proposition isself-apparent, thelatterpartmay beinferred fromthehoma­ loidallaw;for the(r-2)ndminorswill bequadratic functions, and the highestdegree of contactwillcorrespond to those havinga factor in common, which would involvethesatisfaction ofkr(r-1)-1conditions only;but over and above this, thatthecomplete determinant, insteadofcontaining thiscommon factor, asitneedsmust,(r-1) times, shall containitrtimes : thisgives one condition more,makinguptheentirenumbertotr(r-1). Thetotalnumberofdifferent species of singularity forconjugate func­ tionsof agivennumberofletters,can only be expressed by aid of formulee containing expressions for thenumber of various ways in which numbers admitofbeingbrokenupintoagivennumber ofparts. Thecomputation ofthisnumberinparticular cases, upon theprinciple of theforegoing method,isattended withno difficulty. 36] andSurfacesoftheSecondOrder. 239 Wehaveseenthatthisnumber for two, threeand four letters,is respectively one, four, twelve. Ihavefoundthatfor five lettersthenumber istwenty-four, forsix lettersfifty,for seven lettersahundred, and(subjecttofurtherexamination) foreightlettersonehundred andninety-three. Theseries,therefore, as far as I have yettracedit, is 1, 4, 12, 24,50, 100, 193. Thelastnumbermust notberelieduponatpresent. Itwill be observed, thattheforegoing tableforthecontacts ofsurfaces ofthesecondordercontains no formcorresponding to acomplete intersection in twonon-intersecting lines and an undegenerated conic.Infact, if two suchlinesformpartoftheintersection, at leastoneotherrightlineinter­ sectingthemboth,mustgo to make up theremaining part.Thisis easily verified; foritisreadilyseenthatthemostgeneralrepresentation of two conoidsintersecting in twonon-meeting lines will be U=xy+zt, V=axy+bzt+ext+eyz, wherethetwo lines in question are (x=O, z=O), (y=0,t=0). Nowitwill be found thatthefirstminorsofV+)..Uformed from the aboveequation will allcontainthe common factor (a+)..)(b+)..)-ce,showing thatthecontactisquadrilinear orlinear-indefinite, thatisbilinear,according astheroots of aredistinct orequal;whichexplains howitisthatonly one species of bilinearcontact(thatis to say, thecasecorresponding tothetwo conoids agreeing inthetworightlines in which each is cutby a common tangent plane)comes to find a place in thepreceding enumeration. Itmaynotbeuninteresting, underaneuristiepointof view, to statethat theabovetheory,which, as well in whatitaccomplishes as inwhatit suggests (theauthorcannotbutfeel conscious), constitutes asubstantial accession to analytical science, arose out of a theorem which occurred to himaslikelyto betrue,intheactofreviewing forthepress his paper "OnCertainAdditions" inthelastNovember Number- ofthisMagazine, and which he hadonlythentimetothrowintoa foot-note as a probable conjecture. Wishing tosubjectitto ananalytical test,he found itnecessary to obtain thecondensed formswhich serve to characterize theconfluent contact of [*p.148above.] 240 Oontacts ofLinesandSurfacesoftheSecondOrder.[36 conics. In thisway he became awareofthegreatutilityofthesecondensed forms, and of thedesideratum tobesupplied inobtaining acomplete list of themapplicable to allvarieties ofcontact. Thehappythoughtthenoccurred to him of inverting theprocesswhich he had appliedin thetreatment of thecontacts of conics, in theNovember Number- oftheCambridge and DublinMathematical Journal; forwhereas thenatureofthecontacts was thereassumed andtranslated intothelanguage ofdeterminants, he soon discovered thatitwasthemoreeasyand secure course to assume therelations of every possible immutable kindthatcouldexistbetween thecomplete and minordeterminants corresponding tothecharacteristics, by aid of these relations toconstruct thecharacteristics, and from thecharacteristics so obtained, determine thegeometrical character of eachresulting species of contact. Thushehasbeen able to effect theveryresultsstatedbyhimself asdesiderata attheclose ofthepaperinthisMagazine abovereferredto. Note.-It ispropertoremark,thatallthecondensed forms giveninthis paperhaveactually beenobtained bytheauthorin the way above pointed out.Thelimitsimposed bytheobjectsto which theMagazine isdevoted haverestricted him from exhibiting themethodatfull;butanyof his readerswill be able without difficulty to make itoutfor himself. Theprocess consists in finding U+XVbymeansofsolvingfor each case aproblem ofposition (akindof chess-board problem) on asquaretable, containing threeplaces in lengthandbreadth for conics, four places by four for surfaces, and so on (ifneed be) according tothenumberofvariableletters involved. U+XVbeingthusdetermined in form,UandVbecomereadily cognizable. Itisrightalso to add, thatsome of thecondensed forms here setforth have been incidentally noticedand employed by previous authors, asPluckerandMrCayley. Theconditions in each case to which theposition-problem issubject areimmediately deducible fromthelaws which thecomplete determinant, andthesuccessive minorsystems ofdeterminants ofU+AV,arerequired to satisfy. [*p.119above.] 37. oxTHERELATION BETWEEN THEMINOR DETERMINANTS OFLINEARLY EQUIVALENT QUADRATIC FUNCTIONS. [Philosophical Magazine, I.(1851),pp.295-305.] ISHOWED in thepreliminary partof mypaperonContacts intheFebruary Number ofthisMagazine-, bydpriorireasoning, thatif aquadratic function (U)belinearlyconverted intoanother(V),anyminordeterminant of any order ofVmustbe a syzygetic function of alltheminordeterminants ofUof the same order. Theobjectof mypresentcommunication is toexhibitthesyzygy in question, which, asIindicated, islinear;by which Imeanthatadeterminant oftheone function isequal to thesum ofthepari-ordinal determinants oftheotheraffectedrespectively withmultipliers formed exclusively out of the coefficients of theequations oftransformation. In order thata clear enunciation of thetheorem in viewmaybe possible, itis necessary to premise anewbutsimple, and, as experience hasproved to me, a most powerful, becausenatural,methodofnotation applicable toallquestions concerning determinants. Everydeterminant isobtained byoperating uponasquarearrayof quantities, which,according totheordinary method, mightbedenoted asfollows: ~l> ~I'" ~n, as,I'as,I'"as,n, Mymethod consists illexpressing thesamequantities biliterally as below: 8.[.p.221above.] 16 242TheRelation betweentheMinorDeterminante of[37 where of course, whenever desirable, insteadof~,a,...an,andai'a,...an, we may write simply a,b...l,andel,fJ...Arespectively. Each quantityis nowrepresented by twoletters; thelettersthemselves, takenseparately, beingsymbolsneitherofquantity nor ofoperation. butmereumbra!or ideal elements ofquantitative symbols. We havenow a means of representing thedeterminant above given in a compact form;forthispurpose we need buttowriteonesetofumbreeovertheotherasfollows: (~'a~an\.If '11>~a,J we now wish to obtainthealgebraic value of thisdeterminant. itisonly necessary to takeai,~...anin all its 1, 2, 3...ndifferent positions, and we shall have in which expressione1,e2...enrepresents someorderofthenumbers 1,2...11,and the positive ornegative sign is to be takenaccording tothe well-known dichotomous law. Thus, for example, {:;~}willrepresent aaxbfJxcryl +afJxb'Yxea +aryxbelXefJ -afJxbelxcryj'. -aelxfryxefJ -aryxbfJxea Although notnecessary for ourimmediate object,itmay not be inop­ portuneto observe how readilythisnotation lendsitselfto afurthernatural extension of itsapplication. {iibcd}willnaturally denote elfJryo obcdabcdx - x .afJryO'YoelfJI thatis {(aaxbfJ)}{(c-yxdO)}{(aryxbo)}{(eelxdfJ)} -(afJxbel)x -(eoxdry)- -(asxfry)x -(efJxd'J.)• Andingeneralthecompound determinant J~'bl ll. a;:b2~'" artb, l;l lalJfJIAI•~,fJl"'~ Clr,fJrx,,) willdenote 37] Linearly Equivalent Quadratic Functions. 243 where,asbefore, we have thedisjunctive equation (}l'(},....(}r=1, 2...r. Asanexample of the power of thisnotation, I willcontentmyself with statingthe following remarkable theorem in compound determinants, one of themost prolific in resultsof any with which Jamacquainted, butwhich iqderived from a more particular caseofanothervastly more general. The theorem iscontained intheannexed equation {Ut,a,.Ur,arH'aI'a,....a-,Ur-f.'.!·"Ut,a,.a-, ar+ .} all~Glr,Glr+l'allas'"Glr,Glr+2alla,.ar,Glr+. Itis obvious, that,withouttheaid of my system of umbralorbiliteral notation, thisimportant theorem could not be made thesubjectofstatement withoutan enormous periphrasis, and could never have been made theobject ofdistinctcontemplation or proof. Toreturntothe more immediate object of thiscommunication, suppose thatwe have any binaryfunction of two sets of quantities, Xl'X,....xn; EllE,...·E",of which the general term will be of theform Cr,.xXrE.j according totheprinciples ofnotation above laid down, nothing can be morenaturalthantorepresent Cr••by thebiliteralgroupara.;thefunction inquestion willthentaketheform !a,.a•.xrE.; thex'sandE'sdenoting quantities. butthea'sanda'smere umbne, The function maythenbe thrown undertheconvenient symbolical form (UtxI+a,.x,.+ +a,.x,,) x(aIEl+~E2+ + IX,.En). Soifwe confine ourselves to quadratic functions, for which Xllx,....Xn; ~I'E,....Enbecomerespectively identical, thegeneralsymbolical represen­ tationof any such willbe (UtxI+a,.x,.+...+a"x,,)". Thecomplete determinant will bedenotedby {allas..·a,.t, aI'~.-.IX,.) andany minor determinant of therthorder by 16-2 (2)244TheRelation betweentheMitwrDeterminants of[37 where 01,O~•..Orare some certainrdistinctnumbers takenoutoftheseries 1,2,3...r.Suppose nowthatwe have U=(~.xl+a,a;+ ...+a.",xn'f linearlytransformable into V=(blYI+b,y,+...+bnYn)~, by means of thenequations ~=~bl'YI +~b~ .y~+ +a-b«.Ynj. a;=~bl.Yl+a~b2'Y~++a..pn.Yn~. .x~'~~~b~...~~';:~b~...;~'~.::~';~fi'''~J' in which equations. beitobserved, each coefficient arb,isasinglequantity, perfectly independent ofthequantities denoted generally byara,. brb, which enterintoUandV.Ourobjectis to be able to expresstheminor determinant {bkl•bktb1<r} bl..bt.blr' in which theonegroupofdistinct numbers, k., /,..~...k;mayeitherdiffer wholly from. or agreewholly or in partwiththeothergroupofdistinct numbersll'l~...lr.undertheform of !{(ae,. a~.•.aer)xQ}, b."b.~...b.r Th. . I I fQ' di h d bl (Bl•B~Or)eparticuar va ue0correspon mg to eac ou e group. 1/>1'1/>,I/>r' b d dbQ(Bl'89....Br)h bl ,.dmay e enote y 1/>1'1/>9.•.•I/>riso t at our pro em consistsmeter- '..h I fQ(Bl,09....Or),h .mmmgt e va ue01/>1'1>s•.,I/>rint eequation {bkl'bkt..·b1<r}=s{Q(0..B9....Or)X(ael•aftaer)}. bl,.bl,...bl,. 1/>1'1/>2'"1/>,\aol>,,a.,a.r· Accordingly I enunciate that Q(Bl•B9..•.Or){ak"akta1<r}{all'al,'"al,.} 1/>..1/>2...l/>r=bel'bftberxb.,.b+s...b.r (all'al,'"al,.[{akl.Uks,••akr} +lbe..bft...be)xb.1'b.9....b.r' subjecttoonesoleexception in the case of 01>0,'"Orbeingidentical with 1/>1'1/>"•••I/>rinamely, thatfortheterms(for such case) of theform 37J Linearly Equivalent Quadratic Functions. 245 Q(::::::::::),the value to be takenis notthatwhich the generalformula wouldgive, namely, 2{a,tpaa,t,}, bit,b"b'r' hut thehalfof this,thatis simply the square of {a,tpaakr}• a.1,aft a.~ ThIfQ(81'8,...0,.). .'b' .I . .e va ue 04>t,~...</>,.,.It18 0VIOUS.contams on y quantities of the form a,..b"which are coefficients in the equations oftransformation, but none of theforma,.'.a,orb,..b,jshowing thatthesyzygetic connexion between ,the minor determinants ofUandVof the same orderis linear, ashas been alreadyanticipatively announced. The problem which Ihavetreatedabove is only aparticular case of a moregeneralone, which may be statedas follows: given U=(a.Xl+a,x,+...+anx..)', andsupposing m linear equations to beinstituted between Xl'x,...11;", 80thatUmaybemade a function of (n-m)lettersonly, to expressany minordeterminant of the reduced form of Uwithoutperforming the process ofelimination between the given equations. Letthegivenequations be writtenunderthe form a.tIn+11l:1+a,tIn...IX'+ +tIna..+IX..=0, altIn-j-ixl+£l.#nHIl;+ + an~,X..=0, and let it be convened (which takes nothing away from thegenerality of theseequations) thattIn+ra"H shall signify zero for all values of rand 8 concurrently greaterthanzero. Suppose thatXl>X~...Xm,beingeliminated, Ubecomesof theform (bm+lItmH+~Xm+l+...+b..X..)2; and suppose thatwewish to determine the value of the complete determinant ofthislastfunction; it will be found to be (bno+1'bno+,•.•b..){aI'~tIn,tIn+1Un+m}(a.,as...amJI b.+l,bm+l'"b..=a.,astIn,an+ltIn+m+Un+l>a"H'"a..+ I thesquared divisor being, as is obvious, a function only of the coefficients ofthetransforming equations. and depending foritsvalue.upon the particular (3)246 The Relation betweenthe Minor Determinants of[37 mquantities selected forelimination. Thedividend, onthecontrary, is independent ofthisselection, butinvolves thecoefficients of thefunction combined withthecoefficients of transformation. Thisisthesymbolical representation ofthetheorem givenby me in thepostscript to mypaperin theOamlJridge andDublinMathematical, Journal forNovember 1850·. Suppose, now, more generally thatwe wishtofind anyminordeterminant, Thesolution isgiventbytheequation . {b''''+I,b,.......b,.+o} b<t>"'H'b<t>.+t'" b<t>.+. (wherein thetwogroups (Jm+1'(Jm+J'.,.(Jm+l;4>m+1.4>m+'J'" 4>m+lareeach ofthem 8differing, or wholly or in partagreeing individuals arbitrarily selected outofthe(n-m)numbers m+1, m+2,,.,n) _{a.l,a's'"a,... a.....,a+t•••a,..+o).{a.,~,..am}J -a<t>J'a4>J...a<t>..,a<t>..+l'u<t> a<t>..+oI-;-Un+1'Un+2'"a"+m Ifwe make n=2'1andm='1,anda.,.-tra.,.+1=0 for all positive values ofeitherror8,anda.,.-iUn+e =0for all values of iandediffering from one another, and forequalvaluesa,.-.,a.,.-te = -1, it will readilybe seenthatthis lasttheorem reduces totheone first considered; and oncarefulinspection itwill be found, thatthesolution givenofthegeneralquestion includes withinitthatpresented fortheparticular case inquestion. Suchinclusion, however, I oughtin fairness to stateis far from beingobvious; andto demonstrate itexactly, and ingeneralterms,requires theaid ofmethods which my readerswouldprobably find to exceed theirexisting degreeof knowledge orfamiliarity withthesubject. Thetheorem aboveenunciated was inpartsuggested inthecourse of a conversation with Mr Cayley(towhom I am indebted for myrestoration totheenjoyment ofmathematical life) on thesubjectof one of thepre­ liminary theorems in mypaperonContacts inthisMagazine. Itiswonderful thatatheorysopurelyanalytical shouldoriginate in ageometrical speculation. My friend M. Hermite haspointedoutto me, thatsomefaintindications ofthesametheorymay be found in theRecherches Arithmetiques of Gauss. Thenotation which I have employed fordeter­ minants is verysimilartothatofVandermonde, with which I have become acquainted sincewritingthe above, in Mr Spottiswoode's valuable treatise On theElementary Theorems ofDeterminants. Vandermonde wasevidently ontherightroad. I do nothesitate to affirm, thatthesuperiority of his andmynotation overthatin use in theordinary methods isasgreatand almostasimportant totheprogress ofanalysis, asthesuperiority ofthe notation ofthedifferential calculus overthatofthefluxional system. For whatisthetheoryofdeterminants !Itisanalgebra uponalgebra; a. [.p,186sbove.] [tseep.251below.] 37J Linearly Equivalent Quadratic Functions. 247 calculus whichenablesus to combine and foretell theresultsofalgebraical operations, inthesame way as algebraitselfenables us to dispense with theperformance ofthespecialoperations ofarithmetic. Allanalysis must ultimately clotheitselfunderthisform", Ihave inprevious papersdefined a " Matrix" as arectangular arrayof terms,out of which different systems ofdeterminants may beengendered, as fromthewomb of a common parent jthesecognatedeterminants being by no means isolatedintheirrelations to oneanother,butsubjecttocertain simple laws of mutualdependence andsimultaneous deperition. Thecon­ densedrepresentation of any such Matrix, according to my improved Vander­ mondian notation, willbe {~'~...aral al,as·..amJ. Toreturntothetheorems ofthetext.Theorem (2)admitsofbeing presented in a more convenient form forthepurposes ofanalytical operation, so as to become relieved fromall cases of exception appertaining toparticular terms. asidentical withitsequal,Thelimitation tothegenerality oftheexpression forQarises from our treating If,however, we now convene totreatthesetwo forms as distinct, sothat intheorem (2) will.{n(n-l) ..,(n-r+I)}2 ..Icontain I .2...r terms,thenwe may write SImpy •Perhaps themostremarkable indirect questlon towhich the methodofdeterminants has beenhitherto appliedis HeBle's problem of reducing acubicfunction of3letteratoanother COUlIisting onlyof4termsbylinearsubstitutions-a problem whichappearstosetatdefiance IIItheproeeseea andartifices of common algebra. I have succeeded in applying amethod foundeduponthiscalculus to thelinearreduction ofabiquadratic function of twoletterBto Cayley'sformzC+mz2y2+y'.andofa5·function of twoletterstothe newform zI+yD+ (az+by)D. Thislutreduction is effected by meansoftheproperties ofacertainotherfunction ofthe 8thdegreeconnected withthegivenfunotion ofthe5thdegree. See apaperonthissubjectin thelonhcoming MayNumberoftheCambridge andDublinMathematical Journal. [p.191above.) 248TheRelation betweentheMinorDeterminants of[37 whichequation issubjectto noexception forthecaseoftheUsand4>'8 becoming identical. As regards thistheorem, it will not fail to strikethe readerthatitoughttoadmitofverification; forthatUmaybederived fromVin the same mannerasVfromUif weexpressYI'YI'"y..interms ofXl'XI'"X..,by solving the system of equations (2), which thereis no difficulty in doing. Infact, if we write YI=ad31x1+alJIa;++aJ3..xn, YI=aJJlXI+aJ3.1l;++aJ3nXn. we shallobtain t:J_{a,.al···ar-I,'a,.+I, ar+I···an}{ai,~...u",} (lrf-J,-b b ..:..b.,I'"'-1>bH I•b'H'"bn.b..b«...b«. Accordingly we shall find {amI'Um.z...am,.}=IrQ('til'tl'"'tr)x(bo/JI'i;...bo/Jr)}, apl'aPs•..aprl0)1>0)1'"O)r\b..1'b...•..b.r and substituting for thea'sand{J'gtheirsymbolical equivalents given above, andapplying thetheorem given below, weshall easily obtain If,now,in the expression {bkPb",...bl:.}=I{(akl'a",...al:r)(all'alsaz.)(aBI'alit.:aBr)}, b/l•bls..·bz. bBI'b'I'"b'r b~l'b~ b~ra~I'a~...a~r weresubstitute for{aB. 'as....air}itsvalue in the form of a</>t'a</>I•••a~r I{(b..pb b .....)Ql bl/Jpbo/JIbl/JrJ' we shall obtain(bbkl'bbkl...bbl:.)undertheform ofI..Is'"I. 37] Linearly Equivalent Quadratic Functions. 249 andR(~'~~...4~r)must=0,exceptforthecase of"'I,"'.·""'r;"10"."'''r TI'TI···..,.", beingrespectively identical withk«,k....kr;llol....lr,for which case R(~l:~:::7:)mustbeunity.Ihave gone through thiscalculation and verifiedtheresult;inorderto effect which, however, thefollowing important generalization of theorem (I)mustbeapprehended. Suppose two sets of umbra-, andletrbe anynumber lessthanm,and let any r-arycombination of themnumbers 1, 2, 3...mbeexpressed bygOI'gO....-e;whereqgoes throughallthevaluesintermediate between 1andp"p,being m(m-l) ...(m-r+l). 1.2 ...r ' thenIsaythatthe compound determinant, a""~••..all".'am+l,Gm+.'"am+nl ~"a.,a2l/m'am+I,Cl.m+2·"am+nl b2l/"b2l/'»:bm+1,bm+s...b1n.+n isequaltothefollowing product, 1, 1"''' am+l ,U,nH'"am+n'"ai,~...am+n bm+l,bm H•••bm+nbl,b••.•bm+n where(4) and"(m-l)(m-2) ...(m-r+l) p,= 1.2...(r-1) , ,(m-l)(m-2) ...(m-r)p,- .- 1.2...r ' whenr=1, we have thecasealreadygiven in theorem (2), and of course p,"istobetakenunity. Thisverygeneraltheorem isitselfseveral degrees removed from my still unpublished Fundamental Theorem which is a theorem fortheexpansion oftheproducts ofdeterminants. 250OnLinearly Equivalent Quadratic Functions. [37 Obs.The analogy upon which theextension oftheVandermondian notation from simple to compound determinants isgrounded, would be better apprehended ifthebiliteral symbols of simple quantities werewrittenwith theumbralelements disposed vertically, as:'insteadofhorizontally, asabj whichlatteristhemethod forthepurposes of typographical uniformity adopted inthetextabove. The othermode is, however, much to be pre­ ferred, and is what I propose hereafter toadhereto.Formy two general umbras,a, b,Vandermonde uses two numbers, oneseta-cock upon theother, as 5'. The objection to theuse ofnumbers isapparent assoonasit becomes necessary totreatofthemutualrelations of diverse systemsofdeterminants, andhismode of writingtheumbrasmilitates againsttheperception ofthe mostvaluable algebraical analogies. The one important pointin which Vandermonde hasanticipated me,consists in expressing a simple determinant by two horizontal rows of umbrreone over theother.Buttheideaupon whichthisdepends is so simple and natural,thatitwassure toreappear in anywell-constructed system of notation. 38. NOTEONQUADRATIC FUNCTIONS ANDHYPER­ DETERMINANTS. [Philosophical Magazine. I.(1851),p.415.] PERMITme tocorrectanerroroftranscription intheMS. of my paper IIOnLinearly Equivalent Quadratic Functions" inthelastnumber ofthe Magazine. Thetheorem [po246above]marked(3),shouldreadasfollows:- {b '.H 'b..+I•••b,_} b._1•b b .. {Cl:I.~•••am, a...+1•a'.'+1'" a....., an+1•Un+•...Un+rn} =Cl:I,~•••am. a••H,a...+1'"a.".+•• an+1•an+!...an+m I maytakethisopportunity ofmentioning, thatbyextending to algebraical functions generally amultiliteral systemofumbralnotation. analogous to thebiliteral systemexplained inthepaperabovereferred to asapplicable toquadratic functions, I have succeeded in reducing to a mechanical methodofcompound permutation theprocess for thediscovery of thosememorable formsinvented by Mr Cayley. andnamedby himhyper­ determinants, which have attracted thenoticeandjustadmiration ofanalysts all over Europe, and which will remainaperpetual memorial, aslongas the name of algebrasurvives, ofthepenetration andsagacity oftheirauthor. 39. ON A CERTAI~ FUNDAMENTAL THEOREM OF DETERMIN AKTS. [Philosophical Magazine, II.(1851), pp. 142-145.] THEsubjoined theorem, which is one susceptible ofgreatextension and generalization, appears tome, and indeedfrom use and acquaintance (it havingbeen long in my possession) I know to be so important andfunda­ mental,as toinduceme toextractit from a mass of memoranda onthesame subject jand as an actofdutyto myfellow-labourers inthetheoryof determinants, more or less forestall time(thesurediscoverer oftruth)by placingitwithoutfurtherdelay on record in the pagesofthisMagazine. Its developments andapplications must·bereserved for a more convenient occasion, when theinterest intheNewAlgebra (for such, truly,itisthe office of the theoryofdeterminants toestablish), and the number of its disciples in thiscountry,shall have received theirdestined augmentation. In arecentletterto me, M. Hermite wellalludesto thetheoryofdeterminants as"Thatvasttheory,transcendental inpointof' difficulty, elementary in regardtoitsbeingthebasi>!ofresearches inthehigherarithmetic and in analytical geometry." Thetheorem is as follows :-Sllppose thattherearetwodeterminants of theordinary kind.eachexpressed byasquarearrayoftermsmade up ofnlines and ncolumns, so thatin eachsquaretherearen'terms. Now let'11bebrokenup in any givenmannerinto two partspandq.sothat p+q=n.Let, firstly, one of thetwo given squaresbe divided in a given definitemannerintotwoparts,onecontaining pof thengiven lines, and the otherpartqof the same; and secondly, let the otherofthetwogivensquares bedivided in every possible wayintotwoparts,consisting ofqandplines respectively, sothatontackingonthepartcontaining qlines ofthesecond squaretothepartcontaining- plines ofthefirstsquare.and the partcon­ tainingplines ofthesecondsquareto thepartcontaining qof the first, we 39JOn aFundamental TheoremofDeterminants. 253 get back a new couple of squares, eachdenoting adeterminant different fromthetwogivendeterminants jthenumber of such new couples will evidently be n(n-1)...(n-p+1). 1.2...P , and my theorem is,thattheproductofthe given coupleofdeterminants isequalto thesumofthe products (affectedwith the proper algebraical sign) ofeachofthenewcouplesformedasabovedescribed. Analytically thetheorem maybestatedasfollows. Let JUI>at·..an}{ai,a2•..a,1} [bl,b,...bn'131>/32'"/3n' according to the notation heretofore- employed by me in the preceding numbers ofthisMagazine, denoteany two common determinants, each of the nth order, and letthenumbersel,e,...enbedisjunctively equaltothe numbers I,2...nandP+q=n;thenwill [ai, a 2an){al>a2•••~} (bl,i.bJX131,13,...fJn =s+{UI>~... an1{ai,~... anl -i;i. ...bp,/39J>+p fJ8P+2'"fJ,JX/381'fJlj...fJ8p,bp+I'bpH...bnf' Thegeneraltermnuderthesignofsummation may berepresented by aid ofthedisjunctive equations </>1'</>2'"</>n=1, 2n, 'tl.'t2...'tn=1, 2n, under the formof (a.1•blxa4Jy.b2x '" x a~p.b,.)(u",P+1 •bp+1xal/tP+2.bp+'lx...xa",,,.bn) x(~"+1'fJ'P+I.xa~p+,'fJ8I>+tx '" x ~",13,,,)(a",•.B'Ixa",•.(8"x...x~p'fJ8p)' 1st. When </>1'</>2...</>p='tl''t2'""p,itwillreadilybe seen, thatfor given values of </>1,</>2'"</>p,theproduct ofthethirdand fourth factors becomessubstantiaUy identical withthe..generalterm of thedeterminant Sai'~...a,1) t/3I'132'"fJnl' andconsequently, makingthe system </>J,</>2...</>p(or, which is thesame thing.its equivalent 'tl''t2'"'tp)gothrough all its values, we getback for the sum of thetermscorresponding totheequation </>1'4>'1•••</>P="I>"2'""p, [*p.2~2above.] 254 OnacertainFundamental [39 theproductofthedeterminants 2nd.Whenwehave not theequality above supposed between the9's andthe","'s,let thecorresponding termincluded undertheIwillcontainthe factor a.,./3.xa",.13.. Jl+'Jl+' p-{1'-' Nowleaving91'~...9p,and'h.'\fit...'/rpunaltered, we may takea systemof values ()t',()2'•.•(),.',suchthat 8'1'+'1=()I>-'. and O'I>-'=Op+"._ and for all othervalues of qexceptp+T],orp-~.O'q=Oq.Thecorrespond­ ingnew value of thegeneraltermso formed by thesubstitution ofthe 0'forthe0series, will be identical withthatofthetermfirst spoken of, but will have thecontrary algebraical sign, because the0'arrangement ofthe figures1,2, 3...pisdeducible byasingleinterchange from the 0arrange­ mentofthesame,the rule for theimposition ofthealgebraical signplus orminusbeingunderstood to be.thattheterm in which {3'P+1'/3.P+I'"/3••;13'1'/31Jt...{3", enterintothesymbolical forms of-the respective derived couples of deter­ minants, hasthesamesign as, or thecontrary sign to,thatin which {3'P+1'{3,,,......./3,..;/3,1'{3't'"/3(fp 80enter,according asan odd or anevennumberofinterchanges isrequired totransform thearrangement (Jp+1'Op+t...en;01,Ot...()p intothearrangement 0'p+t>8'p+t...0'n;8'1'(J't...0'p- Ihavetherefore shownthatall thetermsarisingfromtheexpansion of theproducts included uuderthesign ofsummation, for which thedisjunctive identity c/>J,9t...9p=";1''\fit•..'/rpdoes not exist, enterintothefinal sum in pairs, equal in quantity anddiffering in sign, which consequently mutually destroy, and thatthetermsfor which thesaididentitydoesexisttogether makeupthesum 39] TheoremofDeterminants. 255 whichproves, upon first principles drawndirectfromthatnotion of polar dichotomyof permutation systemswhichrestsatthebottom of the whole theory of thesubject, thefundamental, and,asI believe, perfectly new theorem, which it is the object of thiscommunication to establish. Inapplying thetheorem thusanalytically formulized, it is ofcourse to be understood that,underthe sign };,permutations within the separate partsof agivenarrangement, ()p+l'()P+2". ()n;OlJ()2'••01" areinadmissible, thetotalnumberof terms so included beingrestricted to n(n-1)...(n-p+1) 1.2...p Thetheorem may beextended so as to become a theorem forthe ex­ pansionoftheproductof anynumberofdeterminants, andadapted soasto takeinthatfarmoregeneralclassof functions known to Mr Cayley and myselfunderthe new name of commutants, of which determinants present onlyaparticular, andthatthe most limitedinstance. 40. ONEXTENSIONS OFTHEDIALYTIC METHOD OF ELIMINATION. [Philosophical Magazine. II.(1851), pp. 221-230.] THEtheoryaboutto bedescribed is anaturalextension ofthemethodof elimination presented by me ten years ago (in June,1841) in thepages of thisMagazine, which I have been induced to review in consequence of the flattering interestrecentlyexpressed inthesubjectby my friend M. Terquem, and some othercontinental mathematicians, andbecause oftheimportance ofthegeometrical andotherapplications of which itadmits,and ofthe inquiries to which itindirectly gives rise. We shall be concerned in the following discussion withsystems of homogeneous rationalintegralfunctions of apeculiarform, to which for presentpurposes I propose to give thename ofaggregative functions, consisting ofordinary homogeneous functions of the samevariables butofdifferent degrees,brought together intoone sum made homogeneous by means of powers of new variables entering factorially. ThusifF, G,H...Lbe anynumberof functions of any numberofletters e,y...tofthedegreesm,'In-t,rn-t'...rn-(t)respectively, F+G>..'+Hp/ + ...+L81.) will be an aggregative function ofthevariables entering intoF',G,&c.andof A.,p....8.I shallfurthercallsucha function binary,ternary, quaternary, and so forth, according tothenumberof variables contained inthefunctions F, G,H, &c.thusbroughtintocoalition. Itwill beconvenient to recall theattention ofthereaderto themeaning of some of thetermsemployed by me in the paperabovereferredto. IfFbe any homogeneous function of:x,y.z...t,thetermaugmentative ofFdenotesanyfunction obtained fromF'of the form :x"ylJzY•••f]xF. Again, if we have any number of such functions F, G, H...Kofasmany 40JExtensions oftheDialyticMethodofElimination. 257 variables 11:,y,Z•••t,and we decompose F, G,H...Kin anymannerso as to obtaintheequations F=:If'"PI+'!Ip.+zep.+&c+td(P), G=af'QI+'!IQi+zeQ.+&c+td(Q), H=:If'"R,.+'!IR.+zeR.+&c+td(R), K=:If'"BI+'!IBi+zeBa+&c....+td(S), andthenformthedeterminant Php.,p•...(P), ,QI'Q., Q•...(Q) R,.,~,R....(R) BI>B., B•...(B) thisdeterminant, expressed asafunction of 11:,y,z...t,is what, in thepaper referred to, Icalledasecondary derivee, butwhich for the futureIshall cite bythe more concise and expressive name of aconnective ofthesystem of functions F. G,H ... K from which it is obtained. Oneprevailing principle regulates all the cases treatedof in this and the antecedent memoir, namely thatof forming linearlyindependent systems of augmentatives or connectives, or both, of the given system whose resultant is to be found, of the same degree one with theother, and equal in number(whenthisadmitsofbeing done) to the numberofdistincttermsin the functions thusformed. The resultant of these functions, treated aslinearfunctions of theseveral combinations of powers of the variables in each term, will thenbethe resultant of the given system clear of all irrelevant factors.Ifthenumber oftermsto beeliminated exceedthenumberof the functions, the elimination of course cannotbe executed. Ifthecontrary be the case, buttheequality isrestoredbytherejection ofacertainnumberof the equations, theresultant soobtained will vary according to thechoice of theequations retained for thepUrp<>!leof theelimination. Thetrueresultant will not thencoincide withany of theresultants 80obtained, butwillenterasa common factor into them all. The followingsimple arithmetical principles will be found applicable and useful for quotation in thesequel:- (a)Thenumberof terms in a homogeneous function of plettersofthe mth degree is 8.m(m+l) ...(m+p-l) 1.2...p 17 258 OnExtensiom oj [40 (b)Thenumber ofaugmentatives ofthe(m+n)thdegreebelonging to afunction ofplettersofthemthdegreeis (n+l)(n+2)(n+p-1) 1.2p (c)Thenumberofsolutions inintegers (excluding zeros) of theequation Ut+a..,+ ... + Up=Ieis (Ie-1)(1e- 2)...(Ie-P+1) 1.2'"(p-1) Tobeginwiththecase ofbinaryaggregatives. Let F",(x,y)+F-.(x, y)"A'+F __..(x,y)p/+&c....+Fm-<LJ(x,y)O('lf ':~~:.:~.~.~':"::.~~:.~?~~~..":':~~'.~~:::'.~.~~:::.:.~.~:~'~.~~'..~~.~:':(A) Kp(x,y)+K~,(x,y)"A'+K~,"(x, y)p/+&C.•.•+K~(,)(x,y)0(" be asystemoffunctions (whoseResultant it isproposed todetermine) equal innumbertothevariables x,y,"A,J.I.•••0,andsimilarly aggregative, thatis havingonlythesame powers of A,J.I.,&c.entering intothem,butofany degreesequalorunequalm,n...p.Letthenumberofthefunctions ber. Raiseeach ofthegivenfunctions byaugmentation tothedegrees,where s=(m+n+ ..,+p)-(,+"+...+(,»-1, thenumberofaugmentatives oftheseveralfunctions will be (s+l)-m, (s+I)-n, (s+I)-p, andthetotalnumberwilltherefore be r(s+l)-(m+n+ ...+p), which =(r-I)(m+n+ ...+p)-r(,+"+...+(,». Again,thenumber oftermsto beeliminated will bethesum ofthe numbers oftermsinfunctions respectively ofthesth, (s - ,)th,(s -,')th,... (s -(t»thdegrees, which are respectively s+1, s+1-" s+1-,', 40] theDialytu MetlwdofElimination. 259 andthenumberofthesepartialfunctions isr-1.Hencethenumberof termstobeeliminated is (r-1){m+n+&c.+P-(£+t'+&C.+(L»)-(£+£'+&c.+(£» =(r-1)(m+n+&C.+p)-r(£+£'+...+(£», whichisexactlyequal to thenumberoftheaugmentative functions. Hence theResultant- ofthegiven functions can be found dialytically bylinear elimination, andtheexponent ofitsdimensions inrespecttothecoefficients ofthegiven functions will be thenumber (r-l)Iom -rIo£, asabove found. Themethodabove given may be replaced byanothermore compendious, andanalogous tothatknown by thename of Bezout's abridged method for ordinary functions of twoletters. Asthemethod is precisely thesame whatever thenumberofthefunctions employed maybe, I shall for the sake ofgreatersimplicity restrictthedemonstration tothecaseofthreefunctions, U, V, W, whose degrees (if unequal, writteninascending orderofmagnitude) arem,n,prespectively. Let U=F".(:x,y)+Fm-..(:x,y)z<, V=Gn(:x,y)+Gn-.(:x,y)z<, W=Hp(:x,y)+Hp-,(e,y)z<. Lete.6)betakenany two numbers which satisfy in integersgreaterthan zerotheequatione+6)=m+1, and let F".(:x,y)=cf>m-t.a:'+cf>-.y-, Gn(:x,y)='Yn-,.a:'+'Yn-...y-, Hp(z,y)="1p-f.:x'+7Jp-<»•y-, wherethe4>'s,'Y's,7]'Smay be always considered rationalintegerfunctions of zandy;for every termin each of the functions Fm,Gn,Hpmusteither containa:'orr,since, if not, its dimensions in a;andywould not exceed (e-1)+(6)-1), thatism- 1, whereas each termis ofmconjoined dimensions, atleast,ina; andy.Hencefromtheequations u=o, V=O, W=O, • TheBeeuUant ofasystemorfunctions meansingeneralthesamethingastheleft-hand sideor thefinalequation (clear or extraneons factors)resulting fromtheelimination ofthe nriables between the equations formed by equating thesaidfunctions severally tozeJOO. 17-2 260 OnExtensions of [40 byeliminating :r!',yBandz'weobtaintheconnective determinant 'Y,....."'Y-,G,....., 'T/p-f,"lp_,Hp-, which will be of thedegree m+n+P-«()+lJ)+s), thatillofthedegree(n+p- , -1)inxandy;andthenumber ofsuch connectives byprinciple (0)isp. Again, by augmentation wecanraise each of thefunctions U, V, W to thesamedegreeastheconnectives, andbyprinciple (b)thenumber ofsuch willbe n+p-m-" P-', n-" fromU, V, W respectively, together makingupthenumber 2n+2p-m-3,. Hencein all we have 2n+2p-3,equations; andthenumberofterms to beeliminated will be,n+P-,arisingfromP,,,,G«. HI" andn+p-2, fromPm-"G........Hp-,;together makingupthepropernumber2n+2p-3,. Eachconnective contains ternarycombinations ofthecoefficients, namely one ofthecoefficients belonging tothatpartofU, V, W whichcontains Z', andtwo coefficients from theotherpart:thedimensions oftheresultant in respectofthecoefficients of theformer will hence be readilyseen to be equal tothenumberofconnectives +thenumberoftermsintheaugmentatives intowhichZ'enters,thatis, willequalm+n+p-2,;thetotaldimensions oftheresultant inrespectto allthecoefficients of U, V, W will be 3m+(2n+!op-m-3,), thatis, 2m+2n+2p-3,; andconsequently, inrespecttothecoefficients of F",;Gn;HI"will be of (2m+2n+2p-3,)-(m+n+p-2,), thatis,ofm+n+p-,dimensions. Thisresult,whichisofconsiderable importance, may be generalized asfollows. Returning tothegeneralsystem(A), for which we have provedthatthe totaldimensions oftheresultant are (r-l)(m+n+...+p)-r(,+,'+...+(,), 40] theDialytic MethodofElimination. 261 letthecoefficients of thecolumn of partialfunctions r: Gn, be called thefirstset;thecoefficients of thecolumn Fm-" Gn-" thesecondset.:and so forth jthenthedimensions inrespectoftheIst, 2nd...(r-l)thsetsrespectively are8, 8 - t,8 -£'•••8 -(r), where 8=m+n+&c.+P-(t+t'+&c.+(t». Theimportant observation remainsto be made, thatalltheaboveresults remaingoodalthough anyone or more of theindices of dimension of the partialfunctions in the system (A), asm-£,m-£/,n-£,&c.,should become negative, provided thatthetermsin which such negative indices occur be takenzero,aswill beapparent onreviewing theprocesses alreadyindicated uponthissupposition. Ifwetake m=n=...=p,and£=£'=&c.=(£)=m-E, theexponent ofthetotaldimensions of theresultant becomes (r-l)rm-r(r-2)(m-E) =rm+r (r- 2)E, when E=0,thisbecomes mr,whichismade up of 2m unitsofdimension belonging tothecoefficients of thefirst column, and ofm belonging to each of the(r-2)remaining columns. Consequently, if we have Fm(x,y)+e>..+r>..'=0, Gm(x,y)+'TJ>"+'TJ'X-'=0, Hm(x,y)+p..+">'"=0, Km(e,y)+0>..+8'>..'=0, or any othernumber ofequations similarly formed, theresultofthe elimination is always of m dimensions only in respectofe,'TJ,,.0,or of E','TJ',r,(J',and of2minrespectof the coefficients in F, G, H, K. I now proceed to stateand toexplainsomeseeming paradoxes connected withthedegreeoftheresultant of suchsystems of defective functions as have been previously treatedof inthismemoir,ascompared with thedegree 262 OnExtensions of [40 (B) all of threeofthegeneralresultant ofacorresponding systemofcomplete functions of the samenumberof variables. Inorder to fix our ideas, let us takeasystem of only threeequations of theform Fm(z,y)+F'1Ir-<(x,y)r=O} Gn(x,y)+Gn-.(x,y)Z'=0 . HI'(x,y)+Hp-.(x,y)Z'=0 Theresultant ofthissystemfound by thepreceding methodis in 2m+2n+2p-3,dimensions. Butingeneral, theresultant of equations ofthedegreesm, n, p is ofmn+mp+npdimensions. Now in order toreason firmlyand validly upon thedoctrineofelimination, nothingissonecessary as to have a clear and precise notion, nevertobelet go from themind'sgrasp,oftheproposition thatevery system of nhomo­ geneous functions ofnvariables hasa single and invariable Resultant. Themeaning ofthisproposition is, thatafunction ofthecoefficients of the given functions canbe found, suchthat,whenever it becomes zero, and lIeverexceptwhen it becomes zero, thefunctions may be simultaneously made zero for some certainsystem of ratios between thevariables. The function so found, which is sufficient and necessary tocondition thepossibility ofthecoexistence of theequality to zero of eachofthegiven functions, is theirresultant, and by analogy theymay be termeditscomponents. It followsthatifRbearesultant ofagivensystem of functions, any numerical mnltiple of any power of Ror ofany root of Rwhen (upon certainrelations beingsupposed to be instituted between thecoefficients of itscomponents) Rbreaksupintoequal factors, willalso bearesultant. Thisisjustwhat happens in system (B) when m=n=p=,;theresultant found by the methodinthetextis ofthedegree3m;thegeneralresultant ofthesystem ofthreeequations to which it belongs is of thedegree3m2;thefact being, thatthelatterresultant becomes aperfectmth power for theparticular values of thecoefficients which cause its components totaketheform ofthe functions in system(B). Suppose, however, thatwe have stillm=n=p,but,lessthanm, 6m-3,will express thedegreeoftheresultant of system (B);butthisis no longer in generalanaliquotpartof3m2,andconsequently theresultant of system (B) thatwe have found is no longer capable in generalofbeing aroot ofthegeneralresultant. Thetruthis,thatonthissupposition the generalresultant iszero;asitevidently should be, because thevalues ~=0,'!1=0satisfytheequations in system (B), exceptforthecaseofm=,;zz consequently theresultant furnished inthetext,although found by thesame process, is something ofadifferent naturefrom an ordinary resultant; it 40J theDialytic MethodofElimination. 263 expresses, notthatthesystemofequations (B) may be capableofcoexisting, butthattheymay becapable ofcoexisting for values of ~,'!Lotherthan0zz andO.'I'hia-iswhatIhaveelsewhere termedasub-resultant. Butthere isyetafurthercase, to which neitheroftheaboveconsiderations willapply. Thisis whenm, n,parenotequal,butp-£=O. Onthissupposition thedegreeoftheresultant of(B) becomes 2m+2n-p, which in generalwillnotbe a factor of mn+mp+np;and inthiscaseit will nolongerbetruethatthevalues ~=0,'!L=0 willsatisfythesystem(B),zz inasmuch asthelastequation thereincannotso be satisfied. Now, calling thegeneral resultant Randtheparticular resultant R',ifR'should breakupintofactors so asto become equalto(r')/lx(S')b...(t')",itmightbe thecasethatRshouldequal(r')".(s')/J...(t')",andtherewould be nothingin thisfactwhich would be inconsistent withthetheoryoftheresultant as abovesetforth;butsuppose thatR'isindecomposable intofactors,then it isevident thatwemusthaveR=R'.R",andconsequently thatthe existence of such a particular resultant asRIwillarguethenecessity of theexistence ofanother resultant R";inotherwords,theresultant so foundcannotbe in a strictsensethetrueandcomplete resultant forthe particular caseassumed, andyettheprocessemployed appearsto givethe complete resultant, oratleastitisdifficult to see how thewanting factor escapesdetection. Tomakethismattermore clear, takeaparticular and a verysimplecase,wherem=2,n=2,P=£=1,soasto formthesystemof equations ..dar+Bxy+Cy'+(D:c+Ey)Z=O} ..d'ar+B'xy+C'y'+(D':c+E'y)z=0 . le+my+nz =0 Byvirtueof mytheorem, thedegreeoftheresultant R'is 2(2+2+1) -3.1=7, buttheresultant Rofthesystem ..dar+B:cy+OJ!+(D:c+Ey)z+Fz'=O} ..d'a;I+B'xy+C'y2+(lY:c+E'y)z+F'zt=0 , la;+my+nz =0 whichbecomes identical withtheformer when F=0,F'=0is of(0) (D) 2x2+2x1+2x1, thatis, of 8dimensions. HenceitisevidentthatwhenF=0,F'=0,Rmust becomeRIxRI'. 264Exteneion» oftheDialytic MetlwdofElimination. [40 Itwill be found in fact",thatonthesupposition ofF=O,F'=O,R becomes equal to NxR:;and accordingly, besides theportionKoftheresultant of system (C), found by the methodinthetext,thereisanotherportion Nwhichhasdropped through jbutit may be asked, is Ntrulyarelevant factor?wereitnot so,thetheoryoftheresultant would be completely invalidated jbutintruthitis;forN=0 will make theequations in system (C), considered as aparticular caseof system (D), capableof co­ existing jthepeculiarity, whichatfirstsightprevents this from being obvious, consisting inthefactthatthevalues of ~,~whichsatisfythe, z z threeequations whenN=0becomeinfinite. Thus, finally, we have arrivedata clear and complete view oftherelation oftheparticular tothegeneralresultant. Thegeneralresultant may be zero, in which casetheparticular resultant issomething altogether different from anordinary resultant; ortheparticular resultant may be a root of thegeneralresultant, or it may be more generally theproductof powers of thesimple factors, which enterintothe composition ofthegeneralresultant jor lastly, it may be an incomplete resultant, the factorswantingto make it complete beingsuchaswhenequatedto zero,will enablethecomponents oftheresultant to coexist, butnot forotherthan infinitevalues of certainoftheratiosexisting between thevariables. Without forthepresentfurtherenlarging onthehitherto unexplored and highlyinteresting theory of Particular Resultants, I willcontentmyself withstatingonebeautiful andgeneraltheorem relating tothem;to wit, "ifF=0,G=0,&c.be a given system of equations withthecoefficients left general, and Rbetheresultant ofF, G,&c.,and if now thecoefficients in F, GbesotakenthatRcomes to containasa factor or be coincident with R'm,thenwillR'=0indicate that(whenthecoefficients are sotaken 8.8 abovesupposed) F=0,G=0,&c.will becapableof being satisfied, not, asin general, by one only, butbymdistinct systems of values of thevariables inF, G,&c.,subjectof course to thepossibility, in special cases,ofcertainof thesystems becoming multiple coincident systems." Ipassonnow·tothemorerecondite andinteresting theoryofthe resultant ofTernary Aggregative Functions, thatis tosay,functions of theform Fm(x,y,z)+Fm-.(x,y,z)t·+&c....+Fm-l.)(x,y,z) tl.), which will be seen to admitof some remarkable applications tothetheory of reciprocal polars. [*BeetheAuthor's remarks below,p,283.] 41. ON A REMARKABLE DISCOVERY INTHE THEORY OF CANONICAL FORMS AND OF HYPERDETERMINANTS. [Philo8ophical Magazine, II.(1851), pp. 391-410.] INarecentlyprintedcontinuation - of apaperwhichappeared inthe Cambridge andDublin.Mathematical Journal, Ipublished a complete solution of thefollowing problem. A homogeneous function of x,yof the degree 2n+1 being given, required torepresent itasthesum ofn+1 powers of linearfunctions of e,y.I shallpreparetheway for themore remarkable investigations which form the proper object of this paper, by givinganew and more simple solution ofthislineartransformation. Letthe given function be aoa;tn+l+(2n+1)~a;my+!(2n+1)(2n) ~:rfl"-l'!l+..•+a9n+lym+l, andsupposethatthisisidentical with (]Jlx+qly)'rtHI+(PtX+qty)'m+l+&c.+(Pn+lX+q"'+ly'!"+l. Theproblem isevidently possible and definite, therebeing2n+2 equations to be satisfied, and (2n+2)quantities PI,ql>&c. forsatisfying the same. Inorderto effectthesolution, let ql=PIAl> qt=Pt~, &c.=&c. [*p. 208above.] 266On a remarkable Discovery 'l,ntheTheoryof[41 we havethen ]>1""+1 Pl"+lAt PIIMIAt'+p;m+l +pr+J~ +p,2f&+I~'+...+P':.:/ =ao• + •.• +P:tJA,,+l=al> +...+p:tl;>"lnH=Ut, Pl2n+lAln+pl2nH~" +...+p:tl;>..nn+1 =an, PI2fl+IAln+l +p;mH'A.}IH +..•+p~~l;>..:t~=an+l. ]>I'lnHAt2n+1+pr+1"Al'&+1+...+P:-;/ ;>":+~I=!ltn+1' Eliminate PI'p,...pnHbetween the 1st, 2nd, 3rd ...(n+2)thequations, and it is easily seen thatweobtain a"'H-an~Al+an-I~AtAI...±aoAt~.••A,,+I=O. Again,eliminating in likemannerPl2n+lAt. p;IJIH~...p:;':;l;>""+l between the 2nd. 3rd ...(n+3)thequations. weobtain anH-u,,+l!;>"l +...+al;>"l~••.A,,+I=0; and proceeding in thesamewayuntilwe come to the combination ofthe (n+l)th...(2n+2)thequations, andwriting ~"AI =81• !At~ =81, AtAt..•~I=8n+l' we find an+l-a.,.81+Un-I81•••±Uo8n+1=0, Un+1-Un+181+Un8,...+Ut8n+1=0, Un+1-an+\l81+Un+l8••.•±Ut8n+l=0, Utn+1-Utn81+am-Is....+an8n+l=0·, Hence it is obvious that (a;+Aly)(a;+~Y)...(a;+;>""+IY) is aconstant multiple ofthedeterminant a;nH,_x"y,x"-Iy'...±y"H Utn+I'a.n.a.m-I"· Un •Theseequations intheirsimplified form arisefrom the ordinary resultofelimination. in thiscasecontaining as afaotortheproductor thedifJerenoos of thequantities ~I'>.w,..."-+.1. 41]Oanonical Forms andofHyperdeterminants. 267 Hence ~,At..•An+1are known, and consequently PI'P2...Pn+I,qloql···qn+1 areknown, by thesolutionof anequation ofthe(n+l)thdegree. Thussupposethegivenfunction to be F=a:r!+5b:ry+lOcx'yI+lOda:y+5ext+IOI!l =(PIX+qly)6+(P2X+q.y)'+(P.x+q.y)', weshallhave, by an easy inference fromwhathas preceded, (PIX+qlY)(PIX+q2Y)(PIX+q.y) equaltoanumerical multiple ofthedeterminant a;3-ary,wyI,-ytI. , d,c,i,aI e,d,c,b I f,e,d, c I Thesolution oftheproblem given by me in thepaperbeforealludedto presents itselfunderanapparently different andratherless simple form. Thus,inthecaseinquestion, we shall find according to thatsolution, (~x+~~(~x+~~(Ax+~~ equalto anumerical multiple ofthedeterminant aw+by, b»+cy.ex+dy• b»+cy,ex+dy,dw+eg ex+dy,dw+eg,ex+fy The two determinants, however, are in fact identical, as is easily verified, forthecoefficients of arand!Iaremanifestly alike;andthecoefficient of ary inthesecond form will bemade up of thethreedeterminants, a,b,d;, Ia,C, C b, b,c , b,c.eI'b,d, d e, c,d o,d,IIe,e, e d, d, e of which thelattertwo vanish, and thefirst isidentical withthecoefficient ofaryinthefirst solution. The same thingis obviously trueinregardofthe coefficients of xytinthetwo forms, anda like method may be appliedto showthatin allcasesthedeterminant abovegivenisidentical withthe determinant of my former paper, namely /loX+~y, ~x+asy,~x+asy anx+an+JY asx+a.y ~IX+anHY 268On aremarkable Di8covery intheTheoryof[41 Thus,then,we seethatforodd-degreed functions, thereduction totheir canonical form of the sum of (n+1) powers depends uponthesolutionof one singleequation of the(n+l)thdegree, and can neverbe effected in more thanone way. This new form of theresolving determinant affordsabeautiful criterion for a function of fE,yof the degree 2n+1 being composed of ninsteadof, asin general, (n+1)powers. In order thatthismay be the case,itis obvious thattwoconditions must be satisfied; butIpointedoutin mysupple­ mentalpaperon canonical forms, thatallthecoefficients of the resolving determinant mustvanish, which appearsto give fartoomanyconditions. Thus, suppose we have aaf+7b:If''!J+21clL"'y2+35tk'y3+35e.X'yf+21f:c"JI+7gfEyl+ky. The conditions of cstalecticism, thatis, ofitsbeingexpressible underthe form of the sum of three(instead of,asin general, four) seventhpowers, requires thatall the coefficients of the different powers of fEandymust vanish in the determinant yf,-'!Ix,y2:tfJ,-yg;l,~ a,s.c,d,e b,c,d,e,I c,d,e,f,9 d,e,f,g,h inotherwords, we musthave five determinants, a,b,c,d!a,c,d,ea,b,c,e,I'b,c,d,eii.d,e,Ib,e,d,I o,d,e,Iic,e,f,9c,d,e,9 d,f,Id,f,hd,f,h e,g; g, e, a,b,d,e,i,o,d,e b,c,e,Ic,d,e,f c,d,f,9d,e,f,9 d,e,g,he,f,g,h allseparately zero.Butby my homaloidal law ",allthesefiveequations amountonlyto(5-4)(5-3), thatis,to 2. I ruay notice here, thatatheorem substantially identical withthislaw, and anotherabsolutely identical with thetheorem of compound determinants given by me in this Magazine, and afterwards generalized inapaperalsopublishedtinthisMagazine. entitled [-p.160above.] [tp.241above.] 41]Oarwnical FormsandofHyper-determinants. 269 "OntheRelations between theMinorDeterminants ofLinearly Equivalent Quadratic Forms," have heen subsequently published asoriginalin arecent numberof M.Liouville's journal Thegeneral condition of mere singularity, asdistinguished fromcata­ lecticism, thatis, ofthefunction of thedegree2n+1,beingincapable of beingexpressed asthesum ofn+1 powers, is thattheresolving resultant shall have two equal roots; in otherwords,thatitsdeterminant shall be zero. MrCayleyhaspointedout to me a very elegantmode of identifying the two forms of theresolving resultant, which I have muchpleasure in sub­ joining. Takeastheexample afunction ofthefifth degree, we haveby themultiplication ofdeterminants, '!I,-'!ltc,yr,-x'1,0, 0,0 a,b,c,dtc,y,0,0 b, d,xc, e0,tc,y,0 o,d, e, 1 0, 0,e,y ?f,a, b, c 0,ax+by, bx+cy, ex+dy 0,b»+cy,ex+dy,dx+ey, 0,ex+dy,dx+ey, ex+ly whichdividing out each side of theequation by'1,immediately givesthe identityrequired, andthemethodis obviously general. Turn we now toconsider themode of reducing abiquadratic function of twoletterstoitscanonical form, videlicet (/x+gy)4+(h:c+lcy)4+6m(fx +gy)l(luc+ky)l. Let thegivenfunction be written a:r;4+4bry+6c:r:y+4dxy'+eyf. Let g=/J"l' k=h~,mphl=p.,~+~=81' ~~=SI' then we have 14+M+6p.=a, 4f4AI+4h4~+6p.(281)=4b, 6/4)"1'+6h4)...1+6p.(sl+2s.)=6c, 4f4~1+4h4>"'1+6p.(islsl)=4d, 14~4+h4~4+6f£8.1=e. 270Onaremarkable Discovery intheTheoryof[41 Letnow andweshallhaveEliminatingfandhbetween thefirst, second andthird;thesecond,third andfourth;andthethird,fourthandfifthequations successively, we obtain as~-be,+C-I-"(8~-2s/)=0. be;-C81+d-I-"(4s18~- 818)=0, C82-ds,+e-I-"(888' -2sl~8s)=0. (28}8-88s)Jl.=II, as,-be,+(c+II)=0, b82-(C-;)81+d=0, (c+II)8,-ds,+e=O. HenceIIwill be found from thecubicequation a,b,C+II 2b,2C-II, 2d=0, C+11,d,e a,b,C thatis, JJ'-II(ae-4bd+3c')+ b,c,d=0, c,d,e in which equation itwillnotfailtobenoticedthatthecoefficient of 1Tis zero, and theremaining coefficients arethetwowell-known hyperdeter­ minanta, or,asIpropose henceforth to callthem,thetwoInvariants of theform a:rr+4bry+6c:rys+4dxya+e!f; beitalsofurtherremarked that .11=8(l8l'-S2)1-", in which equation thecoefficient of 81-"istheDeterminant orInvariant of xl+slxy+S2yt., When IIisthusfound, 8},82,and1-",beinggivenbytheequations intermsofv, are known, and by thesolution ofaquadratic AI,~become known in terms of81J8a,and1.hintermsof~,~,Jl.,andtheproblem iscompletely deter­ mined. Themostsymmetrical mode of statingthismethodofsolution is tosuppose thegivenfunction thrownundertheform (fx+gy)4+(ftx+gly)4+6€(fx+gy)t<ftx+lhy)8. Thenwriting 41]Canonical FormsandofHyperdeterminants. 271 - II,thequantity to be found by thesolutionofthecubiclastgiven, becomes Ishallnowproceed toapplythesamemethod tothereduction ofthe function Uo~+Salx"y+28~~ya+56a,x6y'+70a,u:ey'+56aar!t +28asr!f+8a.,a;y+a,'!t, undertheform of (Pta;+qIY'f+(Paa;+qaY'f+(Paa;+qay)a+(p,a;+q,y)a +70e(Pta;+qly)a(Paa;+q2y)t(Paa;+qay)1(p,a;+q,y'f. Itwillbeconvenient tobegin,asinthelastcase,bytaking ql=PI).,., qa=p2~' qa=pa~, q,=p,X" epI'P2aplpl=m, and (II:+).,.y)(a;+~y)(a;+~y)(a;+X,y)=a;'+slry+S2rcy+sarr:y'+s,y'=U, weshallthenhave nine equations fordetermining the nine unknown quantities of thegeneralform PtaXI'+p2a~,+paa'A,'+p,ax,'+M,m=au where ~hasallvalues from 0to8inclusive, andwhere M,=70.1.2...L1.2 (8 - ,) 1.2b multiplied intothecoefficient of y':r;B-'inU'J. Takingthesenineequations inconsecutive fives,beginning with the first, second,third,fourth, fifth, and endingwiththefifth,sixth,seventh, eighth, ninth, we obtainthefiveequations following:- a"s,-~Sa+~Sa-UsSI+a,so-mNl=0, ~S,-~Sa+a,S2-a,sl+aaSo-mNa=0, a,s,-aaSa+a'S2-aaSl+asso-mNa=0, a,~-~~+a,~-a,~+a.,~-m~=~ a,s,-aaSa+aSs2-a.,SI+a,So-m.N;=0, where N,=Mos,-Mlsa+M'Js2-Masl+M" N,=MIS,-M2s,+M,s2-ill,sl+hfa, N,=lU2S,-Masa+M,s2-Masl+Ms, N,=~"tfas,-M,s,+Mas2-Mssl+jf7, Na=M,s,-M,sa+MsB,.-llf7s1+Ma• 272 On a remarkable Discovery intheTheoryof[41 72mI="J we shall have thefive following equations:-Developing nowU2,we have M.70M35M.. 51M5 5o=, 1=2Sl• 1=<>SI+2Sl,•=2s.+2SIS,,555M.=28.+2s1s.+S2"Me=2SIS.+2s,s.,Me=5Sts.+2sl, 35M7=2s.s.,Ms=70s.'. Hence N1=72s.-188 1s.+6st'J N.188913I I=IS.-2SlS.+2SlS1J Hence we have N1=72I, N2=72I~J Na=72I~, N.=72I~. Ne=72Is., where itwillbe observed thatIillthequadratic invariant ofU. Making now +as=0; "Jwhich 18found from the~S.-CLaS.+(a.-~)s,-a,Sl CLaS.-(a.+i)s.+a,S2-aSs1 (a.- ,,)s.+a,s.-U.SI-a,Sl 80thatthe problem reduces itselfto finding equation ofthefifthdegree:-+u. +a,=0, =OJ a"=0, 41JOanonical FormsandofHyperdete:rminants. 273 thenPJitwillbeobserved, being 72 timesthequadratic invariant of (~x+~0(~x+~~(~x+~~(~x+~~ thefunction beingsupposed to be thrownunderthe form of ~(PJ,x+qIY'!+70E(P 1X+qIY)"(p"x+q"y)"(Pax+QaY'f(P4 X+q4'!I)'. Itis obvious thatin theequation for finding IIJallthecoefficients being functions of theinvariable quantities PhqlJ&c'Jand EJmust be themselves invariants ofthegivenfunction; 80thatthedeterminant lastgivenwill presentunderonepointof view four out of thesixinvariants belonging toa function of theeighthdegree,andthesefour will be of the degrees 2J3J4J5respectively", I shall nowproceed togeneralize .this remarkable law,and todemonstrate theexistence and mode of finding 2nconsecutively-degreed independent invariants of any homogeneous function of thedegree4nJand ofn+1con­ secutively-even-degreed independent invariants of any homogeneous function ofthe degree 4n+2;aresult,whether we look to the fact ofsuch invariants existing, or to thesimplicity oftheformula for obtaining them,equally unexpected and important, andtending toclear up some of themost obscure. andatthesametimeinteresting points in thisgreattheoryofalgebraical transformations. In the first place,let me recall to my readersin thesimplest form what is meant by an invariantt ofahomogeneous function, say of two variables xand'!I.Ifthe coefficients of thefunctionl(xJy)be called aJb,c...l,and ifwhen for xweputlai+my,and for '!I.nx+PYJwherelp-mn=1,the coefficients of the corresponding termsbecomea'Jb'•..l'jand if Lta,b...l)=I(a'Jb'...l')J thenIis defined to beaninvariantoff Let nowI(x,y)beahomogeneous function in XJyofthe2£thdegree, and write (dd)'Edx+'7dy(xJy)+X('7x-Ey)'=PJ (E::C+'7~)'/(lx+mYJ nx+Py)+X('7 x-Ey)'=Pl J whereEand'7areindependent ofXJ'!IJandlp-mn=1. Let x'=le+mYJ y=nx+ PYJ d d dx'd dy' d do;'ddydEdx+'7dy=Edxda!+E-ilxdy'+'7dydx,+'7dydy" • Thereasoning inthisplllagraph seems of doubtful conelueivenesa. nmay beaccepted, however, &8afactofobservation confirmed and generalized bythesubsequent theoremJthatthe coeftlcienta areinvllliants. tDlim,HyperdeterminantJConstant derivative. ~ 18 274Onaremarkable Discovery intheTheoryoj[4:1 and if we now write we findlE+m7]=f, nE+P"1=T/, t:ddt:,d,d ~rk+'I}dy=~ da/+'I}dy" •.Again, from the equations between a;',y',e,y.wefind ,, Px-myx= =p:c'-my'pl-mn ' therefore Hence Again, Hencel' ,y-na;l',Y-yn,x'pl-mn- - , 'l}X-Ey=(p'l}+nE)x'-(m'l}+lE)y'='I}'X'-E'y'. P'=(E'::C.+'I}'d~Jj(X', y')+A('I}'X'-E'y')'. d d d dE=lJt+nd7]" d d d d7]=mdf+Pd,," (d )' (d)' (d)'-1d (d)'d7]P'=m:dE'P'+£m.-lpdE'.d7]'P+&c.+p'd7]'PI. ButPIbeing of £dimensions in rand 'I}',and also in xandy,each oftheequations abovewrittenwill be of £dimensions in xandy,and of no dimensions in E','I}';in fact, the successive termsof theright-hand members oftheabove£+Iequations will bemultiples ofthe(£+1)quantities (x')'.(x'),-1y',(x')'-2y"...(y')'. Consequently alinearresultant may betakenof (d )', (d)'-1dId)'dEP,dEd7]PI...\d"PI, treatingx\x',-1y'...y"asindependent, andasquantities tobeeliminated; andthis, according toawell-known principle ofelimination, will prove 41]Carwnical Forms andofHyperdeterminants. 275 thelinearresultant oftheforegoing equations to beequaltothelinear resultant of (d )' (d)'-1d(d)' rdE'P,dE'dr/P'...dr]'P, multiplied bythedeterminant l', m',ft' p'• This last written determinant may be shown from themethod of ,(.+1) ililformation to be equal to (lp-mn)~2-,thatis, tounity,because lp-mn=1.Again, since :x"=l':r+tl'-Im:r-Iy +&c,+m''!I, :i'-Iy'=l'-I71:r+(l'-In+(£-1)l'--'Lmn):r-Iy+&c.+m'-Ipy', theresultant of(:E)'?' ...(~)'P" obtained bytreating:r, :r-Iy'" y' asthe eliminables, will be equal to theresultant ofthesame functions when IE",IE'·-Iy'...y"aretakenastheeliminables'" multiplied byapower of the determinant l',...,m' l'-In,...,m'-Ip n',...,r whichdeterminant, likethelast, is unity. Thus, then,we have succeeded in showing thattheresultant obtained byeliminating :r,:r-Iy...y' between (d )'(d)'-1d(d)',dEP,dEd1]P...d1]P is equal totheresultant obtained byeliminating (:x')',:x',-I'!!'...y"between • Forthestatement of the general principle of the change of the variables of elimination, _mlpaperintheMarchNumber, 1861,of theCamb.anaDub. Math. Jour. [p,186above]. 18-2 276Ona remarkable Discovery intheThwryof[41 •or, which is evidently thesamething,theresultant obtained byeliminating af,af-1y•••'!tbetween thatistosay,thislastresultant remainsabsolutely unaltered in valuewhen fore,ywewriterespectively l.x+my, n.x+py, provided thatlp-mn=1. Hence by definition thisresultant is aninvariant f(a:,y),andAbeing arbitrary, all theseparate coefficients of thepowers of Ainthisresultant mustalso beinvariants. I proceed to express thisresultant; in terms of Aand the coefficients of (a:,y).Let1D'=1.2.3...&and and1(d)'(d)' --p=-f+i\a:' 111'd"1 dy f(a:,y)=ao:z:l'+2&~:z:I'-ly+i(2,)(2,-1) a..:z:I'~yl+&c.+all.y"'. We find, writing UAforA,where U=2,(2,-1)...(,+1), 1- E1=aoaf+~af-ly+~£(£-1)rLsaf-1yl... U 1- E.+ 1=a,a:'+&C.+i\a:'; U 41]Canonical FormsandofHyperdeterminants. Z77 accordingly, by eliminating .trf,£trf-Iy,1-£(£-l)trf-'.ly' .•.'!J', weobtainastherequired resultant ", a,±A,a'_I, a,-2' ao A a,+I' a,+-,a'-lJ ~£ A a,+I' a'+I' a,±1-£(£-1)' as• a--I,............ ,...a,+A Inasmuch asallthecoefficients of Ainthisexpression areinvariants of f(x,y),andthereare noinvariants of the first order, itis clearthatthe coefficientof A'mustbe always zero, which is easily verified:· . Again, if £is odd, the determinant remains unaltered if we write - Afor Ajhence when f(x,y)is of the degree 4e+2, allthecoefficientsof theodd powers of Adisappear. Thus, then, our theorematoncedemonstrates thata function of e,yof the degree 4ehas2einvariants of all degrees from 2up to2e+1 inclusive, and thata function of e,yof the degree 4e+2 hasf!+1invariants whose degrees correspond to all theevennumbers in the seriesfrom 2 to 2e+2. Butin orderthatthe proposition, asabovestated,may beunderstood in its fullimportand value, it is necessary to show thattheseinvariants are independent ofoneanother, which is usuallya mosttroublesome and difficult task ininquiries ofthisdescription, butwhich the peculiar form of our granddeterminant enables us to accomplish with extraordinary facility. In ordertomake the spiritofthedemonstration moreapparent, takethecase ofafunction of thetwelfthdegree, whose coefficients, divided by the 12.11successive binomial numbers 1, 12, -.-2-'&c.may be called a,b,0,d,e,f,g,h,i,j,k, l,m. •XrCayleyhasmadethevaluable observation, that"(given by equatingtozerotheabove determinant) maybedefinedby means of theequation (d ddd)'dzd-;'-dY~{/(2',y)X~(£,'1)}="~(x,Yl• •beingitselfacertainrationalintegralform of a funotion of the lth degree, theratio ofwhose coe1Iioients wouldbegiven by virtueof the above equations as funotions of "and thecoeffioients ofI(:r.,yl. 278Ona remarkable Discovery intheTheoryof[41 Ourgranddeterminant thentakestheform g-t.X,1.e.d,c,i,ct. h,Xj, d, b g-6' e, 0, h,Xf,d, 'l,g+15'e, 0 h,Xd J,'l, g-20'f,e, le,j,h,Xf, 'l,g~15'.e I,le,J,t;i;g-~,I m,t,le,),'l,h,g+X Hereitwillbeobserved that aandmappearonly1time. band1 2times. c andle dandj eandi landh g Letnowthecoefficientsbe called3 4 5 6 7 n;HIlH"H61u;H7, H,andH,manifestly areindependent. Again, if possible, let H,=pH,',thenaandmwouldappeartwice inH" contrary totherule. HenceH,isindependent ofH"s; For asimilarreasonH.cannot depend on H2JHa• Again, if possible, let H.=pH,a+qH,H,+rHa'll, Ha'willcontainbBl·,which by the rule cannotappearinH'IIH,or inH.'II. Hencep=O. AlsoH,will contain b'lll'x the coefficient of xain 41]Oarwnical FormsandojHyperdeterminants. 279 whichisnotzero. And H.alsocontainsbl;henceHsH4willcontain bill. ButHIwillevidently notcontain b&orll,orbllorW,nor canH8contain b&lJ; henceq=O.Finally, HIswillcontaina8and~,butH8can only containas totheselettersthecombination &k';hencer=O. Consequently H8does not depend on H.,H4,H..As regards H"H., HhHI)H8notvanishing, thismay be made atonceapparent bymaking allthelettersbut9vanish; theH'sthenbecome identical withthe coefficients of (g+X)'(g-~)'(g+~r(g-:0), none of which are zero exceptthatofX8.The same or a similardemonstra­ tionmay beextended toH7and easily generalized; hence,then,this most unexpected andsurprising law is fully made out ", Toreturntothesubjectof canonical forms, I have not found themethod 80signallysuccessful in its application tothe4thand8thdegrees, conduct to thesolution of otherdegrees, such asthe 6th, 12th, or 16th, of all of which I have made trial;possiblyanothercanonical form mustbesubstituted to meetthe exigency of these casest;anditmay beremarked in general, that if we have a function of the (2n)thdegree,thecanonical form assumed maybetaken, I(PIx+qdF+V; whereV,in lieu of being the squaredproductof (PIX+qIY),(P.x+qsY),...,(Pnx+qnY), •Thisdemonstration, however, does notextendto showthatthe coefficients of the powers of "maynotpossibly bedependents, thatis,explicitfunctions of oneanothercombined withother invuianu notincluded e.mongtheirnumber, or oftheselatteralone.Forexample, in thecase of the12thdegree, we know by Mr Ca)'ley's lawthattheremustbe twoinvariants ofthe 4thorder.Ourdeterminant givesonlyone ofthese.CalltheotheroneK4;by theabove reasoning itisnotdisproved butthatwemayhave H8=pHs'+qH,H4+rH3s+sH.K4• I believe, however, thattheH'smay bedemonstrated without muchdifficulty tobeprimitive orfnndamental invariants. Thelawof Mr Cayley here adverted toadmitsof being statedinthe following terms:-Thenumber ofindependent invariants of the4thorderbelonging toa function of:1:,yof thenthdegreeisequaltothenumber ofsolutions inintegers (not less than zero) of theequation 2z+3y=n-8. Vi/khismemorable paper(inwhichseveralnumerical errorsoccuragainstwhich the readershouldbecautioned) ..OnLinearTransformations," vol.I. Cambridge andDublinMathematical Journal, newseries.Thereis nogreatdifficulty inshowing, byaidofthedoctrine ofsymmetrical functions, thattherecan never be morethanonequadratic or onecubicinvariant, andinwhatcasesthereisone or the other,oreach,toanygivenfunction oftwovariables. Thegenerallaw, however, for the number ofinvariants of anyorderother than2,S,4remains to be made out,andis agreatdesideratum in thetheoryoflineartral1lJ­ formations. tSeethePostscript [po283]foraverifioation ofthisconjecture. 280Ona remarkable Discovery in theTheoryof[41 maybeanyhyperdeterminant, or(asIshallinfuturecallsuchfunctions) covariant ofthisproduct, understanding P(e,y)tobeacovariant of f(x,y)whenP(lx+my,n:c+py)standsinprecisely thesamerelation to f(lx+my,na:+py)asP(x,y)tof(x,y),provided onlythatlp-mn=l. Fortherelation anddistinction between covariants andcontravariants, see ashortarticleofmine-intheCambridge andDublinMathematical Journal forthismonth. Inendeavouring toapplythemethodofthetexttothe SexticFunction axe+6b:r!y+15cxCyl+20dWy+15exY+6fr£!f+gy', thrownundertheform where u=(PIX+qlY)(Pax+qlY)(P,X+q,y)=80W+81:rf'y+8aXYS+8,y, Iobtainthefollowing equations: as,-bSI+CB1-dBo=E(162s 01s1-548081Ba+12s11) , bSI-CBI+dBl-880=E(548os ls1+6SIIS I-36sosll) , CBI-dBa+881-fs,=E(-548osls,-6SISI1+36s,SII) , dB,-88.+f81-g80=E(-162soBaI+54818IS,+12811) . Intheseequations, if wecallthequantities multiplied byErespectively L, M, N, P, weshallfind 1 1 81L-382M-381N+80P=0, and s,L-s IM- 81N+80P =I j whereIdenotesthedeterminant, or,asIshallinfuturecall such function (inordertoavoidtheobscurity andconfusion arisingfromemploying the sameword in two different senses),theDieoriminent'[', which is thebiquadratic (andof course sole) invariant ofthecubicfunction soxl+81:rf'y+82:XY+S,y'. Thereduction ofthefunction ofthefourthdegreetoitscanonical form may be effected very easilybymeansoftheproperties oftheinvariants of [*p.200above.] t ..Dieeriminant," becauseisa1fordsthediBcrimrn orteetforallClllnaining whesher ornot equalfactorsenl;erintoafunetion ofswovariables, or more generally ofiliee:DBl;enoe oroilier­ wiseofmultiple pointsin the locus represented orabaracserized byanyalgebraioal funation, the mOBSobviousandfirssobserved species of singularity insuchfunation or locus, Progreea intheseresearches iBimpossible wishoutilieaid ofclearBlI:preBBion; andshefirstcondition ofa goodnomenolature iBshatdifferens iliingBBhall be calledbydifferent names. Theinnovations inmailiematioal language hereandelsewhere (noswithout highBanction) introduoed bythe auilior,have been never adopted exceptunderactualexperience oftheembarraument arising from the wantofthem,and will requirenovindication tothosewhohavereachedthatpains whereilieneceBBisy of somesuchadditioDB becomes felt. 41]Oanonical FormsandofHyperdeterminants. 281 the canonical form, asI have shown in theCamhridge andDublinMathe­ maticalJournal. Accordingly I have endeavoured toascertain whether thereduction ofthesixthdegreemightnotbe effected by a similar method. .. ~ial ing .ons ;ing the =0H,=6m.s,+45m's,+216m'sl+891m', H,=4sa'+120sts,m -{6848a'+432s1s,}m' +(13.27. 64s,-64.8la18t)m'+8.81.169s,m' +7.128.729s1ml+16.729. 239m'.Ifwestartwiththeforma:r;8+by'+ez'+90m.x'y'z', where :r;+y+z=O, whichisonlyanothermode of representing thecanonical form previously given,weshallfindthatthereare fourindependent invariants, ofthesecond, fourth,sixthandtenthdegrees. CallingtheseH" H,. H" H10,andwriting It,St,8afora+b+o,ab+ac+be, abeitwill be found, afterperforming someextremely elaborate computations, that Ht=S,-270m', HI,is tooenormously long toattempt tocompute; butwe caneasi15 proveits independent existence bymakingm=0,in which casethe(deter­ minant, or, to use thenewtermproposed, the)discriminant ofa:r;8+by'+ez' becomestheproductofthetwenty-five forms of theexpression (ab)l+(ac)t.11+(be)l.Ih. Nowingeneralthevalueof such a productforaf+,et.l1+ryl,11is obviously of the form (a+fJ+ry)&+afJry{f(a+fJ+ry)'+9(afJ+ary+fJry)}j for when a=0 or/3=0 orry=0,theproduct mustbecome respectively (fJ+ry)&,(ry+a)&and(a+fJ}I.Moreover, withoutcaringtocalculatef, gt,itis enough for ourpresentpurposetosatisfyourselves that9cannotbe zero,as then the productwould have a factor (a+fJ+ry)'.Hence,then,onputting audIntheII800Ildcase*Bachaproduct inthelanguage ofthemostmodemcontinental analytlis is, I believe, termedaNorm.Hwesuppose thegeneralfanction ofz,yof the4thdegreethrownunderthe formAv'+Bu'+Cw4, wheret.I+tI+to=O, andthegeneralfunction ofe,y,IIofthe8rddegree thrownundertheformAu'+Bt7'+Cw'+DII', whereu+tI+to+8=O, thetheoryofnormswill alfordaninstantaneous and, so to speak, intuitive demonstration oftherespective related theor8lll8, andthediscriminant (alittrdeterminant) ofeachsuchfunction is decomposable into &heII1lDlof asquareaada cube. Eachoftheseforms is indeterminate, ineithercasethere beingbuttworelations fixedbetwesn thecoefficients A,B,C;A,B,C,D;andwe mayeasily establish thefollowing singular speciesofalgebraical pcristn,Inthefirstcase (ABC)':(AB+AC+BC)I, (ABCD)':(2:AIB'et -2ABCD2:AB)1 arei_riab'- ratiol. tf=-626,9,=8126. 282 Ona remarkable DiJJcovery intheTheoryoj[41 D E 11 Go D E 11B o D Eundertheform of Ax'+6Ba!y+150x4y2+20Da;8y'+15Ea;8y4+6Fxy'+Gy', andtakingthedeterminant A B o Da=be,fJ=ac,'Y=ab,we seethatthediscriminant, when m is 0, will beof theform 821+f8l8l+g8228 1• Butwhenmis 0,H4vanishes, and thereis noterm81or83inHi'Hence evidently thediscriminant H10justfoundcannotbedependent onHi>H4, orHo;nor is it possible to make H10+pHil+qHsSHo, thatis, (p+1)S21+fS22sl+gsbl aperfectsquareonaccountofgnotvanishing; sothereis noHIuponwhich H10can depend. Hence,admitting, asthereseems every reasonto do,that thenumberofinvariants of afunction ofe,yofthedegreemism-2. wefindthatthefourinvariants in the case of thefirstdegreearerespectively ofthesecond, fourth, sixth,andtenthdimensions, a determination in itself, as asteptothecompletion ofthetheoryofinvariants, of nominor impOrtance. Butitseems hopeless by means of these forms to arriveatthedesired cs-.J.onical reduction. The forms, however, of Hi'H4,H,areveryremarkable r..Bnotrisingabovethefirst, first and second degreesrespectively inSl.~,Sa­ AlsoH4vanishes when m=0 andH.has been obtained byputting ax'+by'+ez8+90m:r;iy2zi Consequently ingeneralthe'-~'1ishing oftheabove-written determinant will expressthecondition thata function of the sixthdegreemay bedecomposable intothreesixthpowers. Thisalsoiatruemoregenerally. IfF(x,y)be afunction of 2i dimensions, thevanishing oftheresultant inrespectto xi,X'-ly...yi(takendialytically) of (d)i(d)'-1d(d)id:r:F,ikdyF...dyF willindicate thatFadmitsof being decomposed intoipowers of linear functions of e,y•. In consequence of the greaterinterest, at least to theauthor,ofthe preceding investigations, I have delayed theinsertion ofthepromised continuation of mypaperonextensions ofthedialyticmethod, which will •Suchafunction 80decomposable may betermedmeio-catalectic. Meio·catalecticism for even-degreed functions is theanalogue ofsingularity forodd-degreed fllDctions. 41]Canonical FormsandofHyperdeterrninants. 283 appearinasubsequent Number. Itakethisopportunity ofcorrecting a trifling slip of the pen which occurs towards theend- ofthepaperalludedto. Thevaluesof~and'!l.become zero, and notinfinite, whenN=0;andthez z antepenultimate paragraph should end with thewords"anincomplete resultant." The·theorem also, in thelastparagraph butone,shouldbe statedmoredistinctly assubjecttoanimportant exception asfollows. Whenever theresultant ofasystemofequations F=0,G=0,&c. contains afactorR"",thiswillindicate that,onmakingR'=0,thegiven system of equations willadmitof being satisfied by malgebraically distinct systems of values of thevariables, exceptin those caseswherethereisa singularity intheforms of F, G,&c.,takeneitherseparately, or inpartial combination with one another. Anexample will serve to make the meaning of theexception apparent. LetF, G, H denotethreequadratic equations inxand'!I,80thatF=0,G=0,H=0 may be conceived asrepresenting three conic sections. LetRbetheresultant ofF, G, H, and snppose the relations of the coefficients in F, G, H to be such thatR=R'tjthenR'=° willimplytheexistence ofone or theotherofthethreefollowing conditions: namely,eitherthatthethreeconies have achord in common, which is the mostgeneralinference; or, which islessgeneral, thattwo oftheconies touch one another; or,which is themost special caseof all,thatone ofthe COlliesisa pair of rightlines. So,again,if we have two equations ine,andtheirresultant containsFt, this may ariseeitherfrom one of thefunctions containing asquarefactor, or fromtheirbeingsusceptible, oninstituting onefurthercondition, namely ofF=0, ofhavingaquadratic factor in common between them. P.S. The conjecture made in thepreceding pageshasbeen since con­ firmed by thediscovery of amodification in thecanonical form applicable to functions of the sixthdegree, which simplifies thetheoryinaremarkable manner. Assume f(x,y),a.function of thesixthdegree,asequal to au'+lnJ8+ew'±muvw(u-v)(v-w)(w-u), whereu,v,ui,linearfunctions of xandy.satisfytheequation u+v+w=O; then will theproductofuvwbecapableofbeingdetermined by means of the solution of aquadratic equation, ofthesquareroot of whose roots the coefficients ofuvwwill be known linearfunctions. Thusby an affected quadratic, apurequadratic, and a cubic equation, thevalues of u, v,w may becompletely ascertained. The discussion of thistheory, and of a general inverse method for assigning thetrue(in the sense of themost manageable) Canonical Formfor functions of any even degree, will form the subject of asubsequent communication. (*p.264above.] 42. ONTHEPRINCIPLES OF THE CALCULUS OF FORMS. [Cambridge and DublinMathe1TUttical Journal. VIL(1852).pp.52-97.] PART1.GENERATION OFFORMS·, SECTIONI.OnSimpleConcomitance. THEprimary objectoftheCalculus ofFormsisthedetermination of theproperties ofRational Integral Homogeneous Functions orsystemsof functions: this is effected by meansoftransformation; butto effect such transformation experience has shown thatforms or form-systems mustbe contemplated notmerelyastheyare inthemselves, butwithreference to theensemble of forms capable ofbeingderived fromthem,andwhich constitute asitwere an unseenatmosphere aroundthem.Thefirstpartof thisessaywilltherefore bedevoted tothetheoryoftheexternal relations of forms or form-systems; thesecondparttotheanalysisofforms:thatisto say,thefirstpartwilltreatoftheGeneration andaffinities, and thesecond partoftheReduction andequivalences offorms, Initsmostcrudeandabsolute, or, so to speak,archetypal condition a Rational Integral Homogeneous Function may be regarded as alinear function ofseveraldistinct andperfectly independent classes of variables. •Itmay be well at the outsetto give notice to my readersof theexactmeaning tobe attached to the following terms: 1. The Iinear- transformations aresupposed tobealwaystakensuchthatthemodulus. thatis,thedeterminant ofthecoeffioients of transformation, isunity;or, as it may bephrased. thetransformations areuni-modular. 2. The word Determinant isrestricted in allcasestosignifythealternate function formedin theusualmannerfromagroup of quantities arranged insquareorder. 8.ThewordDi.criminant (typified by theprefix.symbol 0)is used to denotethedeter. minant(usuallybut most perplexingly so called) of ahomogeneous function ofvariables. 4.Theresultant of two or more homogeneous functions of asmauyvariables istheleft. handside of the finalequation (in itscomplete formandfree from extraneous factors) which results fromeliminating thevariables between theequations obtained bymakingeachof thefunotions zero. 42J OnthePrinciples ojtheOalculu«ojForma. 285 &c.-a"b"'Y=0 01 a'b'0 oThefirststeptowards thelimitation ofthisverygeneralbutnecessary conception consists in imagining thetotalnumber of classes to become segregated intogroups, and certaincorrespondences toobtainbetween thevariables of aclassin anygroupwith some thevariables in eachother classofthesame group. The investigations inthisandthesubsequent sectionwill be confined exclusively to thetheoryof functions where the severalclassesof variables, if more thanone, all belong to a single group, 80 thatthevariables in oneclasshave each theirrespective correspondents intheremaining classes. Such a groupmayagainbe conceived to become subdivided intosets each of the same number of variables, and thecorre­ sponding variables inthedifferent sets to become absolutely identical. This leads to theconception ofahomogeneous function ofrelatedclasses of variables of various degrees of exponency inrespecttotheseveral classes. Therelation ofthedifferent classes, if containing thesamenumber of variables (in which casetherelationmay betermedSimple)will beunder­ stood to be defined by theirbeingsimultaneously subject tosimilaror contrary operations oflinearsubstitution j80that,for example, if e,y,Zj E,71.?;are two such classes. when e,y.Zarereplaced byax+by+CZ. a':c+b'y+c'z.a"x+b"y+c"z,respectively, E,,,/,?;will be,according tothe 6p6Ciesof therelation, subjectto beatthesame time replaced eitherby ~+bl1+c~,a'E+b',,/+c'~,a"f+b"7]+c"~,orotherwise byaE+f:J7]+'Y~, fl.'E+f:J'7]+'Y'~,a."E+f:J"7]+'Y"~,where a.=1 0 0 f:J=0 1 0 ob'c' a'0c' ob"c" a"0c" &c. &c. Ontheformersupposition therelatedclasses:c,y,s,f,7],~will be said to becogredient, and onthelattersupposition contragredient t.Ifnow we have one or more functions of classes of variables sorelatedt.suchfunction or system of functions may have associated withit aconcomitant, also made up ofdistinctbutrelatedclasses of variables, such classes beingcapable ofbeingeithergreateror fewer in numberthanthe classes of thegiven function or system of functions. Intheprimitive function or system, as also in theconcomitant, the relatedclassesmay be all of thesame species, or some of one and theothers ofthecontrary species. Evenif welimitourselves to theconception of a • Beemypaperin theprevious numberorthisJournal[po199 above.] tThegermorthenotionorcontragredience willberoundintheimmortal.drithmetic orthe greatandrenerable Gau88. :Therelation herespokenorwill beobserved tobeoradynamical character, notrelerring to the&ystema &8theyareinthemselves, buttothemovements towhichtheyaresimultaneously lobject. 286 OnthePrinciples oftheCalculusofForms. [42 whenprimitive function orsystemoffunctions withonly one classofvariables, its concomitant maybe composed of variousclassesofvariables, inrespectto some of which itwill becovariant with,andinrespecttotheotherscontra­ variantto,theprimitive function oreystem", Thisis animmense andmost important extension oftheconception (Ifaconcomitant givenin mypreceding paperinthisJournal, andwillbeshown to have theeffect of reducing the wholeexisting theoryundersubjection tocertainsimpleabstract and universal lawsofoperation. Therelation ofconcomitance ispurelyof form. Abeingawvenform, Bisitsconcomitant, whenA'beingderived fromAbysimultaneous substi­ tutionsimpressed upontheclassofvariables oruponeach of theclasses (iftherebe more thanone) inA,andB'fromBbycorresponding (coincident orcontrary) substitutions impressed upontheclass orclassesofvariables in B,B'iscapable ofbeingderived fromA'afterthesamelaw asBfromA; or,'asitmay beotherwise expressed, "functions areconcomitant whentheir correlated linear derivatives arehomogeneous inpointof formt." Thisdefinition impliesthatoneatleastoftheformsmustbethemost generalpossible of itskind:inasecondary butveryimportant sense, however, functions obtained byimpressing particular values or relations uponthe quantities entering intotheprimitive anditsassociate form,willstillbe calledconcomitant. Thus:c'-yswillbetermedaconcomitant to:c'+ys, notthatwe can affirm that(ax+by)'-(e:c+dy'f: thatis(aJ-eI):c'+3(a2b-e2d)wy+3(ab2-cd'):cyl+(bJ-dJ)'!I, treatedasafunction of'eandy,can bederivedfrom(ax+by)'+(c:c+dy)', thatis (al+o'):c'+3(alb+&d)wy+3(ab'+cd2):cy'+(bS+dJ)y', whenad-be=1 bythesamelawas(w-ys)from(:c'+ys),fortheelements forforming suchcomparison arewanting, butbecause :c'+ysand:c'-ysare' thecorrespondent particular valuesrespectively assumed by ax'+3bwy+3c:cys+dys, anditsconcomitant (adS+2&-3bcd):c'-(6bJd-3e1b-3acd)wy +(6ael-3eb'-3cba):cyJ -(a2d+2bJ-3bca)ys, a=1,b=O, e=O, d=1. Withtheaid ofthisextended signification ofthetermconcomitant (whether itbeacovariant orcontravariant) we can in all cases speak(asotherwise we ingeneral couldnot)oftheconcomitant of aconcomitant. Therelation • And of course the concomitant maybeaninvariant toitsoriginant in respect of one or more systems of variables entering intothe former. tOr, more generally, it may besaidthatconcomitance oonsistsin the persistence ofmorpho­ logicalaffinity. 42JOnthePrinciples ojthe Oalculus ojForms. 287 between systemsofvariables has been statedto beSimple(whether theybe cogredient orcontragredient) wheneachvariable in onesystemcorresponds with some one in eachother.Compound relationarisesasfollows:-Suppose :toy;E,'TJtwoindependent systems of twovariables each,andthatthe systemof fourvariables 'u,v,w,tissubjecttolinearvariations imitating, intheway ofcogredience orcontragredience, those to which :xE,X"TJ,yE,Y'TJ aresubject; thenu, v,w,tmay be said to be cogredient orcontragredient tothecontinued systemse,y;E.'TJ.Ife,y;E.'TJbethemselves cogredient, thena.systemof onlythreevariables u,V,w,may becogredient orcontra­ gredient inrespectto:xE,X"TJ+yE,Y'TJ,andif:x.y;E,'TJbecoincident, u, v,.W may be similarly relatedtow,lEY,y'.Theillustration may easily be generalized. anditwill be seen inthesequelthatitsconception ofcompound­ relation between systems ofadiffering number ofvariables willgreatly extendthepowerandapplication ofthemethods aboutto be developed. Without havingrecourse to aformaldefinition, itis obvious thatthenotion of aconcomitant conveyed in my former paperinthisJournal lendsitself without difficulty to·themostgeneralsupposition which can be made of functions between which any number ofsystems ofrelatedvariables are distributed, whatever suchrelation be,whether simple or compound, and whetherofcogredience or ofcontragredience. Theproposition statedin my lastpaperrelativeto aconcomitant oftheconcomitant of a function being aconcomitant oftheoriginalstillappliestoconcomitants inthewider sense in which we now understand thatterm.andthespecies of each systemof variables in thesecondconcomitant withrespecttothespecies or either species(iftherebesystemsofbothkindsintheprimitive) willbedetermined upon the generalprinciple whichdetermines theeffect of concurrence and contrariety beingmade to operateeach upon itselfor one in eitherorder upon the other. Thehighestlaw and themost powerful in its applications which I have yet discovered in the theoryofconcomitants may beexpressed by affirming thatwhen several relatedclassesofvariables arepresentin anyconcomitant, a newconcomitant, derivedfromtheformerbytreatingoneoranynumberof thesecla8S68asindependent oftheremaining classes,willstillbeaconcomitant of theprimitive. I shallquotethishereafter astheLaw of Succession. This law, to which I have been led upinductively, requires anextended examination and a rigorous proof.Itisthekeystone ofthesubject,and any one who shouldsuppose thatitis aself-evident proposition (asfromthe simplicity of theenunciation itmightbesupposed tobe) will commitno slight error. If4>(:x,y..•z)beany homogeneous form of function ofe,y,..•Z,every homogeneoussum in theexpansion byTaylor'stheorem of 4>(u+u', v+Vi...W+Wi), 288 OnthePrinciples ojtheOalculusojForms. [42 whichinfact, on makingu'=:x,v'=y ...w'=e,becomes identical (toa numerical factorpres)with(u::a:+v~+w;z)'</>'is what I have elsewhere termedanEmanant, and by a partialmethodI haddemonstrated thatevery invariant of such an emanant inrespecttou,tJ...w,in which e,y...zare treatedasconstants, oroiceversa,would give a covariant of</>.The reason ofthisis nowapparent. Foritmay easily be shown- thateveryemanant is in fact itselfacovariant ofthefunction towhichitbelongs with respect toeachoftherelatedclasses of variables which enterinto it, or is as it may betermedadouble covariant. The law of Succession shows therefore that aconcomitant to anemanant from which one of theclasses has disappeared will be a covariant of theprimitive inrespecttotheremaining class. Inapplying thelaw ofSuccession,great use can be made of a function of two classes of letterswhich may be termedaUniversal MixedConcomitant; thisis:xE+YTJ+...+ z~,which has the property ofremaining unaltered when anylinearsubstitution (for which themodulus is unity)is impressed upon e,Y...s,and thecontrary one upon E,"l...~t.. Ifj(:x,y) be any function of :X,y,of the degree m,j+X(:xE+y'1)'" will • Todemonstrate thisit is only necessary toobservethatifv,e, .,.to,v',v',...to'be cogredient withthemselves andwithx,1/,...%, </>(v+}.u', lI+}.V',...te+}.te') willevidently beaconcomitant of</>(x,y....%);and.}.beingarbitrary, thecoetJicients ofthe different powersof},mustbeseparately concomitants of</>(x.y, ...%),butthesecoefficients are theemanants of</>.Q.LD. tThus.if x=ar+by'+cz', ~=(gn- hm.)C+(hl-fn).,'+(fm-gl)t'. lI=fx'+gy'+1&%',.,=(-nb+mc) t+(-Ie+na).,'+(-ma+lb)t', %=lz'+mll' +71%',t=(bh-cg)t+(cf-ah).,' +(ag-bf)r. then ~+!I.,+%t=(; ;~-)X(r't+!I''''+Z't') l7"n =x't+11'.,'+6't'. WhentheeoetJieients oftransformation correspond tothedirection-cosines between one system. ofrectangular axesandanother, thereciprocal systemisidentiea1 with the directsystem; 80 that (1;,1/,Zi~,."t.onthisparticular supposition, mayberegarded indifferently &8contragredient orascogredient; aooordinglytbey maybemadeidentiea1, andthenr+r+%1 remains invariable. whichis thewell·known characteristic oforthogonal transformation. Itmaybeobserved here thatthereenstsaspecialtheoryofconcomitance limitedtosuchspeciesoflineartransform­ ations,whichmaybetermedConditional Concomitance, andIhavefoundinseveraloaseathat theinvariants ofconditional concomitants tumoutto beabsolute invariants of theprimitive. Much more important is theremarkthatthereexistsatheoryofuniveraal concomitants for anindefinite number instead ofmerelytwosystems ofvariables, asused in thetext.In the sequel it will be seen thattheapplication ofthisuniversal concomitant (like the touchof anenchanter's wand) serves totransmute covariants intocontravariants, andbackagain,and causessingleinvariants togerminate andfNctifyintocomplete connected systemsof forms. 42JOnthePrinciples ojtheCalculusojForms. 289 beamixedconcomitant off,itbeingevidentthatevery function of con­ comitants of a function is itselfaconcomitant ofthesame. Suppose now f=aa;'"+mbxm-ly+im(m-1)clJ!'l--'ly2+&c., theconcomitant becomes (a+Xm)IJ!'l+m(b+XE"'-I11)lJ!'l-ly+im(m-1)(c+XEm-2112)+&c. Consequently if Pbe anyconcomitant off,P'obtained fromPbywriting a+).,E"',b+XE"'-l11,&c.fora,b,&c.,will stillbe aconcomitant offjand by Taylor'stheorempievidently equals P+(Emfa+Em-l11~+&c.)P +1\(Em:a+E--l'l~+&c,yP +&c. IfwetakePaninvariant off,we have M. Hermite's theorem- for {(:x,y),andprecisely the same demonstration appliestothegeneralcase off(:x,y...z).JYis, byvirtueofthegeneralrule, acontravariant offin respecttoE,11...~:ifPbetakenafunction containing onesinglesystem, andis also a contravariant tofinrespecttothatsystem,P'will be ll.double contravariant jand if we make thetwosystemsinJYidentical, we have the extension of M. Hermite's theorem alludedto by me in one of thenotes'[ tomy last paper,whereinI havestatedthat"Imay betakenanycovariant ofthefunction": asregardsthepurposeofthatstatement, thewordcovariant wasused inerrorforcontravariant. Thepreceding methodmay be viewed asaparticular application ofthe generalprinciple, thatifUl,U2...Urnbe anymfunctions (whether con­ comitants any of themoftheothersor not), thenanyconcomitant of x"Ul+~U2+ ...+XmUmbeingexpressed as a function of Xl'~...Xm. every coefficient in such expression will be a concomitant ofthesystem UhU2...Um.Thus, for example, if UandVbe twoquadratic functions ofnvariables x,y...s,thediscrimiTJ,ant 0(XU+JI-V)willcontainn+1terms, of which thecoefficients of thefirst and lastwill be 0Uand0Vjand every one of the (n+1) coefficients will be a concomitant (of course an invariant) ofUandV.These(n+1)invariants will in fact constitute thefundamental scaleofinvariants tothesystemUandV;andeveryotherinvariant ofU •Thistheorem wasfirststatedto me by Mr Cayley,who, Iunderstand, derivedit from M.Eisenstein, undertheform of atheorem ofcovariants, whichofcourseitbecomes on inter­ changing a,ywith- y,z.Butasatheorem ofcovariauta itcouldnotbeextended tofnnctions of morethantwovariables. M.Hermite appears tohavediscovered thistheorem, underits moreeligible form,subsequently to,butindependently of, M.Eisenstein. [tp.201above,notev.] & 19 290OnthePrinciples ofthe Oalculus ofForms. [42 andVwill be an explicitrational function ofthe(n+1)termsofthescale. Inconnexion withthisprinciple may bestatedanotherrelativeto any system of homogeneous functions of agreaternumberofvariables of thesameclass, namely,thatif anysetofthevariables one less in number thanthenumber ofthefunctions beselectedatwill,andanyinvariant of agivenkindbe takenoftheresultant ofthefunctions inrespecttothevariables selected, all such invariants so formed will have an integral factor in common, and thiscommon factor will be an invariant ofthegivensystemof functions. Itwill beconvenient tospeakhereafter ofsystems for which themarch ofthelinearsubstitutions iscoincident ascogredient, and those for which themarch is contrary ascontragredient systems. Suppose mcogredient classes of mvariables, thedeterminant formed by writingthemxmquantities insquareorder will evidently be auniversal covariant. Thus,takethetwosystems e,Y;E,"l.XTf-YEis auniversal covariant, andevidently thereforeF,which I use to denote ¢(x,y)x¢(E,"l)+),.(x"l-yE)"', will be a covariant to¢(x,y).Let¢(x,y)be ofmdimensions; anyinvariant ofFwill be an invariant of¢ithus,letthetwosystemse,y;E,"lbetreated asperfectly independent, andtakethediscriminant ofF(viewed as a function . .dFdF dF dFofe,y;E,"l),thatIStheresultant of the four functions d»'dy'dE'd~i thisresultant will be an invariant of4>;and),.beingarbitrary, allthe coefficients of its different powers will be invariants of¢.Wethusfall upon anothertheorem of M.Hermite, namelythatifx=4>(~('~x¢)(E,-"l),the X..-YTf'" coefficients of theequation which will give theminimum values of ),.are invariants of¢.Somoregenerally, anyinvariant off(x,y,E,"l)-),.(xE-Y"l)"', fbeing of thedegreemine,yand inE,"l,will be an invariant offiand amongotherinvariants maybetakenthediscriminant obtained bytreating ai,E,y,"lasabsolutely unrelated. Iffbe a function of various classes each containing ncovariables, and if notlessthannoftheseclasses be covariable classes,andafterselecting at willanynof suchsystems, asXl'Yl•••Zlix2,Y2Z2; xn,Yn'"Zn,the symbolical determinant d d d de,'dYl'"dZl d d d da;IdY2'"dz, d d d dX;.,dYn'" dZ n 42] On thePrinciples ofthe Oalculu« ofForms. 291 beexpanded andwrittenequaltoD,thenP/will beaconcomitant oif ;and, moregenerally, byselecting different combinations ofthecovarisble systems nandntogether in every way possible, and forming corresponding symbols ofoperation E, F ... H, we shall have D'.E"...Bt').f,for all values of ""...(s),acovariant of/ inrespecttotheclasses socombined. Thisexplains andcontains thewholepithand marrow of Mr Cayley's simple butadmirable methodofobtaining covariants andinvariants (or,astermedbytheirauthor, hyperdeterminauts) toafunction CPIofasinglesystem tel'YI..•ZI;he forms similarfunctions CPt...cP"oflXI'Yt...ZII;...IX",Y"...Z",and uses theproduct t/JIxcf>tx...xcP"as afunction/ of""systems: themultiple covariant obtained byoperating thereupon becomes asimplecovariant onidentifying the different classes of covariables introduced intheprocedure. SECTIONII.On Complex Concomitance. We have hitherto beenengaged inconsidering only aparticular case ofconcomitance, thetrueidea of which relatesnotto anindividual associated form(assuch),buttoacomplex of forms capableofdegenerating intoan individual form.Sucha complex may be called aPlexus. A plexusof forms isconcomitant toagiven form or combination of forms underthefollowing circumstances. If(0)betheoriginant, meaning therebytheprimitive form orsystemof forms,andPtheconcomitant plexusmade up of thej,formsPI'P;...P", and if, when by dulyrelatedlinearsubstitutions, 0becomes0',theplexus PbecomesP,made up of theformsPI'P', ...p',,)and iftheplexus'P formed from 0'afterthesame law as Pfrom0be made up of theforms 'PI>'PI'"'P",thenwill each form in eitherof theplexuses'P,Pbe alinear function of alltheforms in theotherplexus, and theconnecting constants in every such linearfunction will be functions of thecoefficients of thesubstitu­ tionwhereby0andPhave become transformed into0'andP. A function forming partof aconcomitant plexusmay be termeda concomitantive. Concomitantives therefore usually have ajointrelation toa common plexusandaconcomitant isonlyanothername for an unique concomitan tive.Everyplexuscontains adefinitenumberofconcomitantives; in place of anyone ofthesemay besubstituted anarbitrary linearfunction of alltherest,butthetotalnumber ofindependent formssufficient and necessary to make thecomplete plexusrespond to therequirements ofthe definition willremainconstant. Ifnow we combine together thewholenumber offunctions contained in one or more plexuses concomitant to anygivenoriginant, all ofthesame degreerelative toanygivenselected systemorsystems of variables, and if thenumber oftheconcomitantives so combined be exactlyequaltothe 19-2 292OnthePrinciples oftheOalculusofForms. [42 numberoftermsin each,arranged as a function of theselectedclassor classes of variables, thenthedialyticresultant (obtained bytreatingeachcombination oftheselected variables as anindependent variable, andforming a deter­ minantintheusualmanner), will beaconcomitant to thegivenoriginant. This, which is only thepartialexpansion of some much higherlaw, may betermedthe"Law ofSynthesis." Letfbe anyfunction of asingleclass of variables a;.IX,•••x".LetX represent anyproduct ofthesevariables or oftheirseveral powers of any givendegreer;thenumberofdifferent values of Xwill bef£,where n (n+1)...(n+r -1) f£= 1.2...r ' and»J.»r...XJ.fwill form acovarianti veplexustof Again,let~represent anyproductof thedegreer of the symbols d d d de,'dX,'"dx"; ~J,~J...~,..fwill also form acovariant plexustof Thecoefficients of connexion between theforms of eitherplexusdepend in an analogous manner uponthecoefficients of thesubstitution supposed tobeimpressed uponthevariables, withthesole difference thatevery coefficient takenfromtheline r and column 8ofthedeterminant of sub­ stitution whichappears in any coefficient of connexion oftheoneplexus isreplaced bythecoefficient takenfromtheline8andthecolumn r in the corresponding coefficient of connexion for the otherplexus. Letf(x.y)be anyfunction ofe,yofth~degree2m;then (d)tn(d)m-ld (d)tn dx'dx dy'......dy will form acovariantive plexus;thus.suppose f(x,y)=~afi'"+2m~x2m-l y+...+a"2t!>-tlylm; omitting numerical factors,theplexuswill be composed of the (m+1)lines following: ~xm+rna.J,xfT>-ly a<,lxm+rnaaxm-1y+ +am+lym, ++amHytn, am+lxtn+rnam+2xm-1y+...+Utm+Iym, andconsequently, bythelaw ofsynthesis, thedeterminant is aninvariantoff 42J OnthePrinciples ojtheCalculusojForms. 293 Whenthisdeterminant is zero, I have proved in my paper-on Canonical Forms,inthePhilosophical Magazine forNovember last,thatjisresoluble intothesum of 1npowers of linearfunctions of xandy.I shall here­ after refer to a determinant formed in thismanner fromthecoefficients ofjasitscatalecticant. Mr Cayley was, I believe, thefirst to observe that allcatalecticantst areinvariants. Again, more generally, letj(x,y,~,"1)beafunction of themthdegree ofe,y,and of a like degree in respectofE,"1,which are supposed to be cogredient withxandy;then j(x,y,~,"I)+A(XTJ-y~)'" (sayF)willbeaconcomitant ofj;andtherefore ifwetakethesystem (d)m(d)m-ld (d)mdxF,dxdyF......dyF, which will be functions of Eand"Ialone, and taketheirresultant, this resultant willbeaninvariantoffAsaparticular case of this theorem, let (dd)mj=~d:-C+"1d;lJ4>, where4>is supposed to be a function of xandyonly and of 2mdimensions, jis aconcomitant of4>,andtherefore theinvariant off,obtained inthe mannerjustexplained, will be an invariant ofcp.Thusthen we have an instantaneous demonstration ofthetheorem giventby me in thepaperof thePhilosophical Magazine before named, namely, if cp(x,y)=a1W'"+2rn.a.",xtr'l-ly+...+Ctzm.tlylm, say, inorderto fixtheideas,=aa;8+6ba!y+15ca;4y'+...+gya;thenthe determinant a, b,b, C,c, d-p.,d+A, e C,d+!A, e,j d-A,e,f, g (andtheanalogously formed determinant forthegeneralrase)will be an invariant of4>.Thegeneraldeterminant so formed is peculiarly interesting, becauseitfurnishes when equated to zerotheone sole equation necessary tobe solved in orderto be able to effect the reduction of4>(x,y)to its canonical form, and gives the means, irrespective of anyotherview ofthe theory of invariants, ofdetermining completely andabsolutely thecondition [- Seep.282above.] tButOlecatalecticant of thebiquadratic function of:1:,ywas first brought into notice asaninvarian* by MrBoole;'andthediscriminant of thequadratic function ofe,yisidentical wi*hibcat&lecticant, as alsowithibHessian. Meicatalectioizant would more completely express &hemeaning oftha*whioh, for the sakeof brevity, I denominate *heca*aleotioant. [tp.277above.] 294 OnthePrinciples oftheCalculusofForms. [42 ofthepossibility of two given functions of thesame degree of e,ybeing linearlytransformable one into the other. This theorem will beobtained inamoregeneralmannerinthefollowing section. I only pausenow to maketheveryimportant observation, thatnot only is thedeterminant an invariant, butevery minor system-ofdeterminants thatcanbe formed from it(thereare of course msuch systems) is aninvariantive plexusto the given function f/J. The form underwhichthistheorem presents itselfsuggests atheorem vastlymoregeneralandofpeculiarinterest,asshowingaconnexion between thetheoryof functions of acertaindegreeandof acertainnumberof variables withotherfunctions of alower degree butof agreaternumber of variables. Hereagain,underadifferent aspect, is reproduced thegreat principle ofdialysis, which, originally discovered in thetheory of elimination, in one shape or another pervades the whole theoryof concomitance and invariants. Letf/Jrepresent anyfunction of the degree pq(of anynumber, or, to fixtheideas,sayofthreevariables e,y,z);letthegeneraltermoff/Jbe represented by pq(pq-1)...1(fl)!I ., (1.2...a)(1.2."fl)(I.2 ••.'Y)a,,'Yrcazr, wherea+fl+'Y=pq,and(a,fl,'Y)represents aportionof the coefficient of a:&'!Iz"l. Let =--;;-~-;:1:;-.----;:2:;- .._.P. Iffy'r=0t(1.2...r)(l. 2...8)(1.2...t) T, I, , wherer+8+t=p,80thatthereareasmanyo'sasthereare modes of *Theseminorsystems mean &8follows:-thesystem of rthminorscomprises all the distinct determinants thatcanbe got by striking out from the squarearray(which I call the Matrix) from which the complete determinant is formed, anyr lines and anyrcolumns selectedatwill. Thelast,or mth minor, is of course asystemconsisting of thecoefficients oft/>(z,y),andit is evidentthatift/>(z,y ...I)be anyfunction of anynumberofvariables z,y...I,the coefficients will form an invariantive plexustot/>. The following remark &8tothe ehanges undergone by the coefficients of t/>when the v&riablell undergoanysubetitution, is notwithoutinterestandimportanoe for thetheory. Let zbecomeJz+ /'y++(J)I, Y.........gz+g'y+ +(g)z, I hz+h'y+ +(h)z. Thenthe coefficient of the highestpower of zbecomes t/>U,g ...hl, andthecoefficient of the termcontaining yT...Zlbecomes (dd d)T{d d d}1l'iij+o'dg +...+h'/ih x&e.x(J)iij+(g) dg+'"+(h)djit/>v.g ... h). 42J OnthePrinciples ojtheOalculWJofForms. 295 subdividing pintothreeintegral parts(zerosbeingadmissible) jthatis 1(p+l)(p+2)(p+3).Thenanyproduct such as a:"'yfJz'lmay be divided inavarietyofwaysintotheproduct ofqoftheseO's,anditmay be shown thattheentirequantity pq(pq-1)...1 (a:a!fz'l) (1.2...«)(1.2...,8)(1. 2..."f) _I{ 1.2...q (0.R,O....0'")}-(1.2...?n:J)(1.2...1nt)...(1.2...m,.)"'I~...I'rr , where1nt+1nt+...+m,.=q.Consequently cf>may berepresented underthe form ofafunction ofthedegreeqofi(p+1)(p+2)(p+3)(says)variables 01,01...0"anditsgeneraltermwill be of theform 1.2q ,8{8"'0 8""(1.2...ffll)(l.21nt)...(1.2...mr)(e,,"f),.......rJ, where I%,,8,"faretheindicesrespectively offe,y,s,whenthelastfactor isexpressed as afunction ofthesevariables". Nowif~be used to denote thisnewrepresentation ofcf>whenOlJ01" ,0,aretreatedasabsolutely independent variables, and if we attachtoitanyuniversal concomitant, as (feE+y1]+z~admitting ofbeingwrittenundertheformro(01,0....OJ, wherein thecoefficients will be functions ofE,1],~;thenanyinvariant to ~andro,treatedas twosystems of,variables, will beaconcomitant to~, theoriginal function ine,y,zt.~andromay betermedrespectively, for facilityofreference, thePa.rticular andAbsolute functions. Thus,forexample, wetakecf>afunction ofe,yofthedegree4n,say ala;-ln+4n~:x"'-1 y+&c.+a.n+ly4", andmakep=2n,q=2, sothat~becomes aquadratic function of(2n+1) variables obtained bymakingwn=01,xm-Iy=O•...y'm=02fl+lt,andthe concomitant ro,formed from (Ex+'T'Jy)ln,becomes 01F'+2nO.rn-11]+...+Om+!'T'Jm; thenif wetakeRthequadratic invariant ofro,thatis R 0ell0&1.2.:3(2n)1(0\J=I2fl+!-2nl111 2nc.±(1.2n)22"n+l}, •SeeNote(1) in Appendix. [po 322below.] tInfact~illaconcomitant tot/>,and",toapower of theuniversal ooncomitant; theIJ's forming asystemofvariables oogredient withtheoompoand systemz',!t,z'"z'.,...z'., &0.: anditmustbe wellobserved thatthesamesubstitutions whichrender ~and'"respeoti1'ely identical witht/>andapower of the universal concomitant, wouldrenderaninfinitenumber ofotherfunctions alsocoincident withthesame;butnone of theseotherfunctions would be conoomitants. Hereinweseetheimportance ofthedefinition andconception ofcompound relation; theIJsystembeingcompound byrelation withthee,y,.Isystem,afterthemanner ofcogredience. :Aslightvariation upon the method &8aboveexplained forthegeneralcasehasbeenhere iDtroduced inadvertently bywritingz2A-Iy- IJlI,&0.,in lieu of 2nz!io-Iy= IJ1,&0.,whioh, &8it doesnotinanydegreeaft'ectthereasoning, Ihavenotdeemeditworthwhiletoalter. 296 OnthePrinciples oftheCalculusofForms. [42 itwillreadilybe seenthatthedeterminant of~+A.R,treatedasaquadratic function of (2n+1) variables, will give an invariant ofcp,andthiswill bethe sameasthatobtained bytheparticular methodabove given. Thus, suppose cp(x,y)=aa:'+4b:x'y+6caryi+4d:xy'+ey4. Let :x'=01,2xy=0.,y'=0" ~=aeli+2b010,+c(),'+2cOlO,+2d(),O,+eO,', Q)=(xE+YTJ)'=W()I+a;y(),+ytO" OitR=01()'-4 . ThenAthediscriminant of~+2A.Rinrespectto()I'(),'OJ a, b, C+A.I, b.c-!A.,dI c+A., d,e andI mayremarkthattherelations between theseveraltransformees ofthe invariantive plexuses formed by theminordeterminant systemsof A (in this, and ingeneralforthecase of an evenly-even index)may be found by treating ~+2ARas aquadratic function ofthevariables (inthiscase01,0"0,)and applying therulegivenby me in thePhilosophical Magazine inmy·paper "Ontherelation between theMinorDeterminants oflinearly-equivalent Quadratic Forms."t Thissecondmethod, however, is notimmediately applicable tothecase ofindicesoddly even, thatis oftheform4n+2,to whichthefirstmethodapplies,equallyas tothecase4niforifwe make 2n+1=Pandq=2,Q)beingof an odd degree,has noquadratic invariant; ithas however a quadratic covariant, which will be of theseconddegree inrespectto01.(),...0p+1as wellasinrespectto',TJiand if we call thisR andtakethediscriminant of~+'ARinrespecttothevariables ()I.0,...01>+1, weshallobtain,as I amindebted to aremarkof my valued friend M. Hermite forbringing undermy notice, a very beautiful andinteresting function of A., ofwhich all thecoefficients will be contravariants ofcp.Thus,let cp=a3f'+6b:rf'y+15cx'y'+20dx'yt+15eWy4+6fxyB+g'!l, [*p.241above.] tMoreover, upon thesupposition made in the text,theparticular andabsolute functions ~and Colmaybetreatedin allrespecte asiftheywerefunctions characterizing quadratic loci, andanysingularity intheirrelation willcorrespond toanddenoteasingularity iu thegiven function <ptowhich ~refers.Thus,forinstance, if<pbeafunction ofx,yoftheeighthdegree. ~and Colwillbequadratic functions of fiveletterseach.Quadratic loci have no othersingularity ofrelationthanwhatcorresponds todifferent species of contact. Thenumberofcontacts between loci,characterized by 5letters,is24(see mypaperfinthePhilosophical Magazim, ..Onthecon­ tactsof linesandsurfaces of the second order"). Consequently thismode of representing ~and.. will give rise to the discovery andspecification of24different kindsofsingularity in<p.andthe analytical characteristics of each of them.Butthereofcoursemay.andinfactwill,exist othersingularities in<pbesidesthosewhichhavetheircorrespondencies intherelations ofthese quadratic concomitants. .[:!:p.237above.] 42] OnthePrinciples ojtheOalculu«ojForms. 297 make 80that ~=aOI!+2b010!+cOt2+2cOlOs+2dOtOs+2d010,+gOl+2/0s0,+eOl+2eO,Ot, Co)=(x,+Y"l'f=Olr+02E'TJ+Os'TJ2+O,TJI, R=I301,+OtTJ,-°t'+OITJI' lOt'+OsTJ,Os,+30,TJ - R=rO!1+TJtOl+'TJOt81-9~010,-3rOl0S-3TJtOtO,. Consequently thediscriminant inrespectto0..Ot,0..0,of~-2>..Rbecomes a, b, c-3~,d-9X'TJ b,c+2Xr, d+X'TJ,e-3X1]t c-3xr,d+x~, e+2>"TJI,/ d-9X'TJ,e-3XTJI,f, g Ifthisdeterminant beexpanded as afunction ofX,all the coefficients of the various powers of Xwill becontravariants te-thegivenfunction cf>.Theterm involving X'is zero.Let,become - yandTJbecomee,thentheremaining terms(abstraction made of thepowers of X)become co variants ofcPoThe firstterm(thecoefficient of XS)becomes cf>itself;thelasttermisthe catalecticant, andthuswe see, in general, thatforfunctions ofxandy of anoddly-even degree, a whole series of covariants maybeinterpolated between thefunction anditscatalecticant, thedimensions inrespectofthe coefficients of cPinarrivingateachstepincreasing by 1unitandthedegree inrespectofthevariables diminishing by 2units.Thisisconsequently a muchsimplerand more available scalethanone with which I have been longpreviously acquainted, and which appliesalike to functions of any even degree. Thus,letcf>(x,y)be of2kdimensions; form alltheevenemanants ofcf>, which will be all of theform(,d:+TJ;yrcf>'andtaketheirrespective cata­ lecticants inrespectto,andTJ.We shall in thiswayobtainaregularscale ofcovariants interpolated between theHessianofcf>(corresponding to,=1) andthecatalecticant ofcP(corresponding to,=k).IfcPbe ofthedegree 2k+1, weshallhave an analogous scaleinterpolated between theHessian ofrpanditscanonizant ;thelattertermdenoting thefunction which is the productofthek+1linearfunctions ofxandy,thesum of whose (2k+l)th powersisidentically equaltorp•. BymeansoftheTheoryofthePlexuswe mayobtainvariousrepresenta- • SeeNote(2)inAppendiL [po822 below.] 298 OnthePrinciples oftheOalculusofForms. [42 tions of thesameinvariant; thus, for example, if we takeFafunction ofe,yof the fifth degree and form its HessianH,thatis dtF dtF dtrf-'da:dy d2Fd'F dyda:'dy' thiswillbeafunction of the sixthdegree in !x,y.and ofthetwoorders in thecoefficients. Ifwe combine thetwo plexuses dFdFrPHrPHdlH de 'dyjda:"d.xdy,dy" we shall have five equations between which:r:',x'y,:r:'yt,a:y'.v:maybe eliminated dialytically; theresultant willbeof the 2+3. 2,thatisthe eighthorder in thecoefficients,and of theform0F-It,where 0FandIf, arerespectively thedeterminant andquinticinvariant ofF,eachaffected withapropernumerical multiplier (the"B-AI"of mysupplemental- essay oncanonical forms) which, asMr Cayley has remarked, may also berepre- sentedbytheresultant ofP;c;:;;~~wherePandQarerespectively the quadratic and cubic invariants inrespecttoEandTJof(E-tx+TJ;y)'F. Itwill be well atthispointtorecapitulate inbriefamethod of elimination applicable tocertainsystems of functions published by me many years since inthePhilosophical Magazine, and to compare thismethodwiththatafforded bythetheoryoftheplexus for finding an invariant for each of theverysame systems, possessing all theexternal characters. formed in a precisely similar manner to, and not impossibly identical with,theresultant of every such system.Ishall devote my first moments of leisure to the ascertainment ofthislastmostimportant point, as to theidentity orotherwise ofthe plexus-invariant withtheresultant. Take the caseofthreefunctions of !x,y,Z(say4>,y.(0)each ofthesamedegreen;to fix the ideas,suppose n=3:thereare twopurelyalgebraical processes (modifications of the same methodandleadingtoidentical results)by which theresultant of4>.y,fA) may be found. Ishall call these processes the first and second respectively. Firstprocess: Write 4>=a:'P +yQ+zR, y=a:'P'+yCl+zR:, 00=a:'P"+yQ"+zR", decompositions which may be effected in an infinitevarietyofmanners, 80thatP,Q,Rshall beintegerfunctions of e,y,zjtakethelinearresultant of4>.y,00,in respect to a:',y,s,which call H2•1.1;thiswillevidently be [- p.205 above.] 42J OnthePrinciples oftheOalcv1'ltBofForms. 299 of9- 4,thatis,of 5 dimensions. Form analogously thefunctions Hl.~,I' HI.I.~; H"l,l'HI,l,1JH1•1,iconstitute anauxiliary system of functions which vanish when.p,y,OJvanishtogether jcombine thisauxiliary system with the augmentative system ~.p, y~.p,zicfl,ICYcfl,yzcfl,z:r.cfl, ~OJ, ~OJ, z~OJ,xyOJ,yZOJ,za;OJ, filt,~, z~,xyt,yzy,za:y.. We shall thushave in all 3 +3x 6,thatis, 21 functions into which the 21termsa;a,:r:'y,a:'z,&c.enterlinearly: the linear resultant ofthese21 functions is theresultant ofcfl,y.,OJ,clear ofall extraneousness. Second process: Write cfl=~P+yQ+zR, y=~P'+y(l+zR', OJ=x'P"+y(l'+zR", and,asbefore,takethelinearresultantH,.l.I,which will however be of 9- 5, that is, of only 4dimensions. Again,take cfl=filL+~M+zN, y.=:x"L'+y'M'+zN', OJ=:x"L"+y~M"+zN", and form thedeterminant H•.•.I;weshallthushavetheauxiliary system Letthisbecombined with theaugmentative system :COJ,YOJ.ZOJ;:ccfl,Ycfl,zcfl;a:y.,yy,zy.. Between these 6 +9,thatis. 15 functions, the 15 termsa:',~y,wz,&c.may belinearlyeliminated, and the resultant thusobtained will be precisely the same asthatgotby thepreceding process. Herewe have 6 auxiliaries and 6augmentatives; theauxiliaries are ofthreedimensions in respectto the coefficients of cfl.y.,OJ;theaugmenta­ tivesof one dimension only;in the former process therewere3auxiliaries and 18augmentatives, 6 x 3+9=27=3x 3+18. Now letthismethod be compared with thefollowing: Firstprocess: Take the18augmentatives aPcfl,aPOJ,aPy,&e.as in the first processofthealgebraical methodaboveexplained; butin place of the 3auxiliaries thereingiven,takeanothersystem of 9 asfollows: 300OnthePrinciples oftheOalculu«ofForms. [42 Writethedeterminant dcJ>dq,dcJ>=R ; de:'dy'dz dyd'frd" da:'dy'dz d(iJ.d(iJd(iJ dx'dy'dz dRdRdRde 'dy'dzformaconcomitantive plexus; the 18augmentatives form another; thelinearresultant ofthesetwo plexuses will be an invariant ofcp,y,(iJ,and of precisely the same dimensions as theresultant lastfound; if they are not identical it will be indeed a matterof exceeding wonder, and even more interesting thaniftheyshould be proved so to be. Secondprocess: Combine the augmentative plexus a;(iJ,y(iJ,ZOO;a;q"ycJ>.zcJ>;a;y,y'fr.Z'Ifr. with the differential plexus d2Rd2Rd2Rd2RrPRd2R d$l'dxdy' d~'dydz'dz2'dsde' wethusobtainalinearresultant in amannerprecisely similar to that afforded by thesecond process of our algebraical method. Ingeneral.ifq,.'fr.(iJbe ofthedegreesn,n,n,asthereare twoalgebraical varieties of thelinearmethodfor finding theresultant, so aretheretwo varieties of theconcomitantive method for finding the resembling invariant. Inbothmethods theaugmentatives areidentical; the only difference being in theauxiliary system. Inthefirst process theaugmentative system will be got by operating upon each of the functions q"".(iJ,withthemultipliers xn-\yn-l,zn-I, and the otherhomogeneous products ofe,y,zjtheauxiliary systemby operating uponRwiththesymbolical multipliers(fa:)n-2,(;)n-2.(~)n-2. d d d !I andtheotherhomogeneous products of(],a;,diJ'dzof the degree n-2. In the second process the augmentative system is formed by theaid ofthemultipliers a;n-2,yn-t,zn-2,&c.,andtheauxiliary system by aid of (~)n-l(~)n-l(!!)n-l, de'dy •dz'&c. For the particular case ofn=2thefirst process of the concomitantive methodis merely an application underitsmostsymmetrical form ofthefirst 42J• OnthePrinciples oftheOalculu«ofForms. 301 processofthegeneralalgebraical method. The second process of the con­ comitantive methodforthissamecase(atleast when9,v,00are thepartial differential coefficients of thesame function of thethirddegree)hasbeen shownbyDrHessetogive the resultant, sothatforthiscase,atall events, weknowthateachconcomitantive auxiliary mustbealinear function of the augmentatives and.thealgebraical auxiliaries. Again, if wego to the system where 9,'0/,00areoftherespective degrees n, n,n+1.Inthealgebraical method(forapplying whichthereareno longer two, butone only process), the augmentative system is obtained bymultiplying 9by the homogeneous products of:en-I,a;n-ty,:en-Iz,&c., ..".bythe like products, and 00bythehomogeneous products :en-t,:en-'Jy,&c. Theauxiliary system is made up of functions of the generalform Hp,q,rwherep+q+r=n+2, H".q.rbeingthedeterminant obtained bywriting 9=L:cP+My'l+Nzr, '0/=L'xP+M'y"+N'zr, 00=L"xP+M"y'l+N"zr. And in like mannerforthecase of9,'0/,00,beingoftherespective degrees n,n,n -1,theaugmentative system is obtained by affecting 9,""eachwith multipliers a;n-t,a;n-ty,&c.,and00with the multipliers a;n-I,a;n-Iy,&c. Thenumber of functions (for eithercase)in theaugmentative and auxiliary plexuses thusobtained' will be found to be exactly equal to the numberof termsin each such function, asshown by me in the paperalluded to.Letthisbe compared with the transcendental method(Iusethisword at thispointinpreference toconcomitantive, because in fact the algebraical and differential auxiliary systems are both alike concomitantive plexuses toq,).Forthecaseofn,71,n+1,theJacobian determinant Rof9,v'00 willbeofthedegree 311-2,and the system (;:er-I R,(;:er-2(;y)R,&c. combined with the augmentative systems :en-lloo, a!'-3yoo,&c. a;fl-Iep,:en-ty9.&c. :en-I",:en-2yV,&C. willgive an invariant resembling (at least in generation and form) if not identical with the resultant of9.'0/,00.For thecaseof9,v'00being of the degreesn, n,n-1,theJacobianRis of the degree 3n - 4and 'cs:(drdd»R,d:cdyR,&c. 302 OnthePrinciples oftheCalcuiu»ofForms. [42 isthesystemwhich, combined with the augmentative systems xn-2c/>,a;'>-Iyc/>,&c. a;n-s.y,a;n-lyV', &c. Xn-l6),:r;n-'Jy6),&c. will produce theresembling invariant. Finally,forthelast and more special case which thealgebraical method appliesto,namelyofc/>.,y,6),0,fourquadratic functions ofx,y,s,t,there can be here littledoubt(uponthefirstimpression) thatin place of the algebraically obtained plexus may besubstituted thedifferential plexus dRdRdRdR dx'dy'dz'dt ' which, combined with theaugmentatives xc/>,X'\jr,X6),xOjyc/>,YV'.y6),yOjzep,zV',Z6),zOjtc/>,tV'.t6),to, willrenderpossiblethedialyticelimination ofthe20 homogeneous products W,wy,a;2z,art,xyz,!t,&c.&c.· Uponprecisely thesameprinciples may be verified instantaneously themethod given by Hesse (without demonstration) for finding thepolar reciprocal of lines of thethirdandfourthorders,atleasttotheextentof seeingthatthefunctions obtained by hismethods arecontravariants (ofthe rightdegree and order) of thefunction from which theyare derived. The polar reciprocal to a surfaceofthethirddegree may be obtainedinthesame manner. Letep(x,y,z,t)bethecharacteristic of such a surface. Ifwe form adifferential plexus of thefirstemanant ofc/>takentogether withthe concomitant w=xE+Y'T'J+z~+to,byoperating with d d d d (d fd,dd)([X'dy'dz'dtuponE'dx+'T'Jdy+~dz+O'dt(c/>+}..w), andcombining thisplexus with xE'+Y'T'J'+zr+to',theresultant takenin respecttoE','T'Jf,r.Of(sayR)will(according tothelaw ofsynthesis) bea *Subsequent reflection induces me torejectu.sveryimprobable the(u.tfirstviewlikely) conjecture of theidentityof theresultant with the invariant whichsimulates itsform, except in theproved eases of threequadratic functions andthestrongly resembling case of four quadratic functions lu.stu.dverted to in the textabove. Did thisidentityobtain,anu.logy wouldindicu.te thu.tthecatalecticant of theHesaian of twohomogeneous functions of the same degree in :E,11 shouldbeidentical withtheirresultant, which is eu.silydemonstmted to bef&lse,exceptwhen thefunctions are of the thirddegree. 42JOnthePrinciples oftheCalculusofForms. 303 contravariant tothesystemq,+AWandw,andtherefore toq"becausewis itselfaconcomitant toq,.Ris ofthethirddegree in a,y,z,t,asalso inthe coefficients of q,.Ifwe form a differential plexusofR+p.wanalogous tothatformed above with q,+AW,and combine thesetwo plexuses with the augmentati ve system eur,yw,zw, tw,therewill be4+4+4,thatis, 12 functions containing the12termsafl,yl,Z2,t2,xy,es,txt,yz,yt.zt,A,p.,and thedialyticresultant, which will be found to be a contravariant ofthetwelfth degreeinE,"1,t,0,and ofthetwelfthorder inrespectofthecoefficients of q" willbe(therecan belittledoubt)thepolar reciprocal to thecharacteristic q,. A fewremarksupontheanalytical character of a polar reciprocal may be notoutof place here. Ifq,be any homogeneous function of thedegree m of anynumber(n)ofvariables (x,y...z),theobjectofthetheoryof polar reciprocals isto discover whatistherelationbetweenE."1...texpressed in the simplest termssuchthat,whenthisequation is satisfied, Ex+'TJY+...+tz=0 will betangential toq,=O.Inorderforthistotakeeffect it is necessary thatwhenanyoneofthevariables zisexpressed intermsoftheothers...y,e, andthisvalueestablished inq"thediscriminant ofq"sotransformed, should bezero.Consequently thecharacteristic ofthepolar reciprocal to q,is thatrational integral function which is common to all thediscriminants obtained byexpressing q,(by aid of theequation Ex+'TJY+...-+tz)as a function of any(n-1)ofthevariables. LetIzbe anyinvariant whatever of theorderrofq,z(meaning bythislast symbol what q,becomes when xis eliminated), andIy...Izthecorresponding invariants wheny...zrespectively areeliminated; Ia;willevidently be oftheform(J:....,thenumerator being anintegerofrdimensions in thecoefficients of q,and ofmrdimensions in respect of E,"1'"tjand bythefundamental definition ofinvariants it may easily be shown that .., ..1.1..1.Iz.Iy•••••Iz..mr'mr·..··-,;;r, E"'....I"In-I ~I andtherefores,s,s,hm(n-2)r ~p='TJP=...=t'P'werep=n-1 . Consequently allthesequotients mustbeessentially integer,andanyone of them will be of the order rinrespectofthecoefficients of q,and of the • Weseeindirectly fromthis,thatforafunction of(n-l),&&Y'Y,variables of the degree m, aninvariant oftheorderTmustbesubjectto thecondition that~=aninteger. Thisis easily 'Y ahowuuponindepeudent grounds; when'Y=2,mTmuetbenotmerelyanintegerbutaneven 'Y illuger,anddoubtless some analogoue lawapplies to the generalcase. 304On thePrinciples ofthe Calculus ofForms. [42 degreemrlinrespectofE,'7••.~.Consequently thepolarcharacteristicn- ofcp,whichit>thecommon factor of the discriminants ofI~,III... Iz(for which species of invariant revidently isequalto(n-l)(m _l)n~,thefunction being in factthediscriminant of a function of themthdegreeof(n-I)variables), will be of theorder(n-l)(m-l)n-1inrespectofthecoefficients of ep and ofthedegreem(m -l)n-2inrespectofthecontragredients E,'7'"~. Astowhatrelatestothereciprocity whichexistsbetween epanditspolar reciprocalV'thisisincluded in a much highertheoryofelimination, one proposition of which may be enunciated somewhat totheeffectfollowing, namelythatifepbe ahomogeneous function ofe,y...Z,andcoofx,y...s, u, v...W,and if, by aid of theequations d~+Ad~=O dz dz ' e,y...zbeeliminated andtheresultant be calledV'thentheeffectof performing asimilaroperation uponV'£I),withrespecttou, v...W,asthat justaboveindicated forthesystemcp,co,withrespecttoe,y...Z,will beto give aresultant, one factor of which will be the primitive function epover again.Thereis some reason for supposing thatpolar reciprocals, which are scarcely ever (if ever, exceptindeedforquadratic functions) thesimplest contravariants to agivenfunction, may beexpressed algebraically bymeans of thesimplercontravariants, inthesame way as discriminants admit(in many, if notin all cases, withthesameexception as above) of being repre­ sentedasalgebraical functions ofinvariants of a lower orderorsimpler form. I closethissection with the remarkthateverycomplete andunambiguous systemoffunctions oftheconstants in agivenform orsetof forms charac­ teristic" of anysingularity absolute orrelativein such form or forms must *Irepeatherethatafunction orsystemoffunctions which severally equated to zeroexpress unequivocally andcompletely theexistence of anyposition ornegation, istermedthecharacter­ isticofsuchposition ornegation. Thusforexample theresultant ofagroup of equations isthecharacteristio of thepossibility oftheircoexistence. Thediscriminant ofafunction of two variables is thecharacteristic ofitspossession of twoequalfactors; theoateJectica.nt isthe characteristic of itsdecomposability intothesumofadefinednumber of powers of linear functions of thevariables, &c. 42]OnthePrinciples oftheCalculusofForms. 305 constitute aninvariantive plexus or set of invariantive plexuses. The system unambiguously characteristic ofasingularity of an order nwill (except when n=1)almostuniversally consist of far more than nfunctions, subjectof course totheexistence ofsyzygetic "relations between any(n+1) of such functions. Theexistence ofmultiple roots of a function of two variables is a specific, but by no means apeculiarcase ofsingularity, and requires, for itscomplete and systematic elucidation, to betreatedin connexion with the general theory of the subject. SECTIONIII.On Commutants. Thesimplest species of com mutantis the well-known common deter­ minant. Ifwe combine each of the nlettersa, b...lwith each of the othern, a,f3...x,we obtain n2combinations which may be used to denote the terms of adeterminant ofnlines and columns, as thus: aa,a/3•.•aX, ba,b/3...bX, t«l/3...tx: Itmustbe wellunderstood thatthesinglelettersofeitherset are mere umbne, or shadows of quantities, and only acquirea real signification when oneletterof one set is combined with one of the otherset.Insteadof theinconvenient form above written,we may denote thedeterminant more simply by thematrix a,b,cl, a,/3,'YX; and to find theexpanded value ofsuch a matrixthe rule is evidently totake one ofthelinesinall its 1, 2, 3 ...ndifferent forms,arisingfrom the permutations oftheletters(or umbree)which it contains; andthenformthe product of thenquantities formed by the combination oftherespective pairs oflettersinthesame vertical column, affecting such productwith the sign of+or - according to therule,thatallproducts corresponding toarrange­ ments of thetermssubjectto thepermutation derivable from one another byan even numberofinterchanges areofthesame,andby an odd number ofinterchanges ofacontrary sign.Ifboth lines are permuted anda similar rule applied, with theadditional circumstance thatthe sign of the products •Rational integerfunctions whichadmitof being multiplied severally byotherrational integerfunctions suchthatthe sum of the products isidentically zero, are said tobe ..syzygeti­ callyrelated." ~ W 306Onthe Prineiplee oftheGalCUlU8ofForms. [42 a,b,is made to depend on theproductof thealgebraical signs due to therespective arrangements in the two lines of umbree,it is evidentthattheresultwillbe thesameaswhen only one line is putinto motion, save and exceptthat anumerical factor 1.2.3 ...nwill affect each term.Ifthetwosets of umbras a,b,C •• 'l;IX,/3,'Y•••>..betakenidentical, and if it be convened thatthe orderofthecombination of any two lettersshall not affect thevalue of the , h b d d a,b,C•••l'11d . I d .quantity t ere y enote 'a,b,C•••lWIenote asymmetnca etermmant. Ifinsteadof two lines of umbree, threeor more be taken,thesame principle of solution will continue to be applicable. Thus, if therebea matrixof any even number rof lines each of n.umbrre, aI'b.i.. as,b2Is, thefirst may be supposed to remainstationary, andtheremaining (r-1) lineseachbetakenin 1, 2...ndifferent orders;every order in eachline will be accompanied by its appropriate sign+or-;andeachdifferent grouping ineachline will give rise to a particular grouping oftheletters readoff in columns. The value of thecommutant expressed by theabove matrixwilltherefore consist of thesum of (1 .2 ...ny-lterms,eachterm beingtheproduct ofnquantities respectively symbolized by agroupof rlettersand affected with thesign+or - according asthenumberof negative signs inthetotalofthearrangements ofthelines (from thecolumnar readingoft'of which each such termisderived)iseven or odd. For example, thevalue of c,d, e,f, g,h, will be found by takingthe (1.2)2arrangements, as below, ~~ ~~ ~~~~ ~~ ~~~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ e,f,e,f, f,e,f,e, e,f,e,f,f,e,f,e, ~~~~~~~~~~~~~~~~ Thesigns of c, d;e,f;g,hbeingsupposed+,those ofd,c;f,eandh,9 will be each -,Consequently thesum ofthetermswill beexpressed by acegxbdfh-adegxbcfh-acfgxbdeh+adfgxbceh -acehxbdfg+adehxbcfg+acfhxbdeg-adfhxbceq. 42J OnthePrinciples oftheCakulusofForms. 307 Commutants thusformed may be termedtotalcommutants, becausethe entireofeachline is made to passthroughall its possible forms of arrange· ment.Intotalcommutants it is necessary thatthenumberof lines rbe even;for iftakenodd, on makingall the rlines to change, insteadof obtaining 1.2...nlines,theresultobtained when all butone are made tochange,itwill be found thatthelatterwill berepeatedHI.2...n) times with thesign+,andt(1.2...n)timeswith the sign -,sothatthe algebraicel sum of thetermswill be zero. Moreover the commutants of the species above described, besides beingtotal, are simple, inasmuch asall the umbras to be termedconsist ofsingle letters. My first proposition in theapplication of the theory of commutants to thatof forms is asfollows: Let4>beafunction homogeneous and linearin respect to an even number rofanysystemswhatever of variables, as Xl'Yl'"tl;.2;,YI'''t.,,;Xr,Yr...tr· Formthecommutant d d d de,'dYl'" dtl' d d d dxt'dYI'"dt.", d d d de;'dYr'" dtr. Let thegeneraltermofthiscommutant,expanded, be called F8lxF8ax ...XF8r, then is IF81•4>XF82•4>x...XF8r•4> acovariant orinvariant s,asthe case may be, of 4>. Beit observed thatthe march of the substitution forthedifferent sets of variables in the above proposition is supposed to be perfectly independent. Allthe systems butone may undergo lineartransformation, ortheymay all undergedistinct and disconnected transformations atthesametime,and the proposition still continue applicable. Itwill however evidently be no lessapplicable should themarch of substitution for any of thesystems becomecogredient orcontragredient tothatof anyothersystems. Ifwe suppose 4>to beafunction of an even degree rof a single system ofnvariables e, Y...t,sothatthersystems tel'Yl>&c.,XI'Yi'&c....ter,Yr,&c. becomeidentical, we can at once infer from theabove scheme theexistence and mode of forming aninvariant to4>of the order n.Thislastappears [*Seebelow,p.824.] 20-2 308OnthePrinciples oftheOalcuiu«ofForms. [42 for the case n=2, and ought, for all othervalues of n,to have been known­ to theauthoroftheimmortal discovery of invariants, termedby him hyperdeterminants, inthesense which, according to thenomenclature here adopted, would be conveyed by the term hyperdiscriminants. Before proceeding to discuss thetheoryof compound total commutants, orenlarging uponthatofpartialcommutants, I shall make an interesting application ofthepreceding generalproposition to thediscoveryof Aronhold's SandT,thetwoinvariants respectively of thefourthandsixthorders appertaining to a homogeneous cubic function (say F)ofthreevariables ai,y, z.Thesemay be termedrespectively H4andH5•As toH5a theoretically possible buteminently prolix and ungraceful methodim­ mediately presents itself, namely to takeFI=G,andafterfonningthe commutant with six lines, d d d dx'dy'dz' ddd dx'dy'dz' ddd dx'dy'dz' d d d de 'dy'dI:' ddd dx'dy'dz' ddd dx'(Iy'dz' tooperatewiththe65ternaryproducts of which thisis made up upon G:the resultbeinganinvariant ofG,will be so to F,and being of the thirddegree inrespecttothecoefficientsof G,will be of thesixthinrespectto those of F. Itwillevidently therefore beH5,oratleast anumerical multiple ofH5,the form of which, inasmuch astheonlyotherinvariant is H4,'weknow inform to be unique. Butthegeneraltheorem affordsanotherand probably the *ThatthisW8.8not known explicitly toandshouldhaveescapedthepenetration ofthe sagacious authorof the theory, and thosewho had studiedhispapers,mustbeattributed tothe imperfection of thenotation heretofore employed for denoting the coefficients of ahomogeneous polynomial function. Theumbralmethodofdenoting suchafunction <pof thedegreerunder the form of (=+by+...+C%)',whichisequivalent to,butamorecompendious andindependent mode ofmentally conceiving and handling therepresentation (d d d)zdX+Ydy+......+ %dz<p, exhibitsthetrueinternal constitution of suchfunctions, andnece88arily leadstothediscovery oftheiressential properties andattributes. 42]OnthePrinciples ofthe Calculus ofForms. 309 mostpractically compendious- solution as regards He,of which the question admits. Gis a function of thesecond order as to 3),y,z,and ofthelike orderin respect to~,"I,t',which two systems will be respectively cogredient and contra­ gredient in respect tothee,y,esystem in F.Inotherwords,which is all weneedtolookto,Gisaconcomitant ofF,and so also will be G+X(x,+Y"l+zt')J, which may be termed H.Form now the com mutant dd d dx,dy'(Tz' dd d dx'dy'dz' ddd d,'d"l,d" ddd d,'d"l'dt" this being applied to Hwill give an invariant (thefactthatthemarch of thesubstitutions forthesystemse,y,z;""I,t'is contrary, being com­ pletelyimmaterial totheapplicability of the general theorem above given) ; •Havingsincethiswasprintedbeen favoured with aview of some of the proof. sheetsof XrSalmon's mostvaluable SecondPartof hisSy.UmofAnalytical Geometry (about to appear, andwhichiscalculated, in myopinion,toawakenahigherideaofandexciteanewtastefor geometrical researches inthiscountry), I findthatI ammistaken inthispoint;the188ssym· metrical method operated with by MrSalmon being decidedly the shortest forpractically obtaining SandTin thegeneralcase.Symmetry, like thegraceof aneasternrobe,hasnot unfrequently tobepurch&8ed attheexpenseof some sacrifice of freedom andrapidityofaction. tG is the mixed concomitant tothe given cubic function, which is halfway(sotospeak) betweenitanditspolarreciprocal. Infact,when the operation isrepeated upon G, which was execmted upon the given function toobtainG(thatis, when webordertheHessian of G in respecttoz,1/,.c,vertically andhorizontally with the columnandline£,'I,rJthedeterminant therebyrepresented becomesthepolarreciprocal to the given function. 310OnthePrinciples oftheCalculu«ofForms. [42 thecommutant so formed will be acubic function of x,in which thecoefficient ofXSisanumerical quantity, thatofX'is zero,thatofXisH4andthe constant termisHI' Thusforexample letF=xl+y'+r+6m:cyz,then X,7nZ,my,E G=7nZ,y,m:x,'1] my,m:x,s,~ iE,'1], ~,0 andtherefore H=I{(X-m')xlr+(X+mil)2yz'1]~+yzr-2m:x"T]~l, theIimplying thesum ofsimilartermswithreference totheinterchanges between e,E;y,'1];z,~. Indeveloping thecommutant above,thefirst line may be keptina. fixedposition; forthesake of brevity, (x),(y),(z);(E),('1]),(~)maybe writtenintheplace of d d d d d d dx'dy'dz;dE'd'1]'d~' andit willreadilybe seen thattheonly effective arrangements willbe asunderwritten: (x)(y) (z)(x)(y) (z) (e)(y)(z) (x)(y) (z)(x)(y)(a)(x)(y)(z) (E)('1])(~)('1])(~)(E)ro(E)('1]) (E)('1])(~)(t)(E)('1])('1])(~)(E) (x)(y) (z) (e)(y)(s)(x)(y) (z)(x)(y) (z) (x)(y)(z)(x)(y) (z) (x)(z) (y) (x)(z) (y) (~)('1])(E)(z) (y)(x)(y)(x)(z)(y)(x)(z) (E)('1])(t) (E)(~)('1])(E)('1])(t) (~)('1])(E)(E)('1])(t')('1])(E)(t') (E)(~)('1])(E)('1])(~) (~)('1)(E)(E)('1])(n('1])(E)(~)(E)('1)(~) (x)(y)(z)(e)(y)(z)(x)(y)(z)(x)(y)(z)(x)(y)(z)(x)(y)(z) (x)(z)(y)(x)(z)(y)(z)(e)(y)(z)(y)(x)(y)(x)(z)ty)(e)(z) ('1])(t)(E)(t)('1])(E)(E)(t)('1])(t)(E)('1])('1])(n(E)(E)ro('1]) (~)('1])(E)('1])(t)(E)(t)(E)('1])(E)(~)('1)(E)(~)(11)('1])(t)(E) (x)(y)(e)(x)(y)(z)(e)(y)(z)(x)ty)(z)(x)(y) (z) (e) (y)(z) (y)(z)(x)(z)(x)(y)(y) (z)(x)(y)(z)(x)(z)(x)(y)(z)(x)(y) ro(E)('1])('1])(~)(E)(E)('1])(t)('1])(~)(E) (E)('1])(~) (~)(E)('1]) <E)(~)('1])('1])(n(E)('1])(~)(E)(E>('1])(n (~)(E)('1])(E)('1)(~) 42JOnthePrinciples ofthe Calculus ofForms. 311 Thesignsofthefour lines in each of thesearrangements are two alike, and twocontrary tothesignsof thecorrespondent lines in the first arrange­ ment;hencetheeffective sign is thesame for all, and the result.after rejecting from each term thecommon factor -16.is seen, from inspection, tobe 4(A-m')'-8ml+6(A-TILl)(>.+ml)l-12m(A+m')+2(>..+ml)'+1. which is equal to 12>,,1+0.>..'-12(m-m4)>..+1- 20ml-8ma; herethecoefficients m-m4and 1- 20ml-8maarethetwoinvariants (Aronhold's SandT)forthecanonical form operated upon;anditwill beobserved that (1-20ml-8ma),+64(m -m4)'=(1+8ml)'. which is easilyproved to bethediscriminant of xl+'!t+zI+6mxyz. Itmay however be observed, thatthisis notthediscriminant ofthe function in >..justfound, as reasons of analogy "mighthavesuggested it probably wouldbe:in order thatthismightbethecase. the coefficient of>,,1should be 4 insteadof 12, and of >..,m-ln4insteadofm4-m.Thereis ground for supposing thatanotherfunction of >..maybefound by a different method, in which thisrelationwilltakeeffect. The theorem above given for simple totalcommutants admitsof an interesting application tothegeneralcase of a function Fof thenthdegree, inrespectto each of two independent systems of twovariables e,y;E."1. LetFbe symbolically represented by(aa;+by)"(aE+fJ-IJ)", sothata"a" represents the coefficient of xlIE",nan-1banofxn-1Yr, &c.&c.;thenthe commutant a, b, (1) e,b. (2) a, b, (n) a,{3, (1) a,{3, (2) a,{3, (n) willrepresent aquadratic invariant ofF,which will contain(n+1)'coefficients. Byexpanding thiscommutant weobtainageneralexpression forthe invariant underaveryinteresting form. *Thebiquadratic function ofx,yhavingonly one parameter, andtherefore twoinvariants, itatheory poseesses strikinganalogies to thetheoryof the cubic function of threeletters. The function in >.which gives these invariants for thefirst-named function. according tothemethod givenin thefintsection, has the same discriminant as thefunction itself. 312 OnthePrinciples oftheCalculusofForms. [42 I nowproceedto give the generaltheorem forcompound totalcommutants asapplicable to the discovery of invariants. Lettherebeafunction of mdisconnected classesof systems of variables ; let the systems in thesameclassbe supposed all distinctbutcogredient with one another. The function is supposed to be linear in respect to each system in each class,andthenumber of systems is thesame for all the classes,and thenumberof variables the same in each system. This function maythenberepresented symbolically underthe form (Ial0I~I+Ibl.IYI+...+ III •I~)(I~•IXil+Ibl0IYI+ 0••+Ilil•I~) ..0(Ian.IX"+Ib"•Iy"+..oIl"•It,,) x(I~•'~+ilbl•'YI+...+ 211•I~)e~.IX2+IIbIolYI+.00+Il, 0~) ...('a".IX"+'b".Iy"+...-i; 0~,,) x&c. x(p~.p~!+pbl•pYI+.o.+PlI0P~)(P~.PXil+pbil 0P'!J2+.. 0+Pl2.P~) .0.(Pan.Px"+pb" 0PYn+0"n,.Ptn). Inthisexpression thee,Y...t'sareall real,butthea,b...fsallumbral; in fact,fag,Ibg,&c.may beunderstood to denote d~,d/-d-,&co XgYg Thensystems of variables in each of thesets above writtenaresupposed to becogredient inter88. Take the symbolical productofthefirst set, first makingforthemoment IXI=loXt=0"IX"=e,&c.&c., I~=I~=...It"=t; and let the coefficientsof the several terms a;n,a;n-Iy...&c., be called wherep.is thenumberoftermscontained in a homogeneons function of the nthdegree of the mvariables ai,'!J...t.Inlikemannerproceed with each ofthelines, and thenwrite down the commutant I111J1112111"., 2111>11111211"., Thiscommutant is aninvariant ofF:it will of course be remembered that, unlesspis even, the com mutantvanishes. 42]OnthePrinciples ofthe Calculue ofForms. 313 Thus,for example, taketwo sets of two systemsof twovariables: in all foursystems, e,Y;E,'TJ:p, q; C/>,V. each couple of systems oneitherside ofthecolon (:) beingcogredient inter86:andletFbe symbolically represented by (ax+by)(aE+fl'TJ)(lp+mq)(Xc/>+JA-,y); thentheinvariant given by thetheorem will bethecommutant aa;afl+ob;bfl, lx;1p.+Xm;mJA-. Thesix positions of thisareasbelowwritten(thefirstthreebeing positive andthe second threenegative) act;afl+ab;bfl,atX;afl+ab; bfl, aa;afl+ab; bfl, lX;lJA-+Xm; mu,1p.+Xm;mp.jlX,mJA-;lX;lp.+Xm. aajafl+abjbfl, aa; afJ+ab; bfl, atXjafl+ab; bfl, lp.+Xm;lX;mJA-,l>..jmp.;lJA-+Xm,mJA-jlp.+Am;tx. 1£wewriteFunderitsexplicitform, .AxEpc/>+BxEp,y+OxEqc/>+DxEqt +.A'X7JPc/>+B'X7JP'Ir+O'x.,.,qc/>+D'X'TJq,y +A"yEpC/>+B"y~pt+O"yEqc/>+D"yEqV +A"'YfJP9+B'''Y'TJPY'+0"'Y'TJq9+D"'Y'T/qV, wehaveidentically therelations following, . aalX=A,aa1JA-=B,atXrnX=0,aamp.=D, afllX=A',afl1p.=B',aflmA=0',aflmp.=D', ba1>..=A",balJA-=B",bam>..=0",bamp.=D", b/31>..=A"',bfllp.=B''',bflmX=0"',bflmp.=D''', andthecommutant expanded becomes A(B'+0"+0'+B")D'''+(B+0)(D'+D")A'"+D(.A'+.A")(B'"+0"') -(B+O)(A'+.A")D'''-A(D'+D")(B'"+0"')-D(B'+0"+0'+B")A"'. Intheforegoing thea/sintheseveral lines were forthemomenttaken identical,inorderthemore easily to explainthelaw of formation of the 314 OnthePrinciples ofthe Calculus ofForms. [42 quantities A.Butsupposethattheybecomeactuallyidentical forthesame line.Fthenbecomes a function of the nthdegree in respectto eachof psystems of variables, and may be represented symbolically undertheform (1alx+Ibly+...+Illt)"X(Ialx+'b'y+ +'lit)" ...x(PaJ>x+PbPy+ +PlPt)". We may still furtherlimitthegenerality ofthetheorem bysupposing IX='X=...Px=X, ly=Iy=...Py=y, Fthenbecomes (ax+by+...+uy« Accordingly, asmanydifferent factors as can be found contained aneven numberof times in the exponent of the function, so manyinvariants canbe formedimmediately from a function of any numberof variables mbythe method of totalcommutation. Ifone of these factors be called n,the commutantcorresponding thereto will be of theorder (n+l)(n+2) ...(n+m-I) 1.2...(m-I) inrespectto the coefficients. Thus takem=2, sothat Thegeneral form of such a commutant will be found by taking ..4.lJ..12'"An+lthecoefficients of the several combinations ofe,yin (ax+by)",from which thenumerical coefficients n, in(n-I),&c. may be rejected, as only introducing anumerical factorintotheresult;thecom­ mutantwilltherefore be expressed by means of theform (1) (2) (P) IfP=2, the compound commutant 42JOnthe Principles oftheOalcuhuofForms. 315 80thatwilleasilybe seen to be only anotherform for thecatalecticant of(a3:+by)'In. Thus,letn=2, (a»+by)4=Ax4+4Bwy+6Cx'y'J+4Dx'!/+Ey; a4=A,a'b=B, a'b'=C, ab'=D, b4=E. Thecommutant (which is of theform of thematrixtoanordinary determinant, withtheexception thattheumbrmentercompoundly instead ofsimplyintotheseveraltermsseparated bythemarksofpunctuation), will be a"ab;»., a"abjb';, this,writteninthesix forms at.ab;b)a"ab; ~}al.ab;b}, , , a"ab;b'a"b" ab;a"bl , ,, , al.ab;b}al.ab; ~}al.abo :b}, , , , b'·ab;atabjbI' b'.al. , ,,, givestheexpression thatis ACE-ADa-ERa-C3+2BCD. Oneimportant observation may here be made of a fact which otherwise might easily escape attention, which is, thatcommutants, wherethesame termssimpleorcompound are found in all or several of thelines,ingeneral give rise toproducts, some ofthemequaland with thesame sign, and others equalbutwiththecontrary sign. Thislastphenomenon doesnotmanifest itselfincommutants appertaining tofunctions of twovariables of the two particular anddifferent species which firstand most naturally presentthemselves, namelywherethereare only two lines or only two columnss-c-I believethatitdisplays itselfin every other case of commutantives tofunctions of two variables. Thusitisthat algebraical expressions derivedfromgivenfunctions disguisetheirsymmetry; tomakewhich come tolightitbecomes necessary to add termsofcontrary sign to such expressions. As an example, thereaderisinvitedtodevelope thecubicinvariant of afunction ofxandy,symbolically expressed by (ax+by.."where •Thesecommutants give respectively the quadrinvariant and thecatalecticant, the former ofwhichalonewasformerly recognised byMrCayleyasaoommutant. 316On thePrinciples ofthe Calculus ofForms. [42 by means of the commutant a2ab,b2,, a2ab,b2,, a2ab,s,, a2ab,b2 .- , Suppose Fto be the generaleven-degreed function of two variables ofthedegree2np. Let H=(E;y-"1:xrp F+X(xE+YTJ)np, andexpressHumbrally underthe form (ax+by)np(IlE+fJTJ)np• *[See p. 846below.] The number oftermsresulting from the independent permutation of each of the8linearlinesis61,thatis216;buttbeactualresultis(usingsmalllettersinstead oflarge)P-Q,where P=aei+8ag2+12beh+8c'i+24cj2+24d2g+lx', Q=4alh+4bid+8bgf+22ceg+ScM+86def, sothattheeffective number ofpermutations isonly164. The difference between thisand 216dividedby 216 may be termedtheIndexofDemolition, whichwe see in thiseaseisMorH; thatis,somewhat lessthan1.Forthecubicinvariant ofthefunction ofthefourthdegreethis indexiszero.allthepermutations being effective. Ifwetakethecubicinvariant of thefunction azl2+12b2:lIy+66czl°y2+&C.+my12undertheformP-Q.weshallfind P=6ahl+Waif+6/{fm+1S4bhk+54cfl+155cii+10ddm+430dgj + 155eek+520ehh+520ffl+280ggg, Q=agm+15ai.l:+80bgl+50bij+15celll+40gle+150chj+80del+210dfk +250dhi+230,:6+555egl+660lgh. Thenumber oftermsinPandQis ofcoursethesame,andwill befoundto be 2200 for each; sothatoutofthe6°,thatis 7776permutations ofthe5 lower rows, only 4400areeffective, and theindexofdemolition becomesHH.thatisHi,orrathergreaterthan/,;.TheIndexoCDemo­ litionthusgoes on constantly increasing asthedegree of the function rises;probably (?)it converges eithertowards ~or elsetowards unity.Inarranging thetermsit will be foundmost convenient toadopt,as I have done above. thedictionary methodofsequence. Thecomputations aregreatlyfacilitated bythecircumstance of the effect of anyarrangement ofeachof the 5 lower linesnotbeingalteredwhentheseline,arepermuted withoneanother; thisgives rise to the subdivision ofthe7776permutations intogroupsasfollows: 6 of 120 identical terms,60 of 60, 36 of 20, 60 of 30, 24of 20, 30 oC10, 30 of 5, and6oC1. Sothatthetotalnumberofpermuta­ tionalarrangements to beconstructed is only 252. Othermethods ofabridging thelabourwill readilysuggestthemselves tothepractical computer. Thetotalnumberofthegroup'ofterms is ofcoursealwaysknowniipl'iOri,and,forinstance, in the case before us, mustbeequaltothe number of waysinwhich ~(12x 8),thatisthenumber 18,can be divided into8parts,noneof whichis to exceed the number 12,thatis26;forthecubicinvariant of thefunction ofthe eighthdegree of two variables it isthenumber of ways in which12canbe divided into8parts, of which noneshallexceed 8, andsoforth,zerosbeingalwaysunderstood to beadmissible; andofcourseingeneralfor aninvariant oftheorderr to afunction ofthedegree n of ivariables, thenumber ofdistincttermsis ingeneralthenumber of ways in whioh ~canbedividedinto t rparts,ofwhichnoneshallexceedn,subjecthowever alwaystothepossibility inparticular oases of a diminution inconsequence of some of thegroupsassuming zero fortheircoefficient. 42JOnthePrinciples ofthe Calculus ofForms. 317 Thecommutant a"-Ib b", an-If) (3", a"-If)(3",a"-Ib'"b", a"-Ib...b", CIo", a",a'"an, (1) (2) (p) (1) (2) a",a"-I~...(3", (p) will beafunction of A,and all the several coefficients will be invariants ofF·. Whenp=1 weobtaintheAgiveninthepreceding section, and origin­ allypublished by me in thePhilosophical Magazine forthemonthof Xovember, 1851. The Aobtained onthissupposition has for its coefficients a series of independent invariants. commencing withthecatalecticant and closingwiththequadratic invariant. Whenphas any othervalue, we observeasimilarseriescommencing withacommutantive invariant of a lowerorderthanthecatalecticant, butalways closing with thequadratic invariant. Thus, for example, when 2np=8.we mayobtainby thepreceding theorem threedifferent quadratic functions; onegivingtheinvariants ofthe orders 5, 40.3, 2,thesecond those of the orders 3, 2, thethirdtheinvariant oftheorder2. Inthiscasetheinvariants ofthesameorderglvenbythedifferent A's arethesame to numerical factorspres.Whether thisisalwa.ysnecessarily thecaseisapointreserved for furtherexamination. Thecommutants appliedin thepreceding theorems have been called by metotalcommutants, because the totalof each line of umbras is permuted ineverypossible manner. It'thelines be dividedintosegments. andthe permutation be local for each segment insteadofextending itselfoverthe whole line, we thenarriveatthenotion of partialcommutants, to which I have also (in concertwithMrCayley) giventhedistinctive name of Intermutants. Inorder to find theinvariants offunctions of odd degrees. thetheoryoftotalcommutants required theprocess of commutation to be applied, not immediately tothecoefficients of theproposed function. butto somederivedconcomitant form. I became early sensible ofthisimper­ fection,andstatedtothefriend above named, to whom I had previously • Bysubstituting the symbols ~,~,&c.in place 01theumbrtZla. b,&c.,thetheorem i.Be&8ilystatedlorcovariants generally. Butinapplying thecommutantive method to obtain covariants, orratherin thestatement of theresultsflowing from eachapplication, it is never neoe&88JY togo beyond the caseofinvariants, because the commutantive covariants 01any given homogeneous function arealwaysidentical withcommutantive invariants 01emanants 01the samefunction. 318OnthePrinciples oftheCalcubu«ofForms. [42 imparted mygeneralmethod oftotalcommutation, myconviction ofthe existence of a qualified or restricted method ofpermutation, whereby the invariants ofthecubic function, for instance, of two and of threeletterswould admit,withouttheaid of aderivedform, ofbeingrepresented. Manymonths ago, when I wasengaged inthisimportant research, andhad made some considerable stepstowards therepresentation oftheinvariant, thatis,the discriminant ofthecubicfunction ofIXandy,undertheform of a single permutant, Iwassurprised by anotefromthefriend above alluded to, announcing thathe hadsucceeded in fixing theform of the permutant of which I was atthatmoment insearch.Itis with no intention ofcomplain­ ingofthisinterference onthepartof one to whose example andconversation Ifeelsodeeplyindebted, (andtheundisputed authorofthetheoryof Invariants,) thatI may be permitted to saythat,independent of theinter­ ventionofthiscommunication, Imustinevitably havesucceeded inshaping mymethod 80as tofurnishtheform in question; andthatwithgreater certainty, aftermytheoryofcommutants hadfurnished me with theprece­ dentofpermutable formsgivingrise totermsidentical invaluebutaffected withcontrary signs. As I have understood thatMr Cayley is likely to developthispartofthesubjectin thepresentnumberoftheJournal,itwill betheless necessary for me to enteratanylengthintothetheoryofpartial comrnutants onthepresentoccasion. Themethod ofpartialcommutation is asimplebutmostimportant corollary from thatoftotalcommutation hereinbefore explained. To fix theideas, conceive a class of pcogredient systems, and thatthereareqr such classes perfectly independent. Proceed to divide theseqrclassesin anymannerwhatever intorsets, each containing qclasses;and form the symbol of thetotalcommutant corresponding to each such set. Now let thesecommutants beplacedside by side againstoneanother, andtranspose thetermsin each compound linethusformed once for all, butinany arbitrary manner. Thenpermute in every possible way all those symbols in each line, interS6,which belong to thesameclass,andoperatewiththe symbolsthusproduced by readingofftheverticalcolumns and attending to therule ofthe+and - signs, asinthecase of a totalcommutant; the resultwillbeacommutant oftheformoperated upon.Forinstance, let p=1,q=3,r=2, and let thenumber ofvariables in eachsystembe2. Formthecommutant operators d d dd da;'dy'dE'd",, d d d d dp'dt'dt/>'de' d d d d dr'ds'dp'du 42] OnthePrinciples ojtheCalculus ojForms. 319 {dddddd}d:cd'"dsUxdddUI."YPP ddd ddd ' xdEdedrUxd",dtanU Ubeing supposed tobe afunction homogeneous in e,y;E,",;p,t;4>,e;r, 8;P,a,Interchange inanymannerbutonceforallthesymbols in each line, asthus: d d d d d:c'dy'dE'd",' d d d d d4>'dpIdtIde' d d d d dsIdp'dr'da' Nowpermute,interBe,thevariables of each system, as d d d d do:'dy;dp'dt'&c.; thetotalnumber oftheoperative formsresulting will be (1 . 2)8,and the sum ofthe(1.2)8quantities, halfpositiveandhalfnegative, -formedafterthe typeof d dy' d· dy'd da;' d da;'willbeacovariant ofU. Theproof of the truthofthisproposition is contained in what is shown intheNotesoftheAppendix for total commutants, itbeingonly neces­ saryto make thesystems which are independent vary consecutively, and thenapplythe inference to thesupposition oftheirvaryingsimultaneously. Itmay be extended to the more generalsupposition of classes of an unequal number ofcogredient systems of unequal numbers of variables ineach,theonly condition apparently required beingthatthenumberof distincttermsshall be thesame in each line of thefinalcommutantive operator. The important remarkto be made is, thatinapplying this theoremthereisnothingtopreventany ofthesystemsbeingmadeidentical; or,inotherwords, a given function of one system of variables may be regarded asafunction of asmany different, although coincident, sets aswe may choose tosuppose. Thus, suppose U=..da;ll+2Ba;y+Cyl, we may takethepartialcommutant formed of the two totalcommutant operators 320OnthePrinciples oftheOalculusofForms. [42 combined with itself. Ifwewritetheminthesame order, d d d d dx'dy'd;e'dy' d d cid do'fly'dx'dy' (where I use thedots and dashes to distinguish those inthesame line which are considered asbelonging tothesame class, and therefore aspermutable, interse),we shall evidently obtain4{AO-.8')'; if we commence with a. permutation, so as to have theform ofoperation d ddd do:'dy'de'dy' dcicid do:'dy'dx'dy' it will be found thatweobtain2{AO-.8')2. Again, suppose thatwe have U=Aa;I+3Bwy+30x~l+Dya. Ifwewrite dddd dx'dy'dx,dy' d d d ci dx' dy' dx'dy' ddcici dx'dy'dx'dil' thevalue of thecommutantwould come out zero;butif we make a permu­ tation,and write d d dci dx' dy' dx'dy' ddcid dx'dy'dx'dy' dcidd dx'dy'dx'dy' theoperation indicated bytheaboveperformed uponU,will give a multiple ofthediscriminant ofU. 42JOnthe Principles oftheCalculus ofForms. 321 In likemannerwe mayrepresent Aronhold's SexticInvariant oftheform (e,y,z'rby means of the partialcommutant ci. ci d d d d ck'dy'dz'ck'dy'ds ' ci ci. d d d d ck'dy'ds 'dx'dy'dz' d d d d d ci ck'dy'dz'ck'dy'dz Ifwe make v=(rd~+TJ'~+r:,)(ftx+TJd~+,~r(x,v.z'f, anduseHto signify thedeterminant x,y'z whichisevidently anuniversal triplecovariant, and make W=V+xH, andapply to Wthepartialcommutantive symbol d d d ddd ax'dy'dz'de'dy'dz' d d d d d ci df'dTJ'd"df'dTJ'd" ciddddci df'dTJ"d'"df'd?'dr' we shallobtaina function of Xof which all the odd powers and thesecond powerwilldisappear, and such thatthe coefficients of XIandtheconstant term willbeAronhold's SandT,andthediscriminant oftheentirefunction inrespect to XI(if not for the distribution assigned tothedots and dashes in the foregoing. atleast for some otherdistribution) maynotimprobably be the discriminant ofthe given function (x,y,z'f. & 21 322OnthePrinciples oftheCaleulu«ofForms. NOTES INAPPENDIX.[42 (1)[po295 above.] More generally, in as many ways asthenumbern can be divided into parts,in so many ways can a given function of one setof variables be as it were unravelled so as to furnish concomitant forms. Forinstance, theformar+3bx'y+3cxy'+dy'has foraconcomitant au.x+buy+bvx+cvy+cwx+dwy, whereu, v,warecogredient withx',2xy,y2;and also auu'x+buu'y+buv'x+bvu'x+cvv'x+cvu'y+cuv'y+dvv'y, whereu,Vju', v'arecogredient with each otherand with xandy;andthe proposition in the textmay bebestderived from thismoregeneraltheorem bydividing the index into equal parts,forming as many systems as thereare suchparts,andthenidentifying the systems so formed. (2)[po297 above.] The following additional example will illustrate the power of thismethod. Letq,=(x,y,z)4be thegeneralfunction of thefourth degree. Form by unratelment theconcomitant form(u,v,w,p,q,r)'(sayP)whereu,v,W,p,q,r arecogredient withx',y',z',2zy,2xz,2yx. Again,theuniversal concomitant (xE+Y"l+zn'will have for its con­ comitant whereE,"1,~arecontragredient tox,y,e.Now take the reciproca.l polarof thislastform with respecttoE,"l,?,";thatis, !(vw-lp') Xl'+2~(lqr-lpu) 'YIZI(sayG), whereXt,Yl>Zl'beingcontragredient toE,"1,"will becogredient withx,y,z. P+xGisaquadratic function of the six variables u, v,w,p, q,r, andits discriminant will give afunction of Xof thesixthdegree, all of whose even coefficientswill be covariants ofq,.Ifwe replace Xl'YI'Zlbyx,y,z,these even coefficientswill be respectively (understanding thatorderrefers to the dimensions quoadthe coefficients of q,anddegreeto the dimensions quoad :r;,y,z)asfollows: 42J OnthePrinciples ojthe Calculus ofForms. 323 Oforder6degree0, "5 2, "4"4, "3 6, "2"8, "1"10, "0 12. Thetwolastcoefficients mustevidently beidentically zero.Itispossible thatsome oftheothersmay be so too:asregardstheone ofthethirdorder andsixthdegree,thisis ofthesame form as,andmay beidentical with,the HessianoftP;asregardstheone ofthefourthorderandfourthdegree,this may be tPitselfmultiplied bythecubicinvariant (whichthetheoryof SectionIII.proves to exist)oftP.Butthecovariants ofthefifthorder andsecond degree, andofthesecondorderandeighthdegree,iftheyare notidentically zero,andifthelatterisnottPl(which a trialor two of some verysimplecaseswill easily establish one way or theother)areprobably irreducible forms.Theexistence of acorrelated conicsectionto a curve ofthefourthorder,ifestablished, would be particularly interesting, andits geometrical meaning would well deservebeingelicited. (3)[po303 above.] Ifany form (f)ofthedegreenbewrittensym­ bolically, (~XI+~X.+...+a,x.)", where XI'a;...X,arerealbut~,~...a,umbral,and ifI;be anyinvariant oftheorderrinrespectofthereal coefficients of (f),itis easily seen by reason ofI;remaining unaltered when XI'a;...e;become respectively fl:r;,f.x. ...j.x"provided thatfl,f•...j.=1,thateachterminI;expressed bymeansoftheumbne,mustcontainanequalnumberoftimes ~,a....a" sothateach such termwillcontainnrof each of them,of course differently £ subdivided andgrouped; hence we have theuniversal condition thatnrmust £ beaninteger; butthisis lessstringent thantheactualcondition, which isthat~mustbeanintegerofacertainform;forinstance, asbefore £ observed, when £=2,nrmustbeaneveninteger. £ (4)[po307 above.] To prove thetheorem given inthetextfortotalsimple eommutants itis onlynecessary tobearinmindthatwhenever twocolumns inanytotalcommutant becomeidentical, thecommutant vanishes. To fix the ideas, takethecommutant formed of lines similarto:X'd~'~,written 21-2 324On thePrinciples ojtheCaltmlusofForms. [42 underoneanother; lettherebersuch lines, thetotalnumber ofterms will be (1. 2. 3Y:the1.2. 3 positions of thelinewrittenabove will corre­ spond to (1.2. 3)~1severalgroupings oftheremaining lines. Now when e,y,zundergo aunimodular linearsubstitution,te,d~'d:willundergo arelatedsubstitution notcoincident withthatofa;,y,z,butstillunimodular; lete,y,Zchange, all theothersystems remaining fixed, and suppose d ddb' Ide'dy'dzto ecome respective y d d dfda;+9 dy+hdz' f' dg'd h' d da;+dy+dz' f" d"dh" d da;+9 dy+dz' theneach ofthe(1.2. 3)~1groupsofthetermsarisingfromthepermutation ofte,~,~willsubdivide into27 groups, of which we may rejectthose in which any of theterms(te,~,~)occurs twice or threetimes;accord­ inglytherewill be left only thesix effective orders of permutations, rfd,dh"d)rfdh,d"d)& ~da:'9dy'dz; ~da;'dz'9dy;c. consequently each ofthe(1.2. 3)~1groupsgives rise to 6 times6products f",g",h" whose sum will be f',g',h'xthesum of the 6 products corresponding f,g,h tothepermutations ofte,~,tz;andtherefore, thetransformations being unimodular, thesum oftheproducts corresponding totheentire(1 .2.3)" permutations remains constant whene,y,zchange. Inlikemanner,allthe systemsmaychangeoneaftertheother, and consequently all ofthematthe sametimewithoutaffecting thevalue ofthecommutant: and in like manner forthegeneralcase. Q.E.D. (5)[p.312 above.] The truthoftheproposition relativetocompound commutants andthemode of thedemonstration willbeapparent fromthe subjoined example. Letthefunction be supposed to be (cue+by)(a'a;'+b'y')(erE+f1-q)(tt'E'+1'",'), 42J Onthe Principles oftheCalculusofForms. 325 wheree,y;:c'.y'arecogredient and ~.'7];~'.'7]'cogredient; thea,b,IX,{:J,&c. areofcoursemereumbne, Nowtakethecompound commutant aa', ab'+ab,bb'. all',a.{:J'+a'{:J,{:J{:J'• Lete,y;:c',y'undergo alinearsubstitution, and,accordingly, letabecomefa+gb, a'"fa'+gb', b"ha+kb, b'" ha'+kb', f,g,h,kbeingofcourseactualand notumbral; thentheabove com mutant will beeasilyseentodecompose into6others,which will be equaltothe originalcommutant multiplied bythedeterminant r.2fg,gl, fh,fk+gh, gk hi,2hk, Ie'- which is equalto(fk-gh)8,thatis=1. Andso ingeneral, which shows, as 10thepreceding note,thatallthe classesofcogredient systems may be transformed successively one after theother,andtherefore simultaneously, without altering thevalueofthe commutant, (6)InthelastMayNumber- oftheJournal, Mr Boole,to whose modest laboursthesubjectisperhapsatleastasmuchindebted as toanyoneother writer,hasgivenatheoremt, (14) p. 94, theexcellent ideacontained in whichthereis nodifficulty inshaping 80as torenderitgeneralizable by aid ofthetheoryofcontraveriants. Itmay beregarded in somesortapendant orreciprocal totheEisenstein-Hermite theorem, presented by meundera wideraspectintheFirstSectionofthispaper. [-Camb.andDub.Math.Joum,Vol.VI.(I8oI),pp.87-106.] tHrBoola applied his theorem toobtainthe cubic invariant of(z,y)4,aayIf>(z,y),by opurating upon its Hessian withIf>(:y'-tx). Moregenerally, whenIf>(z,y)=(z,y)"',the catalecticant oftheantepenultimate emanant ofIf>is alsoofthe degree 2n;andthis,when operated upon by If>(~.-:z).willgiveaninvariant of the order n+I, which isprobably identical withthecatalecticant ofIf>itself.Thereexists.mostinteresting transformation of the catalecticant ofanyemanant of•function of any degree in x,y.whethereven or odd, underthe form ofadeterminant some of the lines of which containcombinations only ofzandy,without anyofthecoeffioients, and all the rest the coefficients only of the given function withoutzory. TheHessian being the caialecticaut of the second emanant isofcourseincluded withinthis lltatement. 326OnthePrincipia ojtheOalculueojForms. [42 Letep(x,y...z)have any contravariant e(x,yz);thenwill ep(tx,~t).e(x,yz) be acontravariant ofep.Fororthogonal transformations thetermscontra­ variantandcovariant coincide, and the theorem forthiscaseappearstohave been known to Mr Boole, see (15), same page. More generally,if"and8 be any two concomitants ofep,thealgebmical product'lr0will also be a. concomitant ofep,provided thatthesystemsofvariables in"andehaveall distinctnames, or thatthose which bearthesamenamesarecogredient with oneanother.Ifthisproviso does nothold good, the productinquestion will evidently be nolongeraconcomitant ofep.Lethowever'I'denotewhat+ becomes, and I;:twhat0becomes, when in place of thevariables x,y...z of every two contmgredient synonymous systems in'Irand0wewrite ::X'd~"''''~,thenwillI;:t"and'1'0be each of themconcomitants ofcfJ, thesynonymous systems becoming cogredient with'Irintheone case and with0intheother. (7)Thereis oneprinciple ofparamount importance which has not been touched upon in thepreceding pages, which I am very far from supposing toexhaustthefundamental conceptions ofthesubject,(indeed,not toname otherpointsofenquiry, I have reason to suppose thattheidea ofcontra­ gredience itselfadmitsofindefinite extension through themedium ofthe reciprocal properties of commutants ; theparticular kindofcontragredience hereinbefore considered havingreference to the reciprocal properties of ordinary determinants only). Theprinciple now inquestion consists in introducing theidea ofcon­ tinuousorinfinitesilnal variation intothetheory. 1'0fixtheideas,suppose Cto beafunction ofthecoefficients of ep(x,y,z),suchthatitremains unaltered whenx,y,zbecomerespectively Ix,gy,hs,provided thatIgk=1. Next,supposethatCdoes not alterwhenxbecomes x+ey+es,wheneand eareindefinitely small:it iseasilyand obviously demonstrable thatifthis betrueforeandeindefinitely small, it mustbetrueforallvalues of eandE. Again, suppose thatCaltersneitherwhenxreceives such an infinitesimal increment, yandzremaining constant, nor when ynorzsepamtely receive corresponding increments, s,xande,yintherespective casesremaining constant; itthenfollows from whathas been statedabovethatthisremains truefor finite increments toxoryorzsepamtely; andhenceitmay easily be shown thatCwillremainconstant for anyccmcurrent lineartransforma­ tionsofe,y,e,whenthemodulus isunity.Thisall-important principle enablesusatonce to fix theform of the symmetrical functions oftheroots ofep~,1)whichrepresent invariants ofep(x, y)whenthecoefficient of the 42J On thePrinciples oftheOalcullUJofForms. 327 highestpower of xis made unity. Italsoiru;tantaneously givestheneces­ saryandsufficient couditious to which an invariant of any given order of any homogeneous function whatever issubject,andtherebyreducestheproblem ofdiscovering invariants to a definite form. Butas theseconditions coincide withthose which have been statedto me as derived from otherconsiderations by thegentleman whose labours in thisdepartment areconcomitant with my own, I feel myself bound to abstainfrom pressing my conclusions untilhe has given his resultstothepress. (8) By aid of the general principle enunciated in Note (6) above,we can easilyobtainAronhold's SandT.LetITbethegiven cubic function of e,y,s,and letG(x,y,z;E,,,,, ~)bethepolar reciprocal in respecttoE,""~ (d d d)2ofEdx+",dy+~dzU,thenG(E,""~je,y,z)as well as theformerG will beaconcomitant toU,butthehomonymous systems of variables in the twoG'swill becontragredient; and, accordingly, (d d d d d d)Gdx'dy'dz;dE'd",'d~.G(E,""~;e,s.z) willbeaconcomitant toU;thisconcomitant is readily seen to be an invariant of thefourthorder;thatis, Aronhold's S.Again, from S,by means of the Eisenstein-Hermite theorem, we may derive a form K(x,y,z) ofthethirddegree in e,y,z,and whose coefficientswill be of threedimen­ sionsjand, accordingly, if the HessianofUbe calledH (U), (d dd)Kdx'dy'dz.H(U) willbe a Sextic Invariant ofU,thatis, Aronhold's T. 43. ONTHEPRINCIPLES OFTHECALCULUS OFFORMS. [Cambridge and Dublin Mathematical Journal, VII.(1852), pp. 179-217.] PARTI.SECTION IV.Reciprocity, alsoProperties andA1ULlogies ofcertainInvariants, cf:c. ITwillhereafter be found extremely convenient torepresent allsystems of variables cogredient withtheoriginal system in theprimitive formby lettersoftheRoman, and all contragredient systems by lettersoftheGreek alphabet; the rules for concomitance may thenbeappliedwithout paying anyregardtothedistinction between thedirection ofthemarchofthe substitutions, the variables at the close of each operation asit weretelling theirowntaleinrespectof being cogredients orcontragredients. This distinction hasnot(asitshould have) been uniformly observed in the preceding sections; as,forinstance, inthenotation foremanants whichhave been derived by theapplication ofthesymbol(Etx+71:y+&cy,instead ofthemoreappropriate one(IV'fa:+y'~+&cJ. The observations in thissection will refer exclusively to points of doctrine which have been startedin thepreceding sections in such order astheymore readilyhappentopresentthemselves. And, first,asto some important applications ofthereciprocity methodreferred to in Notes(6) and (8) of the Appendix [pp. 325, 327 above]. Thepractical application ofthismethodwill be found greatlyfacilitated bytherulethate,y,s,&c. may always in any combination ofconcomitants be replaced respectively by :E'~,f~,&c., andviceversd.I shallapply thisprolificprinciple ofreciprocity toelucidate BOrneoftheproperties and relations of Aronhold's SandT,andcertainotherkindred forms.This SandTarethequartinvariant andsextinvariant respectively of a cubic ofthreevariables. I give the names of sandttothequadrinvariant and cubinvariant ofthequarticfunction of two variables. Furthermore, whoever will consider attentively theremarks made in SectionII.of the foregoing relativeto reciprocal polars, will apprehend without any difficulty thatto everyinvariant ofafunction of any degreeof anynumberof variables will 43]On thePrinciples oftheCalculusofForms. 329 correspond acontravariant ofafunction of thesame degree of variables one more in number, andthatbetween such invariants, whatever relations existexpressed independently ofallotherquantities, precisely thesame relations mustexistbetween thecorresponding contravariants. Thus, then, to8andtthetwoinvariants of(.x,y'twill correspond two contravariants (TandTof(.x,y,z't,and toSandTthe two invariants of(.x,y,z)awill correspond IandI;:ttwocontravariants of(x,y,e,t"f.Callingrtheresultant of(x,'!I't,Rtheresultant of(x,y,z't,pthepolar reciprocal, or, more briefly, thereciprocant of (e,'!I,z't,and(R)thereciprocant of(x,y,z,t't,we have the following equations (presuming thatallthequantities are previously affected with thepropernumerical multipliers), namely r=sa+~,p=aa+r, R=Sa+Ta,(R)=Ia+~. Ipropose in thisFirstAnnotation topointout the remarkable analogies whichexistbetween themodes of generating thefour pairs of quantities 8,t,&c.,thefunctions severally corresponding to whichIshall call u,Cc),U,n. TheHessian corresponding toany of these functions will be denoted by anHprefixed, and when we have to consider, notthepure Hessian, butthe matrixformed from it by addinga vertical and horizontal border of variables, thesameinnumber butcontragredient to thevariable of the function (as, forinstance, theHessian ofubordered with E,71horizontally andverti­ cally, or of UwithE,71,t'),thenIshalldenotetheresultby the ruled symbolH,and iftherebe occasion to add two borders, asE,"l,t';E',71',t", bothrepeated inthehorizontal andverticaldirections, theresultwill be typified by the doubly ruledH. Now, in the first place, asobserved by me inNote(8) oftheAppendix inthelastnumber; if we call thecoefficients of U(10 innumber)a,b,c,d, &c.,we have S-il{d dd.d dd}il1 •f:f'}-dE'd"1'd'Ida'dy'dZ lX,'!I,z,~,"1,!>, alsoasdaH dBouesdaH T=daa;;s+dbdSxdy+dcd,2xdz+&c. Iwill now add thefurtherimportant relation IdTrPHdTdiHdTrPH • S=dada;3+dbd1xdy+dedaxdz+&c., •Itwillbefoundhereafter convenient todesignate contravariants formed in thismanner frominvariants lUIEvtct,of suchinvariants orcontravariants, and according tothenumberof Wnesthatsuchprooes8ofderivation is applied, 1st, 2nd, 8rd, &0.evects. Such evects form apecaliar olaaB.and when considered generally, without reference to thebasetowhich they refer,fueymaybetermedevectants. Evectants willbeagaindistinguishable according as their baleisaninvariant simply or a contravariant. Perhaps thetermapureandaffectedevectauts mayse"etomarkthisdistinctiou. 330OnthePrinciples ofthe Calculus ofForms. [43 d dd.1:''l"}dx'dy'dz'~,"I, \> XX{x,y,ziE,"I,~;f,"I',rj,sothatit will be observed if all the derivatives ofSare zero, Tiszero, andviceversa. Precisely in the same way.using h and Iito denote respectively the Hessian of uandthesamebordered withE,"I,we have (d d d d)-s=IidE'd~;do:'dyh(x,y;E,"I), ded4hdsd4hd.sd4h t=dadaJ4+dbdrdy+ded3f.dr+&c. dtd4hdtd4hdtd4h sI=dadaJ4+dbdx'dy+deC1:Jfdy2+&c. Again,taking(H)the second bordered Hessianofn;thatis,nbordered aswellhorizontally asvertically with the double lines and columns E,"I,~,(J ; E',"I',r,8', ~(li)(d d d d d d d d 1:',J"8') ...= dE'd"l'd'~dOjdx'dy'dz'dt;~,"I,\>' x(H)(x,y,z,t;E,"1,',8;E',"I',~/,8'), ss:dsjJ d~dsjJd'i.dSll«zdSH ~=dadr+dbdardy+de([X2dz+dddxtdt+&c., I,2=~dSi!.+~cJ,3!i+&cdadrdbdardy . Inlikemanneragain =Sd d d a=(h)utE'd"l'd'; dad4(Ii.) T=dadr+&c., 2dTd4}" a=daa:;;.+&c., uandTare the same quantities as arecalculated by Mr Salmon, ..inhis inestimable workOnHiqherPlaneOurves,butarethereexpressed under the names of SandT,withthesole difference thatin place of e,y,z,used by Mr Salmon, the contragredient variablesf,"I',~'areused in the expressions above. Mr Salmon hasalsopointedout to me thatamay beobtained byoperating with (E4:fa+E'7I;+r~tc+&c.) directlyuponIacubicinvariant of the function u,or(x,y.Z)4.This Iisnootherthanthe simple commutant obtained byoperating uponu withthecommutantive symbol formed by takingfourtimesovertheline f::x,d~'tagreeable to theremarkmadein thethirdsectionthat 43J On thePrinciples oftheCalculusofForms. 331 everyfunction of an even degree of 71variables possesses an invariant of thenthorderinextension of Mr Cayley's observation thatevery such function of twovariables possessesa quadrinvariant, thatis aninvariant ofthesecond order. I needhardlyremarkthatais of 2dimensions inthecoefficients and of 4.inthecontragredient variables, Tof 3inthecoefficients and of 5 in the contragredients, :£of 4 intheconstants and 4 in thecontragredients, IJof 6 intheconstants and 6 in thecontragredients, orthatthesingle-bordered Hessians ofuandUandthedouble-bordered Hessians ofwandnare each ofthemquadratic inrespectofthex&c.as wellasoftheE&c.systems. Iftherightnumerical factors be attributed toS, T,Aronhold has shown that H{H(U)} +T.H(U)+&U=O, and in my paperinthelastMayNumber-, I gavetheequation h{h(u)}+s.h(u)+tu=O. Ithinkithighlyprobable thatit will be found thattheanalogous equations obtain,namely H{H(n)}+IJ.H(il)+:£2n=0, h{h(w)}+u.h(w)+TW=O. Theseremarkable equations, if verified (of which I can scarcely doubt),will be most powerful aids to thedissection of theformsw,il,andtherebytothe detection ofthefundamental properties of curves of thefourth and surfaces ofthethirddegree, of which atpresentsolittleis known. Itwill have been observed thatinthepreceding developments thecontravariants ofwandil werederivedin precisely thesame way from wandilasthecorresponding invariants ofuandUfromuandU,withthesoledifference thattheHessian used inthetwolattercasesisreplaced by asingle-bordered Hessianinthe two former cases,and asingle-bordered Hessian inthetwolatterby a double-bordered Hessian inthetwo former. The analogies are not even yet statedexhaustively; foritwill beremembered (asshown in thethirdsection), thatTandScan be derived directlyandconcurrently by means of operating withthecommutantive symbol d d d do:'dy'dz d d d d.'C'dy'dzuponH(U)+>,,(xE+'!1"7+z~)2,d d d dE'd"1'd~ d d d dE'd,,'d' [*p.192 above.] 332OnthePrinciples oftheOalculusofForms. [43 we findwhichgivesaresultoftheformm(}"I+S>..+T),mbeinganumber jand Iconjecture thatif d d d d dx'dy'dz'dt' dddd dx'dy'dz'dt' d d dd dE'dTJ, d~'dO' d d d d dE'dTJ, d~'d(J' be made to operateupon Hn+}"(xE+y."+z~+t(J)2, andtheresultbeputundertheform m(},,4+A}"I+.8},,2+C>..+D), thatAwill be zero, BandCwill berespectively Iand~,andperhaps D(acontravariant, ifiteffectively exist,of8dimensions inthecoefficients ofn,andofalikenumberinthecontragredients f,",',~',(J'),also zero. Butoftheevanescence ofDIdonotspeakwith any degreeofassurance. MrSalmonhasmadeanexcellent observation totheeffectthatif we call d d d(00)what00becomes when E',",',~'arereplaced byde'dy'dz'(oo)h(Cd) willrepresent acovariant toCdof 3+2,thatis, 5dimensions inthecoefficients, andof6 -4,thatis, of2dimensions inx,y,s,h(w)beingof3and6dimen­ sions in theserespectively, and00of2and4dimensions respectively inthe same. Now theseresulting dimensions 5and2precisely agreewiththe formespecially noticedby me in Note-(2) oftheAppendix, whereitwas derived asone ofagroupbythemethod ofunravelment. Therecan belittledoubtthatthesetwo conics eachofthemindissolubly connected witheverycurveofthefourthdegreeareidentical. Theform(00)h(Cd) enablesus to prove readily(thanks to MrSalmon's calculation of00,given in hisHigherPlaneCurves,underthenameofS)thatthisisabonafide existent conic. Forif wetakeaparticular case of Cd,say Cd=~a:'+b.y4+c.z4+6dylZI , h(Cd)= ~:x',0, 0 0,bi!l+dzl,dyz 0, dyz, c.zl+dy' =r~(btCs+ell):x'!fZI+~b.p,a;y+~c.d:x'z4, [.p.828above.] 43JOnthePrinciples oftheCa1c'ldusofForms. 333 and0"becomes andconsequently (0")is andtherefore (0")h(OJ)=4a,.'d(biM+r/J)w, the conic here reducing toapairofcoincident straight lines. This example demonstrates thattheconic is in general actuallyexistent. As I have saidso much upon 8 and Titmay not be irrelevant tostate inthisplace how I obtained theconditions forU,thecharacteristic of the curve of thethirddegree becoming the characteristic ofaconic and astraight line,thatisbreaking up into a.linearandaquadratic factor, which Mr Salmon hasinserted in the notes to his work above referred to.WhenUis ofthisformitmay obviously by lineartransformations be expressed bya:r;'+6du;yz,butwhenstartingwith the generalform, a..,:r;I+b,q+Caz3+&c.+6Dxyz, weform two coutravariants from 8 and T,to wit (r:a...+"ls~2+r~+&c.+E"1~dt)8,say8', (r~+11sd~,+rt.+&c.+'''l~dt)T,sayT', andthenmakea..,=a,D=d,and all the othercoefficients zero, itwill easily beseen on examining theformsof8 andT,given by Mr Salmon, that(8) and(T)(theevectants of8 and T)becomerespectively 4d'""t; 31do",,~; wehavetherefore (T)+A.(8)=0:and (T) and (8), although contravariantive totheirprimitive U,arecovariantive with one another, so that(T)+A.(8)=0 isapersistent relation unaffected bylineartransformations; itfollows therefore-that whenUis of,or reducible to, theform supposed. esasd8es da..,:db,:des:&C.:dD dT dT dT dT =da..,:db,:des:&c.:dD' which is the criterion given in thenotereferredto-. Iamalso able to obtaintheseequations moredirectlybyanothermethod founded upon aNew Viewof the Theory of Elimination, an account of which, •MrSalmonhasremarked iliatilietwoevectanta (S)and IT)interllect inilieninecuspidal pain&lof\hepolarreciprocal toilieClUne. 334OnthePrinciples oftheCalculusofForms. [43 however, I mustreserve for another occasion, butwhich, I may mention, serves to fix notmerelytheconditions, asin theordinary restricted theory, thatagivensetofequations may besimultaneously satisfiable by some one systemof values of thevariables, butthecO'nditions thatsuch set of equations may besimultaneously satisfiable by anygivennumberofdistinctsystems of variables. Mr Salmon hasremarked to me to theeffectthatif in Twewrite fx';y'~,in place of the contragredients, andcallTsoaltered (T), then(T)h(ee)will be an invariant of 6dimensions inthecoefficients of (I). Thissextinvariant I havelittledoubtisidentical withthatobtained by operating upon",withthecommutantive symbolssd:;y'(;yy,;y~,(;z.y,~d:}. (d)2d d(d)2d d(d)2d d ax'axdy,dy,dy dz'dz'dzde. This, like every othercommutantof 2 lines only, is of course capableofbeing expressed undertheform of an ordinary determinant, andtheremarkisnot without interest, asshowing how the proposition known with respectto quadratic functions of any numberof variables, namelyof every such having aninvariantive determinant, lendsitselftothegeneralcaseoffunctions of any even degreeof anynumberofvariables which also have always an invariantive determinant attached tothem,of which thetermsaresimple coefficients of such functions. The only peculiarity (ifitbe one) of quadratic functions inthisrespectbeingthattheyhave each butoneinvariant ofsuch form and no other. In thecase before us, if we write '"=a,.a:'+ba!!+c.z4+4u2x3y+4asx'z+4bJ!lx+4bqz+4cIzllx+4~y +6dy2z2+6ez2w+6fwy2+12lwyz+12ma:y2z+12rw;yz2, thesextinvariant IIIquestion becomes representable undertheformof thedeterminant a,.,ailf,i,e,as• a2,f,blJm,n,l f,bI,i;i;d,m i,111,bl,d,c"n e,n,d,c2,cl•c1 a~,l,l1l,n,ClJe *Thisdeterminant isidentical with the determinant formed by ta.king the second differentia.l ooefficients of the function andarranging in theusualmauner the coefficients of the several powersandcombinations of powers of the variables treatedasifthey were independent quantities, 43) OnthePrinciples oftheOaleulu«ofForms. 335 Beforequitting thesubjectofSandTthetwoinvariants of the cubic function of 3 variables, or, as itmaybetermed,of the cubic curve, itmay notbeamiss to give thecomplete tablewhichIhave formed corresponding toallthesingular caseawhich can befall such curve, which will beseen below tobeeightinnumber; itis ofthehighestimportance to push forward the advanced posts of geometry, and for thispurpose toobtainthe same kind ofabsolute power and authority over, and clearandabsolute knowledge of, theproperties and affections of cubic forms as have been alreadyattained for forms of the second degree. Let U=aa;J+4bwy+4c:rfJz+&c. (1)WhenUhas one double pointS3+T3=O. (2) When Uhas two double points, thatis becomes a conic and rightline dSdT_d8dT=0&c.&c. da db db da ' (3)WhenUhasa cuspS=O,T=O. (4)WhenUhas two coincident double points, thatis, is a conic andatangentlinethereto,which comprises thetwopreceding cases in one, dT dT da=0,db=0,&c. andalsotherefore 8=0. (5)WhenUbecomes threerightlines forming a triangle d3SdJTd2Td'J8 dadbClede-dadbdcde=0,&c. wherea,b,c,eeachrepresent any of the coefficients arbitrarily chosen, whether distinctor identical. Another, and lower in degree system of equations, maybesubstituted fortheabove,obtained by affirming the equality of the ratios between the coefficients of Uandthecorresponding coefficients of itsHessian. (6)WhenUrepresents a pencil of threeraysmeetingin a point dS dSda=0,db...0,&c. and alsotherefore T=O. Also in place of thissystem may be substituted the system obtained by takingallthecoefficients of the Hessian zero. 336 OnthePrinciples ofthe Oalculus ofForms. [43 (7)WhenUbecomes a line, and two othercoincident lines, and alsodS=O da ' d2T da2=0,dSdb=0,&C. diT dadb=0,&c. Ihave not ascertained whether thissecondsystemnecessarily impliesthe firstjIratherthinkthatit does not. Inthepreceding casealso it would beinteresting to show thedirectalgebraical connexion between thesystem formed by thecoefficients of theHessianandthesystemconsisting ofthe firstderivatives ofS. rlJS dadb=O, &C. diT dadb=0,&c.diSdai=0,(8)WhenUbecomes aperfectcuberepresenting threecoincident rightlines and Thefirst of thesesystems ofequations necessarily impliestheequations dT dT°& . bvifrh . da=0,dh= ,c.,asIS 0VIOUSomteequation dSdiH dSdaH T=dada;8+dbda;8dy+&c. butnotnecessarily thesecond and lower system ~::=0,&0.abovewritten. So if we take u=au:'+4bwy+6ca:'1+4d:xy'+ell when 2 roots are equal s'+t2=0, when 2 pairs of roots are equal dedt ds dt da db-db da=0,&c., 8=0,t=O, when 3 roots are equal and when all 4 roots are equal dt=0 da 'dtdb=0,&c. Before closing thisSectionIma.ymake aremark,in reference to thesextic invariant ofCIl,whichadmitsofbeingextended to all com mutants formed byoperating uponthefunction withacommutantive symbolobtainedby d dwritingover one another linesconsisting of powers of.do;Idy'&c.and 43]OnthePrinciples oftheOalculue ofForms. 337 theircombinations (to which, in the ThirdSection, I gave the name of compound commutant«, aqualification which, for reasons thatwillhereafter beadduced, I thinkitadvisable to withdraw). The remarkI have to make isthis, namely thattheinvariant obtained byoperating upon Co)with (ter,te~,(;yY,;y:z'(tzy.:zte}, (d\1d d(d)1d d(d)1d d cia:),do;dy,dy,dydz'dz'dzcia: isprecisely thesameasmaybeobtained byoperating with d d d d d d}du'dv'd1O'dp'dq,dr d d d d d d du'dv'dw'dp'dq'dr uponthe concomitant quadratic function to Co)obtained by the method of un­ ravelment,88 in Note (2) of theAppendix [po322 above] jand so,in general, everycommutant obtained byoperating uponafunction of any number of variables of the degree 2mpwithasymbolconsisting of2plines in which the mth powers of te,~,&c. and theirmthcombinations occur, will beidentical with the commutant obtained byoperating withasymbol d dalsoof2plines, in which only thesimple powers occur of du'dv'&C. (whereu,v,&C.arecogredient withxP,xP-1y,&c.),uponafunction of v,v,&c.,formed by the method of unravelment from the given function. Finally, before quitting thesubjectof reciprocity, Imaystate,itfollows fromthe generalstatement made at the commencement ofthisSection,that inasmuch 88 (xE+y1]+z~+&C.)I isauniversal concomitant form,so also musts:dddd )11dEcia:+d1]dy+d~dz+&c. beauniversal concomitant symbol of operation jaccordingly it is certain that any concomitant in which e,y,z,&c.,E,1],~,&c.enter,operated upon with thissymbol, will remain aconcomitaut: in several cases which I haveexamined, the effect of thisoperation willbe to produce an evanescent form,butI see nogroundforsupposing thatthisisotherthanan accidental, or at alleventsforsupposing thatitisanecessary and universal consequence of theoperation. Itmay also beobserved thatinthecaseofasmany cogredient sets of variables asvariables in each set, asforinstance 3 sets 8. 22 338OnthePrinciples oftheCalculusofForms. [43 of 3variables each,thedeterminant which may be formed by arranging them inregularorder,as iD,y,Z :c/,s'.Z' e",v".Z" isevidently auniversal concomitant, and moreover an equivocal concomitant, possessing the property ofremaining aconcomitant when the variables arerespectively butsimultaneously exchanged for theircontragredients E,7],~;E',7]',r;E",'T/"r';which shows also thatin place of the variables maybewrittenthedifferential operators d dd.d d d.d d d d:r;'dy'dz'd9;"dy" dz" d:r;'"dy'"dz": aremarkwhich leads us to see theexactplace inthegeneraltheory occupied by Mr Cayley's method of generating covariants given in the concluding paragraph of theFirstSection [po290 above]. I may likewise add, that inasmuch as(iD'E+y',,!+z'~+&C.)'isauniversal concomitant, (,d,d&)riD--+y-+c.d:r;dy. will be 80too,byvirtueof thegenerallaw ofinterchange, whichconducts immediately tothetheory of emanation, showing thatthislastsymbol. operating upon any function, furnishes covariants thereunto for anyinteger value ofz. Oneadditional interesting remarkpresents itselfto bemadeconcerning U,the cubic function of iD,y,z,which is, thatcallingasbeforeTitssextic invariant, anda,3b,Sa,d,&c.the coefficients,the formula will give thepolar reciprocal, or, asithasbeen agreed to termit,the reciprocant ofU.I believe the remarkof theprobability ofthisbeing thecaseoriginated with myself, butMr Cayley first verified it by actual calculation, using for thatpurpose the value of T,given by Mr Salmon in his work On theHigherPlaneCurves,alreadyfrequently alluded to, which is an indispensable manualequally for the objects of thehigher specialgeometry asfor the new or universal algebra, being in fact a common groundwhere the two sciences meet and rendermutualaid. Mr Salmon also observed, thatthefirst evect of T,namely (r;a+E'7J:b+&c.)T, 43)OnthePrinciples oftheCalcUlmofForms. 339 wasidentical in formwithwhatmay betermedthefirstdevectofthepolar reciprocal, thatis,theresultofoperating uponthepolarreciprocal with whatUbecomes when :":'TJ't~,aresubstituted inthesteadof:c,y,s. Andinasmuch as,byEuler'slaw, {a(t,r+3b(tEl:'TJ+&c.} x{rfa+~tb+&c.}T {d ' d }=6aria+bdb+&c.T=36T, itfollowsthatTistheseconddevectofthepolarreciprocal, oratleast identical withitinpointof form. But,sincethepreceding matterwas printed,Ihave discovered in thecourse of a mostinstructive andsuggestive correspondence with Mr Salmon,theprinciple upon which theseandsimilar identifications depend, therebydispensing withthenecessity fortheexces­ sivelytediouslabourofverification which, even in thesimpleexample before us, would be found toextendoverseveralpagesof work. Thetheoryin which thisprinciple isinvolved will be given, along with otherveryimportant matter,inthenextnumberoftheJournal: Supplementa.ry Observations on the Method ofReciprocity. Ithasbeenobserved, thatE,n,&c.may always be inserted in place of ~,d~'&c., and triceversa,inaconcomitant form,without destroying itsconcomitance. Accordingly, insteadoftheevectorsymbol dd·rda+E~db+&c., we may employ (d)8d(d)1d d d:cda+d/r:,dydb+&c.i andoperating withthisupon any concomitant, theresultwill be a concomitant. Hencewe see, for example, thatif wetaketheconcomitant SHformed bytheproductoftheinvariant Sandthecovariant H, {(d.8d(d)2d d }d:c)da+d:cdydb+&c.SH difference 22-2{d(d)8 d(d)1d.}Sxdad:cH+dbd:cdyH+&c.,will be acovariant iin factthiswill be found to be T,the between thisandtheexpression before given for T,namely (d8dS(d)2ddS d:c)Hda+d:cdyHdb+&c., being 340OnthePrinciples oftheCalculusofForms. [43 whichiszero,therebeing no invariant to(x,YJZ)Iofthe3rd degree in a,b,c,&c.,asthefactormultiplied bySwould be were it not evanescent. The same observation may be extended to analogous equations given previously. Ihave chiefly, however, made theaboveobservation with a view to makingmore clear theenunciation ofthetheorem which Iam now about tostate,themostimportant, perhaps in itsapplication of anyyetbrought tolightonthesubject,buttheconsequences of which, asIhavebutquite recentlydiscovered it,mustbereserved forafuturenumberof theJournal. Let any function of any number ofvariables be supposed to have for its coefficients thelettersa,b,&c. affected withtheordinary binomial or multinomial coefficients; and let another function be takenidentical with theformer in all respects, exceptinthecircumstance thatalltheirnumerical multipliers are suppressed. Letthisfunction or formbe termedtherespondent totheprimitive: furthermore, bytheinverse of any form understand what thatform becomes when, in place of x,y,s,&c.,E,"1,'J&c., d d d d d d de'dy'ds '&c.,dE'd"1'd"&c., arerespectively substituted (and80for all the systemsofthevariables), and 10 k ' h 00'1 bsti0 adfd d d &1ewiseatt e same time SImIarsutitutions are m e 0da'db'de'c., in place of a,b,c,&c.;thenwe have thisgrandand simple law-The inverse ofanyconcomitant to arespondent isaconcomitant toitsprimitive. When theinverse of any concomitant totherespondent is made to operateupon thesameconcomitant of theprimitive, itwill be found thattheresult isapower of theuniversal concomitant. Iftheconcomitant totherespondent be aninvariant thereof,theruleindicates thaton merely replacing inthe respondent a,b,c,&C.byfa,~,~,&c.,theresultoperating onany invariant orotherconcomitant oftheprimitive, leavesitstillaninvariant orotherconcomitant. Forinstance, if wetakethe function ax'+5b:ry+10cWy-+lOd,rya+56fC!t+fyi, whichhasthreeinvariants L, M, N, ofthedegrees4,8, 12,respectively: and if we call X,p.,vwhatL, M, N become when, in place of a,b,c,d,e,f respectively, we write dId 1d1d1d d da'5'db'10dc'10dd'"5de'df' we shall find that XM=L, p.N=L, and 'AN=alinearfunction of MandLa. 43] OnthePrinciples oftheOalculusofForms. 341 Again,ifinthecaseofanyfunction of :c,y,1&,&c.,wetake,insteadofany otherconcomitant to therespondent, therespondent itself,itsinverse gives thesymbol of operation justpreviously treatedof.Ifagain, in thecase ofafunction of :c,y,say a:r!'+nb:cn-1y+...+nb':cyn-l+a'yn, wetaketheinverse of thepolar reciprocal of the respondent, wegetthe operator d(d)nd(d)n-ld dad'TJ-dbd'TJdE+&c'; andreplacing ~,:Ebyy,e,wefindthat .,ndn-ld& :tda-Ya:db+c, operating onanyconcomitant, leavesitstillaconcomitant, which is M.Eisenstein's theorem beforeadverted to, only generalized bythein­ troduction of anyconcomitant in lieu of thediscriminant. Thisextraordina.ry theorem of respondence will be found on reflection tofavour the notion of treating thecoefficients of ageneralfunction as themselves a system of variables, in amannercontragredient totheterms to which theyareaffixed. Finally,thereis yetanothermode ofapplying the principle of reciprocity, whichmustbe carefully distinguished from any previously statedinthese pages. Ihavesaidthatinplaceof thequantitative symbols of one alphabet, as 11:,y,1&,&c.,we may always substitute theoperation symbolstE,~,:~,&c. oftheopposite alphabet. ButnowIsay,inplaceof thequantitative symbols 11:,y,a,&c. occurring in theconcomitant to any form f,may besubstituted h..(obs I . bib ..)dF dFt equantities 0erve, no onger operative sym0s utquantrties dt:'d'. ~'TJ c::;,&c.,Fbeingitselfanyconcomitant tof.Thus, for instance, takingF identical withf,we seethatf(~,X,~,&c.) isconcomitant tof: oragain,iffbe a function of e,yonly, say f(:c,y),takingFthepolar reciprocal of f,thatisf(-"I,E),we seethatf(-4'j0will be a 342OnthePrinciples oftheOalcuiu«ofForms. [43 concomitant toI:thisconcomitant, bythewayitmay be observed, will alwayscontainIasafactor, because when 1=0,x~+YX=O.Possibly it may be truethat,whenIisafunction of any number ofvariables e,y,z,&c.,andF(E,'IJ,~,&c.)itspolar reciprocal, I(dF(x,y,z,&c.)dF(x,y,z,&c.)& ) dx 'dy,c., which is a concomitant tof,containsIasa factor jbutI have not had time to see how thisis.Itisrathersingular thatMr Cayley andProfessor Borchardt ofBerlinhave both independently made to me theobservation that,whenI(x,y)istakena cubic function of xandy,I(~~,-:!)is equal to the product ofIbythefirstevectant ofthediscriminant off. Thegeneralconsideration oftheconsequences of this new and important application oftheidea ofreciprocity musthe reserved fora futuresection. SECTION V.Applications andExtension 01theTheory 01 thePlexus. If 4>=a:r:'+4bx'y+6Cx'yl+4dxy'+ey4, we canobtain,byoperating catalectically withai,y'upon (d,d)1aIa:x+ydy4>, the twoconcomitants ax'+2bxy+cy!, b:r?+2cxy+dyl,(rd,d)4""xdx+ydy'1'" bxt+2cxy+dy2I' ext+2dxy+ey!(1) a,b,C b,e,d c,d,e(2) theone in fact beingtheHessian, theotherthecatalecticant of4>itself. Again,if 4>=~+5bx4y+IOcx'yt+...+I!I, byoperating catalectically withx',y' uponthesecond and fourth emanants, asinthelastcase,weobtainthetwocovariants Iax'+3b:r?y+3cxyt+dys,bx'+3exty+3dxyt+e'!l1,(I) bx'+8c:rfJy+3dxyl+e'!l,ext+3dx'y+3exy!+I'!I 43] OnthePrinciples oftheOalculusofForms. 343 (2) (1)Supposeax+by, bx+cy, ex+dy b~+cy,ex+dy,dx+ey , ex+dy,dx+ey,ex+fy whicharein facttheHessian andcanouizant respectively of,p.So in general, forafunction ofx,yofthedegree2£or2£+I,wecanobtain£ covariantive forms,thefirstbeingtheHessian, andthelastthecatalecticant onthefirstsupposition andthecanonizant onthesecond: callingtheindex ofthefunction foreithercasen,theformsappearing inthisscale will be ofthedegree(r+1)intheconstants, andofthedegree(r+I)(n-2r) in a:andy. Ithaspreviously" beenintimated thatallthesedeterminants admitofa remarkable transformation. Thistransformation maybeexpressed moreelegantly bydealingnot directlywiththecovariant formsasabovegiven,butwiththeirpolarrecipro­ cantsobtained immediately bywritingEfor -yand"1forx. ,p=ax'+2bwy+3cxy'+dy'; a.2b,c b,2c,d E',2~,"I' winbefoundtobethereciprocant ofitsHessian. (2)Let thereciprocant of itsHessianwill be found to be a,3b,:3c,d b, 3c, 3d, e E',2~,1/', E',2ETJ,"I' (3)Let ,p=aa;D+5bx4y+...+f'!f; thereciprocant ofitsHessianwill be a,4b,6c,4d, e b,4c,6d,48,f E',2ETJI"I', ,',2ETJ,"I', E',2ETJ,"I' [*p.SUabove, DOletl. 344OnthePrinciples oftheOalculusofForms. [43 andthereciproca.nt of its canoniza.nt is a,3b,3cJd. bJ3c,3d,e c,3dJ38,f Ea,3EI.rJ, 3~a,"Ia Thenumerical coefficients in thisandinthefirstcaseareinserted for thesake ofuniformity, butit will of course be readily observed thatwhen thereisbutone line of Eand"I,thatthenumerical coefficients being thesame for each column may be rejected without affecting theformof theresult. So again, if 4>=a:r;S+6lxr!y+ ...+g!le, thereciproca.nt of theHessianis a,5b,10c, 10d, 5e,f, i.5c,10d, 10e,v.g E',2~,"la, Ea, 2~J "la, EIJ2~, "la, Ea, 2~,"II andthereciproca.nt of the second form in thescale,which comes between theHessianandthecatalecticant, is a,b,c,d,e b.e,d,e,f , c,d,e,I,g E',EI.rJJ ~2,"la, Ea,EI.rJ,Efl,"Ia and so in general. The rule of formation is sufficiently plainnottoneed formulating ingeneralterms.Itis easy to see thatalltheseformsarecon­ comitants tothefunction from which theyare formed ifor example, take 4>=a:r;S+6lxr!y+...+g'!l; then d d rkdy4>, form a plexus. 43] On thePrinciples ojtheOalculusojForms. 345 Solikewise if we takey=(a;E+Y"l)', dtdv dE'dTJ fonnaplexus. ButVandcPareconcomitantive, Vbeing auniversal con­ comitant. Hence we may combine together thesetwo plexuses, thatis axe+4ba;lY+6ca;ly't4d:r!f+6Y'1 b:ct+4ca;ly+6d:J:1y'+4ea;y"+fy', e;x4+4da;ly+6e:r:'y'+4fa;y'+gy' E':ct+3E'",a;ly+3E'I'x'y'+"l'a;y" } , Ela;ly+3E'",a;ly'+3E11'a;!t+"lly' and,bytheprinciple of the plexus, :ct,a:'y.a:'y',a;y',y'may beeliminated dialyticaIly, andtheresultant will be the determinant lastgiven, which is therefore acontravariant tocPo• Themannerin which I was led to notice thissingular transformation is somewhat remarkable. Inthesupplemental partofmyessayOnCanonical Forms[p,203 above], mymethodof solution of the problem of throwing thequinticfunction of twovariables undertheformu·+vi+wi,led me to see thatu, v,warethe threefactors of all:+by,be+cy,ex+dy b»+cy,ex+dy,de+ey ex+dy,da:+ey,ea:+fy themoresimplemode of thesolution of the same problem, given by me in thePhilosophieal Magazine forthemonth of November last[po266above], led to a,b,e,d i,c,d,e e,d,e,f s'.-a;y',x'y,-a:' astheproductof the same threefactors;whence the identityofthetwo forms becomes manifest. In the paperlast named I gavetwo proofs,one my own,theotherMr Cayley's, of a like kind of identity for thecanonizant of any odd-degreed function of 1£,yin general. The proof of the identity ofthecorresponding forms in the much more generalproposition above indicated [po325 above, footnote tJmustbereserved untilmore pressing and important mattersaredisposed of. Inthefootnote referred to I oughtto have added, in order to make the sense more clear, thatthedegree of the catalecticant therereferred to in respectofthecoefficients would be n. 846OnthePrinciples ofthe Oalculus ofForms. [43 Iregrettothinkthattherearemanyothertypographical errorsinthe earliersectionsithemostunfortunate oftheseis inthenoteatpage[316], inthevaluesofPandQbelonging tothecubiccommutant dodecadic function ofa;andy,thecorrected valuesof which will be givenin mynext communication. Ioughtalso to observe, in correction oftheremarkmadein thefootnote to page[302],thatitfollowsasaconsequence of arecentpaper byDrHesseinGrelle'sJournal, thatthemethod given by me in thetext applied(according towhatI havetheretermedthe1stprocessforobtaining aninvariant resembling theresultant) to asystemofthreecubicequations (inwhichapplication onlythe1stpowersofd:'~,~enter)produces for thatcasealso,aswellasforthecasesspecified inthenote,notacounterfeit resemblance of,buttheactualresultant itself. Returning tothetheoryoftheplexusof which I am abouttoenunciate amostimportant extension, Ibegtorefersmy readerstothelastparagraph, p. [291], in thelastnumber oftheJournal, whereIhaveshown how to form,undercertainconditions, adeterminant bycombining together various concomitants andeliminating dialytically onesetofthevariables, which determinant will beconcomitantive totheconcomitants outof which itis formed Iandof course also therefore totheircommon original. Nowtheextension ofthistheorem, to which I wish to call attention, isthis,thatnotonlysuchdeterminant asa wholeisaconcomitant tosuch original, buteveryminorsystemofdeterminants thatcanbe formed outofit willform aconcomitantive plexuscomplete withinitselftothesameoriginal, But,much more generally, itshouldbeobserved thatthereis nooccasion tobeginwith asquaredeterminant iitissufficient tohave arectangular arrayoftermsformed by takingtheseveraltermsof oneplexusor ofseveral plexuses combined, provided thattheyareofthesamedegreeinrespect tothevariables (or totheselected systemofvariables, iftherebeseveral systems), andforming outofsuchrectangular arrayanyminorsystemof determinants atwill.Everysuchsystemwill beaconcomitantive plexus. Thesimpleillustrations which follow will makemymeaning clear. Suppose ¢=a:r:8+6b:r:8y+15ca;40y+21da:'y+15ew!t+6fa;!f+gf!. Ihavepreviously remarked, intheforegoing sections, thata,b,c,d,e,f,g, thecoefficients form an invariantive plexusto¢;so also we know thatthe catalecticant a,b.c,d b,c,d,e c,d,e,f d,e,f,9 43]OnthePrinciples oftheOalculu»ofForms. 347 isaninvariant tocp.Butwe are now able tocoupletogether these facts andseethelaw which is contained between them;for if we take (~)'cp,(~)'-l~ cp...(;y)'cp, ,being any number, asfor instance. if we take,=3, weshall have asaplexus aW+3bx'y+3exyl+dyl, ba;a+3ca!y+3d:cyS+eyl, ex'+3dx'y+3exy!+lyI, dx'+3ea:8y+31X'!!+gyl; accordingly not only is thedeterminant a, b,e,d b,c,d,e c,d,e,I d,e,f,9 aninvariant, butalsothesystemobtained bystriking outanyoneline and one column, beingwhat I term the first minors, will beaninvariantive plexus, so too will thesystem ofsecond minors ac-b',bd-&,ee-dS,ad-be, ae-bd, be-cd,&c. form an invariantive plexus,aswellasthelastminors,thatis, the simple termsa,b,c,d,e,f,g.Again, we mighthavetakenthe plexus d d dxfIycp, which would give thearray a,b,c,d,e b,e,d,e,I c,d,e, f,g; buttheminor systems of determinants herein comprised will be found to be identical with those lastconsidered, with the exception thatthehighest system,containing a single determinant only, will now be wanting. So in generalit willeasilybes,enthatasimilarmethod in general, when cpis of2,dimensions, will lead to,+1invariantive plexuses comprising the given coefficients grouped together at oneextremity of the scale, and the catalecticant aloneattheother;and ifcpis of2,+1 dimensions, therewill still be,+1 such plexuses, commencing with the coefficients asone group andendingwith a system of combinations of the(,+l)thdegreeinregard tothe coefficients, which system accordingly takestheplace of the cats­ lectica.nt of the former case,which for thiscaseisnon-existent. 348OnthePrinciples oftheOalculusofForms. [43 thenAs a profitable example of theapplication ofthislaw of synthesis, in itspresentextended form, let it be required todetermine theconditions that a function of e,yof the fifth degree may have threeequal roots. Ingeneral, letq,=aa!+5b:c'y+10c:cayl+10d:c1t+5ete'/t+jy-,thenq,hasaquadratic and cubic covariant of which I have writtenatlarge in my supplemental eBBayabove referred to, being in fact the 8andt(thatisthequadrinvariant andcubinvariant) inrespectto:c',y'(x,ybeingtreatedasconstants) of (,d,d)',.I..ed:c+Ydy'f" Letthesecovariants respectively be called A:cI+2B:cy+Cyl=U, azI+3fJ:cIy+3-yx!f+oy=v; Ax+BY} Bx+Cy forms a plexus, and ar+2fJ:cy+W} fJa;I+2ry:cy+oy will form another. Now when a=0,b=0,C=0,4>will have threeequal roots, and (dd)':e-+y'- q,d:cdy becomes 6dy.X'1y'1+4(d:c+ey):ey'l+(ex+jy)Y", of which the quadrinva.riant inrespectto:c',y'iseasilyseen to be c/,Iy2 andthecubinvariant dly'.Accordingly the grouping A,Bb 0,°}B, Cecomes 0, d'' and thegrouping a,fJ,rybe 0, 0,°} Q !:'comes°0dl .~,ry,0 • , Accordingly, we see thatthedeterminant I~:~Iand allthefirst minors of I~,~:rI,thatisary-{JI,fJo-ryl,ao-fJry.become zero; butthe former singlequantityIi~Ibeinganinvariant, andthislastsystembeing aninvariantive plexus, all thequantities so affirmed to be zero will remain zero,notwithstanding anylineartransformations to which q,may be subjected; thusthenweobtainanimmediate proof of thetheorem tha.t 43J OnthePrineiples oftheOalculusofForms. 349 whenafunction of a;and'!Jofthefifth degree contains threeequal roots thedeterminant of itsquadratic covariant, which in factisitssolequart­ invariant, andthefirst minors of its cubinvariant will be all separately zero. Thistheorem may bemadestillmorestringent; for bycombining A:.&+2Ba;y+Oyt, azI+2/3a;y+'ff, /3:.&+2rya;y+fJ!!, itbecomes manifest thatInthecasesupposed all thefirstminordeter­ minants of A,B, 0 a,/3,ry /3,ry,fJ will be zero, showing in addition to thetheorem lastenunciated thatalso A:B:0::ex:/3:ry::/3:ry:a Itis curious and instructive toremarkthatthis last set of equations, stringent astheyappear,and far more thanenoughtoexpres!.'aduplex condition, are notsufficient toimply unequivocally the existence of three equalroots, unless we havealsoAO-B'=0;for suppose rpto take the form azI+f'!/(b,e,d,eallvanishing); thenit willeasilybe seenthat a=O,/3=0,ry=O,fJ=O, A=0,B=af,0=0.· •IfwetakeL, M, N a system of fundamental invariants tot/>.of whioh all the other invariants of'"arerationalintegerfunotions, then L=l.d,BIand thesimpl88t forms for Mand Nare B, C M=.d,B,CandN= 1&,2fl,'Y ...fl,'Y 1&,2fl,'YI fl,'Y,~ fl,2'Y. ~.Ifl,2"f, ~I whereLandNare thedisoriminants of thequadratio andoubiocovariants of'"rellp8Ctively, and alinearfunction ofM,L'isthedisoriminant of'"itself(L, M, N being of 4.8, and 12 dimen­ .ionsrespectively in thecoefficients oft/». FOl'manypurposes of thecalculusof forms itisdesirable to have the command of_for whichanytwooutofth_threeinvariants may be made to vanishwithout thethirdvanishing; anditwillbe found thatwhen'"is of the form y'J(cz3+111),L=0,M=0;when",is of the form 1/("'+/1'). N=O,L=O;and when ",isof the form ~+eg", M=O,N=O;and of course when '"isoftheformr(fhI+Ifl),L=O,M=0,N-O;itbeing obviously truein general, &8remarked byMrCayley,thatwhen not 188sthanhalftherootsof afunotion oftwovariables are equal, allitsinvariants mustvanishtogether. 350OnthePrinciplee ojtheOa1cu1usojForms. [43 Consequently we shall still have all thefirst minors of A.E,C Il,fl,"{ fl."{.~ zero,although thereis not even 1;0muchasapairofequal roots in </>;AC-lJt however, it will be observed. isnot zero in thissupposition. Thetheoryof Hessians. simple or bordered, may beregarded as one among the infinitediversity ofapplications of theprinciple oftheplexus. LetU, V, W, &c.be anynumberofconcomitants having the common system of variables x,y...s.LetXrepresent ,d,d Jd xdx+Ydy+ ... + ~dz' andtake then'JtU+).XV+&c.+ ~XW=S; dSdSdS dx"dy'"dz' forms a plexus; and this, combined with XV.&c....XW,enables us to eliminate dialytically x',y,z',A...~.Theresultis aHessian ofU, bordered with dVdVdV dx'dy'"dz- horizontally and vertically, and also with dWdWdW de 'dy"'([Z' &c. &c. similarly dispersed; which Hessian, so bordered, ISthus seen to be a concomitant toU,V...W.The Hessian, asordinarily bordered with E,TJ··· ~,is derived by takingforVthe universal concomitant xE+YTJ+ ... + z~, and forW(iftherebe a double border) xE'+YTJ'+ ... +zr. and so forth. IfVbetakenidentical withU,theresulting form,consisting ofU bordered with ~.~i...:~.has been shown-in mypaper"Oncertain generalProperties of Homogeneous Functions," inthisJournal, to be equal to theproductof the simple Hessian of Uand ofUitselfmultiplied by a [*p.173 above.] 43JOnthePrinciples oftheCalculusofForms. 351 numerical factor. The theoryof the bordered Hessian maybeprofitably extended bytaking a,b,e,E' b,c,d,E'TJ c,d,e,7J' E',E'TJ."1', Soagain.if U=a:r:D+5bry+...+lye,asaconcomitant toU.S=xtrU+xXry+...+J£XrW, andcombining with Xry...XrWthe plexus obtained byoperating upon Swiththerthpowers and products of:ic,;y'":z'andeliminating dialytically therthpowers and products ofte',y'...z'.Thus if U=au:'+4brry+6c:r:'-y1+4d3:yJ+e'!fandY=(teE+Y"l)', we obtain, by takingS=X4U+XX'Y,and proceeding asindicated 10the preceding, we find ate+by,bte+cy, ce+dy.E' be+cy, Cfi!+dy,dx+ey,E'TJ c»+dy,dx+ey, etc+Iy,"1' ~, E'TJ, "1', aconcomitant toU. Theseextensions of the ordinary theoryof Hessians will be found to beof considerable practical importance inthetreatment of forms,for which reasontheyare here introduced. SECTION VI.OnthePartialDifferential Equations to Concomitants, Orthogonal and Plagiogonal Invariants, J:c. Inthe7thnote of the Appendix tothethreepreceding sections- Ialluded tothepartialdifferential equations by which every invariant maybe defined. This method may also be extended toconcomitants generally. M.Aron­ hold,asIcollect from privateinformation, wasthe first to thinkof the application of this methodto thesubject jbutitwasMr Cayley who com­ municated to me the equations which define the invariants of functions of [.p.826above.] 352OnthePrinciples oftheCalculusofForms. [43 twovariables ".Themethodby which I obtaintheseequations and prove theirsufficiency is my own,butI believe hasbeenadoptedby Mr Cayley in amemoirabouttoappearinOrelle'sJournal. I havealsorecently been informed of apaperabouttoappearinLiouville's Journal fromthepen of M.Eisenstein, where it appearsthesameidea and mode of treatment have beeu made use of. Mr Cayley's communication to mewasmade in the earlypartofDecember last, and my method(theresultof aremarkmade long before) of obtaining theseandthemoregeneral equations, and of demonstrating theirsufficiency, imparted a few weeks subsequently­ I believe between January andFebruary ofthepresentyear. Themethodwhich I employ, in fact, springsfromtheveryconception of whataninvariant means, and does butthrowthisconception into a concise analytical form. Suppose, to fix theideas, ep=a.x"+nbxn-1y+ in(n-l)~y'+ ...+lyn, andletI(a,b,c ...l)be any in varianttocp. Now suppose xto become x+By,butytoremainunchanged jthe modulus of thetransformation,I~::I,beingunity,Icannotalterin con­ sequence ofthissubstitution; buttheeffect ofthissubstitution is toconvert cpintotheform (Un+nfJxn-1y+in(n-1)ryxn~!I+...+Xyn, where a=a,fJ=b+ae,'Y=c+2be+rufJ,&c.&C. X=l+ ...+nbe"-l+ue". Consequently, if wemake ~b=ae,~c=2be+rufJ,&c.&c., we have by Taylor's theorem, observing that~a=0, (d d )1 (d.d)2~I=~bdb+~cde+&c.I+1:2~bdb+~cde+&c.I 1(d)'.+1.2.3 ~bdb+&c. I+&c.=Oj •Itisextremely deeirable toknowwhether M.Aronhold's equations arethesameinform asthoseheresubjoined. Itisdifficulttoimagine whatelsethey can beinsubstance. Should thesepagesmeettheeye ofthatdistinguished mathematician he will confer a greatobligation ontheauthorandberendering aservicetothetheorybyoommunicating withhim onthe f1ubject: andItakethisopportunity ofaddingthatIshallfeelgrateful for theoommunication of anyideaaorsuggeetions relating tothisnewCalculus fromsnyquarterandin any of the ordinary mediums oflanguage-French, Italian,LatinorGerman, provided thatitbeinthe Latincharacter. 43]OnthePrinciples oftheCalculusofForms. 353 andthisbeingtruefor allthevaluesof e, every separate coefficient of e inI1Imustbezero:hence we obtainndifferent equations byequating tozerothecoefficients ofe,~...enrespectively. Thefirst oftheseequations will be (d d d )adb+2b de+3edd+&c.ep=0, anditisobviousthatthiswillimplyalltherest;for, when eistaken indefinitely small,I(a,b,e...)does not alter(whenthisequation issatisfied) bychanging a,b,c ..:intoa',b',c'...;consequently I(a',b',c',&c.) willnotalter,when in place of a', b',c' wewritea", b", c", &c.,obtained froma',b', e',&c.,bythesamelawasa',b', e',&c.,from a,b,c,&c. Thuswe may go on givinganindefinite numberofincrements, Bytoe, without changing thevalueofI.Consequently, iftheequation above writtenbesatisfied,apriorialltherestmustbe80too.Butthereisnot anydifficulty inshowing thesamethingbyadirectmethod", Forwe have (d d d )adb+2bdo+3edd+&c.I=0, anidentical equation. Hence (a~+2b:c+3ed~+&c.){(a:+2b:c+3efet+&c.)I}=0; hence thatis{(a~+2b~+3cd~+&c.)(a~+2b~+3c~+&c.)}I {d d d }I+adb+2bde+3cdd+&c.I=0, {(d d d )(d d d )I}2a de+3b dd+6cde+&c.+adb+2bde+3e dd+&c.I=0; repeating theapplication ofthesymbolic operator (d d )a db+2bde+&c., • ThemethodaboveKivenhastheadvantage however of being immediately applioable to flVeryspeciesofconcomitant, and we learn from it thatconcomitanoe, whether absolute or conditional, issuffioiently determined when affirmed to existforinflnituimal variations; it cannotexistforinfinitesimal variations without, bynecessary implication, existing forfinite variations a180; amostimportant consideration thisinconducing toatrueidea of the nature ofinvariance andtheotherkinds of concomitance, and incuttingoff allsuperfluous matterfrom thestatement of theconditions by which theyare defined. 8. 23 354 weobtainOnthePrinciples oftheCalculus ofForms. [43 uponandsoon;thenumerical multipliers ofthetermsoftheseveralseries withintheparentheses fonning theregularsuccession of figurate numbers 1, 2, 3, &c. 1, 3, 6, &c. 1, 4, 10, &c. Itiseasyto seethattheseequations correspond to theresultsofmaking thecoefficients of thesuccessive powers of eequalto zero. Imayremark,thatthefirstinstance asfarasIknow on recordofthis, (assome may regarditratherbold)butinpointof factperfectly safeand legitimate methodofdifferentiating conjointly operator andoperand, occurs in apaperbymyselfinthisJournal, Feb.1851,"OncertainGeneral Properties ofHomogeneous Functions" [po165above];where I have applied itinoperating with {(Xl-ale):fal+(XI-a..e)d:+&Cr CeI. which, as I have therenoticed, gives theresult {(Xl-~e)~+&cr+l CeI - re{(Xt-ale)d~+&cr CeI. Theequation(a~+2b~+&c.)1=0isevidently notenoughtodefine Iasaninvariant; itmerely serves toshowthatIdoesnotalterwhen in place of Xwewrite X+ey,butthisistruefor any function of the differences of theroots of theformmultiplied byasuitable power of a. namelythatpower which is justsufficient to causetheproduct tobecome integer. Butif we now, for convenience, write tP=axR+nba;n-ly+in(II.-1)Ctl!'-'Jy'+... +in(II.-1)C'x'yR-'J+nb'xyn-l+a'yR, 43J Onthe Principles oftheOalculusofForms. 355 andformthesimilarequation fromtheotherside, namely (a'~,+2b'te,+30'd~+&c.)1=0, thesetwoequations together will suffice to define any invariant, asIshall proceed toshow-these arethetwoequations alludedtobrought under my notice by Mr Cayley. Iftheycoexist, it followsfrom the method by whichIhave deduced themthata;may bechanged intoa;+ey,oryinto Y+la;,withoutIbeingaltered,eandfhavingany values whatever: and itis obvious thatthesesubstitutions may be performed, not merely alter­ natively butsuccessively, because theequations between the coefficients areidentical equations, and depend only on theform ofI. Letnowa;become a;+ey,andthenybecomey+Ia;jthe result of these substitutions is to convert a;intoa;+efa;+ey, and yinto Ia;+y. Finally,leta;become a;+gyjthena;isconverted into(1+ef)(a;+gy)+ey, andyintoy+I(a;+gy), thatis a;becomes (1+ef)a;+(eg+elg)y, and ybecomes Ia;+(1+Ig)y. The modulus of substitution it isevident,apriori,alwaysremains unity, andnothingwould be gained by pushing thesubstitutions anyfurther,asit isclearthatwe may satisfy the equations 1+el=p, e+g+efg=q, I=p', 1+Ig=q', for all values of p,q,p', q', which satisfy theequation pq'-p'q=1, and for none otherexceptsuch values jhenceIremainsunaltered forany unit-modular lineartransformation ofa;,y,and is therefore aninvariant by definition. Iff/Jbetakenafunction of threevariables, e,y,s,andbethrownunder theform az"+(ala;+bly)zn-l+(~+2b,;cy+cq)Z_2+&c., andIbe anyinvariant off/J,by supposing a;to become a;+ey,and glVlllg b.,b21~,&c.,thecorresponding variations, and takingeindefinitely small, weobtain {a1~l+(~d~s+2b2d~J+(~d~3+2bl~1+3o.d0+&c.}1=0, {~~+ (~i1~~+2bs~)+&c.&c.}1=0: 23-2 356Onthe Principles oftheOakulusofForms. [43 andin like manner, by arranging epaccording to thepowers of yandoffl:,we obtain two otherpairsofequations: itis clear, however, thatthreeequations (it would seem any threeout ofthesix) would sufficeand imply the other three.Themethodofdemonstration will be the same as in theinstance of twovariables: First,it can be shown by the methodof successive accretions, thatIremaining invariable when fl:receives an indefinitely smallincrement ey,oryan indefinitely small increment EZ,orzanindefinitely smallincrement ee,it will also remain invariable when these increments aretakenofany finitemagnitude. Secondly, by eightsuccessive transformations, admissible byvirtueof thepreceding conclusion, a;,y,Zmay be changed into any linear functions of fl:,y,s,consistent with the modulus of transformation beingunity. And in general for a function of mvariables, mpartialdifferential equations similarly constructed (butnot however arbitrarily selected) willbenecessary andsufficient to determine anyinvariant: anditis clearthatall thegeneral properties ofinvariants mustbecontained in and be capable of being educed out ofsuch equations. The same methodenables us also to establish thepartialdifferential equations for any covariant, or indeed any concomitant whatever. Thuslet ep=ag;n+nba;_ly+in(n -1)CtJ!l-y+...+rib'a;y_l+a'yft=0, andletK(a,b,c, &c.jfl:,y,ai,s'.&c.jE,'TJ,&c.)represent anyconcomitant, e,yje',y'beingcogredient, andE,"',&c.contragredient systems jwhen fl:,ybecome a;+ey,y,any such system a;',y'becomes a;'+ey',y';andany such system as E,'"becomes E,'"-eEjandtakingeindefinitely small,the second coefficients a,b,c,&c.becomea,b+ae,C+2be,&c.asbefore;hence theequation to theconcomitant becomes {d . d d d d} •.adb+2bde+...-yda;-y'da;'+...+Ed",-&C.=0 , andin like manner, by changing yintoY+ee,resultsthecorresponding equation {,d 2b' d did d&c.}K0a db'+dc'+...- fl:dy-fl:dy'+...+'"dE- = . These two equations define in a perfectly generalmannereveryconcomitant (withany given numberofcogredient andcontragredient systems) to the formep;andtheduenumber of pairs of similarly constituted equations will serve to define theconcomitant toafunction of anygivennumber of variables']. •Forwehave K(a,b+cu,c+2be.&0.;:1:,II,&0.; ~,'I,&0.) =K(a.b.c.&c.;z,z+ty,&0.; ~,'1-t~, &0.; &~.). tVideNote (10) [po861 below]. 43J On thePrinciples ofthe Oalculus ofFor'TIUJ. 357 Inlikemanner we may proceed to form the equations corresponding towhatmay be termedconditional concomitants, whether orthogonal or plagioganal. Theconcomitants previously considered may be termedabsolute, thelineartransformations admissible being independent of anybutthe one generalrelation, imposed merely for the purpose of convenience, namely of theirmodulus beingmade unity. An orthogonal concomitant is a form whichremainsinvariable, not forarbitrary unit-modular, butfor orthogonal transformation, thatis forlinearsubstitutions ofe,y...e,which leave unchanged :Ir+y'+...+ z~:in likemanner,a plagiogonal concomitant may bedefined of a form which remains invariable for alllinearsubstitutions ofx,y...s,which leave unaltered any given quadratic function of e,y...s: Thus,letitberequired to express the condition ofQ(a,b,0...e,y;E,.,,), beinganorthogonal concomitant to the form ax"+nba!Hy+...+nb'xyn-l+a'y". Letxbecomex+ey, ebeingindefinitely small,thenymustbecomey-ex, andthevariations ofa,b...b',a'will be the sum of thevariations produced bytakingseparately x+eyforxandy-exfory.HencetheoneBole condition forQbeingoftherequired form becomes {(a~+2b;0+-ytx+E~)}Q=O' -(a'~,+2b'~'+-x;y+.":E) or,asitmay be written,8Q-ooQ=0, where 8Q=0,ooQ=0 arethetwo equations expressing theconditions ofQ,beinganunconditional orabsolute concomitant; and so in generalifrpbe a function of mvariables, we may obtainlm(m-1)equations of the form L-M=0 fortheconcomitant, of which however (m-1) only will be independent. Supposing, again,thesubstitutions to which x,yaresubjectto be conditioned byIi+2mxy+ny'remaining unalterable, or which is amore convenient and only in appearance lessgeneralsupposition by :Ir+2mxy+y' remaining unalterable, thegeneraltype of an infinitesimal system of substi­ tutionswill be rendered bysupposing e,yto become (1 +me)a;+ey, -ex+(1 -me)y,respectively, for thena;2+2mxy+y'becomes (I-mte')a:t+[2m+(2m-2mt)e'}xy+(1-mte')y', whichdiffersfroma:t+271WCY+ytonly by quantities ofthesecond order ofsmallness which may be neglected, andEand."will therefore become (I-1M)E-671,-ex+(1+me) y,respectively: then,asto thecoefficients ofrp,inaddition tothevariations whichtheyundergo when miszero,there willbethevariations consequent upona;assuming theincrement mea:,andy 358 OnthePrinciples oftheOalculmofForms. [43 theincrement -mey:butbymakinga:becomea:+me», a,b,C,&c.,b',a' assumerespectively thevariations n.mea, (n -1)meb,'"rneb',0,respectively j andbymakingybecomey-mey,thecorresponding vari~tioD8 become 0, -meb,...-(n-1)meb',-n.mea',respectively. Hencetheequation becomes OQ-fA)Q+m(XQ-p.Q)=0, where0and fA)havethesamesignification asbefore,andwhereAdenotes d d,dd d nada+(n-1)bdb+...+b db'+a:da:-EdE' andp.denotes d d ,dd d bdb+2cde+ + nacia'-Ydy+'7dT}. Iftherebeseveralsystemsofe,yor ofE,'TJ,or of both, theonlydifference intheequation ofcondition will consist in putting I(yf.x), I(x~),I(a:f.x),I(Y~), I('TJ:E)'I(E~),I(E~),I('TJ~), insteadof thesinglequantities included withinthesign ofdefinitesum­ mation. Fearing to encroach too much on thelimitedspace of theJournal, Imustconclude for thepresentwithshowing how to integrate thegeneral equation totheorthogonal invariant oftP,thegeneralfunction ofe,y. Beginning withtP=a,g;2+2bxy+cyl,theequation becomes {d . d d d d}- 2bda+(a-e)db+2bde+Yax-a:dyQ=o. Ada+p.d~+vdc=dO{p.a+2(v-A)b-p.e}. p.="A,2(v-A)=xu,-p.="."; dlog(Aa+p.b+ve)="dOi Aa+ub+vc=1Je'c'.Writenow we have then Let then orda=-2bdO, db=(a-e)dO, de=+2bdOjda:=yd8, dy=-xdO, 43] OnthePrinciples oftheOalculus ofForms. 359 Tofind"wehavethedeterminant Ie,-1,01 2, Ie,-21=0, ! 0,I," thatis, '"+4,,=0, andcallingthethreerootsofthisequation "1' "2' "s, wehave "1= 0,"'.I=2£,"a= -2£; accordingly we mayput ,,=0, X=1,1"=0, 11=1, or "=2£,X=I,I"=2£,11=-1, or "= -2£,X=1,I"= -2£,11= -1. Again, pdx+qdy=(py-qx)dO; andputting -q=ep,p=eq,sothatpx+qy=Erf', ell= -1,6)=£,~= -£; andwe mayput e=t,P=1,q= -£, or e=-£,P=1,q=+£. Consequently thecomplete integral ofthegivenpartialdifferential equation is found by writing a+c=I, x-£y=Ee", a+2Lb-c=l'e'A',x+£y=E'e-", a-2£b-c=l"e-2<'. Bymeansofthesefiveequations, aftereliminating 0,we may obtainfour independent equations between a,b,c;e,y.Suppose Ql=O, Q,=O, Qa=O,Q,=O; thenQ=F(QI'Q"Qa,Q,)isthecomplete integral required. Pursuing precisely thesamemethodforthegeneralcase,itwillbefound that,callingthedegreeofthegivenfunction nwhennis even,theequation in"tobesolved will be ,,(Ie'+4)(Ie'+9)...(,,'+n')=0; andwhennisodd(say2m+I),theequation in"to solvewillhe ("+1)(Ie'+9)...(Ie'+n')=0; 360 OnthePrinciples oftheOalcuiu«ofForms. [43 andperforming the necessary reductions, and calling theroots of the equation, arranged inorder of magnitude, "1£'"2£.••"n£,respectively, it will be found thattheequations containing theintegralbecome LI=IIe""S L",=l",e""s La=lae".<8x-£y=Ee's} x+£Y=E'e's' Lt>+1=In-rle"a+l'S wherell'II...In+t;E,E'arearbitrary constants, and where LI,La...L..+t are the values assumed by the 1st, 2nd ...(n+l)thcoefficientsof the given function 4>,or ax"+nbxn-Iy+...+nb'xy..-l+a/yn, when it is transformed bywritingx+£yin place of x,andy+£xin place of y. £is of course employed in theforegoing according to theusualnotation to represent ..,1(-1). The same method applies to the generaltheoryof plagio­ gonal concomitants, where the linearsubstitutions are supposed such as to leavelrcl+2m:cy+nylunaltered in form, and the equations inewhich containtheintegralpresentthemselves underasimilaraspect. Buta more full discussion of theseinteresting integrals must be reserved untilthe ensuingnumberoftheJournal. NOTES INAPPENDIX. (9) The scale of covariants to a function of (x,y)obtained bythe method ofunravelment [on p. 297 aboveJ, may be otherwise deduced in a form more closely analogous to thatofthecorresponding theorems for thecorresponding invariantive scale[on P: 295 aboveJ, by amethod whichhastheadvantage ofexhibiting thescale equally well for thecase of functions of thedegree 4£+2 or4£+4, the only difference being that inthelattercasethe coefficients of the odd powers of Xwill be found all to vanish, so thatthedegrees of thecovariants will rise by stepsof 4instead of bystepsof 2,justconversely towhathappens intheinvariantive scale; whereas in theinvariantive sealealludedtotheformscontaining odd powers ofXvanish when thedegree ofthefunction is of the form 4£ +2,butdonot vanishwhen it is of theform4£.This method in the form here subjoined isaslightmodification of one suggested to me by my friend MrCayley. LetFbe the given function of e,yof the degree 2n;takethesystems a/,y';XI'YIcogredient with one anotherand with e,y.Thenformthe concomitant K=(:c'f:x+y'~rF+X(x'y-y/x)n-I(:c'YI-y'fXt)(txyl -yxJ) . 43J OnthePrinciples ofthe Oalculus ofForms. 361 andmakingThen(by what may be termed theDivellent method, which has been pre­ viously applied by me in thePhilosophical Magazine for Nov. 1851) calling00,OIl01" ,On,the coefficients of x'n,x'n-\y/,•.•y'ninK, weshallhave 00=Ao:z:R+BOXR-Iy+ +Loyn, 01=AI:z:R+BIXR-Iy++Llyn, On=An:z:R+Bn:z:R-Iy+...+Lnyn, thecoefficients being functions of thecoefficients of fand ofquadratic combinations ofXl'Yl>affected with themultiplier x; andthedeterminant ..10,B;...Lo AI>BI•••LI An,B; ...I« willgiveafunction of }"in which the coefficients of theseveral powers of }" will be all zero or covariants of F. Theactualform of this determinant is not here given for want of space andtime,butwillbeexhibited hereafter. Precisely an analogous method applies to obtainthescale to(x,y,z)'given in Note(2)[po322 above]. CallingF=(x,y,z'f,letthesystemsx',y',zl;Xl'YI>Zl'betakencogredient with oneanotherandwithe,y,z.Then, using Rto express thedeterminant ai,y',z' X,y,z K(Id,d Id)1F R=Xdo:+Ydy+zdz+}", and proceeding as aboveby thedivellent method, we obtainthe scalerequired. (10)[po356 above.] Itis obvious thatthese defining equations ought to givethemeansof discovering and verifying all theproperties of con­ comitants; butit is very difficult to see how in thepresentstateof analysis many of thegeneral theorems thathave been stated,readilyadmitof being deduced from them. Thecomparatively simplebuteminently important theoryof the evector symboldoeshowever admitofaveryprettyverification by aid of these equations. Thus, suppose0anyconcomitant; supposeacontravariant to a function Fofe,y,say axn+nbx-1y+ ...+nb'xyn-l+a'yn. 362OnthePrinciples oftheGakul1.t8ofForms. [43 Then8mustsatisfythetwoequations (L+,:11)8=0,(L'+11:,)8=0, where L=a~+2b~+ ...+nb':fa,, L',d2b,d bd=adb'+dc'+...+nda' Now let I/>=X(8) where f:ndbI-1d1:,,-2,d "d .X=1Oda+ lOl1db+ lO11dc+"'+ l1da" then L(X8)=X(L8)-(XL)8 =X(L8)-(,,,;b+2,n-111~+ ...+nEl1"-1d:')8, ,~(X(J)=X(E~8)+(,~X)(J =X(,~(J)+(,,,~+2,"-111:c+...+nE11n-lfa.) (J. Hence (L+,:11)X«(J)=X{(L+,:11)(J}=X(0)=O. Similarly (L'+11:,)X«(J)=O. Hence if (Jisanintegralofthetwoconditioning equations, so also isX(8). Inlike manner, if (Jbeacovariant or anyotherkind ofconcomitant ofF. it may be proved thatitsevoctant X«(J)is thesame. (11)[po331above.] Very much akinwith the supposed equations isthe following most remarkable equation, which can be proved to exist. LetI/> be a function of a;andyof the5thdegree. LetPandQbe thequadratic and cubic covariants of1/>.Pis of two dimensions in thecoefficients and also in the variables, and Qofthreedimensions in both;theyare in fact the8andt(inrespecttoa;'andy')of(a;'fa:+y';y)'1/>.Then,givingPand Qpropernumerical factors, it will befoundthat H,I/>+PHI/>+QI/>=o. I believe thatasimilarequation connects any function of a;and'!/above the3rd degree with its first and second Hessians. The proof will be given inasubsequent Section, where also I shall give a complete proof, which occurred to me immediately aftersendingthepreceding note to the press. ofthecomplete Theory of theRespondent by means of thegeneralequations of concomitance. 43] On thePrinciples oftheOalculusofForms. 363 P.S. Since thepreceding wasin type, I have ascertained theexistence andsufficiency of ageneralmethod for forming thepolarreciprocal and probably also thediscriminant to functions of any degree of threevariables by anexplicit process of permutation anddifferentiation. Inparticular Iamenabled to givetheactualrule for constructing thepolar reciprocal andthediscriminant curves of the 4th and 5th degrees. So far asregards thepolarreciprocal of curves of the4thdegree M. Hesse hasalreadygiven amethod of obtaining it,butmineisentirelyunliketo this, and restsupon certainextremely simple and universal principles ofthecalculus of forms. The only thingnecessary to be done in orderto carry on the process to curves of the 6thorhigherdegrees, is to ascertain therelation ofthe discriminants of functions of two variables of those respective degrees to such ofthefundamental invariants asare of an inferiororder tothediscriminant. Thetheoryappliesequallywell to surfaces and to functions of any number of variables, and may, I believe, withoutany serious difficulty be extended soasto reduce to an explicitprocess the generalproblem of effecting theelimination between functions of any degree and of any number ofvariables. Themethodaboveadverted to willappearin asubsequent Section. [Oontinued pp.402and411below.] 44. SUR UNE PROPRIETE NOUVELLE DE L'EQUATION QUI SERT A DETERMINER LESINEGALITES SECULAIRES DES PLANETES. [Nouvelles AnnalesdeMathimatiques, XI.(1852), pp. 438-440.] [Extract.] 6.Soitledeterminant carresyml.triqruJ tls,1l ~,I'"a.,,, daMlequelona, a:apresladefinition, Elevantledeterminant dlapuissance p, on obtient ledeterminant Al,1IAI,I'"AI,,, AI,l'AI,I...AI,"(M) (N) stcsdeterminant estsymetrique aussipar"apportdladiagonale Al,l' AI,I...A",ft. Retranchant de chaque termsde la diagonale symetrique de(M)lamerne quantiU ~onobtientledeterminant ~,I-}", ~,I'" ~,,, an,l'an,I••.an,,,-},,(P) 44] Surunepropriete nouoelle. 365 (1) (2) iquationsontDeveloppant eedeterminant etordonnwnt parrapportax,onobtientune ezpr688ion qui,etantegalieazero,donnel'equation xn-!Xn-l+gxn-t+ ...(_I)nt=0, iquationquianracinesreellea(voirt.x.p.259). Retranehant de cheque termsdeladiagonale symitrique dudeterminant (N)laquantite ,.,.,etoperanteom1/'l8oi-deseue, onparvienta1:equation ,.,.n_F,.,.n-l+G,.,.n-t+ ...(_I)nT=0, equation quiaaussinracinesreeUea.Learacinesdeeette learacinesde(equation (1),eleveeschacuneellapuissance p. Demonstration. Representons par PI'plI'P,··,PI" lespracinesdel'equation pi'-1=0. Ecrivons Ie determinant ~,1-pqA, ~,lI'" ~,n a"l'a,,'il-pqx•••a"n a",l an,n-pqX et faisons qegalsuccessivement a.tous les nombres de180suite1, 2, 3...p, onaurapdeterminants; Ieproduitdetouscesdeterminants resteevidem­ mentIe meme dans quelque ordre qu'on prennecesdeterminants, et,d'apree lesproprietes connues des racines de l'unite,tous les termesenpqui ne serontpaselevesa.unepuissance pdisparaitront, etXaccompagnant toujours p,il nerestedone que des All,etIedetermiuant-produit sera (Q) An,!>An,l>......An,n-Xl' OU,faisant,abstraction deX,on a Iedeterminant (N).Ainsi ,.,.=Xl'. 7.Application. determinan tn=2,etp=2; Ia,bI b,e 'C.Q.F. D. (M) elevantcedeterminant au carre, on a Ia'+hi,ab+beI. ab+be,b'+c''(N) 366 determinant determinant FaisonsSurunepropriete nouvelle. Ia-X,bI b,c-X' XI-(a+c)X+ae-b'=0; Ia'+b'-,.,.,ob+beI abwbc,bl+C'_,.,. I ,.,.'-(at+et+2b'),.,.+(ae-b')'=0,ou,.,.=X'. n=2,p=3,[44 (P) (1) (2) (M)nechangepas,et l'on a Ia'+2ab'+blC, atb+abc+b'+betI alb+abc+lJI+blc,ab'+2blc+et' (N) Iedeterminant (P)etl'equation (1)restentlesmemes; marsl'equation (2)devient OU ,.,.=X', car,AletXaetantles deux racines de l'equation (1), on a ~I+Xal=a;I+et+3abl+3Gb',XI'Xal=(ae-lJI)3. 8. M.Sylvester faitobserver que son theoreme est uncasparticulier d'untheoreme plusgeneral, demontre parM.Borchardt, pourdesdetermi­ nantsquelconques, etquidevient Ietheoreme demontre ci-dessus, lorsque ledeterminant estsymetrique (Journal deMathbnatique8, t.XII.p. 63,1847). 45. ON AREMARKABLE THEOREM INTHETHEORY OFEQUAL ROOTS ANDMULTIPLE POINTS. [Philosophical Magazine, Ill.(1852), pp. 375-378.] INorderthatthetheorem whichIpropose to statemay bethemore easilyunderstood, and with theleastambiguity expressed, I shall commence withthecaseofahomogeneous function of two variables only,:xandy. Let 4>=a:r:"+n1J:x-1y+in(n-1)ca;R-Iy'i+...+nb':cy"-'+a'yR, and let the resultofoperating withthesymbol .....d_"-1d _1d d ;Ci-do,+wydb+...+!I:xdb'+y"da" on any function of a,b,c...b',a'be called theEvectant of such function, andtheresultofrepeating thisprocessrtimes the rthEvectant. Understand bythemultiplicity oftheequation thenumberofequalities between the roots thatexist;sothata pair of equal roots will signify a multiplicity 1, two pairs of equal roots, or threeequal roots amultiplicity 2; apairof equal roots and aset ofthreeequal roots, amultiplicity 1+2 or 3, and80on. Now suppose thetotalmultiplicity of4>to bem:thefirstpart of the proposition consists in the assertion thatthe 1st, 2nd, 3rd ...(m-1)th Evectants ofthediscriminant of4>,thatis oftheresultofeliminating :xand ybetween ~,~:(aswellasthediscriminant itself), will all vanish in whatever waythemultiplicity isdistributed; thesecondpartof the proposition aboutto bestatedrequires thatthe mode should be taken intoaccount ofthemannerin which the multiplicity (m)is made up. Suppose, then, thattherearergroups of roots, for one of which the 368 On a remarkable Theorem inthe [45 multiplicity is~,forthesecondmi.&c.,and for the rthm,.,80that ml+'1rL.I+...+m,.=m.Then, I say,thatthemthevectant of thedeter­ minantoftf>is oftheform (a,,/I:+b1!l)"""(a.;r;+bi!!).........(a,.a:+bry)mr.., wherea":bl,a.:bl•••a.,:b;aretheratiosof/I::ycorresponding totheseveral sets ofequal roots. Thislatterpartofthetheorem forthecaseof m=1was discovered inductively by Mr Cayley, by considering thecaseswhentf>isacubic, or abiquadratic function. Iextended the theory to functions of any number of variables, and supplied ademonstration, thatis forthecase of onepairof equal roots. Mr Salmon showed thatmydemonstration could be applied to the caseof twopairsof equal roots, or two double points, &c..and very nearly atthe same timeImadethelikeextension tothecase ofthreeequal roots, cusps, &c.,andalmostimmediately afterIobtained ademonstration forthetheorem in its most generalform. This demon­ stration reposesupon a very refined principle, whichIhadpreviously discovered buthave not yet published, intheTheory of Elimination. Ihave here anticipated alittleinspeaking ofthetheorem asapplicable to curves and otherloci. Suppose tf>(/I:,y,s)=0tobetheequation to a curve expressed homo­ geneously. Let tf>(/1:,y,s)=Q.tI:'&+(na'g;n-Iy+nb'/I:..-IZ) +in(n-1)a":r!"'"""Y+n(n-1)b"tI:";z+in(n-1)C"/I:"-'et, +&c.&c.,. andunderstand by theevectant of anyquantity theresultofoperating upon it withthesymbol ..did -n-Id-"-1.,,1d&c /I:da+g;n-yda'+'"zdb'+"'-:Jda"+ . Suppose, now, thecurve to have double points, the (r-l)thevectant (andof course all theinferiorevectants) of thediscriminant oftf>(meaning thereby theresultofeliminating IX,y,zbetween ~,~~,~)will all vanish, and therthevectant willbeof the form (al/l:+b1!l+clz)"x(a.;r;+bt!!+CtZ)"... x(a,.a:+bry+erz)", wherea,,:bl:C;,a.:b,:Ct...a.,:b;:c;aretheratiosofthecoordinates at therespective double points. Iftherebe cusps themultiplicity ofeach 45] TheoryofEqualRootsandMultiple Points. 369 suchwill be2;andcallingthetotalmultiplicity m.to every cusp will correspond a factor of the2nthpower in the mth evectant jand80on in generalfor various degreesofmultiplicity atthesingular pointsrespectively. Theliketheorem extends to conical and othersingular pointsofsurfaces; 60thatthereexistsamethod, whenalocus isgivenhavingany degree of multiplicity, of at once detecting theamountanddistribution ofthismulti­ plicity,andthepositions of theone or more singular points.Inconclusion Imaystate,thatprecisely analogous results(mutatis mutandis) obtain, when, in place of a single fnnction havingmultiplicity, wetakethemore general supposition of anynumber of homogeneous functions beingsubject tothecondition ofpluri-eimultaneity, thatisbeingcapableofbeingmade tovanishbyeachofseveral different systemsof values for theratiosbetween thevariables. Multiplicity inasinglefunction is, in fact, nothing more nor lessthanpluri-simultaneity existing between thefunctions derived from itbydifferentiating withrespectto each of thegivenvariables successively. ButasI purpose to give these theorems andtheirdemonstration, which Ihavealreadyimparted to mymathematical correspondents, in apaper destined forreadingbeforetheRoyal Society, I need not furtherenlarge uponthemonthepresentoccasion. P.S.Intheabovestatement I have spoken only ofcusps ofcurves which aretheprecise and unambiguous analogues ofthreecoincident pointsin point-systems. inorderto avoidthenecessity ofentering intoanydisquisition 81!tothespecies of singularity in curves or otherlocicorresponding to higherdegrees of multiplicity inpoint-systems. asubjectwhichhasnot hitherto beencompletely madeontoI may here also add aremark,which gives astillhigherinteresttothetheory, which is (to confine ourselves, for thesakeofbrevity,to functions of two variables), thatifany root of x:'!I, saya:b,occur1+P.times,thetotalmultiplicity oftheequation being supposed m,and itsdegreen,thentaking, anyintegernumbernot exceed­ ingp.,the(m+,)thevectant ofthediscriminant willcontain thefactor (ax+by)(p.-.)II. Sothat,for instance, if therebebuta single groupof equal roots, and theybe1+P.innumber. everyevectant up tothe(p.-l)th inclusive will vanish, and from thep.thtothe(2p.-,)thwillcontaina power of(ax+by)n. 8. 46. OBSERVATIONS ON A NEW THEORY OF MULTIPLICITY. [Philosophical Magazine, III.(1852), pp. 460-467.] INthePostscript to mypaperinthelastnumber oftheMagazine, Imis-stated, or to speak more correctly, Iunderstated thelaw ofEvection applicable to functions havinganygivenamountofdistributive multiplicity. Thelaw may be statedmore perfectly, and atthesametimemore concisely. asfollows. Everypointrepresented bythecoordinates a..f31...'Y1>for whichthemultiplicity is'lnl,will give rise in everyeueciant" ofthediscrimi­ nantofthefunction to a factor (a1x+f3IY+...+'YIZr·R, nbeingsupposed to bethedegreeofthefunction. Henceiftherebersuch points, for which theseveralmultiplicities arem..'Ins...'lnr,everyevectant mustcontain (~+~+...+mr)nlinearfactors;and asthelthevectant is of the degree &n,itfollowsthatalltheevectants belowthe(11~+~+...+mr)thevectant mustvanishcompletely. andthisEvectant itselfbecontained asafactor in all above itt.Whenafunction of only two variables is in question, there is no difficulty in understanding whatproperty ofthefunction it is which isindicated bytheallegation oftheexistence ofmultiplicities ~,~...rnr; *Frequent use being madeinwhatfollows of the word Evectant, Irepeatthattheevectant ofanyexpressicn connected withthecoefficients ofagivenfuncnon (supposed to beexpressed inthemoreusualmanner withlettersfor thecoefficients affected withtheproperbinomial or polynomial numerical multipliers) meanstberesultofoperating uponsuchexpressions witha symbolformedfromthe given function bysuppressing allthebinomial orpolynomial numerical partsofthecoefficients to besuppreBBed, andwritinginplaceoftheliteralpartsofthecoeffi- cientsa,b,c,&0.thesymbols ofdifferentiation -fa,~,~I&c.;inallthatfollowsitisthe successive evectants ofthediscriminant alonewhich come underconsideration. I needhardly repeat,thatthediscriminant ofafunotionistheresultoftheprOOOBBofelimination (clearfrom extraneous factors)performed between thepartialdifferential quotients ofthefunction inrespect totheseveralvariables whiohitcontains, or tospeakmoreaccurately. istheoharacteristic of theirooevanescibility. tTheconstitution ofthequotients obtained bydividing alltheotherevectants ofthe discriminant by the ftrs~non-evanescent one,presents manyremarkable features whichremain yettobe fullystudiedout,andpromise awideextension oftheexisting theory. 46J OnanewTheoryofMultiplicity. 371 asalreadyremarked, thissimply means thattherearerdistinctgroups of equal roots, such groups containing 1+nlot,1+m,...1+m,.roots re­ spectively. So for curves andhigherloci,thetotaldistributive multiplicity isthe sum of the multiplicities atthe several multiple points.Butthetrue theoryofthehigherdegrees of multiplicity separately considered atany pointremainsyet to be elaborated, and will be found to involvethe considera­ tionofthetheoryofelimination from apointof viewunderwhich it has neverhitherto beencontemplated. Confining our attention forthepresentto curves, we have aclear notion ofthemultiplicity 1:thisiswhatexists at an ordinary double point. As well known, itsanalytical character may be expressed by sayingthatthe function ofa,y,z,whichcharacterizes thecurve, is capable, when proper lineartransformations aremade, of being expanded undertheform of a series descending according to the powers of z,suchthattheconstant coefficient ofthehighestpower of e,andthelinearfunction of e,y,which is the coefficient of the nextdescending power of e,may both disappear. Again, whenthemultiplicity is 2,thethirdcoefficient,which is a quadratic function ofxandy,will become a perfectsquare. This is thecaseofacusp, which, as I have said, is thepreciseanalogue tothatofthreeequal rootsfor afunction of two variables. Before proceeding to consider what itis which constitutes amultiplicity 3 for a curve, it will be well topause for amoment to fixthe geometrical characters oftheordinary doublepointand the cusp. Ifwe agree to understand by a first polar to a curve the curve of one degreelower which passesthroughall the points in which thecurve is met bytangents drawn from an arbitrary pointtakenanywhere in its own plane, wereadilyperceive thatat anordinary doublepointall the infinite number of first polars which can be drawn to thecurve will intersect oneanother atthedouble point. Again, at acusp all these polars will not only all intersect, theywill moreover all touch one anotheratthecusp. Now we may proceed to inquireastothemeaning ofamultiplicity of the third degree, which,strangetosay,I believe has never yetbeendistinctly assigned by geometricians. Thisis not the case of aso-called triplepoint,thatisapoint where three branches ofthecurveintersect. Supposing x=0,y=0, torepresent sucha point,thecharacteristic ofthecurvemustbereducible totheform (f}r+hx'y+kxy2+ly')Zn-I+&c., which,asis well known, involves theexistence of four conditions. This, however, would not in itselfbeatall conclusive againstthemultiplicity ata triplepointbeing only of thethirddegree;for itcanreadily be shown that theremayexistsingular pointsof any degree of sin!J1darity (asmeasured bythenumber of conditions necessary to besatisfied inorderthatsuch 24-2 372 On anewThenryofMultiplicity. [46 singularity may come into existence), butfor which the multiplicity may be aslowasweplease;as,forinstance, ifata double point(which is not acusp) therebe apointofinflexion on one branchoronboth, or apointofundulation, oranyothersingularity whatever, stillprovided therebe no cusps, the multiplicity will stick atthefirst degree and neverexceedit;for onlythe discriminant itselfwill vanish on thesesuppositions, butnoevectant ofthe discriminant. Thereason,onthecontrary, why a so-called triplepoint mustbe said to have a multiplicity of the degree 4, and not merely of the degree 3, springsfrom the fact tbatthelet,2nd and 3rd evectants ofthe discriminant all vanish atsuch a point. Itis clear, then, thatthereoughtto exist aspecies of multiplicity for which the 1stand 2nd evectants vanish,butnotthe3rd. In fact, asata. doublepointthefirst polars all merely intersect, butatacusphaveall acontactwith one anotherofthefirst degree, so we oughttoexpectthat thereshould exist a species of multiple point such thatallthefirstpolars should have with eachotheracontactofthesecond degree (or if we like so to say,thesamecurvature) atthatpoint.Whenthecurvehasatriplepoint, all its first polars will have thatpointuponthemasadoublepoint;andit is not at thefirst glance, easyapriorito saywhatisthenatureofthe contactbetween two curves which intersect atapointwhich is adouble pointto each of them:we know upon settledanalytical principles, thatwhen one curve havinga double pointis crossed therebyanothercurve not having a double point, thatthetwo must be said to have with one another,acontact ofthe1stdegree;and we now learnfrom our theoryof evection, thatifeach haveadoublepointatthemeeting-point, the degree of thecontactmust fromprinciples of analogy be considered to be of the3rddegree". Now,then, we come to the question ofdeciding definitely what is a multiple point for whichthedegree of multiplicity is 3.Itis,adopting eithertest,whether of first polar contactor of evection, acuspsituated or having its nidus,soto say, at a pointof inflexion. Inotherwords,x=0,y=0 will be apoint whosemultiplicity isintermediate between thatof the cusp and thatof a so-called triplepoint, when the characteristic of the curve admitsofbeing writtenunderthe form zn-2:rfJ+Z,,-3(g:rfJ+h:J;2y+ircy2)+Z"-4&c.; or inotherwords,when over and above the vanishing oftheconstant and linear coefficients, and thequadratic coefficient being a perfectsquare, asinthecaseof an ordinary cusp, thissquare has a factor in common with thenext(thecubic)coefficient; or again, in otherwords,a curve hasapoint •Thismayeasilybe verified by directanalytical means; asalso the more generalpro­ position, thattwo curves meeting atapointwheretherearembranches of the one andR branchee of theother,mustbeconsidered tohavemncoincident pointsin common, thatis,ifwe like sotoexpress it, tohaveacontactof the degree mn-1. 46J Ona newTheoryofMultiplicity. 373 for which themultiplicity is 3 when its characteristic functionadmitsofbeing expanded according to thepowers of one of thevariables, in such a manner thatthefirst coefficient and thesecond(thelinear)coefficient vanish, and thatthediscriminant ofthethirdandtheresultant ofthethirdand fourth arebothatthesametimezero.Thisbeingthecase, it may be shown that thefirst polars will all have with each otheracontactoftheseconddegree; andmoreover, thatall theevectants of thediscriminant will have as a common factor a linearfunction of the variables, raised to a power whose indexisthreetimesthatof thecharacteristic function. As, then,thereisbut one kind of ordinary double point, and butone kind of pointwithmultiplicity 2,sothereis one, and only one, kindof point with a multiplicity:l Acusp is apeculiar doublepoint;a flex-cusp (asforthemoment I callthepoint lastabove discussed) is a peculiar cusp. This law of unambiguity, however, appears tostopatthethirddegree. A so-called triplepoint(whichought infact to be called a quintuple point)is apointfor which themultiplicity, as shown above, is of thefourthdegree;butit is not theonly point of that degreeofmultiplicity. Without assuming to haveexhausted every possible supposition upon which such adegree of multiplicity maybebrought into existence, it will be sufficient to takeas anexample acurve whose character­ isticis capable of assuming theform z......-JzI+zt>-l(gzl+hrcly)+z-.(k:ca+lx'y+maflyJ+n:cy')+zR-a&c. Itmayreadilybedemonstrated thatthefirst polars of thiscurve have allwithoneanotheratthepointe,yacontactofa.degreeexceeding the 2nd,thatis ofatleastthe3rddegree(and, I believe, in generalnothigher). Nowthepointe,y isevidently not atriple-branched point,butacusp with threeadditional degrees of singularity; sothatwe have evidence of the existence ofapointwhosedegreeofsingularity is 5,and whose multiplicity isatleast4,butwhich is in no sense amodified triplepoint.Itisprobably true(buttodemonstrate thisrequiresafurtheradvance to be made thanhas yetbeenrealized in thetheoryoftheconstitution ofdiscriminants) thata CQSPmay be so modified by thenidusatwhich it is posited, as, withoutever pa88ingintoatriplepoint, to be capableoffurnishing anyamountofmul­ tiplicity whatever, curiously inthiscontrasting with anordinary double point, DOamountwhatever ofextraordinary singularity imparted to which, or so to speak,to itsnidus,caneverheighten itsmultiplicity soastomake itsurpass thefirstdegreewithout firstconverting itintoacusp. I may illustrate the natureofaflex-cusp by what happens toacurve of thethirddegree. When itbreaksup intoaconicand a rightline,thereare twoordinary doublepoints; fortheexistence ofthesedouble points, as for theexistence ofacusp, two conditions arerequired. When, however, therightline and conic touch one another (acasusomissusthisintheworks of thespecialgeometers), the characters ofthecusp and thepointof inflexion are combined at thepoint 374 On anewTheoryofMUltiplicity. [46 ofcontact; themultiplicity is ofthethirddegree, and thesingularity also ofadegreenotexceeding this;threeconditions onlybeingnecessary to be satisfied in orderthatagivencubic may degenerate intosuchaform; anditwill be found thatthediscriminant andthefirst and second evectants thereofvanish for thiscase, and thatthethirdevectant ofthediscriminant will beaperfect9thpower;whereas inorderthatthecubic may have a so-called triplepoint.thatis maydegenerate intoatridentofdiverging rays, fourconditions mustbesatisfied, anditwill be found thatwhen this is the case,thefirst. second. and thirdevectants ofthediscriminant will allvanish, andthefourthwill beaperfect12thpower of 0.linearfunction ofthe variables. I may mention, bytheway,atthisplace,thatthelaw ofa. discriminant andthesuccessive evectants up tothemthinclusive, all vanishing, maybeexpressed otherwise (not inidentical, butinequivalent orequipollent terms),bysayingthatthediscriminant and all its derivatives of'adegreenotexceeding themthwill allvanish-understanding by a derivative ofthediscriminant anyfunction obtained fromthediscriminant bydifferentiating itany specified number oftimeswithrespecttothe constants ofthefunction to which itbelongs, thesameconstants being repeated ornotindifferently", And very surprising itmustbeallowed tobe,statedasabareanalytical fact,that(m+1)conditions imposed upon thecoefficients of afunction of anynumberofvariables andof anydegree should suffice to make theinordinately greaternumber offunctions which swarmamongthederivatives ofthemth and inferiordegreesof thedis­ criminant each and all simultaneously vanish. Without pushing theseobservations toofarforthepatience ofthegeneral reader,it mayberemarked by way of settingfoot with our new theoryupon thealmostunvisited regiun of thesingularities of surfaces, thatbythelight of analogy we may proceed with a safeand firm stepas farasmultiplicity ofthethirddegreeinclusive. Thefunction characteristic ofthesurfacebeingsupposed to beexpressed intermsofthefourvariables x,y.Z,t,andexpanded according todescending powers of t,thenwhene,y,Zis anordinary doublepointofthefirstdegree ofmultiplicity, theconstant andthelinearcoefficient disappear; whenthe pointhasamultiplicity 2,thediscriminant ofthequadratic coefficient will be zero, thatisthiscoefficient will be expressible by means of due linear transformations undertheform of a;2+y';and when themultiplicity is tobe ofthedegree3,thecubic coefficient will, atthesametimethatthequadratic coefficient is putunderthe form a;2+y',itself(forthesamesystem of xand y)assumetheform ofacubicfunction ofe, y,E,in which the highestpower ofE,thatiszS,willnotappear; or inotherwords(restoring toa,y,ztheir • Or. to speak more simply, the discriminant and its saceeasive differtntia18 up to the mth exolusive mustallvanishsimultaneously. 46] OnanewTheoryofMultiplicity. 375 generality), not only will the first derivatives of thequadratic function be nullifiable simultaneously with each other, butlikewiseatthesametime withthecubic function itself. These threecases will be for surfaces, the analogues 80far,butonly so far asregardsthedegree of the multiplicity, tothedouble point, cusp, and flex-cusp of curves", Theanalogue tothe so-called triplepointofthecurves will be a pointwhose degree of singularity, depending uponthevanishing of the six constants in thethirdcoefficient (which is a quadratic function of :x,y,z)atthesametimeasthethree constants in the linear factor, would seem to be but6 morethanfor a double point,thatis in all 1+6 or 7,butwhosemultiplicity, asinferred from thenatureof thecontactof its first polars, which will be of the 7thorder, wouldappeartobe 8 (aseeming incongruity which I am not atpresentin a condition toexplainjj ; sothattherewillapparently be4stepsofmultiplicity tointerpolate between thiscase and the case analogous (submodo)tothe flex-cusp, lastconsidered. Whether theseintervening degrees correspond tosingularities of anunambiguous kind, no one is atpresentin a condition toofferan opinion. I will conclude with a remark,theresultof my experi­ ence inthiskind ofinquiryasfarasI have yet gone in it, namely that itwould be most erroneous to regarditasa branch of isolated and merely curiousorfantastic speculation. Every singularity in a locus corresponds totheimposition ofcertainconditions upon the form of its characteristic i by aid of the theoryof evection we are able to connect the existence of these conditions withcertainconsequences happening totheform of the discrimi­ nant,andtherebyit becomes possible, upon known principles of analysis, to infer particulars relating to theconstitution of thediscriminant itself initsabsolutely generalform, very much upon the same principle aswhen thevalues of a function for particular values of itsvariable or variables are known,thegeneralform of the function therebyitself, to some corresponding extent,becomes known. Thus, for instance, I haveby the theory of evection initsmost simple application, been led to a representation of thediscriminant •Atanordinary conicalpointofasurfacefor which the multiplicity is 1, every section ofthesurfaceisacurvewithadoublepoint.Whenthemultiplicity is 2, the cone of contact becomes apairofplanes,through theintersection ofwhichanyotherplanethatcanbedrawn entethesurfaceinasectionhavinganordinary cusp of multiplicity 2,butwhichthemselves cutthesurfaceinsections, havingso-called triplepoints,sothatforthesetwoprincipal sections (whichisrathersurprising) themultiplicity suddenly jumpsup from 2 to 4. All otherthings remaining unaltered when the multiplicity of theconicalpointis 3, the cusp belonging toany sectionofthesurfacedrawnthrough anyinterseetion ofthetwotangentplanespasaea from an ordinary cusptoa flex-cusp. tSo,too,ataso-called quadruple pointinacurve,the degree of the contactof the 1st polars is 8,andtherefore themultiplicity of the curve atsuchpointis9;butthenumber ofconstants whichvanishforthiscase(namely allthose of the cubic coefficient in x,y)overandabovewhat vanishforthecaseofaso-called triplepointis only 4, which is aunitlessthanthe difference between themeasures of themultiplicities attherespective points;andthisdifference continues toincrease aswepasson to so-called quintuple andhighermultiple pointsin the curves. 376 OnanewTheoryojMultiplicity. [46 ofafunction of two variables underaform very different andvery much morecomplete and fecund in consequences thanhas ever been supposed, orthanIhadmyself previously imagined, to be possible. According to theopinionexpressed byananalystoftheFrenchschool, ofpre-eminent force and sagacity, it isthrough thistheoryofmultiplicity, here forthefirst time indicated, thatwe may hope to be able to bridgeover forthepurposes ofthehighesttranscendental analysis, theimmense chasm which at presentseparates our knowledge of the intimate constitution of functions of two from thatofthree,or anygreaternumber of variables. Itis, as Itakepleasure inrepeating, toahintfrom Mr Cayley", who habitually discourses pearls and rubies, thatI amindebted fortheprecious andpregnant observation ontheformassumed by the first discriminantal evectant ofabinaryfunction with a pairof equal roots, out of which, combined with some antecedent reflections of my own, this newtheoryof multiplicity hastakenitsrise. The idea of theprocess of evection, and the discovery of itsfundamental property ofgenerating what, in my calculus of forms (Camhridge andDublinMathematical Journal), I havecalled contravariants, is due to my friend M. Hermite. The polar reciprocals of curves and otherloci arecontravariants and, as I have recently succeeded in showing, for curves at least, evectants, butof course not diecriminsntal evectants iand I am alreadyable to give theactualexplicitrule for the formation of thepolar reciprocal of curves as high as the5thdegree, which withalittlelabourandconsideration can becarriedon to the 6th, and in fact to curves of any degreenwhen once we are acquainted with any mode ofdetermining all suchindependent invariants of a function of two variables asareofdimensions not exceeding 2(n-1)inrespectof the coefficients. Bythespecialgeometers (by whom I meanthose who, unvisited bya higherinspiration, continue toregardand tocultivate geometry asthe science of mere sensiblespace)thisproblem has only been accomplished, and thatbutrecently, for curves whose degrees do not exceed the4th. Mr Salmon hasmadethehappyandbrilliant (and bythecalculus of forms instantaneously demonstrable) discovery. communicated to me in thecourse of amost instructive andsuggestive correspondence, thata certain "eadilyascertainable •MrOayley'etheoremstood thus:-If ="+nbx"-I y+...+nb'zy,,-I +a'Y" have two equal roots, aud 1ll'beitediecriminant, then will {IiIiIi}Y"-_ytl-IZ-&c zZ"- 1ll'dadb' da' beaperfectnthpower.Itwilleaeilybeseenthatthie theorem ie convertible into atheorem of evection by interchanging in the reeult zandywith11and-a: 46J Ona newTheon)ofMultiplicity. 377 evectantofeverydiscriminant ofany function whatever isanexactpowerof itspolarreciprocal", I believe thatit may be shown, that,with the sole exception of odd­ degreedfunctions of two variables, the polar reciprocal itself(asdistinguished from a power thereof)of every function is an evectant, not (of course) of the discriminant, butofsomedeterminable inferiorinvariant. P.S. The termspluri-simultaneous and piuri-simultaneity, used or suggested by me in my last paperintheMagazine, may beadvantageously replaced bythemoreeuphonious andregularly formed words eonsimul­ taneous,consimultaneity. Multiplicity and all its attributes and consequences areincluded as particular casesinthegeneralconception and theoryof consimultaneity, thatis ofconsimultaneous equations, or, which isthesame thing,ofconsimulevanescent functions. •NlUIIely, forafunction of degree n,andvariability (thatis,havinganumberofvariables) p.the(11-1)P-1th evect of the discriminant isthe(11-l)thpower of the polarreciprocal. 47. A DEMONSTRATION OF THETHEOREM THAT EVERY HOMO­ GENEOUS QUADRATIC POLYNOMIAL IS REDUCIBLE BY REAL ORTHOGONAL SUBSTITUTIONS TOTHEFORMOF A SUM OF POSITIVE ANDNEGATIVE SQUARES. [Philosophical Magazine, IV.(1852),pp.138-142.] ITis well known thatthereduction ofanyquadratic polynomial (I,I)a;2+2(I,2)xy+(2, 2)y'+...+(n, n)t' totheforma1r+ ~7l+...+a"tJ2,where ~,7J•••earelinearfunctions of X,y...t,suchthata;2+ys+...+t'remains identical withr+"It+...+ (Jt (whichidentity isthecharacteristic testoforthogonal transformation), depends uponthesolution oftheequation (I,I)+;\.,(I,2) (I,n)=O. (2,I), (2,2)+;\.(2,n) (n,I), (n,2).........(n,n)+A Theroots of thisequation givea"~...an;andiftheyare real,itiseasily shownthattheconnexions between X,y ...t;~,"I'"e,are also real. M.Cauchyhassomewhere givena.proofofthetheorem·, thattheroots of x, intheaboveequation mustnecessarily always be real;buttheannexed demonstration is, I believe, new;andbeingverysimple, and reposing upon atheorem ofinterest in itself, and capable nodoubtofmanyotherapplica­ tions, will, I think,beinteresting tothemathematical readers ofthis Magazine. *JacobiandM.Borchardt have also given demonstrations; thatofthelatterconsistsin showingthatSturm's funotions forascertaining thetotalnumberofrealrootsexpressed bymy fcrmulse (many years agogiven in thisMagazi7U!) areall,inthecaseofI(X),representable asthe sumsofsquares,andaretherefore essentially positive. 47] OnHomoqeneous Quadratic Polynomials. 379 Let f(A)=(1, 1)+A, (2, 1), (3,1),(1, 2) (1,n) (2, 2)+A (2,n) (3, 2), (3, 3) +A(3,n) (n,I), (n,2) (n, n)+A itiseasilyprovedthatf(A)xf(-A) =[1,1]-AI,[1,2] [1,n] [2, 1], [2, 2]- At[2,n] [n,I], [n,2].........[n,11]-AI where [t,e]=(t,1) x (1,e)+(t,2) x (2,e)+...+(t,n)x(n, e). If,now, for all valuesofrands, (r, s)= (s,r),thatis, iff(O)becomes the complete determinant to asymmetrical matrix,theneveryterm[r, s]in thederived matrixbecomes a sumofsquares, and isessentially positive, and(-l)"f(A)xf(-A)assumes theform (AI)"-F(A,)n-l+G(A')"~+ ...±L, whereF, G,...Lwillevidently be allpositive jforitmay beshownthatF will bethesumofthesquaresoftheseparate terms,thatis, ofthelast minordeterminants ofthegivenmatrix,Gthesumofthesquaresofthe lastbutoneminors,andso on,Lbeingthesquareofthecomplete deter­ minant. Forinstance, if f(A)=a+A,ry, f3 ry,b+A,a f3,a,c+A -f(A)xf(-A)=A6-F>..4+GAt-H, where F=a'+b'+e'+2a'+2{3'+2'Y', G=(ab-ry')'+(be-a')'+(ac-f3')' +2(aa-f3'Y)'+2(bf3-rya)'+2(cry-af3)', H=a,ry,f3' ry,b,IX B,a,c Henceitfollowsimmediately thatI(A)= 0cannothaveimaginary roots; for,ifpossible, letA=P+q.j(-1),andwrite a+p=a', b+p=b', e+p=e', A+P=A', 380 On Homogeneous Quadratic Polynomials. [47 a C'+A'a,"I. b'+A',a'+A', "I, fl. or sayep(A'),andtheequation ep(A')xep(-A')=0 will be of theform A'6-F'A"+G'A"-H'=O. whereF', G'.H'are allessentially positive, Hence,byDescartes' rule, no value of A"can benegative, thatis,(A-p)'cannotbe oftheform -q'; thatis to say, itisimpossible foranyoftheroots ofI(A)=O to beimaginary, or, as was to be demonstrated, alltheroots are real. Imaytakethisoccasion toremark,thatbywhatever linearsubstitutions, orthogonal orotherwise, agivenpolynomial bereduced totheformIAIr', thenumber ofpositiveandnegative coefficients is invariable: thisis easily proved.Ifnow we proceed toreducetheform(expressed undertheumbral notation) (~XI+a,x,+...+anxn)'totheform Al'I'+A,','+...+An-Irn-I+An'n', by firstdrivingoutthemixedtermsin which XIenters.thenthose in which x,enters,and so forth untileventually onlyXnoftheoriginal variables is left,itmayreadilybe shown that Al=(~),A,=(~~)+(~).A.=(~~~)+(~~)......f(A)becomes A(~a,...an).(~a,...an-I)...,,= -s- •ala,...an ,~a,... an-I Itfollows.therefore, thatinwhatever orderwearrangetheumbne ~a,...an, thenumberofvariations and ofcontinuations ofsignintheseries I,(~), (~a,)...(~a,an), Ut ~a, Uta,an will beinvariable, andinfactwill bethesameasthenumber ofpositive andnegative roots in thegenerating function inAabovetreatedof,thatis, since all theroots are real,will bethesameasthenumber ofvariations andcontinuations intheseriesformed by thecoefficients of theseveral powers of A,thatis I,I(al) .I(Uta,)...(Uta,an\. Ut Uta, ~a,o-J The first partofthistheorem admitsof aneasydirectdemonstration; for by my theoryofcompound determinants, given in thisMagazine-, we knowthat ~a,a,....lo.,.l Uta,..•ar-Iar+11 ala,0.,.-1a,.UtaJlar-Io.,.+l =(~a•...0.,.-1)x(ala,o.,.-Io.,.o.,.+l) . ala,...ar-I ~a,a,....1ar~1 [*Cf. pp.241,252abcve.] 47] OnHomogeneous Quadratic Polynomials. 381 Thefirstmemberofthisequation isequivalent to (a"tItCLr-1a,.)x(ala.aa,.-lar+1)_(a"a.aCLr-lar)1. a"asCLr-1a,.a1UsCLr-1ar+1a"asCLr-1a,.H Henceitfollows,thatifthetwofactors on theright-hand side ofthe equation have the samesign, (ala.aCLr-1a,.)and(a1asa,.-Iar+l) altitar-Ia,. a"a.aa,.-Iar+1 have also the samesigninterS6,andconsequently thetwotriads [a"asUr-I][a"as..,CLr-1a,.J [a"as...ar_la,.ar+l] , a"a.aCLr-1' a"a.aCLr-lar' a"tIt.. ·a,.-lara,.+1 and [a"ayar-I],[ala.aar-Ia,.+I],[a"a.aar-Ia,.+Ia,.] a"asCLr-l a"a.aCLr-1a,.+1 Q1a.aa,.-Ia,.Ha,., willinallcasespresentthesamenumber of changes and continuations, which proves thatthecontiguous umbree, Ur,ar+l'maybeinterchanged without affecting thenumberofvariations andcontinuations in theentire series;but,asis well known, anyoneorder of elements is always convertible intoanyotherorderby means of successive interchanges ofcontiguous elements, whichdemonstrates that,inwhatever order the elements a",as...an bearranged, thenumberofcontinuations andvariations in 1,(a,,),(a"a.a\ ...(ala.aan), a"a"tit)a"a.aUn isinvariable. Butthatthesamethingistrue(as we know it to be), for the relationbetweenanyoneof these unsymmetrical series and thesymmetrical series(resulting fromthemethodoforthogonal transformation) 1,~(a,,),I(Qla.a),'"(a"a.aUn), a" a"asa1asUn is by no means 80easilydemonstrable inthegeneralcase by a directmethod, andtheattention ofalgebraists isinvitedtosupplysuchdirectmethodof demonstration. My knowledge of thefact ofthisequivalence is, asI have stated,deduced from thatremarkable butsimple law to which I have adverted, which affirms the invariability ofthenumberofthepositiveand negative signsbetween all linearly equivalent functions of the form I±crw (subject, of course, to thecondition thattheequivalence is expressible by means of equations into which only real quantities enter);alaw to which my view of the physical meaning ofquantity ofmatterinclines me, upon the ground of analogy, to give thename of the Law of InertiaforQuadratic Forms,asexpressing thefact oftheexistence of aninvariable number inseparably attached to such forms. 48. ONSTAUDT'S THEOREMS CONCERNING THECONTENTS OF POLYGONS ANDPOLYHEDRONS, WITH ANOTEON A NEWANDRESEMBLING CLASS OFTHEOREMS. [Philosophical Ma.gazine, IV.(1852),pp.335-345.] THEbeautiful andimportant geometrical theorems ofStaudtare, I believe,little,ifatall, known to English mathematicians. Theyoriginally appeared inCrelle'sJournal fortheyear 1~43,and have been recently reproduced in M.Terquem's Nouvelles Annales fortheAugustNumber of thepresentyear. Thesetheorems may be summed up, in aword,asintended to showthe possibility and method of expressing theproductof any two polygons or any two polyhedrons asentirefunctions of the squaresof thedistances ofthe angularpoints of the two figures from one another. The well-known expres­ sion forthesquare of the area of atriangleintermsofthesides (in which, when expanded, only even powers of thelengthsofthesides appear), is but aparticular caseofStaudt's theorem for polygons, for it may be considered asthe case of two equal and similartriangles whoseangularpoints coincide. So in like manner, as observed by Staudt,asimilarexpression in termsof itssides may be found for thesquareof a pyramid. This expression had, however, been previously given (although, byastrangle negligence, not named for what it was) by Mr Cayley in theCambridge Mathematical Journal fortheyear1841-,in hispaperon therelations between the mutualdistances to oneanotherof fourpointsin a plane and five pointsin space;thesingularly ingenious (andassingularly undisclosed) principle of thatpaperconsisting inobtaining an expression for thevolume of apyramid in terms of its sides, and equating this, orratherits sqnare, to zero asthe conditions of the four angularpointslying in the same plane. •Query, Is Dot thisexpression forthevolumeof&pyramid intermsofits sides Wbefound in someprevious writer?Itcanhardlyhaveescapedinquiry. 48] OnStaudt' 8Theorems. 383 Theanalogous coudition for fivepointsin space is virtually deduced by goingoutintorational space of four dimensions, andequating to zerothe expression obtained forthevolume of aplupyramid; meaning therebythe figure which standsinthesamerelation to space of four asapyramid to spaceofthreedimensions. Mr Cayley's method, ifithad been pursued astepfurther,would have led him to acomplete anticipation oftheprincipal partofStaudt's discovery. Themethod heregivenisnotsubstantially different from Mr Cayley's, butis made to restupon a more general principle oftransformation thanthatwhich he has employed. .A.Bto Staudt's ownmethod,itisasclumsy and circuitous ashisresultsaresimple andbeautiful. Geometry, trigonometry andstatics,are laid undercontri­ butiontodemonstrate relations which will be seen to flowasimmediate and obviousconsequences from the most elementary principles inthealgorithm ofdeterminants. Perhaps, however, M. Staudt's methodisasgoodascould befound in theabsence oftheapplication ofthemethodofdeterminants, thepowers of which, even so recently astenyears ago, were not so well understood or so freely appliedasatthepresentday. Thefollowing new butsimpletheorem, of which Ishall have occasion to makeuse, will be found to be a very useful addition totheordinary method forthemultiplication ofdeterminants. "Ifthedeterminants represented by twosquarematrices are to be multiplied together, any number ofcolumns maybecutofffrom the one matrix,and acorresponding numberofcolumns fromtheother.Eachofthelines ineitherone ofthematrices soreduced inwidthasaforesaid beingthenmultiplied by each line of theother,and theresultsofthemultiplication arranged asasquarematrixandbordered withthetworespective sets of columns cutoffarranged symmetrically (the onesetparallel tothenew columns, theothersetparalleltothenew lines), thecomplete determinant represented bythenewmatrixsobordered (abstraction made of thealgebraical sign)willbetheproduct ofthetwo original determinants." Thus(~)x(~)may beputunder~nyone ofthethreefollowing forms:- Iaa+b{3,wy+b8j ca+d{3,C'y+dB or aa,wy,b 2, 2,a, b • ca,C'y,dor 2, 2, c, d {3,8,0 a,{3,0, 0 'Y,8,0, 0 •Anyquantities mightbesubstituted insteadof 2 in the places occupied by thefigure in the aboTedeterminant, as suchtenDSdo not influence the result;thisfigureis probably, however, theproperquantity arisingfrom the application of the rule, because (as allwho have calculated withdeterminants areaware)the value of the determinant represented byamatrixofnoplace. is notzerobutUDity. 384 OnStaudt'sTheorems concerning the [48 And ingeneralfor twomatrices of nltermseach,thisruleofmultiplication will give(n+1)distinctformsrepresenting theirproducts. Thus,asafurtherexample, a,b,ca,{:J,'Y, a',b',e' xa',e.'Y' a",b",e" a."Je:7" besidesthefirstandlast forms, will be representable bythetwointermediate forms aa+b{:J,aa'+bfJ',ai'+bd",c a'«+b'{:J,a'a'+b'{:J',a'a:"+b'fJ",c, a"a+b"fJ,««+b"fJ',a"a"+b"{:J",e" ,"0 'Y. 'Y, 'Y, and all,,ao",i, aa, c,a'a',a'a",b',e' aa, +a'«,alia',ItIIb"e" aa,, {:J,s,s;0,0,"0,0 'Y,'Y,'Y, 0,'Y 0,'Y' s:0,'Y" 0, I, 0a,fJ, a',fJ', 0,Toarrive,forinstance, atthelatterof these two forms, we have only to writethetwo given matrices undertherespective forms a,b,c,0,0,a,0, 0,{:J,'Y a',b',e',0, 0 a',0, 0,fJ','Y' a",r,c",0, 0 a",0, 0,e:'Y" 0, 0, 0, I, 0 0, 1, 0, 0, 0 0, 0, 0, 0, 1 0, 0, 1, 0, 0 andthenapplytheordinary rule ofmultiplication. So, again, to arrive atthefirst oftheabovewrittentwo forms, we mustwritethetwogiven matrices undertherespective forms a,b,c,0 a',b',d,0 and -a",r,«,0 «", 0, 0, 0, 1 and proceed asbefore. Thisrule isinteresting asexhibiting, asabove shown, acomplete scale whereby we may descend from theordinary mode ofrepresenting theproduct of twodeterminants totheform, also known, where thetwooriginaldeter- 48] contentsofPolygons andPolyhedrons. 385 minants are made tooccupy opposite quadrants of a square whose places in one of the remaining quadrants are leftvacant,and shows us thatunder oneaspectatleastthislatterform may be regarded asamatrixbordered bythetwo given matrices. A second butobvious theorem requiring preliminary noticeisthe following, namely thatthevalue of the determinant to thematrix lIt,1lCZt,,··· CZt,n,1, as,lla"I1'" lZ.a,n,1, Un,llan,II ' "Un,n,1, 1, 1,...1, 0, isthesameasthevalue of the determinant tothematrix AI,I'AI,II•••AI,n,1, A2,I'AII,II...A2,n,1, ........................... An,llAn,II ' "An,n,1, 1, 1,...1,0, where in general Ar"=ar,,+h;+k" ~,~...hnand~,k,...knbeing any two perfectly arbitrary seriesofquantities. Thissimpletransformation is of course derived by addingto therespective columns in the first matrixthelastcolumn (consisting of units)multiplied respectively by~,~...!Ln,0;and totherespective lines, thelastline (consisting ofunits)multiplied respectively bykllk,...kn,O. Suppose, now,thatwe have two tetrahedrons whose volumes arerepre- sentedrespectively by one-sixth oftherespective determinants ~,YI'ZI'1EI'1'Jll ~ll1 a;,y"Z,I,1 E,l,1'J1I' ~,1,1 ,X..y"Z"1 E"1'J"~"1 ,,x.,s:Z.,1E.,1'J., ~4'1 'xr,Yr,Zrrepresenting the orthogonal coordinates ofthepointr in one tetrahedron, andEr,1'Jr,~rthesame for the pointr intheother. Bythefirst theorem theirproductmay berepresented (striking off the lastcolumn only from each matrix)bythematrix Ia;EI> ~~EIII I~E"Ia;E.,1 ulIEI>I'xIIEII,uIIE"u,E.,1 ~l> ~E,IJI'x,E"I'x,E.,1 u.~, u.~, I~~, u.~,1 1, 1, 1, 1, 0 B. 25 386 OnStaudt' 8Theorems concerning the [48 where, in general,anysuchtermasurEarepresents xrE.+Yr'TJ.+zr~•. Again, by virtueofthesecondtheorem. adding -ruII,-lUll,-lu.',-lu,1 totherespective lines,and -i!EI',-l!E.',-lIE.I,-p:U 1,!(XI-E,)', !(xl-E,)', !(x.-E,)',1 !(x,-E,)',1 °totherespective columns, theabovematrixbecomes (afterachangeofsigns notaffecting theresult)the-ithof !(XI-~I)"!(XI-E,)I,!(Xl-E,)'. !(X,-EI)',!(XI-EI)',!(x.-E.)" !(x,-EI)I,!(x.-~I)"!(x.-E.)', II(x,-EI)'.!(x,-EI)',I(x,-E.)', 1, 1, 1, orcallingtheangular pointsoftheonetetrahedron a,b,c,d,andofthe otherp, q,r, s,8 x 36,thatis 288times,theirproduct isrepresentable by -1xthedeterminant (ap)',(aq)l,(ar)l,(as)&,1 (bp)',(bq)',(br)l,(bs)',1 (cp)',(cq)',(cr)l,(CS)I,1 (dp)',(dq)',(dr)',(ds)',1 1,1, 1,1,° andof course ifp, q,T,8coincide respectively witha,b,e.d,576timesthe squareofthetetrahedron abedwillberepresented underMrCayley's form, 0,(ab)',(ac)l,(ad)',1•, (baY. 0,(be)', (bd)', 1 (ca)l,(eb)', 0,(cd)',1 (da)l,(db)',(de)', 0,1 1,1,1,1,°fouroutofthesixteendistances vanishing, andtheremalDlDg twelve reducing to sixpairsofequaldistances. Thedemonstration ofStaudt's •Thecorresponding quantity to the above determinant for the case of thetriangle(hereafter given) is identical with the Norm tothesum of the sides. I have succeeded infindingthe Factor(oftendimensions inrespectof the edges), which,multiplied bytheaboveDeterminant itself,expresses the Norm to the sum of theFaces,thatis, thesuperficial area oftheTetrahedron. 48] contentsofPolygons andPolyhedrons. 387 theorem fortriangles isobtained in precisely the8&IIleway bythrowing the productof the two determinants a;,'!Ill1 EI''Ill1 1liI,'!I"1andE"1].,..1 a:,,'!I"1E"1]"1 undertheform of - ithof I(a;-'I)',I(a;-E,)',1:($1-,,'I,1 I(a:,-'I'f, 1:(a:,-EI)I,I($1-E,)',1 I(a:,-'I'f, 1:($1-E.)',I(a:,-,,'I,1 I, 1, 1, 0 Whenthetwotriangles coincide, calling theirangularpointsa,b,e theabovewrittendeterminant becomes 0,(ab)',(ac'f,1. (ba)l, 0,(be'f,1 (00)',(eb'f, 0,1 1,1, 1, or (ab)4+(ac)4+(be,!-2(ab'f.(ae'f-2(ab'f.(be'f-2(ac)'.(be'f, thenegative of which is thewell-known form expressing thesquare of four timestheareaof thetriangleabe. Thereisanotherand more general theorem ofStaudtfor twotriangles not in the sameplane, which may be obtained with equal facility. Infact, ifwestartfrom the determinant (ael)',(afJ'f,(art)',1 , (bel'f,(bf3)I,(bty'f,1 (Cel'f,(cf3'f,(e-y)',1 1, 1, 1, and add to eachcolumn respectively thelastcolumnmultiplied byeEl',eU, 6E,'respectively, we arriveatthe form (aa'f+eEt',(afJ'f+eE.',(art)'+eft',1 (ba'f+e'll,(bf3)'+eU,(bty)'+eft',1 (Cel'f+e'I',(cfJ)'+eEt',(e-y)'+eU,1 1, 1, 1, andconsidering '111]1j'I'1].j'I,1],asthecoordinates of el,fJ,,,/,the 25-2 388 OnStaudt's Theorems concerning the [48 projections upon theplane of abcofatriangleABC,whose plane intersects theformer plane in theaxisofy,and makes with thatplane an angle whose tang~ntise,itis easily seen thatthisdeterminant istermfortermidentical withthedeterminant I(aA)I,(aB)I,(aC)l,1I, (bA)',(bB'f,(bC)l,1 (cA)I,(cB'f,(cC)',1 1, 1, 1,0 which therefore expresses - 16 timestheproductofthetriangles abcand afJ"'f,thatisabcxABCx cosine of theangle between thetwo. A similar method, if we ascend from sensible torational geometry, may be given for expressing intermsofthedistances theproductofanytwopyramids (in ahyperspace) by the cosine of the angle included between thetwoinfinite spsces"in which theyrespectively lie. Topassfrom the caseswhich have been considered of two triangles to two polygons, or of two tetrahedrons to two polyhedrons, generally presents nodifficulty; andfor Professor Staudt's method of doing so, which is simple and ingenious, and does not admitof material improvement, thereaderis referred tothe memoir inCrelle'sJournal orTerquem's Annales alreadyadverted to.Itis, however, to be remarked (andthisdoes not appearto be sufficiently noticed in thememoirs referred to),thatwhilsttheexpression for theproductof any two polygons interms of the distances given by Staudt's theorem is unique, thatfor theproduct of two polyhedrons given by the same is not so, butwilladmitofasmany varieties ofrepresentation asthereareunitsin theproductof thenumbers respectively expressing thenumberof ways in which each polygonal face of each polyhedron admitsof being mapped out into triangles. I cannot help conjecturing (andit is to be wished thatProfessor Staudtor some other geometrician would consider this point) thatin every casethereexists, linearlyderivable from Staudt's optional formulas (butnot coincident with anyoneof them), some uniqueandbest, because most symmetrical, formula. for expressing the productof two polyhedrons in terms of thedistances of theangularpointsof the one from those of the other. Inconclusion I may observe,thatthereis a theorem for distances measured on a givenstraight line, which, although notmentioned byStaudt,belongs to precisely thesame classashis theorems for areasin a plane and volumes in space;namely atheorem which expresses twice therectangle ofanytwo such distances underthe form of anaggregate of four squares, two takenpositively and two *Inrational oruniversal geometry, thatwhich is commonly termedinfinitespace(asif it weresomething absolute and unique, andtowhich, by the conditions of our being, the repre­ sentative powerof thennderstanding islimited), isregarded as a single homs.loid related toa plane,precisely in the same way as a plane is to a rightline.Universal geometry bringshome tothe mind with an irresistible forceofconviction thetruthoftheKantian doctrine oflocality. 48J contentsofPolygons andPolyhedrons. 389 negatively; thatis tosay,ifA,B,C,Dbeanyfourpointsonarightline 2ABxCD=AP+BCS-ACS-BD'.I know not whether thistheorem benew,butit is one which evidently mustbe of considerable utilitytothe practical geometer. Nateontheabove. Thefundamental theorem indeterminants, published by me in the Philosophical Magazine inthecourse of last year·,leadsimmediately toa classoftheorems strongly resembling, anddoubtless intimately connected with,thoseofStaudt. Thusfortriangles we have by this fundamental theorem XI'X"a:, ~ll~"~, Yl'y"y,x 1, 1, 1 .1;,Ell ~2'7]11'7]"'7]'1 1, 1, 1. -Yl''7]11'7],x 1, 1, 171.,Y2'y.+Yl''I"'I. 1, 1, 1 1, 1, 1X'7]1'y"y. 1, 1, 1 1, 1, 1 1, 1, 1 andconsequently, ifABC,DEFbeany twotriangles, ABCxDEF=ADExFBC+AEFx DBC+AFDxBCR Thismay be considered a theorem relating to twoternarysystems of pointsina plane. The analogous andsimilarly obtainable theorem for two binarysystemsofpointsinthesamerightlineis ABxCD=ACxDB-AD xCB. Asinapplying thislasttheorem toobtaincorrectnumerical resultswemust givethesamealgebraical sign to anytwolengthsdenoted bythetwo arrangements XY;ZT,according as thedirection fromXtoYisthesame 88thatfromZtoT,orcontrary to it, so in thetheorem fortheproducts oftriangles, theareasdenoted by any two ternaryarrangements XYZ,TUV mustbetakenwiththelike orthecontrary sign, according as thedirection oftherotationXYZis consentient with or contrary tothatofTUVj 80 thatthreeofthesix possible arrangements ofXYZmay be used indifferently for oneanother, buttheotherthreewould imply a changeof sign.Ifwe [*Seepp.249,253above.] 390 OnStaudt' 8Theorems concerning the [48 analyse what we mean by fixing the direction of therotation ofXYZ,and reducethisformofspeech to itssimplest terms, we easily see thatitamounts toa.scertaining on which side of B, 0lies,thatiswhethertoitsrightor left, to aspectator stationed atAon a given side of the plane ABO. Letus nowpasstothecorresponding theorems for twotetrahedrons put respectively underthe forms 11;,Xs,X..X4,EI>t;Es,E4 YI>y..,y.,Y4 "11>"1i'"1a,"14 ZI>Zs,Z.,Z4'1''I',.,'4 1, 1, 1, 1 1, 1, 1,1 We may represent thisproductineitherof two ways by the application of ourfundamental theorem, namely as a;,El'E..E. E4'Xs,Xa,X4 Yl'"11''1..,"1. "14'Ys,Ys,Y4+&c. X Zl''I>'I',.'4'Zs,E.,E4 1,1, 1, 1 1, 1, 1, 1 oras IXI'Xa,EI>E.. E.,E4'Xa,X4 Yl>Ys,'11,"1s 'la,"14'Ys,Y4+&C. X 1'1'Es,'I>,..'a,'4'Za,Z4 1,1, 1, 1 ,1,1, 1, 1 therebeing four products to be added together inthefirst expression and six inthelatter;and the rule, ifwe wish thatalltheproducts may be additive, beingthaton removing the sign of multiplication thedeterminant to thesquarematrixformed by theGreeklettersin situshall always preserve thesamesign. Hence we derivetwogeometrical formuhe concerning the products of polyhedrons, namely (1)ABODxEFGH=ABOExFGHD-ABOFx GHED +ABOGxHEFD-ABOHxFGED. (2)ABODxEFGH=ABEFx GHOD+ABGH xEFOD +ABEGxHFOD+ABHFxEGOD +ABEHx FGOD+ABFGxEHOD. Theseformulee giveriseto an exceedingly interesting observation. In orderthatthey shall be numerically true,we must have a rule for fixing the sign to be given to the solid contentrepresented by anyreadingoff of thefourpointsof atetrahedron, thatis wemusthave a rule for determining 48] contentsofPolygons andPolyhedrons. 391 thesign of solid contents of figures situated anywhere in space analogous tothatwhich,asappliedtolineardistances reckoned on a given rightline, isthetruefoundation of thelanguage oftrigonometry, and the condition precedent forthepossibility of any system of analytical geometry such as exists, and which, not altogether without surprise, I have observed in the pages of thisMagazine one of the learnedcontributors hasthoughtit necessary tovindicate thepropriety ofimporting into histheoryofquaternions. Variousrulesmay be given for fixing the sign of a tetrahedron denoted by a given orderof fourletters. One is the following: the contentofABCD is to be takenpositive ornegative, according as to a spectator atAthe rotation ofBCDis positive or negative. Another, again, is to consider.AB andCDasrepresenting, saytwo electrical currents, and to suppose aspectator so placed thatthecurrent.ABshall pass throughthelongitudinal axis of his body from thehead towards thefeet, and looking towardstheothercurrent CD;thesign of the solid contentof thetetrahedron (and, indeed, also the effect,in a general sense, of theaction of the two currents upon one another) willdependupon the circumstance ofthislattercurrentappearing toflow from the righttotheleft, orcontrariwise inrespectofthespectator. Last andsimplest mode of all, thesign of the solid contentofABCDwill depend upon the nature(inrespectto its being a right-handed orleft-hand ed-screw) of anyregularscrew-line (whether thecommon helix or one in which the increase or decrease of theinclination is always in thesamedirection) terminating atBandC,and sotakenthatBAshall be the direction ofthe tangentproduced atB,andCDthedirection ofthetangentproduced at C. Inasmuch as ofthetwenty-four permutations of aquaternary arrangement a defined twelve have one sign, and theothertwelve the contrary sign, these various definitions of the direction, or, asit may be termed,polarity, of a tetrahedron corresponding toagiven reading, whetherastakeneach initself or compared one with another, give rise to, or ratherimply a considerable number ofinteresting theorems included in our intuitions of space, and probably belonging to the, in my belief, inexhaustible class ofprimary and indemonstrable truthsof theunderstanding. '49. ON A SIMPLE GEOMETRICAL PROBLEM ILLUSTRATING A CONJECTURED PRINCIPLE INTHETHEORY OF GEO­ METRICAL METHOD. [Philosophical Magazine, IV.(1852), pp. 366-369.] THE following theorem deserves attention asillustrating aprinciple' of geometrical method which will bepresently adverted to.Itis curious, also, from the fact ~fits solution being by no means so obvious and self-evident asone would expect from the extreme simplicity ofitsenunciation. It appeared, and for the first time, it is believed, attheUniversity ofCambridge aboutatwelvemonth back, where itexcited considerable attention among some of themathematicians of the place. The proposition, asoriginally presented. wasmerely to prove thatifABObeatriangle, and ifADand BEdrawnbisecting theanglesatAandBandmeeting the opposite sides inDandEbe equal, thenthetrianglemust be isosceles. Itisparticularly noticeable thatall thegeometrical demonstrations yetgiven of thistheorem are indirect. Thusthe first and simplest(communicated to meby a promising younggeometrician, MrB.L.SmithofJesusCollege, Cambridge), wasthe following :-Assume one of the angles atDABto begreaterthanthe corre­ sponding angle EBA;it can easily be shown that,uponthissupposition, Dwill be higher up from ABthanE;sothatifDFandEGbedrawn paralleltoAB,DFwill be above EG;it istheneasily shown thatDF=AF, EG=BG,andconsequently DFand.ill'are each respectively less thanEG 49] OnalfimpleGeometrical Problem. 393 andBO;andalsoDFA,which is thesupplement of twiceDAB,will be less thanEGB,which is the supplement of twiceFBA;from which itis readily inferred, byaneasycorollary to a proposition of Euclid, thatDAwill be less thanFB,whereasitshould be equal to it;sothatneitherofthehalfangles atthebasecan begreaterthantheother, and the triangleis proved to be isosceles. Another andindependent demonstration by thewriterofthis articleis less simple, buthasthe.advantage of lending itselfatonce to a considerable generalization of the theorem asproposed. Assuming, as above, thatDABisgreaterthanEBA,it is easily seen thatDEproduced will cut BAatXontheside ofit:also ifADandBEintersect inH,it is readily demonstrable, by asuitably constructed apparatus ofsimilartriangles, that AH:BH::CE:CD. ButasHBAis lessthanHAB,AHis lessthanBH,and therefore CEis lessthanCD,and therefore CEDisgreaterthanCDE;thatis to say, CABlessXisgreaterthan'CBAplusX,andtherefore DABlessXis greaterthanEBA,thatisADEisgreaterthanABE,andtherefore the perpendicular fromAuponDEisgreaterthanthatfromEonAB,which iseasilyprovedtobeabsurd. Hence,asbefore,thetriangleis proved to be isosceles. This proof,it is obvious, remainsgood for allcasesin whichEB andDA,drawn on eitherside ofthebase, divide the angles atthebase proportionally, provided thatthese lines remainequal, and make positive ornegative angles with the basenotlessthanone-half of therespective corresponding angles which thesides of thetriangleare supposed to make withit. The analytical solution of the question, asmightbe expected, extendstheresultstillfurther. To obtain this, let BAC=n.BAD, ABC=n.ABE, nforthepresent being any numerical quantity, positive or negative; callingBAC=2na,ABC=2n/3,we readily obtain,by comparison of the equaldividing lines with thebase of the triangle, sin(2na+2/3) sin2nasin(2n/3+2a) sin2n/3 orsin(2na+2/3)sin2na sin(2n/3+2a)=sin2nfJ; andby an obvious reduction, tan(n-l)(a-/3)_tan(n+l)(tt+/3) tann(a-/3) -tann(a+fJ) Whenthisequation isputunderanintegerform, it is of course satisfied bymaking a==/3;on anyothersupposition thana=/3itevidently cannot besatisfied by admissible values of theangles for any value of nbetween 394 Ona simple GetJmetrical Problem. [49 + 1and+00;for onthatsupposition, since(a-{3)and(a+{3)areeach less thanI:nO,thefirst side of theequation will benecessarily aproperfraction andpositive; butthesecond side, eithera.positive improper fractionif (n+1)(a+{3)be less,andanegative properor anegative improper fraction if(n+1)(a+{3)begreaterthanarightangle. Ifnbenegative, letitequal -v,then tan(v+I)(a-{3)tan(v-I)(a+{3) t&nv(a-{3) =tanv(a+{3) ; and forthesame reason asbefore, if vliesbetween 00and1,thisequation cannotbe satisfied. Hencethetheorem is proved to be truefor allvalues ofn,exceptbetween +1and-1.Forthesevaluesitceases to be true; in fact, for such values for any givenvalues of (a-{3)therewill bealways, asitmay be easily proved, one or more values of (a+{3);thusifn=j,the equation becomes andifn=-i,tan3(~) tana+1:3 ;l-1; tan3C~~ --";""....."..";"=-1a-{3 ,tan--;l showing thata+{3=90anda-{3=±90 intheserespective caseswill afford a solution overandabovethesolution a={3,which is easily verified geometrically". Itwould be an interesting inquiry(for those who have leisure for such investigations) todetermine forR.nygivenvalueofnbetween + 1 and - 1 the superior andinferiorlimitstothenumberofadmissible values ofa+/3corresponding to anygivenvalue of a-M. Myreaderwill now be prepared to see why itisthatallthegeometrical demonstrations givenofthistheorem, even inthesimplest caseof all,namely whenn=2, areindirect, I believe I may ventureto saynecessarily indirect. Itisbecausethetruthofthetheorem depends onthenecessary non-existence of real roots (between prescribed limits)oftheanalytical equation expressing theconditions of thequestion; and I believe thatitmay be safelytaken asan axiom in geometrical method,thatwhenever thisisthecasenoother • In the first of theseeases, if the base of the triangle issupposed given,thelocus of the vertexisarightlineandaeirele; inthesecondcase,arightlineandanequilateral hyperbola. tWhen ,!,n liesbetween 2.~1and2.~1 (.beinganypolitive integer), it'iseasilyseenthat theauperior limitmustbeatleastasgreatas•. 49] Ona simple Geometrical Problem. 395 form of proof thanthatofthereductio ad absurdum is possible in the nature ofthings.Ifthisprinciple is erroneous, it mustadmitof aneasyrefutation inparticular instances. Asan example, I throw out (not a challenge, but)aninvitation to discover adirectproof,if such exist, of thefollowing geometrical theorem, assimple aoneasit isperhaps possible to imagine:-"To prove thatif fromthe middleof acirculararctwo chords bedrawn, and theremotersegments of these chords cutoff bythelinejoiningtheend of the arc be equal, the nearersegments will also be equal." The analytical proof depends upon the factof theequation :r;'+a:r;=lJ2(whereais the given lengthofeachsegment, andbthelengthof the chord of halfthegivenarc)havingonly one admis­ sible root jand iftheprinciple assumed orpresumed tobetruebe valid, no otherform of pure geometrical demonstration thanthereductio adabsurdum should be applicable in thiscase.For the converse case,wherethenearer segments are given equal, thereducing equation isa(a+x)=b',indicating nothing tothecontrary ofthepossibility of therebeing a directsolution, which accordingly is easily shown to exist. The indirectform ofdemonstra­ tion,itmay bementioned, issometimes liable to be introduced in amanner toescape notice. As, for instance, if it should betakenforgrantedinthe course of an argument, thatonetriangleuponthesamebaseandthesame sideof itasanothertriangle, andhavingthesamevertical angle, musthave itsvertexlying on the samearcjthiswould seem to be immediately trueby virtueofthewell-known theorem, thatangles in thesamecircularsegment are equal, butinrealitycanonly beinferred from itindirectly by showing theimpossibility of its lying outside or inside thearc in question. To go one stepfurther,I believe itto be the case,thatgranted to betrueall those fundamental propositions in geometry which are presupposed in theprinciples upon which thelanguage ofanalytical geometry isconstructed, thenthatthe reductio adabsurdum not only is of necessity to be employed, butmoreover in propositions of an affirmative character neverneed be employed, except whenasaboveexplained theanalytical demonstration is founded on the impossibility orinadmissibility ofcertainroots due to thedegree of the equation implied in theconditions of thequestion.Ifthissurmiseturnout tobecorrect, we are furnished with auniversal criterion fordetermining whentheuseoftheindirectmethodofgeometrical proofshould be considered. validandadmissible andwhennot-. •Irreportmaybebelieved, intellects capable ofexiending ilieboundsoftheplanetary systemandlightingup newregionsof theuniverse with the torchofanalysis, have been baffied by the difficulties of the elementary problem statedat theoutsetofthispaper,inconsequence, it is tobepresumed, of seeking a form of geometrical demonstmtion of which the question from itsnaturedoesnotadmit.Ifthisbe so, no betterevidence could be desired to evince the imporiance of such a criterion 88iliatsuggested in thetext. 50. ONTHEEXPRESSIONS FORTHEQUOTIENTS WIl;ICH APPEAR INTHEAPPLICATION OFSTURM'S METHOD TOTHE DISCOVERY OFTHEREALROOTS OFANEQUATION. [Hull British Association Report (1853),PartII.,pp.1-3.] MANYyearsago Ipublished expressions fortheresidues whichappearin theapplication oftheprocess of common measure tofxandf'e,and which constitute Sturm's auxiliary functions. These expressions arecomplete functions ofthefactors of fxand of differences of theroots offe,andare therefore in effect functions ofthefactors exclusively, sincethedifference between any two roots may be expressed asthedifference between two corresponding factors. Havingfoundthatinthepractical applications of Sturm's theorem thequotients may beemployed withadvantage toreplace theuse oftheresidues, I have been led to consider theirconstitution; and havingsucceeded inexpressing thesequotients (which are of course linear functions ofx)underasimilarform to thatoftheresidues, thatis,as complete functions ofthefactors and differences of theroots offx,Ihave pleasure insubmitting theresulttothenoticeof theMathematical Section oftheBritishAssociation. Let~,hs.h,h"bethenroots offo: Let~(a,b,at)ingeneraldenotethesquaredproductofthedifferences ofa,b,a...l. LetZ.denoteingeneral 'i.~(h6Ihe.'" hBi),where81182•••8.indicate any combination ofiout ofthenquantities a,b,0,.••t,with the convention that Zo=I,Zl=n;andlet(i)denotef{I+(-I)i},beingzero when iis odd,and unitywheniiseven;thenI findthattheithquotientQ.may bewritten underthe form Q.=.PII(x-~)+.PII(x-hs)+...+.p"I(X-h,.), whereingeneral 50]OnSturm's Method ofRealRootsofanEquation. 397 f'xIfwe suppose fx'bymeans of thecommon measure process, to be expanded undertheform of an improper continued fraction, thesuccessive quotients will bethevalues of Ql>Q2...Qnabove found, thatis f'x1 1 1 1 . fx=Ql-Q2- Q.-...Qn' thesuccessive convergents ofthisfraction will be 1Q, Q2Q.-1 f'x Ql'QlQ2-1'QIQ,Q.-QI-Q.' ...,fx' Thenumerators anddenominators of these convergents willconsequently also be functions of thefactors exclusively. Theyarethequantities thesum oftheproducts of which multiplied respectively byfxandf'xproduce(to constant factors pr~)theresidues. The denominators are expressible very simply in termsof the factors and the differences of the roots;andtheir valuesundersuch forms were published by meaboutthesametimeasthe valuesoftheresidues in thePhilosophical Magazine; theexpression for thenumerators is much more complicated, butis given in my paper, " The Syzygetic Relations," &c.,inthePhilosophical Transactions. [po429below.] Bycomparing the expression for anyquotient with the expressions for thetwo residues from which itmay be derived, we obtainthe following remarkable identity: Zi-IxZ"thatis In~~...hi-I)xIn~~·..~)=.PI'+,Pl+,Pa'+ ...+,Pn• Whenthe roots are all real, we have thustheproductof one sum of squares bytheproductofanothersum ofsquares(thenumberin each sum depend­ ingupon the arbitrary quantity i),brought undertheform of asum of aconstant numbernof squares, which in itselfis aninteresting theorem. The expression above given for Q,leads to a remarkable relationbetween thequotients andconvergents toj:. Letit be supposed, asbefore,that f'x 1 1 1 1 fx=Qlx-Q2X-Q.a:-...Qna:' and letthesuccessive convergents tothiscontinued fraction be NI(x)N,(e)N.(x)Nn(x) DI(x)'])2(x)'.D.(x)'...Dn(x)' wherethenumerators anddenominators arenot supposed to undergo any reductions, butareretained intheircrude forms asdeduced from thelaw N,=Q,N&-1-Ni-t, D,=Q,Di-I-Di-t. 398OnSturm's Method ofRealRootsofanEquation. [50 N,(x)being 1, and D,(x)beingQ,(x);thenitmay be deduced from the published resultsaboveadverted tothat D.(x)=~;Z;·--,·z;·(,) Inh"h....."-..)(x-h,,)(x-h..)...(x-heJ). >\-I'" 11)+, Hence and we have therefore andconsequently ZS._,ZS--. Z8(i) Q.=z.~zs.'"z--s--I{(Di-'(h.)'t(x-h.»),,>-. (1)+, which is thegeneralequation connecting theform of each quotient with thatofthedenominator to theimmediately preceding unreduced convergent intheexpansion-:undertheform ofan improper continued fraction. Ifinsteadof thedenominator of theunreduced convergents, thedenom­ inatorsoftheconvergents reduced totheirsimplest forms be employed, thepowers of Zintheconstant factor will undergo adiminution. The essential partofthistheorem admitsof being statedingeneraltermsas follows:- "Ifthequotient of analgebraical function of xbyitsfirstdifferential coefficient be expressed underthe form of acontinued fraction whose successive partialquotients arelinear functions of e,anyone ofthese quotients may be found (to a constant factorpres)bytakingthe sum of the products formed by multiplying each factor (x-h)of the given function by thesquare of whatthedenominator oftheimmediately antecedent conver­ gentfraction becomes aftersubstituting in it for xtherootcorresponding to suchfactor." P.S. Since the above wasreadbefore the BritishAssociation, the theoryhasbeenextended by theauthorto comprise thegeneralcaseof theexpansion ofany two algebraical functions undertheform ofacontinued fraction, and hasbeenincorporated into the paperin thePhilosophical Transactiom abovereferred to. 51. ON A THEOREM CONCERNING THECOMBINATION OFDETERMINANTS. [Oambridge andDublinMathematical Journal, VIII.(1853), pp. 60-62.] Let1..1.represent the line of terms I~,I~,'"lam• ...... LetIAxIBrepresent !(Ia,.XIbr),where of course therearemterms withinthesymbol of summation. Again. let 'Arepresent thelineI~.I~,...lam, .... andletI:~IxI:~Irepresent!I:~::~:IxI:::::::I, liar.la,Idenoting thedeterminant (Ia,..la, -la,.lar).la,.,la, .... .. therebeing of course lm(m- 1) termscomprised withinthesign of summation; and so, in general. let 1..1.IB 2..1.'B 'AxIB,nbeing less thanm, 400 Ona Theorem concerning the [51 (and where in generalrAdenotes r~,r/Zt,•.•r~ trepresenandrBdenotes rbi,rbs,...rb lalli'lalls'...lalls Iblll:Iblls,...»; Isalll,Salls,...Salls'bill'»;...»;x........................ ........................ nalll'"alls'...nahto"bill'"blls,...nblls Now let rbeanyintegerlessthanm, and let m(m-l) ...(m-r+I) ~=1.2...r and, supposing 01,Os>...Ortobernumbers oftheset1, 2,...m,let GI,GI,...G,.denotethe~rectangular matrices of theforms ',Ll. respectively, and letHitHs,..,H,.denotethe",rectangular matrices oftheforms 'IB respectively. Nowform the determinant GlxH,., GlxH,. G,.XHI,G,.XHs,...G,.xH,. then, if we give rthesuccessive values 1, 2,3 ...m (in which lastcasethe determinant in question reduces to asingle term), the values of thedeter­ minantabovewrittenwill beseverally in theproportions of K,s».Klm(~l', ...s»,K; thatis tosay,thelogarithms of these several determinants will beasthe coefficientsof thebinomial expansion (1 +:c)m. Whenwe make r=m, andequatethedeterminant corresponding tothis value of rwiththatformed by makingr=1,thetheorem becomes identical with a theorem previously given by M. Cauchy, for theProduct ofRect­ angularMatrices. 51] Combination ofDeterminants. 401 Itwould be tedioustosetforththedemonstration ofthegeneraltheorem indetail.Sufficeitheretosaythatit isadirectcorollary from the formula marked (4)in mypaperinthePhilosophical Magazine forApril1851, entitled "OntheRelations between theMinorDeterminants ofLinearly Equivalent Quadratic Functions "," when thatformula isparticularized by making {ltm+l'ltm+1t'"am+n} bm+l'bm+2t•••bm+n represent adeterminant all whose termsarezerosexceptthosewhich lie in one ofthediagonals. theselatterbeingallunits.which comes, in fact, to defining that I~\=1.andI~/=0. Theimportant theorem herereferred to is made almostunintelligible by anunfortunate misprint ofqO",.10",.JOm,"0",.in place of qOr.lOr,JOr,"Or' Imayheretakenotice of anotherandstillmoreinexplicable blunderinthe samepaper,formula(3)t.inthelatterpartoftheequation belonging to which {a,,,alJ....as..a,..+1'a,........as-} a"'l'a""•...a.....a....+1,£t+_, a.... iswrittenin lieu of {~.a...,.••am, a....."a......., a +.~lanHan+_}. ~,CIt,•••am, a~l'a",-, aan+lanH~ s.[-p.249 above.] [tBeepp. 2411.251above.] 26 52. NOTEONTHECALCULUS OFFORMS. [See pp. 363 and 411.] [Cambridge andDUblinMathematical Journal, VIII.(1853), pp. 62-64.] ACCIDENTAL causes have prevented me from composing theadditional sections ontheCalculus of Forms, which I had destined forthepresent Number ofthisJournal. Inthemeanwhile thesubjecthas notremained stationary. Amongtheprincipal recentadvances may be mentioned the following. 1. The discovery of Combinants; thatis tosay,ofconcomitants to systems offunctions remaining invariable, notonly when combinations of thevariables aresubstituted forthevariables, butalso when combinations ofthefunctions aresubstituted forthefunctions; and as a remarkable first­ fruitofthisnewtheoryof double invariability, therepresentation ofthe Resultant of anythreequadratic functions undertheform ofthesquareof acertaincombinantive sexticinvariant addedtoanother combinant which isitselfabiquadratic function of 10 cubic invariants. Whenthethree quadratic functions arederivedfrom the same cubic function, thisexpression mergesin M.Aronhold's forthediscriminant ofthecubic.Thetheory ofcombinants naturally leads to thetheoryofinvariability fornon-linear substitutions, andI havealready made a successful advance inthisnew direction. 2.Theunexpected andsurprising discovery of a quadratic covariant to anyhomogeneous function in e,yofthenthdegree,containing (n-1) variables cogredient witha!'--'J.,xn-ay.••yfl--'J.andpossessing theproperty of indicating the number ofreal and imaginary rootsinthegivenfunction. Thiscovariant, onsubstituting forthe(n-1)variables thecombinations of thepowers of x,ywith which theyarecogredient, becomes theHessian ofthegivenfunction.". *Thiscovariant furnishes, if we please, functions symmetrica.l inrespectto the two ends of a.nequation fordetermining thenumber ofitsreal a.nd ima.gina.ry roots. The ordina.ry Sturmia.n functions, it is well known, ha.ve not thissymmetry. Asa.nother exa.mple ofthe successful application of the new methods tosubjects which ha.ve been long before the mathe­ ma.tica.l world a.nd supposed to beexhausted, I maynoticethatIobtainwithout a.neffort, bytheiraid,amuch more simple, practica.l, andcomplete solution of thequestion ofthesimul­ taneous tra.nsformation of twoquadratic functions, or the orthogona.l tr&.nsformation ofone suchfunction, thana.nypreviously given, even by the great ma.sters Ca.uchy a.nd Ja.cobi, who havetreatedthisquestion. 52] ifote on the Calculus ojForms. 403 3.Thedemonstration due to M. Hermite of a law of reciprocity connect­ ingthedegreeordegreesof anyfunction orsystemoffunctions withthe orderorordersoftheinvariants belonging tothesystem. Thetheorem itselfwasfirstpropounded by meaboutatwelvemonth back, and com­ municated to Messrs Cayley, Polignac, and Hermite, asservingto connect together certainphenomena which had presented themselves to me in the theory: unfortunately itappeared tocontradict another law too hastily assumed bymyselfandothersasprobably true,andIconsequently laid asidetheconsideration of thisgreatlaw ofreciprocality. To M.Hermite, therefore, belongsthehonourofreviving andestablishing,-to myselfwhat­ everlowerdegreeofcreditmayattachtosuggesting andoriginating,­ thistheorem ofnumerical reciprocity, destined probably to become the comer-stone ofthefirstpartof our new calculus; thatpart,I mean, which relatestothegeneration andaffinities offorms", 4. I may noticethattheCalculus ofFormsmay now withcorrectness betermedtheCalculus ofInvariants, byvirtueoftheimportant observation thateveryconcomitant of agivenform orsystemof forms may be regarded asaninvariant ofthegivensystemand of an absolute form or system of absolute formscombined withthegivenform orsystem. Asregardsthat particular branchofthetheoryofinviriants whichrelatestoresultants, or, inotherwords, to thedoctrine ofelimination, I may here statethetheorem alludedto in apreceding Number oftheJournal, to witthatifRbethe resultant ofasystemofnhomogeneous functions ofnvariables, written outintheircomplete and most generalform (so thatbydefinition R=0 isthecondition thattheequations got bymaking thengivenfunctions zero.shallbesimultaneously satisfiable by onesystemof ratios), thenthe condition thattheseequations may be satisfied by,distinct systems of ratiosbetween thenvariables iso'R=0,thevariation 0beingtakenin respectto every constant entering intoeach ofthenequations . •Thistheorem ofnumerical reciprocity promises toplayasgreatapartin the Theory of FormsasLegendre's celebrated theorem ofreciprocity inthatofNumbers. Another demonstra­ tionof it, which leaves nothing to be desired for beautyandsimplicity, hasbeensince diacovered by Mr Cayley, which ultimately restsuponthatsimplelaw(essentially although not onthefaceof italaw ofreciprocity) given by Euler,which affirms thatthenumberof modes in whichanumber admitsof being partitioned isthesamewhether thecondition imposed upon themode of partitionment bethatnopartshallexceedagivennumber, orthatthenumberof parl8constituting anyonepartition shallnotexceedthe same number. 26-2 53. ON THE RELATION BETWEEN THEVOLUME OF A TETRA­ HEDRON AND THE PRODUCT OF THE SIXTEEN ALGE­ BRAICAL VALUES OF ITS SUPERFICIES. [Oambridge andDublinMathematical Journal, VIII.(1853), pp. 171-178.] THEareaofatriangleisrelated(asis well known) in averysimple manner to theeightalgebraical values of its perimeter:Ifwe callthevalues of the squaredsides ofthetrianglea,b,c,therewill benothing todistinguish the algebraical affections of sign of thesimplelengthssoastoentitleone toa. preference overtheother. The areaof thetrianglecan onlyvanish byreason ofthethreevertices coming intoastraight linejhence, according to the generaldoctrine of characteristics, wemusthavetheNorm of ';a+';b+';c, containing asa factor some root or power of theexpressions for theareaof thetriangle. The Norm inquestion beingrepresentable as-NtwhereN istheNorm of at±bt±ct,which is of four dimensions in the elements a, b,C, and undecomposable into rationalfactors, weinfer thatto anumerical factor presthesquare of theareamustbeidentical withtheNormN.andthus, by a logical coup-de-main, completely supersede all occasion for the ordinary geometrical demonstration given of thisproposition, which in itsturn,with certainsuperadded definitions, would admitof being adopted asthebasis of anabsolutely pure system of Analytical Trigonometry thatshouldborrow nothing from the methods and resultsof sensuous or practical geometry. Butintothisspeculation itis not my presentpurpose to enter:whatI propose to do is to extendasimilarmode of reasoning to space of three dimensions, and to pointout ageneraltheorem indeterminants whichis involved asa consequence in thegeneralization of theresultoftheinquiry when pushed forward into theregions of what may be termedAbsolute or Universal Rational Space. LetF, G,H, K bethefoursquaredareas of thefaces of a tetrahedron, andVthevolume; then, since Vonly becomes zero in thecaseofthefour vertices coming intothesame plane, which is characterised bytheequation .,IF+';G+';H +';K=0 53]Relation betweenthe Volume oja Tetrahedron, etc.405 subsisting, weinferthatNtheNorm of mustcontain apower of Vasarational factor. VI is rational andof threedimensions inthesquared edges;theNormabove spoken of is of eightdimensions inthesame.Consequently thereisarational factor, sayQ,remaining, which is of five dimensions inthesquared edges, and this factorInow proceed to determine, theotherfactor VI being, asis well known,anumerical productofthedeterminant 0,abl,a,c2,adl,1 ba',0,bc',bd',1 ca',cbl,0,cdJ.1 da',db-,dei,0,1 1,1,1, 1, 0 a,b,e,dbeingthefourangular pointsofthetetrahedron. SeeLondon andEdinlYurgh Philosophical Magazine, 1852.[p,386 above.] ThequantityQpossesses aninterestofageometrical character; forifwe calltheradiioftheeightsphereswhichcanbeinscribed inatetrahedron rlJrior.,r"r.,r"r"r"weevidently haver1rlr.r,r.r.r,r, xN=(3Vr.Hence (R)hrodfh. hradii .3'V'3'V',te puct0 t e elg t 11m question,=N=Q' Consequently Qisthequantity whichcharacterises thefactof one or more of theradiioftheinscribed spheresbecoming infinite. Forthetriangle thereexistsnocorresponding property; thiswe knowapriori,andcan explain alsoanalytically fromthefactthatif we call Ptheproductofthe radiiofthefourinscribable circles,vtheNormoftheperimeter, andA thearea,we have andPv=2'A'. 2'A'v=p =A', whichcontains nodenominator capableofbecoming zero, sothataslong asthesidesremainfinitethecurvature oftheinscribed circlesisincapable ofvanishing. Todetermine Nasafunction of theedges,and.thento discover by actual division thevalue of ~,would be thedirectbutanexcessively tedious andalmostimpracticably difficult process. Ihave ever felt apreference fortheapriorimethodofdiscovering forms whose properties are known, and neveryethavemetwith an instance whereanalysis hasdeniedtogentle 406 OntheRelation betweenthe [53 Similarly,solicitation conclusions which she wonld be loth to granttotheapplication of force. The casebefore us offers no exception tothetruthofthisremark. Qis afunction of fivedimensions intermsofthesquared edges:letU8 begin by finding thevalue of thatpartofQin which atmost acertain setof four of theseedges make theirappearance, and to find which con­ sequently theothertwo edges may be supposed zero without affecting the result. We may make two distincthypotheses concerning thesetwo edges ; we may suppose thatthey are opposite, thatisnon-intersecting edges,or thattheyare contiguous, thatisintersecting edges. Tomeetthefirsthypothesis supposeob=0,C8=O. Forconvenience sake, use F, G, H, K todenote16timesthesquare of each area, insteadofthesimplesquareoftheareas. Call 16(abc)'=K, 16(abd)'=H, 16(acd)2=G, 16(bcd)J=F. Then - K=(alJ)'+(ac)'+(be)'-2(alJ'f(ac'f-2(alJ'f(be'f-2(ac)1(be)1 =ac'+bc'-2(ac)1(be)2. - H=ad'+bd'-2(ad)2(bd'f, - G=ca'+da'-2caldal, - F=cb'+db'-2cbtdbl• Henceone value of ..;F+..;G+..;H+tv'Kwill be ..;(- 1){(act-bet)+(bdt-ad-)+(dal- acl)+(bcJ-bd')}=o. Hence,onthisfirstsupposition, theNorm vanishes. ButVIdoes not vanish whenab=0,cd=0, foritbecomes, savinganumerical factor, 0, 0, 0, 0,ae',ad',1 bet,bdI,1 0, 0, 1 dal,dbl,0,0, 1 thatisI,1, 1, 1, (ac'.bd'-adl•be')(cb'+ad'-cal-bd') +(bet-act)(cal.db'-cb'.dal) +(adl-bdl)(cal.db'-cb'.dal) =2(act.bdl-adl.be')(adl+be'-acl-bdI); andconsequently, sinceNvanishes butVIdoes not vanish, Qvanishes, showing thatthereis noterminQbutwha.tcontains oneatleastofany 53J Volumeofa Tetrahedron, etc. 407 twoopposite edgesasa factor jor, inotherwords,thereis noterminQ of which theproduct ofthesquareoftheproductof allthreesides of some one orotherofthefour faces does not form a constituent part. Next,letussupposeab=0,ac=0,then XI=I6abe'= -be, HI=I6abdl= -(adl-bOJ)", ~=16aad'= -(adl-cd')", FI=I6becP= -be'-bd4-ed4+2be'.btl'+2be'.ccP+2bcP.cdl• Fourofthefactors of Nwill betherefore {,(be'+ccP-bcP)±F},{£(be'-cd'+bcP)±F}, ,denoting.j(-I),andtheproductofthesefour factors willbe {(be'+cOJ-bcP)'+F'}x{(be'-cd'+btl')"+]'I}, which is equalto 16f>c4.bdt•cd'j andsimilarly, theremaining partoftheNormwillbe {(2alP-btl'-cd'+bel,!+]'I}x{(2adt-btl'-cdl-bel)'+F'}, thatis {4ad4-4ad,t(bcP+cd'+bet)+4bc'.bd,t+4bd,t.ccP+4edt.bet} x{4ad4-4adt(bd,t+cdl-be')+4bdt.cdl} . Again,sinceact=°andbe'=0,V,becomes 0,0, 0, ad',1, 0,0,be',bd,t,1 0,ebl,0,cd',1 da',db',de'0,1, 1, 1, 1, 1,°whichisevidently equalto 0,0,alP,1 0,ad'10,ebt,cdl1,, 2betdb',-f>c4da'0, 1,dal, 0,I, 1,I,I,1, 0, =2be'{2betalP+ad4-ad'bdt-edtalP+bd'ed'}-2bc'adl =2bet{ad4-alP(bd,t+cd'-bet)+bd,t.ccP}. 408 On theRelation betweenthe [53 Hence, paying no attention to any mere numerical factor, we have found that Nwhenac=°andbe=0,QorVIbecomes be'.bdt.cd'Iad·-ad'(bd'+cd'+be')+be'.bd'+bd",edt+ccP.bet}. Hence, with theexception of the termsin which five out of thesix edges enter,the complete value of Qwillbe I(be'.bd'.cd'){ad·-ad'(bd'+ed'+be')+bet.bd'+bd'.cd'+edt.he'}, or more fully expressed, and stillabstracting from terms containing five edges, =Ibe'.bdt.cd'{(ab·+act+ad·)-(ab'+act+he')(bd'+be'+cd') +be'.bdt+bd'.cd'+cd'.bet}. Itremains only to determine thevalue of thenumerical coefficient affecting eachofthesix terms of the form ab' .act.ad'.be'.bdt. To find this, let ab'=ac'=adt=he'=bdt=cd'=1j thenevidently, since all thesquaredareasare equal, severalofthefactors ofNwill become zero, butV,evidently doesnot become zero for a regular tetrahedron jhenceQbecomes zero: andifwecallthenumerical factor soughtforA,wemusthave (observing thattheIincludes four partscor­ responding toeachofthefour faces) 4 {3- 9+3}+6X=0, therefore -12+6X=0, orX=2. Hencethecomplete value of Qis Iab'.be'.ca'{(dat+db·+det)-(dal+db'+de')(ab'+he'+ca') +ab'.be'+he'.ca'+cat.ab'} +2I(ab'.he'.cd'.dat•act); or, which isthesamequantity somewhat differently andmoresimply arranged, Q=I(ab'.be'.cat){(da·+db·+det+da'.db'+db'.de'+de'.dat) +(ab'.he'+be'.ca'+cat.ab2)-(da'+db'+de')(abl+be'+cat)}, andthisquantity equatedto zero expresses the conditions of a radiusofan 53J Volumeoja Tetrahedron, etc. 409 inscribed spherebecoming infinite. The directmethodwould have involved, asthefirststep,theformation oftheNormofanumerator consisting of ,.;F±,.;a±vB±";K, thevalue of which is andcontains 4+6+12,thatis 22positiveterms,and 12,thatis 13negative terms,together 35terms,eachof which mightbeanaggregate of6'or 1296quantities, andthusinvolve in all the consideration of 45360 separate parts,foreachofthequantities F,G,H, Kbeingaquadratic function of threeofthesquarededges, will containsixterms.Itis notuninteresting to noticethatinaddition tothecasealreadymentioned of twoopposite edges beingeachzero,asab=0,cd=0,Qwill also vanishforthecase ofab=cd, be=ad;thatis forthecase of two intersecting edgesbeingeachequalin lengthtotheedgesrespectively opposite tothem.Thisisevidentfrom thefactthatonthehypothesis supposed thefaceacb=acdandtheface bde=bda;henceN=0, andtherefore, Vnotvanishing, ~.,thatisQ,will vanish. We may moreover remarkthatsinceab=0 andcd=0 doesnotmake Vvanish,theperpendicular distance ofobfromcd,which,multiplied by abxcd,gives six timesthevolumes, mustonthissupposition becomeinfinite. Whenthreeedges lying in thesameplaneallvanishsimultaneously, Q vanishes, since one edge atleastineveryfaceofthepyramid vanishes, andValso vanishes, asisevidentfrom the expression for V,, when ob=0, ac=0,be=0,becoming amultiple of 0, 0, 0, ad-,1 0, 0, 0, bdJ,1 0,0,0,adJ,1 ad',bdl,cdt,0,0 1,1, 1, 0,0 which is evidently zero. Itappeared to menotunlikely, fromthesituation and look of Q(the characteristic of one of theinscribed spheresbecoming infinite), thatitmight admitofbeingrepresented asadeterminant, butI have not succeeded in throwing itunderthatform. I have astrongsuspicion thatifwetake (jafunction corresponding toatetrahedron a'b'e'd',in thesamewayas Qcorresponds toabed,Q(j,andnotimprobably ";(QQ'),will be found to be 410Relation betweenthe Volume oja Tetrahedron, etc.[53 (aswe know from Staudt's Theorem of ~CV-.V'I»arational integral function of thesquares of the distances of the points a,b,e,dfrom the points a', b',e',d'. ThatNshould divide out by VIis initselfananalytical theoremrelating to 6arbitrary quantities 0»,acI,ad',bel,bd'-,rxP,whichevidently admits of extension to any triangular number 10, 15,&c.ofarbitrary quantities. Thus we may affirm, apriori,thatthenorm of ~L±tiM±~N±~p±~Q, where (for thesakeofsymmetry, retaining doubleletters,asA.B,A.C,&c., todenotesimpllJquantities) 0,A.B,A.C.AD,1 0,A.B,A.C,A.E,1 A.B,0,BC, BD, 1 A.B,0,BC,BE,1 Q=A.C.BC,0,CD,1,p=A.C,BC,0,CE,1 A.D,BD,CD,0,1 A.E,BE,CE,0,1 1,1,1,1,°1,1.1, 1, 0 N=&c., M=&c., L=&c., will contain asafactorthedeterminant 0,A.B,A.C,A.D,A.E,1 A.B,0,BC, BD, BE,1 A.C,BC,0,CD, CE, 1 A.D,BD,CD.0,DE,1 AE,BE.CE,DE,0, 1 1, 1, 1, 1, 1,° andasimilartheorem mayevidently beextended tothecaseofanyn(n2+1) arbitrary quantities whatever. 54. ONTHECALCULUS OF FORMS, OTHERWISE THETHEORY OFINVARIANTS. [Continued fromp.363above.] [Cambridge and Dublin Mathematical Journal, VIII.(1853), pp. 256-269.] SECTION VII.OnCombinants. REASONS of convenience have induced me todepartfromtheplan to which I originally intended toadhereinthedevelopment ofthistheory, andI shallhereafter, from time to time, continue toaddsections on such partsofthesubjectas may chance to be most presentto my mind or most urgentupon my attention, without waitingfortheexactplacewhichthey oughtto occupy in amore formal treatise,andwithouthavingregardtothe separation ofthesubjectinto the two several divisions statedattheoutset ofthefirst section. The presentsection will be devoted to abriefand partialexposition of thetheoryofOombinants ",with a view to theapplica­ tion ofthistheoryto the solution of theproblem of throwing theresultant ofthreegeneralhomogeneous quadratic functions underitsmost simple form, being analogous to thatgiven by Aronhold in theparticular case where thethreefunctions arederived from thesamecubic, and becoming identical therewith whenthecoefficients areaccommodated tothisparticular supposi­ tion[.I shall confine myself for thepresent tocombinants relating to systems of functions, all of thesame degree. Ift/>t,4>2'...4>r,be homogeneous functions of any numberof variables,any invariant orotherconcomitant of the system which remainsunchanged, not onlyforlinear substitutions impressed upon thevariables contained withinthe functions, butalso forlinearcombinations impressed uponthefunctions them­ selves,is whatItermaCombinant. ACombinant isthusaninvariant orother concomitant ofasystem in its corporate capacity (qull.system),being in fact •Discovered by theAuthorofthispaperinthewinterof 1862. tAsimilarmethodwillsubsequently beappliedto therepresentation of theresultant of two cubicequations asafunction ofCombinants bearingrelations tothequadratic andoubio invariants ofaquarticfunction of zandy. preoisely analogous tothose which the Combinants thatenterintothesolution abovealludedtobeartotheAronholdian invariants ofaoubio function. 412 Onthe Calculus ofForms. [54 common to thewhole family of forms designated by~4>t+~q,1+...+"A.,.q,r , whereX,,~,..."A.,.,arearbitrary constants. Ifthecoefficients of q,I'4>1'...4>ro be supposed to be writtenoutinrlines(thecoefficients of corresponding termsoccupying thesame place in each line), so as to form a rectangular matrix,anycombinantive invariant will beafunction of thedeterminants corresponding totheseveralsquaresofrtermseachthatcan be formed out of suchmatrix,or, astheymay betermed,thefllUdeterminants belonging to such rectangular matrix.Ifwe call any such combinant K,then.over and above theordinary partialdifferential equations which belong to it in its character ofaninvariant, it will be necessary and sufficient, in order to establish itscombinantive character, thatKshall besubjecttosatisfy(r-1) pairsofequations of the form wherea,b,c...; a',b',c'..., arerespectively lines in thematrixabove referredto. So anycombinantive concomitant will beafunction ofthefulldeter­ minantsofthematrixformed by thecoefficients of thegivensystem of forms and ofthevariables, and will be subjecttosatisfytheadditional differential equations justabovewritten. Itwillreadilybeunderstood furthermore, thataninvariant orother concomitant may becombinantive inrespecttoacertainnumber of forms ofasystem,and not in respectofotherformstherein; or more generally, may becombinantive inrespectof each,separately considered, of aseries of groupsinto which agivensystem may be considered to be subdivided, without beingso inrespectof the several groupstakencollectively. Inthefourthsection of my memoir [po429 below] on a"Theory ofthe Conjugate Properties of tworationalintegralAlgebraical Functions," recently presented totheRoyalSocietyof London, thecaseactually arises of an invariant ofasystem of threefunctions, which is combinantive inrespect only to two of them. Forgreatersimplicity, lettheattention forthepresentbekeptfixed uponcombinants which are such in respectofasinglegroupof functions, all ofthesamedegreeinthevariables. (Itwill of course have been perceived thatwhenthesystem is made up of severalgroups,therewould benothing gainedbylimiting thegroupsto be all of thesamedegree interse;itis sufficient thatall ofthesamegroupbe ofthesamedegree per8e.) 54] OntheOalculusofForms. 413 All such combinants willadmitofanobvious and immediate classification. Let\1Ssupposethatacombinant is proposed which is in its lowest terms, thatis tosay,incapable ofbeingexpressed asarationalintegral algebraical function of combinants ofaninferiororder.Suchacombinant may,notwith­ standing this,admitofbeingdecomposed intonon-combinantive invariants ofinferiordimensions toitsown,andin sucheventwill betermedaeomple» combinant; oritmay beindecomposable afterthismethod, in which event itwillbetermedasimplecombinant, Itwillpresently be shown, thatthe resultant ofasystemofthreequadratic functions is made up of acomplex combinant of twelve dimensions, andofthesquareof asimplecombinant of sixdimensions, expressible asabiquadratic function oftennon-com­ binantive invariants, eachofthreedimensions inthecoefficients. There isanobvious mode of generating complex combinants; according to which theyadmitofbeingviewed as invariants ofinvariants. Supposing CPh~,'"cP,.,to bethefunctions ofthegivensystem, AtCPI+~cP,+...+">""cP,. mayconveniently betermedtheconjunctive ofthesystem: if now one or moreinvariants orotherconcomitants betakenofthisconjunctive, there resultsaderivative function or system offunctions ofthequantities At,x,., ...A,.,in which everytermaffecting any power or combination of powers of theAseriesisnecessarily aninvariant orconcomitant ofthe givensystem.Ifnowaninvariant orotherconcomitant betakenofthe newsysteminrespecttoAI.A"...X,.,(theoriginalvariables (supposing them toenter)beingtreatedasconstants). thissecondarily derivedinvariant will beitselfanInvariant, oratalleventsaConcomitant inrespectofthe originalsystem,andbeingunaffected bylinearsubstitutions impressed upon theAsystem, is bydefinition acombinant of such system. A similar methodwill obviously applyiftheoriginal systembe made up of various groups jeachgroupwillgiverise toaconjunctive, and one or more con­ comitants beingtakenofthissystemofconjunctives andtreatedasinthe case first supposed, (theonly difference being,thattherewill on the present supposition beseveralunrelated systems insteadofasinglesystemof new variables, thatis,several Asystems insteadof one only) theresult,when all theAsystemshave been invariantized out(thatis, made to disappear byany process for forming invariants), will beacombinant inrespecttoeachofthe groups,severally considered, of thegivensystemof functions. Hereletitbepermitted to me to make amomentary digression, inorder tobeenabledto avoid for thefuturetheinconvenience ofusingthephrase "invariant orotherconcomitant," andso to be enabledat one and thesame timetosimplify thelanguage andtogiveamorecomplete unitytothe matterofthetheory,byshowing how every concomitant may in fact be viewed as asimpleinvariant, sothatthecalculus of forms may hereafter admitofbeingcited,as I propose tociteit,underthenameoftheTheory ofInvariants. 414 Onthe Oalculus ojForms, [54 Thus,to begin withthecase ofsimplecontragredience andcogredience, ifE,'TJ,~..0arecontragredient toe,y,eo'0,any form containing E,'7,~..., whichisconcomitantive to agivenform orsystemof formsS,whichcontains x,y,Z...,may beregarded asconcomitantive to thesystemS',made up of Sandthesuperadded absoluteformEx+'TJY+~z+ ...,say':)-;whereE,'TJ,~..o aretreatednolongeras variables, butasconstants. Inlikemannerevery systemofvariables contragredient toa;y,Z•• "or to any othersystemof variables inS,will give rise to a superadded form analogous to ':)-,thetotality of which may be termedSI;andthusthevarioussystemsE,'TJ,~'"will no longerexistasvariables inthederivedform,butpurelyasconstants. Again, ifScontainanysystemofvariables q",y,':)-,&c.,contragredient tox,'!I,Z,&c., thesystemofvariables ?t,v, w,&c.,cogredient withx,y,Z,&c., may be considered asconstants belonging to thesuperadded formq,u+'I/rV+~... ; butifSdo not contain anysystemcontragredient toe,y,z,&c.,then u, v, w,&c.may betreatedasconstants belonging tothesuperadded system of forms xv-yu, yw-zv, zu-ana,&c.; and so in generalanyconcomitant containing any sets of variables insimplerelation, whether ofcogredience orcontragredience, with any of thesets inthegivensystemS,may in all cases be treatedas aninvariant ofthesystemS',made up of Sanda. certainsuperadded systemSI'all the forms contained in which areab­ solute,by which I mean, thattheycontainnoliteralcoefficient. Thesame conclusion may be extended tothecaseofconcomitants containing setsof variables incompound relationwiththesets inthegivensystemof forms S. Thus, suppose Ul>~,..0Un,to be in compound relationofcogredience with xn-l,a;n-'y,xn-sy.,...yn-l; Ut,~,..,Un,mayberegarded asconstants belonging tothesuperadded form Utyn-l_(n-1)~yn-'J.x+l(n-1) (n-2)Uayn-s:r;'J :+,..±Una;n-\ sayn.Andthusuniversally we areenabled to affirm, thataconcomitant ofwhatever natureto agivensystemof forms, may be reducedtotheform ofaninvariant ofasystemmade up of thegivensystemandacertainother superadded systemofabsolute forms:without, therefore, abandoning theuse ofthetermsconcomitant, cogredience, contragredience, &c.,which for many purposes arehighlyconvenient and save much circumlocution, we may regardeveryconcomitant asadisguised invariant, andunderthe name of theTheoryofInvariants comprise thetotalityofthetheoryofConcomitance. I havealreadyhad occasion to make use of thesuperadded formnin discussing thetheoryof theBezoutiant (aquadratic formconcomitant to twofunctions ofthesamedegreeine,y,which plays amostimportant part inthetheoryoftherelations oftheirreal roots), in thememoir for theRoyal Societypreviously adverted to. I nowreturntothequestion ofapplying thetheoryofcombinants to thedecomposition oftheresultant ofthreegeneralquadratic functions of 54] Onthe Oalculus ofForms. 415 1&,y,e.Itwill of course be apparent thateveryresultant of any system of n functions ofthesamedegree of asingle set of n variables is acombinantive invariant ofthesystem. This is animmediate andsimple corollary to the theorem given by me in thisJournal, in May, 1851. Accordingly, in pro­ ceeding to analyse thecomposition of theresultant ofthreequadratic functions, I may, besides impressing linearcombinations uponthevariables, impresslinearcombinations uponthefunctions themselves, in any way most conducive to simplicity and facility of expression andcalculation; and whatever relations shallbeprovedtoexistbetween theresultant andother combinants for such specific representation, mustbe universal, and hold good forthefunctions in theirmostgeneralform. (1) The system, by means of linearsubstitutions impressed upon the variables which enterintothefunctions, may be made to assumetheform Xi+!f+Z2, ar+b!f+cz'l, lll?+m!f+nz'il+2pyz+2qzx+2rxy. (2) By means of linearcombinations ofthefunctions themselves the systemmayevidently be made to takethe form (c -a)Il?+(c -b)y', (a-b)!f+(a-c)Z2, ley2+2pyz+2qzx+2rxy; and finally, by takingsuitable multipliers ofX,y.zin lieu of X,y,z,itmay be made to become P(Il?-y2), a(!f-Z2), !f+2fyz+2gzx+2hxy. We have thusreduced the number ofconstants inthesystem from eighteen to five; and as itwillreadilybe seenthatin anycombinant of the system in its reduced form pandacan only enteras factors of the simple quantity, (pu)i,for all purposes of comparison of the combinants of the system of like dimensions with one another, pandamightadmitof being treatedas being each unity, and accordingly, practically speaking, we have onlytodeal with threein place of eighteen constants, amarvellous simplifi­ cation, and which makes it obvious, apriori,or atleastaffordsapresumption almostamounting to and capable of being reduced to certainty, thatthe numberoffundamental combinants ofthesystem, of which alltherestmust beexplicitrationalfunctions, will be exactly four innumber; which, for the canonical formhereinbefore written, onmaking pandaeachunity,will correspond to 416 OntheOalculusofForms. [54 and will be of the3rd, 6th, 12th, and 9th degrees respectively. Thereason why the squaresoff,g, h,insteadof the simple termsf,g, h,appearinthe 2nd and 3rd of these forms is, because, on changing a;into -e,yinto -y. orzinto -s,two ofthequantitiesf,g,hwillchangetheirsign,butthe formsrepresenting theinvariants ofevendegreesoughttoremainabsolutely unaltered for such transformations. Ishall in the course of thepresent section set forth themethods for obtaining thesefour combinants, which. although oftheregularly ascending dimensions 3,6,9,12, belong obviously to two different groups, the one of threedimensions forming a classin itself, and thenaturalorder of the threeothersbeingthatdenotedbythesequence 6, 12. and 9,and notthatwhich would be denoted bythesequence 6,9,12, thecombinant oftheninthdegree being properly toberegarded asin some sort anaccidentally rationalsquareroot of a combinant of 18 dimensions. Letnow p(w-y')=a, a(yJ-r)=W, yJ+2fyz+2gza;+2Jucy=v: Theresultant will be found by making a;=±y, z=±y, when a;=+y}, z=+y a;=+y},z=-y a;= -y}, z=+y a;=-y},z=-y Hencetheresultant RV=(1+2f+2g+2h)yJ, V=(1-2f-29+2h)yJ, v=(1+2f-2g-2h)y', V=(1-2f+2g-2h)yJ. =pt~(1+2f+2g+2h)(1-2f-2g+2h)(1+2f-2g-2h)(I-2f+2g-2h) =(pu)t{(I+2h)'-4(f+g)2}{(I -2h)'-4(f-g)2} =(pu)t{(I+4h'-4,P-4g')2-(4h-8fg)2} =(pu)t(l- 8(,P+g'+hl)+16{(P+gt+ht)-2(glhl~+hSP+'pg')}+64fgh]. Lcloow K=~U+~V+v~ Kbeingwhat Itermalinearconjunctive ofa,V,~W.Theinvariant ofK. in respect to a;,y,z,will be the determinant p~, h~,sr«. h}l-,}l--p~+au, f~ g~, fp., -UJI 54] OntheOalculusofForms. 417 thatis =(2lgh-!f),.,..+U(hi-ff),.,.111-P(/1-gl),.,.'A-pU,.,.AII+plUAIII-pulAv'; or,multiplying by 6, we may write Is,II,'K=flA",,1I+3b,!J.III+3~,.,.IA+3a,AIII+3c1Av'+b2,.,.' . where d= -pu, b2=12fgh-6gl, bl=-2p(I'-gl),113=2u(h'-gl), a,=plu, ~= -2pu2. thenotation beingaccommodated tothatemployed by Mr Salmon in The HigherPlaneCurves, A,,.,.,IIinIKbeingcorrespondent toai,y,zin MrSalmon's form.Ifnow we employ Mr Salmon's expression for theS (thebiquadratic Aronholdian ofIK),observing that a,=0,CI=0,~=0,C,=O. we have thecomplex combinant SA,,.,.I S,II,.K=d4_2cP(bl~+alb,)+da,b2cl-a,~blb,+bll~1+fla'ba' _4(1-8(P+hl -2gl )+4(121gh-6gl ) ) - Pu'-16(P-g')(h'-ff)+16(I'-,ql)1+(hi-g')' =p4u'{I- 8(I'+9'+hI)+16(I'+9'+h4-hlg2-g2fl-f'hl)+48Igh}. Hence,callingtheresultantR,we have -3R+4SA,I',.Is,II,'K=1 - 8(f'+gl+hI)+16(/4+9'+h4) +32(f'gl+gth'+hlf')={I - 4(/1+gl+hl)}1=Pt. Letnbetakenthepolar reciprocal to theconjunctive -AU+,.,.V+IIWj and for greatersimplicity, as we know, apriori,fromthefundamental definition of acombinant, which (save astoa factor) mustremainunaltered by anylinearmodification impressed uponthefunctions to which itapper­ tains,thatpandUcanenterfactorially only in any combinant, letpandU beeachtakenequal to unityinperforming theintermediary operations. Then -A,h,.,.,g,.,.,E h,.,.,A+,.,.+II,If£,"In=g,.,.,I,.,.,-II,~ " "I, ~,O E'(v'+II,.,.+IIA+1',.,.1) +"II(-All+g2,.,.1) +r(AI+Af£+All+Itl,.,.l)=-2"1~(ft.f£+hg,.,.1) +2~,{g(,.,.A+""11)+(g-fh),.,.I} -2",(h""l1+h,.,.l) s, 27 418 On theOalculusofForms. [54 Uponn,whichisaquadratic function in respectofeachofthetwo unrelated systems ~,"',t;X,p.,u,and also inrespect of thecoefficients in (U,V,W),we mayoperatewiththecommutantive symbol d d d d~'d",'dt d d d d~'d",'dt d d d A'd,.,,'dv d d d iii'dp,'dv which, for facility of reference, I shall term8E. Considering the first line asstationary, we shall obtain, for thevalue of BE(0),216commutantives, which may beexpressed underthefollowing forms: d d d dE'd",'dt d d d dE'd",'dt [d'dldlJ dXI'dp,I'dvl d d d dE'd",'d~ d d d dE'~'dt [til'd d ddJ dXI'd,."dv'dp.dll d d d dE'd",'d~ d d d dE'd",'dt [d d til'ddJ dxdv'dp.I'dxdv d d d dE'd",'dt dd d d,'d",'dt [~d~'txt,.",~J 54J OntheCalcul·usofForms. 419 2d d d dE'd",'d~ d d d dE'd",'d~ [d d d d d d J d'Adp.'d,.,.d,,' d"d>.. Inthis expression thefirst lines may be considered stationary, the second lines aresubjecttotheusual process of commutation, which makes threeofthesixpermutations positiveandthreenegative; andthethird orbracketed linesaresubjectto the simple process which makesallthe permutations ofthesame sign. Inthethreemiddle groups two of the termsinthefinal line arealwaysidentical; itwilltherefore bemore convenient tointroduce themultiplier 2,andthento consider eachsuch line torepresent thethreedistinctpermutations, takensingly. Letnow Andlet [d~2':;2'~J=L, [dt!:-~s.!:-J=L' d:>..2'dp.d,,'dp.d" ' [d ddtd d]L" d:>..dv'dp.I'd'Xd"= , [d d d d dlJL'" d:>..dp.'d'Xdp,'d~= , [t'X:,.,.,t,.,.t",t";'XJ=t; Then,attending to the convention justpreviously explained, we shall have E(n)=(L-2L'-2L"-2L'"+2Ll) x{(n)-2(n)'-2(0)"- 2(n)'"+2(nhl. 27-2 420 OntheOalctdusofForms. [54 asymbolical product. anytermin which such l)8L'fl"will mean ([~":p.;v':P.d:J}Id dd'd dan, ldEd"d"l"dEd' andasimilarinterpretation mustbeextended to each of the25partial products; we havethen L(fi)=8[f. -2L'(fi)=0, -2L"'(fl) =0. - 2L"(fl)=-4ft.24(fl)=-2, - 2L(fl)'=0, -2L(fl)'"=0, 4L'(fiY=0,4L"(fl)'"=0, 4L"(fl)'=0,4L'"(flY"=0, 4L'"(fi)'=8p,4L'(fl)'"=8h', - 2L(fl)"=0,4L'(fi)"=0,4L"(flY'=0.4L'"(fl)"=0, - 4L 1(fir=0, - 4L 1(fiY"=0, -4L1(fi)"=4gl; and.finally,thefivetermscomprised in 2L(fl)l'...,4L1(fi)l' each=O.Alltheaboveequations can be easily verified by directinspection, itbeingobserved that8(fl)represents v2+AV+P.V+flp.I,-AV+ftp.I,AI+AP.+Xv+hlp.', that8(flYrepresents vi+P.V+AV-t-f'p.I,-fAp.-hgp.I,-fA,.,.-hgp.', that8(fl)"represents -AV+glp.l,9(P.A+p.JI)+(g-fh)p.I,9(P.A+p.v)+(g-fh)p.', that8(fiY"represents AI+P.A+VA+hlp.I,-hp.v-fgp.I,-hu»-fgp.I, andthat(fihrepresents -fAp.-hgp.I,9(P.A+p.v)+(g-f h)p.I,-hp.v-fgp.l. We have thus E(fi)=8g'-4ft- 2+8f'+8hl+4ft =2{4f'+4ft+4h1-l}. Hence (A) 54J OntheOalculusofFWmB. 421 .Ifwerestore toU, V, W theirgeneralvalues,andmake U=aa;I+byl+ez2+2fyz+2gzx+2h:cy, V=a'w+b'y2+e'z2+2f'yz+2g'zx+2h'xy, W=a"w+b"yl+e"zI+2f"yz+2g"zx+2h';xy,· andconstruct thecubic function ~=(atz+a'y+a"z)(b.x+ b'y+b"z)(cx+e'y+e"z) -'(ax+a'y+a"z)(fx+/'y+f"Z)I- (bx+b'y+b"z)(g;+g'y+iz,! -(cx+e'y+e"z)(hx +h'y+h"z), +2(fx+f'y+j"z)(gx+g'y+g"z)(hx+h'y+h"z), tha.t is I(abc-ap-bgl-chi+2fgh)w +I{a'be+ab'e+abc'-(a'll+2aff')-(b'gl+2bgg')-(e'hl+20M') +2f'gh+2fg'h+2fgh'}cc'y +{a'b"e+a'be"+a"b'e+a"be'+ab'e"+ab"e'-2a'jf"-2af'f"-2a"fj' - 2b'gg"-2bg'g"-2b"gg'-2e'hh"-2eh'h"-2e"hh' +2f"g'h+2j'g"h+2fg'h"+2f"gh'+2f"gh'+2fg"h'}xyz, SA,,.,.Is,II.•Kinthepreceding equation becomes simply theAronholdian Sto~,which may be calculated by Mr Salmon's formula previously quoted. nmay betakenequal to the determinant atz+a'y+a"z, 1uJ:+h'y+h"z, gx+g'y+g"., E• 1uJ:+h'y+h"e,bx+b'y+b"z,fx+f'y+f"z,'rJ gx+g'y+g"z, fx+f'y+f"z, E, 'rJ,cx+e'y+e"z, ~ ~. 0 And the cubic commutant of this,obtained by affecting itwith the com­ mutantive operator, d d d dx'dy'dz d d d Ck'dy'dz d d d dE'd'r}'~ d d d dE'd'rJ'd~ 422 OntheOalculusofForms. [54 will give48E(n)ifeachofthefour lines of theoperator undergoes permuta­ tion, or8E(n),if one of thefour lines is keptstationary. Thusitfalls withinthelimitsofpractical possibility tocalculate explicity, by theformula (A),thevalue of theresultant. I give to the Sof~theappellation ofthe Hebrew letterc;(shin),and to the commutant ofntheappellation ofthe Hebrew letterto(teth).Theselettersare chosen with design;for I shall presently showthatwhen the threegivenquadratic functions arethe differential derivatives of thesamecubic function y,thetobecomes the Aronholdian Ttothecubic function, or, aswe may write it, Ty,andthe c;becomes the Aronholdian Sof the Hessian thereto,thatisSHy. Thus for thefirsttimethetrueinwardconstitution oftheresultant of threequadratics isbroughttolight.The methods anteriorly given by me, andtheonesubsequently addedby M. Hesse for finding this resultant. adverted toin Section II., lead,itis true, to theconstruction oftheform, butthrow no lightupontheessential mode ofits composition. 55. THEOREME SUR LES LIMITES DES RACINES REELLES DES EQUATIONS ALGEBRIQUES. [NouveUes AnnalesdeMatMmatiques, XII.(1853), pp. 286-287.] SOIT f(a;)=0 uneequation slgebrique dedegren,etsupposons qu'enoperantsurf(a;) et!,(a;)comme dans Ie theoreme de M.Sturm,onobtienne leenquotients CL:ia;+bita,rc+bi,a,rc+bl...ana:+b«; ilfautremarquer seulement qu'onobtientIen1bmequotient, a.,;r;+b",en divisantl'avant-dernier residuparledernierresidu, Formons 10.serie de2nquantitea ±2-b 1±2-b. CL:i a- il n'ya.o.ucuneracinedel'equation±2- bl a-±2-b" an f(a;)=0 entre10.plusgrandede cesquantites et+00 ,nientre10.pluspetitede ces quantites et-00•• •Proohainemeni, nned6moneiration deceUulorime g~Mralul. [po424below.] 56. NOUVELLE METHODE POURTROUVER UNELIMITE SUPERI­ EURE ETUNELIMITE INFERIEURE DESRAOINES REELLES D'UNE EQUATION ALGEBRIQUE QUELCONQUE. [Nouvelles AnnalesdeMatMmatiques, XII.(1853), pp. 329-336.] 1.LEMME. Soient 01>010OJ'"Or-I' Or unesuitedequantitee positives, assujettiea 8.eetteloi 1 1 101=Jl.J.,02=P1+-t0.=J.'a+-...a,=p.a+- ...Or=p..., Jl.J. P1 ""'-1 OUles,.,.sont desqusntites positives quelconques, Si,danslafractioncontinue11111 ~+it+q.+ ...+qr-I+qr' (leaquantites ql'q2...etantdesquantites positives ou negatives), on alea inegalitee [ql]>01>[qJ>o;[q.]>a•...[qr-l]>Or-I>[qr]>Or (les crochets indiquent laracinecarreepositive du carrede laqusntite que ces crochets renferment), ledenominateur de la fraction continue aurameme signe que Ie produitqlq2q••••qr-lqr' Demonstration. POSODS ql=ml, 1q.+-=ma,m1 56J Nouvelle Methode, etc. 425 ilestai.sede verifier que les denominateurs suecessifs de 10.fractioncontinue sont fnta.meme signe que ql: 111111 1 1 ql==~'[ql]<~'~<~,[q.]>1'-1'[q.]>~,etc. ; done 9, a.meme signe que ?nt,et aussi ~'71I.sIestde meme signe que qlq.:111m,>P1+-, n~>fl.,-<- , 1'-1 ?ntP1 doncq,a.meme signe que ?ntjainsimlm.?ntestde meme signe que qlq,q., et, encontinuant, onparvient 8.demontrer queml?nt?nt ...m,..-Im.-,o'est-a-dire ledenominateur de10.fractioncontinue, est de meme signe que Ie produit QI9.q•...qr-Iqr' 2.THEORDE. Bif(x)eatunefonctionalgebrique entierededegren, etBil'onFendarbitrai1'ement une autre 4>(a;)algebrique eteniiere,et d'un degremoindrequen,etqu'ondeveloppe lafraction j(~1enfraction continue 4>(x)1 1 1 1 f(x)==XI+X,+...+Xr-1+X/ aUXI'X,...XrSOfltdesfonctions rationneUes dee,etsiIonformeCequation (8) (XI'-Ol)(X,'-0.')...(XI"-l-Olr_l)(Xr'-Or')=0, laracinedellesuperieure decetteequationseraplus grande,etlaracine1'ulle inferieure decetteequation seramoindre qu'aucunedesracinesrullesde fiquation f(x)==O; etBitouteslesracinesdel'equation (0)somimaginaires, l'equation f(x)==0, auraauasitoutesS88racine«imaginaires. Dbnonstration ..Tous les quotients de10.fractioncontinue qui suivent le premierquotient, savoir:XIIX•...Xr,sontengenb'aldes fonctions Iineaires dex,etXlsera aussi lineaire, si 4>(x)est de degre n-1jlescasparticuliers nechangent pas10.marche de la demonstration jmaisilfautremarquer que lorsque f(x)et4>(x)ont des racines communes, Ie dernierquotient aura 1&forme [~],[xJetantl'avant-dernier terme, et alors, dans l'equation (8), au lieu de Xr'-Or',onecritsimplement Xr'. 426 Nouve1leMethode, etc. [56 SoientL10.plusgranderacineetA10.pluspetiteracinedel'equetion (8)j alorsaucunfacteurde (8) ne peutdevenirnulpourdesvaleursdexcomprises entre+QOetL,etentreAet-QOjdoneonauratoujours [XI]>o; [X,]>0., [Xr_l]>Or-I' [Xr]>Or' OrI(x)estevidemment egalaudenominsteur de10.fraction continue multiplie parunfacteurconstant. Done,en vertudu lemme, le denominateur de10.fractioncontinue estdememesigneque IeproduitXIX,X•...Xr-IX r pourlesvaleursdexcomprises entre+QOetL,etentreAet-QOjmaisdans cesintervalles 10.fonction generaleXin'etantpascomprise entre+0,et-0, nepeutdevenirnulle,et,parconsequent, nepeutchanger designejdone le denominateur de10.fraction continue conserve le meme signepourtoute valeurdexrenfermee entrecesintervalles, etdem~me/(x); Lestdoneune limitesuperieure etAunelimiteinferieure desracinesdel'equation I(x)=o. Lenombredesracinesreellesdel'equation (8) estevidemment pair,zero compris jdans ee derniercas,c'est-a-dire (8)n'ayantaucuneracinereelle, {(x)nechangera donepasdesignepourdesvaleursdexcomprises entre +QOet - QOjautrement touteslesracinesdeI(x)=0sontimaginaires. Le theoreme estdoneeompletement demontre, 3. Siq,(x)est dedegren-1,10.fractioncontinue renferme tmgeneral (sauflescasOUquelques-uns des coefficients deviennent nuIs), comme il aeteditplushaut,nquotients lineaires de10.forme done,d'apresIetheoreme, 10.plusgrandeet10.pluspetitedes2nquantites i.±01i.±0,bn-I±Cn_1i;±o;----- CZ:1 a, an-I an sontrespectivement unelimitesuperieure etunelimiteinferieure des racinesdel'equation I(x)=o. Si1'0nprend(r=n) /JrJ=,.".=...=P-n-I=1,!J-n=2, onvientautheoreme enonee[po423]. 56] Nouve1leMethode, etc. 427 4.Lora meme que lesquotients XI' XI'etc., ne sont paslineaires, on n'aurapourtant jamaiaa.resoudre que des equations dupremierdegre.En effet, soient les 2requations de degre quelconque XI -01= 0, X I-01= 0Xr-Or= 0, XI+01= 0,XI+01= 0Xr+Or=O. nsuffit de trouverunequantitelsuperieure aux racines de ces equations, etuneq,uantite Ainferieure a.cesmemesracines,letAserontdeslimites pourl'equation f(:r;)=O. Si done une de ces equations est de degre p>1, onappliquea.cetteequation Ieprecedeei-dessus, en choieissant une fonction cf>(e) de degre p-1,et, en agissantainsi, on arriveraparune sorte de trituration a.n'avoira.traiterque desequations dupremierdegrd 5. Ona 1Oi=p.o+-j P-o-I plus10.valeurdeI-'iestpetite,et plus on aurade chances a.resserrer les limi d b,±O·ites ans les deux fractions-'--';parcontre, on auraundesa.vantage sous ai rtdI d f· . bi+1±Oi+1a 1ce rappo ans es eux ractions suivantes jcarHI-1-"+1+-j ~I p.o plus",..diminue, et plus Oi+Jaugmente. Cetinconvenient n'apaslieu pour 10.derniere fraction; onpeutdoneprendreu;=0 etOn=_1_. """-I 6. II est a.remarquer que tous les raisonnements precedents subsistent enrenversant 10.suitedesI-'etl'ecrivaut ainsi: 1 1 1-,-+I-'r-h...-+1-'1.P-r-II-'r-'l ,.". 7. IIyalieua.des recherches interesaantes sur10.formea.donnera. cf>(:r;),et sur les valeurs a.donner aux quantites I-'pourobtenirles limites les plus resserrees, et je erois etreparvenua.demontrer que10.forme10.plus avantageuse estf'(:r;),precisement 10.forme que M. Sturmaadoptee. 8. Dans 10.reduction enfractioncontinue dej(~;,nOUBn'avons eon­ sidereque des quotients binomes jmais on peutpousser les divisions plus loinetobtenirdesquantites de10.forme cd l a:r;+b+:x+a;s+'''+ wi 428 Nouvelle Methode, etc. [56 Ie restecorrespondant sera de la forme a'rl+b':r;"+C'a;r-I+...+~. Enoperantainsi, le nombre de termesdanschaquereste ira en diminuant, comme dans le precede ordinaire, et Iedernierreste sera de 18.formeC:d', ,.,.etantunentierpositifounegatif,et Iedernierquotient de la forme PteP+Q,xP-J,Petantunentierpositifounegatif;nommant lesquotients ainsiobtenus qloq2...qr,on voitaisement qu'onaura J(a;)=Ma;±iD, ouMest uneconstante, iun nombre entierpositifounegatifdontIll.valeur dependdeIll.manieredont on a operedans les divisions successives, et Dest ledenominateur deIll.fractioncontinue 1 1 1 1 1-----ql+ql+ql+...+qr-l+qr Done, si 1'0necrit, comme ci-dessua, x=(q12-Cll)(q 12-C22)•••(qr2-Crl)=0, nommant Let A les rscines extremes decetteequation, si zeron'estpas compris entre+00etL,nientreA et - 00,18.demonstration donnee ei­ dessussubsiste encore pour Ie casgeneral. Etlors meme que zeroest comprisentreoes limites, Let Arestenttoutde meme les limites pour las rscines,abstraction faite de la racine zero. 57. ON A THEORY OF THE SYZYGETIC· RELATIONS OF TWO RATIONAL INTEGRAL FUNCTIONS, COMPRISING AN APPLICATION TOTHETHEORY OF STURM'S FUNCTIONS, AND THAT OF THEGREATEST ALGEBRAICAL COMMON MEASURE. [Philosophical Transactions oftheRoyalSocietyofLondon, CXLIII.(1853), PartIII.,pp.407-548.] INTRODUCTION. IIHowcharming is divine philosophy I Notharshandcrabbed Ilo8dull fools suppose, Butmusicalas is Apollo's lute, Andaperpetual feaat ofnectar'd sweets, Where no crude surfeitreigns! "-COllUS. INthefirst section of the ensuing memoir, which is divided into five sections, I consider thenatureandproperties oftheresidues which result fromtheordinary process of successive division (such asis employed for the purposeof finding thegreatest common measure) appliedtof(:r;)andf/J(x), twoperfectly independent rational integral functions of e.Every such residue, aswill beevidentfromconsidering themode in which it arises, isasyzygetic function of thetwo given functions; thatis to say, each of the given functions being multiplied byanappropriate otherfunction of a given degree in x,theBumofthetwoproducts will express acorresponding residue. Thesemultipliers, in fact,arethenumerators anddenominators to the successive convergents to~expressed undertheform ofacontinued frac­ tion.Ifnow we proceed dprioriby means of thegivenconditions asto •CtmJugate would imply somet.hing veryditJeren* fromSyzygetic, namely, a *heoryofthe Invariantive propenies ofasystem of ~oalgebrsical functions. 430 OnaTheuryoftheSyzygetw Relations [57 thedegree in xof themultipliers andof any residue, to determine such residue, we find, as shown in Art.2,thatthereare as many homogeneous equations to be solved as thereareconstants to bedetermined; accordingly, with the exception of onearbitrary factor which entersintothesolution, the problem is definite jand if it be furtheragreedthatthequantities entering into the solution shall be of the lowest possible dimensions in respectof the coefficients of fandq"and also of thelowestnumerical denomination, thentheproblem (save as to the algebraical sign of plusor minus)becomes absolutely determinate,· and we can assignthenumbers ofthedimensions for therespective residues and syzygetic multipliers. The residues given by themethodof successive division are easily seen not to be of these lowest dimensions; accordingly theremustenterinto each ofthemacertainunnecessary factor, which, however, as itcannotbe properly called irrelevant, Idistinguish by the name of theAllotrious Factor. The successive residues, when divested of these allotrious factors, ItermtheSimplified Residues, and in Arts.3and4I express the allotrious factor of each residueintermsoftheleadingcoefficients of the preceding simplified residues offandq,.InArt.5 I proceed to determine byadirectmethodthese simplified residues in termsof the coefficients offandq,.Beginning with the case where fandq,areofthesame dimensions (m)ina,I observe thatwe may deduce, from fandq"mlinearly independent functions of xeach ofthedegree(m-1) inx,all ofthem syzygetic functions of fandq,(vanishing when these two simultaneously vanish), and with coefficients which are made up of terms, each of which istheproductof one coefficientof fand one coefficient of q,.These, in fact, are the very same mfunctions HSareemployed in themethodwhich goes bythename of Bezout's abridged method to obtaintheresultant to(thatis, theresultoftheelimination ofxperformed upon) fandq,.Asthesederived functions are of frequent occurrence, I find itnecessary to give themaname, and ItermthemthemBezoutics or Bezoutian Primaries; from these m .primaries mBezontian secondaries may be deduced by eliminating linearly between themintheorder in which theyaregenerated,-first, thehighest power of xbetween two, thenthetwohighestpowers of xbetween three, and finally, all thepowers of xbetween themall:along with thesystem thusformed it is necessary to include thefirstBezoutian primary, and to consider it accordingly as being also the first Bezoutian secondary; thelast Bezoutian secondary is a constant identical withtheResultant offand¢. When·themtimesm coefficients of theBezoutian primaries are conceived asseparated from the powers of xandarranged inasquare, I term such squaretheBezoutic square. This square, as shown in Art. 7, is sym­ metrical aboutone of its diagonals, and corresponds therefore (asevery symmetrical matrixmust do) to a homogeneous quadratic function of m variables of which it expresses thedeterminant. Thisquadratic function, 57J oftwoA1gebraical Fuuction«. 431 which plays a greatpartin the last section and in thetheory of real roots, ItermtheBezoutiant; itmay be regarded asa species of generating function. Returning to the Bezoutic system, I prove thattheBezoutian secondaries are identical in form with the successive simplified residues. InArt. 6 I extendtheseresultstothecase ofjandcf>being of different dimensions in flJ.InArt. 7 I give a mechanical rule for theconstruction of the Bezoutic square. InArt. 8 I show how thetheoryofj(flJ)andcf>(a:), wherethelatteris of an inferior degree to f,may bebrought underthe operation oftheruleapplicable to two functions of thesame degree atthe expense of theintroduction of a known and very simple factor, which in fact will be a constant power of theleading coefficient in j(a:).InArt. 9 I give another method of obtaining directly the simplified residues in all cases. InArt. 10 I presenttheprocess of successive division underits mostgeneral aspect. In Arts. 11 and 12 I demonstrate theidentity of thealgebraical signof the Bezoutian secondaries with thatofthesimplified residues, generated by a process corresponding tothedevelopment ofj;underthe form of an improper continued fraction (where thenegative sign takes the place of the positive sign which connects theseveraltermsof anordinary continued fraction). As thesimplified residue is obtained bydrivingout anallotrious factor,thesigns of the former will uf course be governed by the signs accorded by previous convention tothelatter;the convention made is, thattheallotrious factors shall be takenwith a sign which rendersthem alwaysessentially positivewhen the coefficients of thegivenfunctions are real. I close the section with remarking therelation ofthesyzygetic factors and the residuestotheconvergents ofthecontinued fraction which expressesj:,and of the continued fraction which is formed by reversing theorderof thequotients inthefirst named fraction. Inthesecond section I proceed to express the residues and syzygetic multipliers intermsoftheroots and factors of thegivenfunctions; the methodbecoming as it may be said endoscopic insteadof being eeoscopic", asinthefirst section. I begin in Arts. 14 and 15 with obtaining in this • These words admitof anextensive andimportant application inanalysis. Thusthe methods forresolving anequation (or to speak more aocurately, formakingoneequation depend uponanotherof asimplerform)furnished byTschirnhausen andMrJerrard(although not so presented bythelatter)areessentially exoscopie; on the otherhand,themethods ofLagrange andAbel for effecting similarobjectsare endoscopic. So again,thememoir ofJacobi,..De Eliminatione," hereinafter referred to, takestheezoecoplc, andthevaluable" NotaadElimina­ tionempertinens" ofProfessor Richelot inCrelle',Journal, the endoscopic viewof the subject. Inthepresentmemoir(inwhich the twotrainsofthoughtarisingout ofthesedistinctviewsare brought intomutualrelation) thesubjectistreated(chie8y but notexclusively) under its endoscopic aspectinthe second, thirdandfourthsections, andexoscopically inthefirstandlast sections. 432 On aTheoryofthe Syzygetic Relations [57 way,underthe form of asum or double sum of termsinvolving factors and roots of fandr/J,andcertainarbitrary functions of theroots in each term,ageneralrepresentative, or tospeakmore precisely, a groupofgeneral representatives foraconjunctive of anygivendegreein:&tofandr/J,thatis, arational integralfunction of e,which is thesum oftheproducts offand r/Jmultiplied respectively byrationalintegral functions of :&,80astovanish of necessity when fandr/Jsimultaneously vanish. This varietyofrepresenta­ tives refers not merely to theappearance ofarbitrary functions, butto an essential andprecedent difference of representation quiteirrespective ofsuch arbitrariness. InArts. 16. 17, 18, 19, 20,21,I show how the arbitrary form offunction entering intotheseveraltermsofanyone(atpleasure) oftheformulasthat represent aconjunctive of anygivendegreemay beassigned, 80astomake suchconjunctive identical in form with asimplified residueof thesame degree. The form of arbitrary function so assigned, itmay be noticed, isafractional function of theroots, so thattheexpression becomes a sum or double sum of fractions. I first prove in Arts. 16, 17 thatsuch sum is essentially integral, and Idetermine theweightof itsleadingcoefficient in respectoftheroots offandr/J(thisweightbeingmeasured bythenumber of roots offandr/Jconjointly, whichappearin anytermof such coefficient). Now inthesucceeding articlesIreverttotheBezoutic system of thefirst section,andbeginning withthesupposition ofmandnbeingequal, Idemon­ stratethatthemostgeneralform of a conjunctive of anydegreein:&will be alinear function of theBezoutics, from which it is easyto deduce thatthe simplified residues ofanygivendegreein:&aretheconjunctives whose weightinrespectof the roots is aminimum; sothatallconjunctives having thatweightmust be identical (toanumerical factorpres),and any integral form of less weightapparently representing aconjunctive mustbenugatory, everytermvanishing identically. Theseresultsare then extended tothe caseof two functions of unlikedegrees. The conclusion is, thattheweight oftheformsassumed in Arts. 16 and 17 beingequal to theminimum weight, theymust(unlesstheywere to vanish, which is easilydisproved) represent thesimplified residues, or which is thesamething,theBezoutian secondaries. Wethusobtainfor each simplified residueanumber ofessentially distinct forms of representation, butall of which must be identical toa numerical factorpres,aresultwhich leads to remarkable algebraical theorems. Thenumber ofthesedifferent formulse depends uponthedegreeofthe residue; therebeingonly one for thelastorconstant residue,two forthe lastbutone,threefor thelastbuttwo, and so on. The formulas continue to have ameaning whentheirdegreein:&exceedsthatofforr/J;butthen, asalthough alwaysrepresenting conjunctives, theyno longer represent 57J ojtwo Algebraical Function», 433 residues, thisidentityno longer continues tosubsist. InArts. 22, 23, 24, 25, Ienterintosomedevelopments connected withthegeneralformulas in question; these,itmay be observed, are all expressed by means of fractions containing inthenumerator anddenominator products ofdifferences; the differences in thenumerator products beingtakenbetween groups of roots offand groups of roots of t/J;and in the denominator between roots off inter88and roots of t/Jinterse.Agreatenlargement isthusopened out to theordinary theoryofpartialfractions. InArt. 26 I find thenumerical ratiosbetween thedifferent formulee whichrepresent (toanumerical factorpres)thesame simplified residue, and in Arts. 27 and 28 I determine therelations ofalgebraical sign of these formulee tothesimplified residues orBezoutian secondaries. InArt. 29 Idetermine thesyzygetic multipliers corresponding to anygivenresidue intermsofthefactors and roots of thegivenfunctions; buttheexpressions forthese,which are closely analogous to those for theresidues, cease to be polymorphic. Theyareobtained separately fromthesyzygetic equation, anditis worthy of notice, thattoobtaintheone we use the first of the polymorphic expressions for theresidue, and to obtaintheothertheopposite extremity ofthepolymorphic scale. Inthesubsequent articles ofthis section, by aid of certaingeneralproperties ofcontinued fractions, I establish atheorem ofreciprocity between theseries of residues and eitherseries of syzygetic multipliers. SectionIII.isdevotedtoadetermination ofthevalues of thepreceding formula! for theresidues andmultipliers inthecaseapplicable to M.Sturm's theorem, wheret/Ja:becomes thedifferential derivative offa:Itbecomes ofimportance to express theformulas for this case in termsoftheirroots and factors of fa:alone,withoutthe use of theroots and factors of I'a:,which will of course be functions of theformer. Byselecting aproperform out of thepolymorphic scale,thefractional termsoftheseries for each residueinthiscase become separately integral, and weobtainmy well-known formulas forthesimplified residues (Sturm's reduced auxiliary functions) in termsofthefactors and thesquared, differ­ ences of partialgroupsof roots. Thisisshown in Art. 35. InArt.36the multiplier ofI'ICinthesyzygetic equation is expressed by formulas of equal simplicity, and inacertainsensecomplementary totheformer. This method,however, does not apply to obtaining expressions for themultiplier offa:inthesameequation intermsoftheroots and factors of fe;forthe separate fractions whose sum represents anyone ofthesefactors, it will be found, do not admitof being expressed asintegralfunctions of theroots and factors. To obviate this difficulty I look to thesyzygetic equation itself, whichcontains fivequantities, namely,thegiven function, itsfirstdifferential derivative, theresidueofagivendegree,andthetwomultipliers, all of & ~ 434 On a Theory ofthe Syzygetic Relations [57 which,exceptthemultiplier offx,areknown, or have been previously deter­ minedasrationalintegral functions of theroots and factors of fe.I use thisequation itself for determining thefifthquantity, themultiplier in question. To perform thegeneraloperations byadirectmethodrequired forthiswould be impossible; the difficulty is got over by finding, by means of the syzygetic equation, the particular formthattheresultmustassume whencertainrelations ofequality springup between theroots offx;and then,by aid of these particular determinations, thegeneralform isdemon­ stratively inferred. Thisinvestigation extendsover Arts. 38, 39, 40, 41, 42, 43. Itturns outthattheexpressions for themultipliers offxare of much greater complexity thanforthemultipliers off'xor fortheresidues. Any such multiplier consists of a sum of parts,each of which, asinthecaseofthe residues and the factors of f'x,is affected with a factor consisting ofthe squareddifferences of a group of roots;buttheotherfactor,insteadofbeing simply (as for theresidues and factors before mentioned) aproductofcertain factors oflx,consists of the sum of a series of products of sums of powers byproducts ofcombinations of factors of fx,each of which series is affected with the curious anomaly of itslasttermbecoming augmented in acertain numerical ratiobeyond what it should be in order to be conformable to the regularflowofthepreceding termsintheseries". Thefourthsection opens with theestablishment of twopropositions concerning quadratic functions which are made use of in thesequel. Art.44 contains the proof of a law which, although ofextreme simplicity, I donot remember to have seen, and with which I have not found thatanalysts are familiar: I meanthelaw of the constancy of signs (as regards thenumber of positive and negative signs) in any sum of positive and negative squares into which a given quadratic function admitsof being transformed by substituting forthevariables linearfunctions of the variables with real coefficients. This constant numberof positive signs which attaches to aquadratic function underallitstransformations, which is atranscen­ dentalfunction of thecoefficients invariable forrealsubstitutions, may be termedconveniently itsinertia,untilabetterword be found. This inertia it is shown in Art. 45, by aid of atheorem identical with one formerly given by M. Cauchy, is measured by the numberofcombinations of sign in the series of determinants of which the first is the complete determinant ofthe function, thesecond,thedeterminant when one variable is madezero,the next, the determinant whenanothervariable as well asthefirst is made zero, and so on, untilall the variables are exhausted, andthedeterminant • Thesyzygetic multipliers areidentical with the numerators anddenominators (expressed in their simplest form) of thesuccessive convergents to the continued fraction which expresses f:z.JZ 57] ojtwoAlgebraical Functions. 435 becomes positive unity.InArt. 46 I give some curious and interesting expressions fortheresidues and syzygetic multipliers, undertheform of determinants, communicated to me by M.Hermite jand in Art. 47I show how, by theaid ofthegenerating function which M. Hermite employs, and ofthelaw ofinertiastatedattheopening ofthesection, an instan­ taneousdemonstration may be given of theapplicability of myformulee for M.Sturm'sfunctions for discovering thenumberof real roots of fe,without any reference to the rule of common measure jand moreover, thatthese formules may beindefinitely varied, and give the generating function, out of which theymay be evolved, in itsmostgeneralform.Hadthelaw of inertiabeen familiar to mathematicians, thisconstructive andinstantaneous methodof finding formulas fordetermining thenumberof real roots within prescribed limitswould, in all probability, have been discovered long ago, 88an obvious consequence of such law. I thenproceed in Arts. 48and 49, toinquireasto thenatureoftheindications afforded by the successive simplified residues to two generalfunctionsIandrf>jand I find thatthe succession of signsof these residues serves to determine thenumberof roots ofIorrf>comprised between given limits, afterall pairs of roots of either function contained withinthegivenlimitsand not separated by roots of the otherfunction have been removed, and the operation, ifnecessary, repeated totiesquotiesuntilno two roots of eitherfunction are leftunseparated by roots oftheotherjor inotherwords,untilevery root finally retained in one function is followed by a root of theother, or else by one of theassigned limits. The system of roots comprised between givenlimitsthusreduced I calltheeffective seale of intercalations jsuch a seale may begin with a root ofthenumerator or ofthedenominator ofj.~;and upon thisand the relativemagnitudes of thegreatest root of rf>xandIxitwill depend whether intheseries of residues (among which Ixandrf>xare forthispurpose to be counted) changes will belost orgainedasxpassesfrom positive infinity to negative infinity. InArt. 50 I observe thatthetheoryof real roots of a single function given by M. Sturm's theorem is a corollary to thistheory oftheintercalations of real roots of two functions, depending uponthewell­ known law, thatodd groups of thelimitingfunction!,x liebetween every twoconsecutive real roots of Ix.InArt. 51 I verify thelaw of reciprocity, alreadystatedto existbetween theresidues of Ixandrf>:r;,by anaposteriori methodfounded on thetheoryofintercalations. InArts. 52, 53, 54,Iobtain aremarkable rule, founded upon theprocess of common measure, for finding asuperiorand inferior limitinan infinite variety of ways to theroots of any given function. Thismethodstandsin asingular relationofcontrast to those previously known. All previous methods (including those derived through Newton's Rule) proceed upon theidea of treating thefunction whoseroots are to be limitedasmade up of thesumofparts,each of which 28-2 436 On a Theory ofthe Syzygetic Relations [57 retainsaconstant sign for all values of thevariableexternal tothequantities which are to be shown to limittheroots, My method,ontheotherhand, proceeds upon theidea oftreating thefunction astheproduct offactors retaining aconstant sign for such values of thevariable. InArt.5.5,the concluding articleofthefourth section, I pointoutaconceivable mode in whichthetheoryofintercalations may beextended tosystems ofthreeor more functions. InSectionV. AI,ts. 56, 57, I show how thetotalnumber of effective intercalations between theroots of two functions of thesamedegreeisgiven bytheinertiaofthatquadratic form which we agreedtotermtheBczoutiant toIand4>;and inthefollowing article(58)theresultisextended to embrace thecasecontemplated in M.Sturm's theorem; thatis to say, I show,thatonreplacing thefunction ofIXby a homogeneous function of IXandy,theBezoutiant tothetwo functions, which are respectively the differential derivatives ofIwithrespecttoIXand with respecttoy,will serve to determine by its form or inertiathetotalnumberofrealroots and ofequalroote inI(IX).Thesubjectispursued inthefollowing Arts. 59, 60. Theconcluding portionofthissection is devoted to aconsideration ofthe properties oftheBezoutiant underapurelymorphological pointofview; forthispurposeIandq,aretreatedas homogeneous functions of two variables e,y,insteadofbeingregarded asfunctions of xalone.InArts. 61, 62, 63, it is proved thattheBezoutiant is aninvariantive function of the functions from which itisderived; and in Art. 64theimportant remarkis added,thatit is an invariant ofthatparticular class to which I have given thenameofCombinants, which have theproperty ofremaining unaltered, not only for lineartransformations ofthevariables, butalso for linear combinations ofthefunctions containing thevariables, possessing thusa character of double invariability. InArts. 65, 66, I consider therelation oftheBezoutiant tothedifferential determinant, so called by Jacobi,but which for greaterbrevityI calltheJacobian. Onpropersubstitutions beingmade in theBezoutiant forthemvariables which it contains (m beingthedegree in x,yofIand4»,theBezoutiant becomes identical with theJacobian toIand4>;butasit isafterwards shown,thisis not aproperty peculiar totheBezoutiant; in factthereexists a whole family of quadratic forms of mvariables, linea-linear (liketheBezoutiant) inrespectofthe coefficientsin Iand4>,all of which enjoy thesameproperty. Thenumber ofindividuals of such family mustevidently beinfinite,because any linear combination of any two of themmustpossess a similarproperty; I have discovered, however, thatthenumber ofindependent forms of thiskind islimited,beingequal to thenumberof oddintegers notgreaterthanthe degreeofthetwo functions Iand4>.InArts. 67 and 68, I give themeans ofconstructing thescale of forms, which I termtheconstituent orfunda- 57J ojtwo Algebraical Functions. 437 mentalscale,of which all othersof thekindare merely numerico-linear combinations. This scale does not directlyincludetheBezoutiant withinit, anditbecomes an object of interesttodetermine thenumbers whichconnect theBezoutiant withthefundamental forms;thiscalculation I havecarried on (in Arts. 69, 70, 71) from 1n=1 tom=6 inclusive, and added an easy methodofcontinuing indefinitely. Inthismethod thenumbers inthe linearequation corresponding to any value of 1naredetermined successively, and each made subjectto a verification before thenextisdetermined, there beingalways pairs of equations whichoughttobringout the same resultfor each coefficient. Inthenextandconcluding Art. 72,I remarkupon the different directions in which a generalization may besoughtofthesubject-matter of the ideas involved in M. Sturm'stheorem, and of which the most promising is, in my opinion,thatwhich leads throughthetheoryofintercalations. Some of the theorems given by me in thispaperhave been enunciated by me many years ago, butthedemonstrations have not been published, nor have they ever before been puttogether and embodied in thatcompact and organic orderin which theyarearranged inthismemoir,-the fruitofmuch thought andpatienttoil, which I have now thehonour of presenting totheRoyal Society. P.S.Inasupplemental parttothethirdsection I have given expressions intermsoftheroots ofcf>xandfa;forthequotients whicharisein developingtundertheform ofacontinued fraction, and some remarkable properties concerning thesequotients. Inasupplemental parttothefourth section I have given an extended theoryof my new methodof finding limitstothe real roots of any algebraical equation. This method, so extended, possesses 00. markedfeatureofdistinction from all preceding methods used for the same purpose, inasmuch asitadmitsin every case of thelimitsbeingbroughtup intoactualcoincidence with theextreme roots, whereas in othermethods a wide and arbitrary interval is ingeneralnecessarily leftbetween the roots andthelimits. 438 On a Theory ofthe Syzygetic Rdations SECTION I.[57 Onthe completeand simplified residuesgenerated inthe process ofdeveloping undertheformofa continued fraction, anordinary rational algebraical fraction. Art. 1.LetPandQbe tworationalintegralfunctions of :1:,and suppose thatthe process of continued successive division leads to the equations sothatP-MoQ+RI=O Q-Ml~+~=O ~-M2R2+Ra=O I Q1 1 1-=------&c.P Mo-.AIl-M2-(1) (2) which is what I propose to call an improper continued fraction, differing from aproper only in thecircumstance of the successive termsbeingconnected bynegative insteadof positive signs. Mo.Ml>MOl,&c.,Rl,~,Ra,&c. are, of course, functions of :1::thelatter we may agree to call the1st, 2nd, 3rd, &c.residues (in order to avoid theuse ofthelongerterm"residues withthesignschanged"); and by way of distinction from what theybecome when certainfactors are rejected, we may call R I,~,Ra,&c.thecomplete residues. Each such complete residue will ingeneralbe oftheform~jf-',N,andD,beingintegralfunctions of the, coefficients only of PandQ,butp,anintegralfunction of these coefficients, and of :1:;p,maythenbetermedthe sth simplified residue, and ~'the sth, allotrious factor. Suppose Pto be of mandQofndimensions in :1:,and m-n=e,the process of continued division may be so conducted, thatallthe residues maycontain onlyintegerpowers of :1:;and we may upon this supposition makeMoofedimensions, and Ml>M2•Ma,&c.each of one dimension only in :1:;sothat~,~,Ra,•••willberespectively of(n-1), (n-2),(n-3),&c.dimensions in :1:. 57J oftwo Algebraical Functions. 439 PandQare supposed to be perfectly unrelated, and each the most general functionthatcan be formed of thesame degree. From (1) we obtain R1=MoQ-Pl ~=MIR.-Q =(MoMI- 1)Q- M 1P ,(3) u;=(MoMIM 2+Jlo+M2)Q-(M1M',l-1)pJ' &c.=&c. and ingeneralweshall have R,=Q,Q+P,P, (4) where it is evidentthatQ,will be of e+(£-1),andP,of(£-1) dimensions Ine. Art. 2. Hence itfollowsthatthe ratios P,:Q.:R,may beascertained bythedirectapplication of the method of indeterminate coefficients,for Q, willcontaine+"andP,will contain £disposable constants, makinge+2£ disposable constants in all. Again, Q,QandP,Pwill each rise to thedegree n+e+£-1 in:r:jbuttheirsumR,is to be only of n-£dimensions in a: Hence we have to make (n+e+£- 1)-(n-,),thatise+2£- 1quantities (which are linear in respect to the given coefficients in PandQ,as well as in respecttothenew disposable constants inP,andQ,)all vanish, thatis to say,therewill bee+2,- 1linearhomogeneous equations to be satisfied by means of e+2,disposable quantities; theratios of these latterare, therefore, determinate, sothatwe may write P'=~'(P')} Q,=X,(Q,) j R,-x,(R,)(5) andwhen(P,), (Q,),(R.)aretakenprimeto oneanother,itis obvious that (R,)will be in all of e+2,dimensions in thegiven coefficients, thatis of£in respectofthecoefficients of P,and ofe+,in respect of those of Q;~,will correspond to what I have previously called theallotrious factor;being in fact foreign to thevalue ofR,asdetermined by means of the equation (4), andarisingonly from the particular methodemployed to obtainitthrough themedium of the system (1):it becomes a matterof some interestand importance todetermine thevalues of thisallotrious factor for different values of ,•. • Theseareidentical withwhatItermedquotients ofBUCC888ion in theLondonandEdinburg" Philowphical ..vagaziru (December, 1839)[po43above]; butbyaue&Ililyexplicable errorof iD8dvertence, thequantities QI'Q2'&c.thereinsetoutarenot&Btheyarethereinstatedtobe, 440 OnaTheoryoftheSyzygetic Relations [57 Art.3.Thismay be done by thefollowing method, which is extremely simple,andwouldadmitof aconsiderable extension initsapplications, were itnotbesidemyimmediate purpose todigressfromtheobjectssetoutin thetitletothememoir, byentering upon an investigation ofthespecialor singular cases which may ariseintheproce8Sofforming thecontinued fraction, when one 01'more of theleading coefficients in anyoftheresidues vanish; such an inquiry wouldrequire ttmoregeneral character tobe imparted tothevaluesofthequotients andresidues thanIshallformy presentpurposes care tosuppose. Letusbeginwithsupposing e=1,andwrite f=ax"+biXn-1+CX'H+&c. }(6) <f>=cu;n-l+f3iXn~+,,/iXn-3+&c. . Let"bethefirstresidueofi,andCc)of$,andtherefore ofa~'sothat Cc)isthesecondresidueofi· LetCc)=A(ee),Cc)beingentirelyinteger,andAa.function ofthecoefficients infand<f>.Ifwe make A=~,NandDbeingintegerfunctions, Dwill evidently beLt,whereLdenotesthefirst coefficient in thesimplified residue a.~,and isevidently of twodimensions ina.,f3,&c.,andof oneina,b,&c.; DCc)istherefore of 2 x 2+1,thatis fivedimensions ina,f3,&c.,andof two dimensions ina,b,&c.;butCc)(byvirtueofwhathasbeenobserved ofthe equations insystem (.5»is ofthreedimensions ina,f3,&c.,andof twoin a,b,&c.HenceNis of two dimensions ina,f3,&c.,andof none in a,b,&c. Thisenablesusatonce toperceivethatN=a'. For",is oftheformf-(piX+q)<f>,} • andCc)is oftheform<f>_(p'iX+q')'"I (7) thequotients ofsnccession orallotrious factorsthemselves, buttheratiosofeachsuehtothe onepreceding, if intheaeries; 80that- Q1is>'1 Q.>...z 2IS>. 1 Q•>'s 3IS>., &c•...• Thiserror iscorrected by mydistinguished friend M. Sturm(Liouville', Journal,t.VID.1842, Sur unthem'eme d'Algebre deM.Sylvester), who appears, however, tohave overlooked thatI was obviously well acquainted with the existence andnatureofthesefactors,andtheirell8ential character, of being perfectsquares in thecasecontemplated in hismemoir and my own. MM.Borchardt, Terquem, andotherwriters,inquotingmyformu18ll for M.Sturm's auxiliary functions, havethusbeen led intotheerrorofalludingtothemascompleted byM.Sturm. 57] ojtwo Algebraical Functions. 441 butN=0makes Q)vanish, and therefore, upon this supposition,fandcI> wouldappearto have a common algebraical factor't,thatis to say, N vanishing wouldappearto imply thattheresultant offandrf>mustvanish, sothatNwouldappearto becontained asa factor in thisgeneralresultant, whichlatteris, however, clearlyindecomposable into factors-a seeming paradox-the solution of which mustbesoughtfor inthefact,thatthe equation N=0 isincompatible withtheexistence of the usual equations (7) connectingf,rf>,'tandQ):butthisfailure of the existence oftheequations (7)(bearing in mindthatNhas been shown to be a function only of the set of coefficients a,fl,&c.),can only happenby reason of avanishing when­ everNvanishes; amusttherefore be a root of N,or whichis thesamething, Na power of aand hence N=a2• The same resultmay beobtainedaposteriori byactuallyperforming the successive divisions; if the coefficients of any dividend bea,b,c,d,&c.,and of the divisor a,fl,'Y.0,&c., the first remainder, forming the second divisor, willbeeasily seen to have for itscoefficients- The compound determinant abovewrittenwillbethefirst coefficient intheremainder underconsideration; thesubsequent coefficients willbe represented bywritingf,rf>;g,'Y,&c.,respectively in lieu of e,e.Omitting the common multiplier ~.thedeterminant abovewrittenis equal tom 442 On aTheoryoftheSyzygetic Relaiion« [57 .{a,b,c a,b,e a,b,d a,b,d}0,a,{3x0,a,S 0,a,"1x0,a,"1 a,{3,"1 a,{3,e a,{3,S a,{3,S a,b,c x{~a,b,d a,b,c }. +0,a,{3 0,a,"1-"10,a,{3 a,{3,"1 a,{3,S a,{3,"1 Thelastwrittenpairoftermsaretogether equal to a,b,ci 0,a,{31x{-d{3a'+crya'+aa({3S-r)], a,{3,"1 which is of theforma'A-a'{J2(f3S-"1')ajandthesum ofthefirstwritten pairis oftheforma'B+(af3'af30-a'Yf3ary(3)a.Hencetheentiredeter­ minantis of the form a'(A+B),showing thata'willenterasafactor into thisand every subsequent coefficient in thesecondremainder, aspreviously demonstrated above. Itmay, moreover, be noticed, thatthisremainder, whena'hasbeen expelled, will for generalvalues of thecoefficients be numerically as well asliterallyinitslowest terms, asevinced by thefactthatthereexistterms (for example aa'rye)having±1 fortheirnumerical part.Thesameexplicit methodmightbeappliedto show, thatifthefirst divisor were edegrees insteadof being only one degree in xlowerthanthefirst dividend, «HI would:becontained in every termof the second residue: thedifficulty, however, of the proof by thismethodaugments withthevalue ofejbutthe sameresultspringsasanimmediate consequence from themethod first given, which remains goodmutatismutandis forthegeneralcase,asmay easily be verified by the reader. Applying nowthisresultto the functions PandQ,supposed to be of therespective degreesnandn-einx,andcalling thecoefficients of theleadingtermsinthesuccessive simplified residues aI'a"aa,&c.,anddenoting byatheleadingcoefficientin Q,andasbeforedenot­ ingthesuccessive allotrious factors by AI,A-"&c.,itwill readily be seen that 1 1 1 1 A1=-e+l' A-,~=2' A-,>"=i' A4A,=i' &c.,a al a, aa thatis 1 >'1=ae+l' and ingeneral (8) 57] oftwoAlgebraical Functions. 443 Art.4.Strictlyspeaking, we have notyetfullydemonstrated thatthe complete allotrious factorsarerepresented bythevaluesabovegivenforX, butonlythattheselatterarecontained asfactors in theallotrious factors; wemustfurtherprovethatthereexistnoothersuch factors. Thismay be shownasfollows: itis obvious from thenatureoftheprocessthatthe complete residues will always remainof onedimension inrespectofthegiven coefficients, thatis, first of one dimension intheseta,b,c,&c.,and of zero dimensions intI,f3,'Y,&c.;thenconversely, of one dimension in(x,f3,'Y,&c., andof zerodimensions ina,b,c,&c.,andso on,theresidues beingevidently required to conform in theirdimensions tothoseofthefirstdividend andthe firstdivisoralternately. Thesecoefficients thenare always of unitdimensions inrespecttothegivencoefficients; whereas it has been shown (Art.2)that thesimplified residues inrespecttothesecoefficients are successively ofthe dimensions 2+e,4+e,6+e,&c. Letthecomplete residuecorresponding to~beMx.m.a2m, thatis or sayML;inpassingfrom~toa2q+lthedimensions rise2unitsfor all values of qexceptzero, and when q=0thedimensions increase persa1tum from 1 to 2 +e;hencethetotaldimensions ofLinthejointcoefficients will be [(e+1)-2(e+2)) -4(m-l)+4m+e= 1, andtherefore Mis of zero dimensions, and~isthecomplete allotrious factor.Inlikemannerifthecomplete residuecorresponding to~+lbe M~la.m+l>thatis or sayML,thedimensions ofLwill be - (e+I)-4m+{e+2(2m+I)},thatis, 1, andhence,asinthepreceding case,Mis of zero dimensions, andx",,+listhe complete allotrious factor. Art.5. Iproceed toshow how thesimplified residues may be most conveniently obtained by adirectprocess, identical withthatwhich comes intooperation inapplying tothetwogivenfunctions of:cthemethod familiarly knownunderthename of Bezout's abridged methodofelimination. Letuscallthetwogivenfunctions UandV,andcommence withthecase whereUandVare ofequaldimensions (n)inz,Thesimplified ,thresidue willthenbeafunction ofn-,dimensions ine,andof,dimensions inrespect of eachgivensetof coefficients, andmay betakenequaltoV,U+U,V,where V,andU,areeachof(,-1)dimensions ina: 444 LetOn aTheoryofthe Syzygetic Relations [57 U=aox"+a1x'l-1+a,Xn-t++an, V=boxn+b1xn-1+b2xn-2+ +bll; we may writeingeneral, mbeingtakenanypositive integernot exceed­ mgn, U=(aoa:"'+ ~Xf1l-1+ +a",)x'l-m+(am+lxn-m-l +am~x'l-m-2+ +a,,), V=(boxm+ b1xm-1+ +bm)xn-m+(bm+ixn-m-l +bm+2x'l---m-2++bn). Hence (boxm+b1Xf1l-1+...+bm)U-(aoxm+a1a;'"-1+...+am)V =,.,.1(IXn-1+mKtxn-t +mK,xn-,+...+mKn, (9) where if we use (r,8)todenotearb,-a,brfor all values of rand8,wehave mKl=(O,m+l), mKt=(O, m+2)+(I,m+l), ,"K,=(0,m+3)+ (1,m+2) + (2,m+1), audingeneralmK,=I(r,8),thevaluesofrand8admissible withinthesign ofsummation beingsubjecttothetwoconditions, onetheequalityr+s=m+i, theothertheinequality r lessthan i.Bygivingtomallthedifferent values from 0 to m-1 in succession, andcalling boa:'"+b1Xm-l+...+bm,aoxm+a1a:"'-1+...+am respectively QmandP,",we have QoU-PoV=K1x'l-1+K2,xn-t++K; Q1U-P1V= lK1x'l-1+ lK2x'l-2+ +.K; Q2U-P,V=tK1Xn-l+2Ktxn-2+ + tKn (10) Qn-lU-Pn-lV=n_1K1Xn-1+n_1Ktx'l--'l+ ...+ n-1Kn Theright-hand members ofthesenequations Ishallhenceforth termthe Bezoutians toUandV. Thedeterminant formed by arranging in asquarethensetsof coefficients ofthenBezoutians, andwhichIshalltermtheBezoutian matrix,gives,as is well known, theResultant (meaning therebytheResultinitssimplest form ofeliminating thevariables out)ofUandV. Eliminating dialytically, firstxn-1between thefirstandsecond,thena;'l-l andxn-2between thefirst, second andthird,andso on,andfinally, all the powers of a:between thefirst, second, third,...nthoftheseBezoutians, and repeating thefirst of them,weobtainaderived setofnequations, the right-hand members of which Ishalltermthesecondary Bezoutians toU andV,thissecondary systemofequations being 57J oftwoAlgebraical Functions. 445 QoU-PoV=Klx"-I+K1xfH+K,x"-a+...+K; eKIQ.- KIQI) U-(IKIPo-KIPI)V=Llxn-'J+L,xn-a+...+L"'_I {(IKlaK, - 2KllKa) Qo+(2KIK2-KI$1)QI+(KIIK,-IKIIKa)Q2}U -{(IKlaK,-aKIIK2)Po+(2KI[(I-KI$2)PI+(KIIKa- IKIIKa)PalV =Mlxn-a+Maxn-4+...+Mn-a &c.=&c.(ll) And we can now alreadywithoutdifficulty establish theimportant proposition, thatthesuccessive simplified residues to~,expanded underthe form of an improper continued fraction, abstracting fromthealgebraical sign(the correctness of which also will be established subsequently), will berepre­ sentedbythensuccessive Secondary Bezoutians tothesystemU, V. Forif we write thesystem of equations (11)underthegeneralform ~,U-H,V=A,xn-,+B,xn-'-I+&c., thedegreeof~.andH,inxwill bethatofQ.-Iandp._1>thatis£-1jand thedimensions ofA"Bll&c.,inrespectofeach set ofcoefficients is evidently s;consequently, byvirtueof Art. 2,A,xn-I+B,x"-'J+&c.,which is the rthBezoutian, will (saving atleastanumerical factor of a magnitude and algebraical sign to be determined, butwhich, when properconventions are made, will be subsequently proved to be +1)represent thesth simplified residue to ~.,as was to be shown. Art. 6. More generally, supposeUandVto berespectively ofn+eand ndimensions in a: Let U=a.xn~+~xn~-I+aax"+O-:l+&c. V=boxn+blxn-l+&c. Making U=(ao:c"+m+al:c"+m-I+&c.+aHm)xn-m+(ao+m+1a;n-m-1 +&c.+anH), V=(boxm+b1Xm-1+...+bm)xn-m+(bm+lxn-m-I+&c.+b,..), weobtaintheequation QmU-PHmV=mKI:r;n+o-1+mKax"+e-a+&c.+".Kn+" (12) •Vissupposed tobetaken as the first divisor, andthe term residue isused, ashithertoin thispaper,throughout in thesenseappertaining to theexpansion conducted, 80as to lead to an improper continued fraction, inthatsenae,in fact,inwhich it would, more strictlyspeaking,be entitled to the appellation oftUU.ratherthanthatofrtrid~. 446 whereOn aTheoM)ofthe Syzygetic Relations [57 Qm=(boxm+ ... +bm),P'+m=(ao:r;"+m+...+aH"'); ",K1=aob",+!;".l{,=aob",+!+a1b",+!;•..",K.=aob",+o+a1bm+o-l+&c.+a.b...; mK.+!=aObm+O+1+&c.+at+lb",-a'+m+lbo;&c.=&c. Bygivingtomeveryintegervaluefrom0to(n-1)inclusive, we thus obtainnequations oftheform of (1~),each ofthedegreen+e-1ine,and of onedimension inregardto eachsetof coefficients. Inaddition totheseequations we have the8equations oftheform a;/"V=bo:r;"+"+b1:r;"+,.-1+&c.+bnX", (13) in which p.may be made to assumeeveryvaluefrom 0 to (e-1)inclusive, andtheright-hand side oftheequation for all such values of p.willremain of adegreeinxnotexceeding n+e-1,thedegreeoftheequations ofthe systemabove described. Therewillthusbeeequations in which only the (b)setof coefficients appear,andnequations containing in every termone coefficient outof each of thetwo sets. Thetotalnumber ofequations isof course n+e.Between the8 equations ofthesecondsystem(13) and theroccurring first inorderofthe firstsystem(12),we may eliminate dialytically thee+r-1highestpowers ofe,andtherewillthusarise anequation oftheform Br-1U-Q)o+r-IV=Lxn-r+L':r;"-f"-l+&c.+(L), (14) whereBr-1andQ)t+r-larerespectively ofthedegreesr -1and8+r-1ine, andL, L'...(L)are of r dimensions inthe(a)set, and of (8+r)dimensions inthe(b)setof coefficients, and consequently Lxn-1+L':r;"-f"-1+ •.,+(L) mustsatisfytheconditions necessary and sufficient toprove its being(toa numerical factorpres)asimplified residueto(U, V). Thussuppose Then,corresponding tothesystemof which equation (13)18thetype, we have V=boaf+b1x+b" XV=bow+b1af+b2IX. Again, to form thesystemof which equation (12) isthetype, we write boU-(aoaf+a:.x+a,)V=bo(~x+u4)-(aoaf+a:.x+~)(b.x+b,) = -aOb1w-(ao~+a:.bl)af+(boa,-a1b2-a,bl)x+(bOa4-~b2)' (boX+bl)U-(aoW+alaf+¥+a,)V=(boX+b1)a4-(aO''t'+a1af+~+~)b2 = -a.b,w-a:.b,af+(bOa4-a,b2)x+(b1a4-b,~), 57] oftwo Algebraical Functions. 447 Combining thetwoequations of the first system with thefirst ofthesecond system, we obtainthe first simplified residue Le+L',where 0,i; i. - L=bo,bi> b2 aobi>aob2+alb!> ~b!+Utbl-boUt and aob!>aob2+~bl>Cl:ib!-bOa4 By again combining thetwoequations of the first system with both of the second system, we have thedeterminant 0,i: bi> i. o aObl,aOb2+albl>alb2+Utbl-biUt,Cl:ib2-bOa4 aob!, albl,a2b2-boa4, ao~-a4bl which is thelast simplified residue, or in otherterms,theresultant tothe systemU, V. Art. 7.Itis mostimportant to observe thattheBezoutian matrixto two functions of thesame degree (n)is asymmetrical matrix,the terms similarly disposed in respect to one of thediagonals beingequal. Thusretaining thenotation ofArt. 5,so that (0, 1)=afJ-b«,(I,2)=b'Y-cfJ,(2, 3)=c~-d'Y, (0, 2)=a-y-ca,(I,3)=b~-dfJ, &c. (0, 3)=as-da, &c. &c. whenn=1 theBezoutian matrixconsists of a single term(0,1); when n=2, it becomes (0, 1) (0, 2) (0, 2) (I,2); whenn=3,itbecomes (0, 1) (0, 2) (0, 3) (0,+3))(0, 2) (1,3) (1, 2) (0, 3) (1, 3) (2, 3); 448 On a Theory ofthe Syzygetic Relations [57 whenn=4,itbecomes (0, 1) (0, 2) (0, 3) (0, 4) whenn=5,itbecomes(0, 2) (0, 3) (0, 4) (0+3)(0+4) (1, 4) (1, 2) (1, 3) (0,4)(1,4) (1~3)(2~S)(2, 4) (1, 4) (2, 4)(3,4); (0, 1) (0, 2) (0, 3) (0, 4)(0, 5) (0, 2)C\.3»)-(0+4»)(0+5») (1, 5) ,(1, 2) (1, 3) (1, 4) (0, 3)(0+4» )(::~:))(1+5») (2, 5) (1, 3) +(2,4) (2, 3 (0, 4)(0+5»)(1+5») (2+5») (3, 5) (1,4)(2,4)(3,5) (0, 5) (1, 5) (2, 5) (3, 5) (4,5), andso forth. Everysuchsquareitisapparent maybeconceived as asort ofslopedpyramid, formed by thesuccessive superposition ofsquarelayers, whichlayerspossessnotmerely II.simplesymmetry aboutadiagonal (such as isproperto amultiplication table),butthehighersymmetry (suchas existsin anaddition table),evinced in allthetermsinanylineofterms parallel tothediagonal transverse totheaxisofsymmetry beingalike-. Thusforn=5,thethreelayersorstagesinquestion willbeseentobe, thefirst- (0, 1) (0, 2) (0, 3) (0, 4) (0, 5) (0, 2) (0, 3) (0, 4) (0, 5) (1, 5) (0, 3) (0,4)(0, 5) (1, 5) (2, 5) (0, 4) (0, 5) (1, 5) (2, 5) (3, 5) (0, 5) (1, 5) (2, 5) (3, 5) (4,5); • A square arrangement having this kind of symmetry, namely, such asobtains in the so-called Pythagorean addition tableasdistinguished fromthatwhichobtainsin themultiplica­ tiontable,maybeuniversally calledPersymmetric. 57J oftwoAlgebraical Functions. 449 thesecond- andthethird-(1, 2) (1, 3) (1,4)(1, 3) (1,4) (2,4)(1,4) (2, 4) (3,4); (2, 3). Ingeneral, when 71.is odd, say 2p+1,thepyramid will end with lLsingle term(p, (p+1)),andwhen even, as2p,withasquareof fourterms, u»-2),(p-1)),«p-2),p) «p-2),p),«p-l),p). Eachstagemay beconsidered asconsisting ofthreeparts,adiagonal setof equaltermstransverse totheaxis ofsymmetry, andtwotriangular wings, one totheleft,andtheothertotherightofthisdiagonal; thetermsin each suchdiagonal fortherespective stageswillbe (0, n), (1,n-1),(2,(n-2))...(p, (p+1)), Pbeing~-1whenniseven,andn;1whennisodd. Ifwechangetheorderofthecoefficients in each of thetwogivenfunctions, itwill be seen thattheonly effect will betomaketheleftandrighttriangular wings to changeplaces,thediagonals in each stageremaining unaltered. Themode of forming thesetriangles isanoperation ofthemost simple and mechanical nature,too obvious to need to be furtherinsistedon here. Art. 8. Whenwe aredealing with two functions ofunequal degrees. nandn+e,we canstillform asquarematrixwiththecoefficients of the twosystems. ofeand11equations respectively, butthiswillnolongerbe symmetrical aboutadiagonal; itis obvious, however, thatif wetreatthe function ofthelowerdegree,asifitwere ofthesamedegreeastheother function, which we maydo by filling upthevacantplaces with terms affected withzero coefficients, thesymmetry will berecovered; anditis somewhat important (aswillappearhereafter) tocompare the values of the Bezoutian secondaries asobtained, first intheirsimplest form by treating each ofthetwofunctions ascomplete in itself,andsecondly, astheycome out, whenthatofthefunctions which is of thelowerdegreeis looked upon asa defective form of a function ofthesamedegreeastheother.Asingle example will suffice to makethenatureoftherelationbetween thetwosets ofresultsapparent. Take s.fx=axC+b:r;S+c:r;I+dx+e, cf>a:=0xC+0:r;S+'Yw+ax+e. 29 450 On aTheoryoftheSyzygetic Relation« [57 ThegeneralmethodofArt.7thengivesfortheBezoutian matrix 0,ary,ao, ae a-y,(Z).(~). ao,(~),(CO~d~)' ae,be,ce-e~,be o.ary, ao, ae a~,ao, ae, 0 ao,ae+bO,(co~~),ce-~ ae,be, ce- e~,de-eo. Again,adopting themethodofArt.6, weshouldobtainthematrix O,~, 0, e ~ ~ ~ 0 S.a.-IS.Cs~,.). C<-'" ~~ ~-~ ~-~ Henceit isapparent thatthesecondary Bezoutians obtained bythe symmetrizing method will differ from thoseobtained bytheunsymmetrical method by aconstant factora';and80ingeneralitmayreadilybeshown thatthesecondary Bezoutians, bytheuse ofthesymmetrizing method, will eachbecome affected withaconstant irrelevant factoraOl,where CI)isthe difference ofthedegrees ofthetwofunctions, andatheleadingcoefficient ofthehigherone ofthetwo.Whenaistakenunity,theBezoutian secondaries, asobtained byeithermethod, will ofcoursebeidentical. Art.9.Thereisanother method- ofobtaining thesimplified residues toanytwofunctions. UandVofthedegreesnandn+erespectively, which, •Originally given by myselfin theLondonandEdinburgh Philosophical Magazine, Il8long agoas1889 or 1840[po54above]; andsomeyearssubsequently inunconsciousness ofthat fact,reproduced by myfriendMr Cayley, towhom the methodissometimes erroneously ascribed, andwhoarrivedat thesameequations byanentirelydifferent circle of reasoning. 57J oftwoA1gebraical Functions. 451 although lesselegant,oughtnotto bepassedover insilence. Thismethod consistsinforming theidentical equations (ofwhichforgreaterbrevitythe right-hand members aresuppressed) V=&c. xV=&c. a;"-lV=&c. U=&c. afV=&c. eU=&c. af+lV==&c. wU=&c. afHV=&c. &c.=&c. xn-1U=&c. af+n-1V=&c. Ifweequatetheright-hand members of(e+2t)oftheaboveequations to zero,andtheneliminate dialytically theseveralpowers of a:fromx,,+e+'-l toXn-,H(bothinclusive), theresultofthisprocess will evidently be of(e+t) dimensions inrespectofthecoefficients inV,oftdimensions inrespect ofthecoefficients inUandofthedegreexn- ,inx;itwill also be of the form (A+Ex+...+Laf-1)U+(F+Gx+...+Qa;<+'-l) "J.~ andbyvirtueof Art.2,mustconsequently betherthsimplified residueto thesystemU, V. Art.10.Themostgeneral view of thesubjectofexpansion bythe method ofcontinued division, consists intreating theprocess as having reference solely to thetwosystems ofcoefficients inUandV,whichthem­ selvesareto beregarded inthelightofgenerating functions. Tocarryout thisconception, weoughttowrite U=ao+a,.y+a,yt+a,y3+&c.adin! V=bo+b1y+biyi+bq+&c.ad inj., andmightthensuppose theprocess of successive divisionappliedtoUand V,so as toobtainthesuccessive equations U-M1V +~=O, V-MiR..+R...=0, u;-MaR...+Ra=O, &c. &c., 29-2 452 On aTheoryoftheSyzygetic Relations [57 MlJM2,M"&c.beingeachseverally ofanydegreewhatever iny,andin generalthedegreeofyinM,beinganygivenarbitrary function 4>(I.)of". Thevalues of thecoefficients of the residues ~,~,R,...,or oftheseforms simplified bytherejection ofdetachable factors, become thenthedistinct objectoftheinquiry,and will, of course, dependonly upon thecoefficients inUandVandthenatureofthearbitrary continuous ordiscontinuous function 4>(I.),whichregulates thenumber ofstepsthrough whicheach successive process of division is to bepursued. Following outthisidea ina particular case,ifweagainreduceourtwoinitialfunctions totheforms previously employed, andwrite U=aox"+a1a;"-1+&c. V=box"+b.x"-l+&c.; andif,insteadofmaking, according tothemoreusualcourse of proceeding, thedivisions proceedfirstthrough onestepandeverafterthrough twosteps atatime,which is tantamount tomaking 4>1=1,4>(1+fA»=2, we push each division through onesteponlyatatime,and no more (so thatin fact4>(£) is always 1), we shallhave U-m1V+~=o, V-rn,.;x~+R,=0, tu-.«; ~+R,=o, ~-m.x~+R4=0, &C.&c., mlJ'1n:I,m.,&c.beingfunctions ofthecoefficients only of UandV;anditis notwithoutinteresttoobserve(which is capableof aneasydemonstration) thatthesimplified residues contained inRlJ~,&c.,foundaccording tothis mode of development, will be thesuccessive dialytic resultants obtained byeliminating the(I.-1)thhighestpowers of a;between the£first ofthe systemofannexed equations (supposed to beexpressed intermsofa;) U=O, V=O, xU=0, xV=0, aflU=0, aflV=0, &c.=&c a;fl-1U=0, x"-lV=O. 57J oftwoAlgebraical Functions. 453 Ifwe combine together 2i+1 oftheaboveequations, thehighestpower of xentering on theleft-hand side will be :c"+',and weshall be able to eliminate 2i of these factors, leavingxn-·thehighestpowerremaining unelimino.ted. Ifwetake2i,thatis ipairsoftheequations, thehighestpower of a:appear­ ing inanyofthemwill be,x,,+H,and weshallbeable toeliminate between themsoasstill to leave ,x"+'-l-II'-ll, thatisx"-iallbefore,thehighestpower ofxremaining uneliminated janditwill bereadilyseenthatsuch ofthe simplified residues corresponding tothismode of development asoccupythe odd places in theseries of such residues, will beidentical withthe successive simplified residues resulting fromtheordinary mode of developing ~under theform ofacontinued fraction. Art.11.Ithasbeen shown thatthesimplified residues off,xandcfJx resulting fromtheprocess of continued division are identical inpointof formwiththesecondary Bezoutians ofthesefunctions, butitremains to assignthenumerical relations between any such residueandthecorre­ sponding secondary. Todetermine thisnumerical relation, itwillof course be sufficient to compare themagnitude ofthecoefficient of anyone power of xintheone, withthatofthesamepower in theother;and forthispurposeI shall make choice of theleadingcoefficients in each. Inwhatfollows,andthroughout thispaper,itwill always be understood thatincalculating thedeterminant corresponding to anysquaretheproductofthetermssituated inthediagonal descending from left to rightwill always betakenwiththepositive sign, whichconvention will serve to determine thesign of all theotherproducts entering intosuchdeterminant. Nowadopting theumbralnotation for determinanta", we have, by virtueof a much more general theorem for compound determinants, thefollowing identical equation:- (ll-ta,cz,...am-I)X(alcz,a,;..am+!) alelsa•••.a.n-l alelsClaClm+l (ll-ta,a am-lam) (ll-tcz,am-lam+l) =\ala..aClaClm-lamXalelsa'''-lam+! _(ll-ta,a,...a"t-la",)X(ala, alR-l~l), alelsa•...am-lam+l alelsa"'-lCl". andconsequently (ll-ta,a,am-I)X(alcz,cz, alll-la",a"'+l) alelsa,«'''-1 alelsa 1%,'1-1ama"'+l ..BeeLondonandEdinburgh PhilOlophical Magazine, April1861[po242above]. 454 Ona Theory ofthe Syzygetic Relations [57 andconsequently when will have different algebraical signs, itbeingof course understood thatallthe quantities entering intothedeterminants thusumbraUy represented above are supposed to be real quantities. Thistheorem, translated into theordinary language ofdeterminants, maybestatedasfollows:-Begin with any square oftermswhethersymmetrical orotherwise, say ofr lines and r columns: letthis squarebebordered laterally andlongitudinally bythesamenumber l'of new quantities symmetrically disposed in respectto one of thediagonals, theterm common to thesuperadded line and column beingfilled up withanyquantity whatever; wethusobtainasquareof(r+1) lines and columns; letthis beagainbordered laterally andlongitudinally by(r+1)quantities symme­ tricallydisposed above thesamediagonal asthatlastselected, theplace in which thisnewline and columnmeetbeingalsofilled up withanyarbitrary quantity; andproceeding in thismanner, letthedeterminants corresponding­ tothesquarematrices thusformed be called Dr.Dr+lIDr-t'J... :this series of quantities will possess theproperty, thatnoterminitcanvanish withoutthetermsoneitherside ofthatsovanishing havingcontrary signs. Thusif we begin with a squareconsisting of onesingleterm,we maysuppose thatbyaccretions formedaftertheabove rule ithas been developed into thesquare(M) below written, and which of course may be indefinitely extended:- a,l,11t,p,8, t,b,n,q,t, m,n,o,r,u, (M) p,q,r,d,v, 8,t,u,v, e. HereDo,u; o;o; D4,D~willrepresent theprogression i.a,i,?It,p,8 l,a,m,pt,b, a,m n,q,ta,lI,t.b,n,q1,a,i,bi.b,n m,n,C,r,u- m, n, C,r m, n, C dp,q,r,d,v p,q,r, 8,t, u,v, e (II) 57J oftwoAl{jebraical Functions. 455 80if we use the matrix a,t,m,p,8, l'h,n,q,t, , m,n,c,r,'u, p,q,r,d,v, 8,t,u,v,e, thedeterminants DlJDi,D"D.,representing i,a,i,ln,p I,a, Ina,l l'i,n,ql'i,,a, nl'b,, m,n,c,rm,n,c p,q,r,d will possess the property in question jtheline and column l, bjl', bnot beingidentical, the first determinant Dorepresenting unitymustnot be included intheprogression. We shall have occasion to use thistheorem asapplicable tothecase of a matrixsymmetrical throughout, and we may termtheprogression (II),above written, aprogression of thesuccessive principal determinants aboutthe axisofsymmetry ofthesquarematrix(M), and so in general. Now it is obviousthatthe leading coefficients of thesuccessive Bezoutian secondaries arethesuccessive principal determinants abouttheaxis ofsymmetry of the Bezoutian squares; theywilltherefore havetheproperty which has been demonstrated of such progressions jto wit,ifthefirst ofthemvanishes, the second will have asigncontrary tothatof+1;ifthesecond vanishes, thethirdwill have a sign contrary tothatofthefirst, and so on. Art.12.NowletIxand¢Xbe any two algebraical functions of a;with theleading coefficients in each, for greatersimplicity, supposed positive: andinthecourse of developing j;undertheform of an improper continued fraction by the common process ofsuccessive division, let any twoconsecutive residues (theword residue being used in thesameconventional sense as employed throughout) be Ar+Br-1+Or-'/.+&c. B'r-1+O'r-I+D'r-I+&C. The residue nextfollowing, obtained byactuallyperforming thedivision and dulychanging thesign of the remainder, will be {(All)(.A.0')O'}B'-0-B'-BB'r-2+&c., 456 OnaTheoryofthe Syzygetic Relations [57 which is of theform;s{B'M-AC't}:rf-'l+&c" Thus the leading coefficientsin thecomplete unreduced residues will be A, B',;s{B'M-AC's), and when reduced by the expulsion of the allotrious factor will become A,B',UM-AC's,andconsequently, whenB'the leading coefficient of one of the simplified residues vanishes, theleading coefficients of theresidues immediately preceding and following thatone will have contrary signs. First,letI:r;andcf>:r;be of the samedegree. ABregardsthenumerical ratioof eachBezoutian secondary to thecorresponding simplified residue, ithasbeenalreadyobserved thatthereare always unitcoefficients in the latterof these, andthesameis obviously trueoftheformer ; hence if we calltheprogression of theleadingcoefficients of thesimplified residues s;s;R"R"&c., andthatoftheleadingcoefficients of theBezoutian secondaries B»e;B,.B"&c., we have Itmay be proved by actualtrialthatBI=R,andB;=R,.Moreover, sincethesignsareinvariable, anddo not depend upon the values of the coefficients, we may suppose Bs=0 (which may always be satisfied by real values of the quantities of whichBsis afunction); we shall also, therefore, haveR,=0, andconsequently B,hastheopposite sign to thatof BI,andR, theopposite sign to thatof RI,which is equal to BI:hence when BI=0, B,andR,areequal, and consequently are always equal;in likemannerwe can prove thatR,andB,havethesamesign when R ,nnd B,vanish,and consequently are always equal, and so on adlibitum,which proves thatthe series B I,BI,•••Bnisidentical with the series R,R"...Rn,andcon­ sequently thattheBezoutian secondaries areidentical in form, magnitude andalgebraical sign with the simplified residues. Secondly, when I:r;andcf>:r;arenotofthesamedegree, it has been shownthatthe secondaries formed from thenon-symmetrical matrixcorre­ sponding tothiscasewill bethesameasthose formed from thesymmetrical matrixcorresponding toI:r;and<I>:r;(where <I>:r;iscf>:r;treatedby aid of evanescent termsasofthesamedegreeasI:r;),withtheexception merely ofaconstant multiplier (a power of theleading coefficient of I:r;)being introduced intoeachsecondary. By aid of thisobservation, theproposition 57J oftwo Algebraical Functions. 457 established forthecaseof two functions of thesamedegreemay be readilyseen to be capableofbeingextended, fromthecaseoffand4J beingofequaldimensions inX,tothegeneralcase oftheirdimensions being anywhatever. Art. 13. Before closing thissection, it may be well to call attention to thenatureof therelationwhichconnects thesuccessive residues offxand 4Jxwiththesefunctions themselves, andwiththeimproper continued fractional form into which ~is supposed to be developed in theprocess of obtaining theseresidues. IfcfJxbe ofndegrees,andfxofn+edegreesinX,we shall have epa;1 1 1 1 fx=Ql-q2-q.- ...q;.' whereQlmaybe supposed tobe afunction ofXofthedegreee,and qhqllq",are alllinearfunctions of Xjthe total numberofthequotients Ql'q"q"being of course nwhentheprocess of continued division is supposed to becarriedoutuntilthelastresidueis zero. Upon thissupposi­ tionthelastbutoneresidueis aconstant, thepreceding one afunction ofx ofthefirst degree, theonepreceding thata function of xofthesecond degree,and so on. Letus calltheresidueofthedegree £ine,~,jit willreadilybe seen thatthesuccessive complete residues arranged in anascending orderwill be ~o,1;)-0'l»,1;)-0(q"-lq"-1),1;)-0(q"-'Jqfl-lq"-qn-ll-q,,),&c., being in theratios Again, we shall have in general AJ-L,4J=1;)-" (15) A,beinganintegral function of xofthedegreen-£-1,andL,anintegral function of xofthedegree(n+e)-£-1janditiseasyto seethatthe successive convergents tothecontinued fraction _1__1_.~&c Ql-q.-q.-. havetheirrespective numerators anddenominators identical with those of thefractions 458 On a Theory oftheS1Jzygetic Relations [57 Adopting thelanguage which I have frequently employed elsewhere, I call ~,asyzygetic function, or more briefly a conjunctive offandep,and A,andL,may betermedthesyzygetic factors to ~,soconsidered. Ifwe divide each termof theequation (15) bytheallotrious factor(M),we have ~'f-L'''''=RMM'f'" whereR,istheethsimplified residueto(f,ep);and if we call ~=T"and L, b . h .M=t.,so as to 0tamt eequation T,f-t,ep=R" (16) we seethat\thefractionformed by thecomponent factors to anysimplifiedt, residueof(f,ep),willbeidentical in value(although nolongerin itsseparate terms)withone ofthecorresponding convergents toj,exhibited underthe form of an improper continued fraction. I shall in thenextsectionshow how, not only thesuccessive simplified residues, butalsothecomponent syzygetic factors of each of them,andconsequently thesuccessive con­ vergents, may beexpressed intermsof the roots of thetwogivenfunctions. Sincethepreceding section was composed thevaluable memoirofthe lamented Jacobi,entitled"DeEliminatione VariabiliseduabusEquationibus Algebraicis," Crelle,Vol.XVI.,has fallen undermynotice. Thatmemoiris restricted totheconsideration of twoequations ofthesame degree, and the principal resultsinthissectionasregardstheBezoutic squareandthe allotrious factorsapplicable tothatcase will be found contained therein. Themode of treatment however is sufficiently dissimilar tojustifythis sectionbeingpreserved unaltered underitsoriginal form. SECTION II. On the general solutionin terms ofthe roots, ofany two given algebraical functions ofe,ofthe syzygetic equation, whichconnects them with a third function, whosedegreein xisgiven, but uihoseformistQbedetermined. Art. 14. Letfand4>be twogivenfunctions inxofthedegreesmand nrespectively ine,and forthesake ofgreatersimplicity letthecoefficients ofthehighest power of xinfandepbe each takenunity,and letitbe proposed to solve thesyzygetic equation T./-t,4>+~,=0, (17) 57J oftwoAl{jebraical Functions. 459 where ~,isgivenonly inthenumberof itsdimensions ine,which I suppose tobe,;buttheforms of or..t..~,areall to be determined intermsof hi.~...h".theroots ofIand1'J1.1'J,•••1'Jntheroots of 1/>. I shall begin with finding ~,;and before givinga moregeneralrepresen­ tation of~..I propose now to demonstrate thatwe may make ~,=I{P'1"'1,...'1,X(x-h'}.)(x-h'1J...(x-h'})}. (18) wherePq"q•... '1,is used to denote J(hq'+1-1'J1)(h'l'+1-1'J,)(h'l'+1-1'Jn) x(h'1.-+t-1'J1)(h'l,+'l-1'J,)(h'l'H-1'Jn) t~.(h~:~.~.":)(~.~.~~. ~.).r-..":)Ro..'"....h,). x(It'l,,.-1'J1)(h'l••-1'J,)••.•..(h'l".-1'Jn) R(h'}l'h'1•...h'1)denoting anyrational symmetrical function whatever of thequantities preceded bythesymbolR.andql>q2••.q"q'+l...qmbeingany permutation ofthemindices 1, 2 ...'11'. Suppose1=0andI/>=0,thenxis equal to one of theseries of roots hi.i,...hm. andalsoto one of theseries of roots 1'J1,1'J2...''In· Suppose thenthat x=h.=1'J.., and consider any termof~,. Ifin any such term Gl:is found in the series 'l»q2...q..then (x-h'l)(x-It'l.)...(x-1t'1)=O. Butif not,thenxmustbe found in thecomplementary series h'l'+I' h'l,+!•••i,.... andconsequently P'l"q....q,willcontaina factor h.-1'J..andP'l,,'l....'1,=0; in every case therefore P'1,,'l....q,x(x-h'l)(x-Itq,)...(x-It'l)=O. Therefore ~,as expressed in equation (18) is a syzygetic function ofI and1/>;and we have found a function ofthe,thdegreeinx,and of course expressible bycalculating thesymmetric functions as a function only of xand of the coefficients oifand1/>.which will satisfytheequation or.!-t,1/>+~,=o. 460 On aTheoryoftheSyzygetic Relations [57 Itwill beremembered thatbyvirtueof Art. 2 we know apriorithatall thevalues of~.satisfying thisequation areidentical, saveasto anallotrious factor, which is a function only ofthecoefficients in fandcp. Itisclearthatwe may interchange thehand TJ,m and n,andthus anotherrepresentation of a value of~,satisfying theequation (17)willbe 9,=IR('I]q"TJq,...TJq)x (TJq,+1-hI)(TJq,+I-h2)...(TJq,+1-hm) (TJq,H-hi)(TJq,H-112)•••(TJq'+<J-hm) (h )(1.)(h)(a:-TJq)(a:-TJq,)...(a:-TJqJTJq,+3-I T/q'+3-I"J•••TJq,+3-m Art. 15.Ifwe employ in generalthecondensed notation [l,m,np], A,,.,., II todenotetheproduct ofthedifferences resulting fromthesubtraction of each ofthequantities A,,.,....vinthelower line from all of thoseinthe upperlinel,m,n...p,thetwo values above givenfor~,may be written undertherespective forms ~ [hq'+l' hq'+3'"hq",]...R (hq"hq,•••hq) . (a:-hqJ(a:-hq,).••(a:-hq), 1]1. 1]2...1].. [1]e,1]t...TJE]and ~R(TJe,,1]e,·"TJe)· h'+!h'+2h"(a:-1]el)(a:- TJe,)..·(a:-'1e)' 1, 2:••• m in each of which equations disjunctively and in some orderofrelationeach witheach qllq2'qs...q".=1, 2, 3 m, and tlJt2't3'"t..=1,2,3 n. Thesetwo forms are only thetwoextremities of a scale of forms allequally welladapted toexpres.'3 ~,;for letvandIIbe any two integers sotakenas tosatisfytheequation v+ v= t, andletR(~; ~).wherethedotsdenoteanyquantities whatever, be used todenotearationalfunction whichremains unaltered in value when any two ofthequantities undereitherofthetwobarsaremutually interchanged, thenwe maywrite (19) 57J oftwoAlgebraical Functions. 461 Forif, as above, we suppose a:=h.="1..,anytermof~,in which ql'q2'"q" comprise amongthema,or in which ~I'E2'" ~ocomprise amongthem Cel, willvanishbyvirtueofthefactors (x-hq)(x-kg,)...(x-kg,,)x(x-"1£,)(x-"1£1)•••(x-"1£); butifneitheranor fA)is socomprised, thenamustbe one of theterms inthecomplementary seriesqw+I'qw+"...qm,andCelone ofthetermsinthe complementary series ~0+\JEO+l...En'andtherefore one ofthequantities hg~,hg~...hqwillequalone ofthequantities "1.•"1...."1.,andcon-_,I_" '" '0+1'.+2 'II sequently thetermof~,inquestion willvanishbyvirtueofthefactor [hg.i,...hg]HI w+1 '"vanishing. Ineithercasetherefore everytermincluded "1£.+1'"1to+2•••"1tll withinthesignofsummation vanishes whenx=h.="1..,thatis,whenever fx=0and~=O.Hence ~"asgivenbyequation (19), will satisfy thesyzygetic equationTJ-t,4J+~,=0 for all valuesoft'andvwhich make tI+V=£,andfor allsymmetrical forms of thefunction denoted bythe symbolR(~; ~). Art.16. Ishallnowproceed to show how to assignthearbitrary function whose form is denoted bythissymbol in suchamanneras tomake ~.becomeidentical withasimplified residuetofand4J.Tothisend Itake forR(hg"hq,...hq,, ;"1£""1EI•.."1t.)thevalue [hg"-;...hg,,] "1.,"1.•••"1.R- 'I"'. . (20)-[hq"-:•••kqJ["1t""1tl "1t] , hqv+I'hqt>+t...hg:x"1t.+1'"1£.+1"1t: 462 OnaTheoryoftileSyzygetic Relations [57 Onreducing thefractions contained withinthesign of summation toa common denominator, ~,willtaketheformD~a'whereDwill bethe product ofthetm(m-1) differences of EIJE,...Emsubtracted each from each, andathecorresponding product ofthedifferences inter se of HI,H~'"Hn.Hence,unlessthesum inquestion is anintegral function oftheE'sandH'sit will become infinitewhen any two of theEseries, or anytwo oftheHseriesofquantities are made equal. Suppose nowE.=Et; thetermsin (22) which contain EI-E~inthedenominator willevidently groupthemselves intopairsoftherespective forms, thesum ofthispairoftermswill be of theform {[E~J[-EIJ}'!:-~_ Htt'_l!tt-""Ht•xHt.+1'Ht.H...Htn+QE~-E I. .E~ -..--- - , [EqfJ+t'EqfJ+~'"E'1mJ whereQ,itmay be observed, does notcontain HI -H~,80that~remains finite when HI=H~. The above pairoftermstogether make up a sum of theform p1~(EI,E.)VE~-~(E2,E1)VEl Q1:1--E~ vElx"/rE,'2 which, as thenumerator ofthethirdfactorvanishes whenEI=E "remains finite on thatsupposition. Hencethewhole sum of termsin (22) which is 57J oftwoAlgebraical Functions. 463 made up of such pairsofterms,andofothertermsin which EI-E2does notenter,remains finite when EI-E.=0,andtherefore generally when D=0,andsimilarly whenHI-H,=0, andtherefore also when a=0 ; hencetheexpression for1;)-,in (22)isanintegral function oftheEandH seriesofquantities, aswasto be proved. Art. 17. Letus now proceed to determine thedimensions ofthecoeffi­ cientof:If,thehighestpower of xinthisvalueof1;)-"whensupposed tobe expressed undertheform of an integralfunction (asithasbeen proved to be capableofbeingexpressed) oflill~...lim;1'Jh1'J•...1'Jn;e, Thiscoefficient is thesum offractions thenumerators of each of which consist of two factors, which are respectively ofvxvandof (m -v)x(n-v) dimensions inrespectofthetwo sets of roots takenconjointly, andthe denominators of two factors respectively ofv(m-v)andv(n-v)dimen­ sions inrespectofthesame. Consequently, theexponent of thetotaldimensions ofthecoefficient in question =vv+(m -v)(n-v)-v(m -v)-v(n-v) =(m -v-v)x(n-v-v) =(m-t)(n-t), andthusis seen to dependonly onthedegree tinxof1;)-"and not upon the mode ofpartitioning tintotwopartsvandv,for thepurposeofrepresenting ~"by means of formula (19). Art.18. Ishallnowdemonstrate thatevery form in thisscale (to a numerical factorpres)isidentical with asimplified residuetof,rp,ofthe samedegree tinx.Any such simplified residueis, like 1;)-"asyzygetic function, or to use a brieferform of speech a conjunctive off,rp;and if we agreetounderstand bythe"weight" of anyfunction ofthecoefficients of fandepitsjointdimensions inrespectoftheroots offandrpcombined, Ishallprove,-first, thatanysimplified residueoffandrpof agivendegree inxisthatconjunctive, whoseweightinrespectoftheroots offandrp islessthantheweightof anyothersuchconjunctive; and second, that~" asdetermined above (in equation 22), is of thesameweightasthesimplified residue, and can therefore only differ from itby some numerical factor. Forthepurpose ofcomparison ofweights, itwill of course be sufficient to confine our attention tothecoefficients of thehighestpower in a:(or anyother,thesame for each) of theforms whose weightsare to be compared. Supposefto be of mdimensions, andrpto be of ndimensions IIIx; andletm=n+e. 464 OnaTheoryoftheSyzygetic Relations [57 Suppose Af+Lrp=Ax'+Br-I+ '" +K, (23) A=Xo.x'1+>--l'x'l-I+ ... + }"q, L=lo.x'1+·+~.x'1+o-1+ ... +lq+., thenumberof homogeneous equations to besatisfied bytheq+1quantities >'0.>'1"'~'andtheq+e+1quantities lo,lI'"lq+ewillbem+q-t,and therefore q+1 andq+e+1takentogether mustbe not less thanm+q-t+I, thatis2q+e+2mustbe notlessthanq+m-t+I,thatisqnotlessthan m-t-e-1;and ifthisinequality besatisfied2q+e+2-(q+m-t+1)+I, thatisq+t+e-m+2 will be thenumberofarbitrary constants eutering intothesolutionofequation (23). Ifqbegreaterthan(n-1),letq=(n-1)+t;andlet (A)=Xox'H+>--Ix"-2+ ...+:>""-1' (L)=lOx"+·-l+~x"+e--1+...+le+n--I; andlet(A),(L)be sotakenas tosatisfytheequation (A)f+(L) rp=Ax'+Bx'-l+...+K; and make E=(A)+(f+gx+'" +hxt-I)rp, X=(L)-(f+gx+ ... +hxt-I)f, f,g...hbeingarbitrary constants ithen Ef+Xrp=(A)f+(L)rp=Ax'+Bx'-I+...+K. Nowthetotalnumberofarbitrary constants inthesystem(A) and(L) will ben-1+t+e-m+2,thatist+1; hence thetotaluumberofarbitrary constants inEandXwill be t+1+t,thatisq-n+t+2, which isequalto q+t+e-m+2,thenumberofarbitrary constants inthemostgeneralvalues of A and L.Hence{A=2,L=Xlisthegeneralsolutionoftheequation Af+Lq,=Ax'+Bx'-l+...+K;andconsequently themostgeneralform ofAx'+Bx·-I+ +K,which is evidently independent ofthe(t)arbitrary quantitiesf,gh,willcontainthesamenumber ofarbitrary constants asenterintothesystem (A) and (L),thatist+1. Art. 19. Letus nowbeginwiththecase ofgreatersimplicity when 'In=n,thatise=0;andletusreverttothesystem of equations marked(10) inSection1.,in which UandVare to be replaced byfandrp. First,lett=n- I,thent+I,thenumberofarbitrary quantities inthe conjunctive, isn. Fromthe system of equations (10) we have, for all values of PI'P2'PI...p", . (PIQO+PIQl++p"Qn-I)f -(PIPo+PaPI++p"Pn--I)rp =(PIKI+PIIKI++p"r&-IE1)x"-I+&c., 57J oftwoAlgebraical Functions. 465 andconsequently themostgeneralvalue of ~n-lintheequation Tn-I!-t..-lt/>+~'H=0, where ~n-l=Ax"-l+Bxn-I+...+L, will beobtained bymaking Tn-I=PIQO+PIQl+ ... +P..Qn-l, tn-I= -plPO-PsPl'"-PnPn-l, whichsolutioncontains n,thatisthepropernumberofarbitrary constants. Again,if£=n- 2, £+1=n-1, which will therefore bethenumber ofarbitrary constants inthemostgeneralvalueof~'Hintheequation T-J-tn-It/>+~ft-2=0. Thismostgeneralvalue of ~'Histherefore found by making TtH=p\QO+P'IQl++P'..Q..-l> tn-I= -P'lPO-p'aPl-p'nPn-l' whereP'l'P'I'"p'nare nolongerentirely independent, butsubjecttothe equation p'lKl+P'IIKI+ ...+p'nn-lK1=0, soasto leave(n-1)constants arbitrary. Wethusobtain ~n-'l=(p'lKa+P'.lKI+ ... +p'"n-lKa),x"-'l +&C.Inlike manner, and forthesame reasons, themostgeneralvalues of ~n-sinthe equation Tn-sf-tn-at/>+~n-I=0, will be found by making Tn-s=P"lQo+P".Ql+ ... +p"nQn-l> tn-.= -P"lPo-P"2PI."-p"nPn-l> wherep"ltP"2'"p"naresubjecttosatisfying thetwoequations P"lKl+P"21Kl+ +p"nn-lKl=0, P"lKI+p"21Ka+ +p"nn-lK2=0, so as to leave (n-2)constants arbitrary; and wethusobtain ~n-s=(P"lKa+p"'lKa+...+p"nn-1Ka)x·o-a+&c., and so on, thenumber ofindependent arbitrary constants in~decreasing (asitought)eachtimeby oneunitasthedegree of ~descends, untilfinally, ifTo!-tot/>+~o=0,~obeingaconstant, thegeneralvalue for ~oisfound by making s,TO=(PI)Qo+(P.)Ql+...+(Pn)Qn-l' to=-(PI)Po-(Pa)PI-...-(Pn)Pn-1, 30 466 OnaTheoryoftheSyzygetic Relations [57 where(PI),«(JJ)••.(p,,)aresubjecttosatisfythe(n-1)equations (PI)KI+&c.=0, (PI)K,+&c.=0, which gives(PI)K"-1+&c.=0, ~o=K"(Ph.+IK"(p),+...+n-IK"(P),.. Nowevidently thelowestweightinrespecttotheroots ofUandVthat canbegivento(PIKI+PUKI+...+P"n-IKI)x"-1 +&c.,whenthemultipliers PI'p,...p"areabsolutely independent, is found by taking PI=1,p,=0,ps=0...p"=0, whichmakestheweightoftheleadingcoefficient in ~n-"thesameasthat of KI,thatis 1. Again, when one equation, P'IKI+p'uKI+ ... +p'....-IKI=0, existsbetween the(p)'s,thelowestweightwill be found by making p\=IK I,p',=-K I,p's=O, P'4=O...P'..=0, which makes theweightoftheleadingcoefficient in ~n-,dependon IKIK,-KIIK" which is of theweight1+3,thatis 4, inrespectoftherootsoffandl/J. Similarly, ~n-swillpaveitslowestweightwhen its leadingcoefficient isthedeterminant K"K"K,, IKI,IK"IK, ,KI,,K"sK, theweightof which is 1 +3+5=9;and finally, thelowestweighted value of~oisthedeterminant represented bythecomplete Bezoutian square;the weightingeneralof~n--ibeing1+3+...+(2i - 1), thatis~"',orwhich isthesamethingotherwise expressed, theweightoftheleadingcoefficient ofthelowest-weighted conjunctive offandy/>ofthedegree,inxis (n-,)(m-,)..Itwill of course have been seen in theforegoing demon­ stration,thattheweightof,x,[whichmeans ~(a,.b,-a,br),a,.,a,beingthe coefficients of x"-r,x..-.inf,andbr,b,of thesameinl/J]hasbeencorrectly takento ber+8inrespectoftherootsoffandl/Jconjoined. *nand m are supposed equal and 1=n-i. 57J oftwoAlgebraical Functions. 467 Art. 20.Ifnow we proceed in like mannerwith the generalcaseof m=n+e,itmaybeshown, in precisely thesamewayasinthepreceding article,thatthemostgeneralvalue of any conjunctive offandepwillbea linearfunction of efunctions, Il!'+axa;n-l+asll!'--2+ +a", Il!'H+axil!'+a,x"-I+ +a"x, Il!'H+axx"+l+u,x"+ +a,.w, a;m-I+a1xm-,+&c. and of the nfunctions, K1a;n-1+K21l!'--2+ + K", lK1x"-1+IK2Xf1-2+ + lK", &c. &c. n-1K1a;n-1+"_IK 2x"--2+ ... + 'B-IK", andthatconsequently, if the degree of such conjunctive ina:be(n-i), itwillbeofthelowestweightwhen it is alinearfunction of the entire eupperset of functions, and iof the lower set;and consequently, the coefficient of the highest power of xin such conjunctive willbe the determinant K,K, K,+e lKslK, IKiH 2K,$, 2K'H I, ~a'_I,a•...a,+e-I 1, axa'_1>a,u.H- , 1a'-2'a,a,+e-, 1 u, theweightofwhich is evidently thatof K1xIK,X$,X'-lK,x(a,)", thatis 1+3+5+ +(2i-1)+ei, thatisi2+ei,ori(e+i), which is(n-t)(m-t) ift=n-i. 30-2 468 On a Theory oftheSyzygetic Relations [57 Hencethe.weightoftheleadingcoefficient in thelowest-weighted conjunctive of/ and 4>ofthedegree £inIXis(m-t)(n-t),mbeingthe degreeof/and nof4>. From this we infer thatanyconjunctive offand4>of the degree L, of which the leadingcoefficient is of the'weight(m-I.)(n-I.),allthe coefficients being of course understood to beintegral functions of theroots of / and 4>,must, to anumerical factorpres,beequivalent to anyother of thesameweight jandfurthermore, any supposed function of IXof thetth degree which possesses theproperty characteristic ofaconjunctive of vanish­ ing whenfand4>vanishsimultaneously, butof which the weightofthe leadingcoefficientwould be lessthan(m-£)(n-I.),must beamerenugatory form and have all itstermsidenticaUy zero", Art.21.We have previously shown, Art. 16,thatI;},asdefined by equation (21),is anintegral function oftheroots / and 4>,and vanishes whenfand4>vanish. Moreover, itsweightintherootshasbeen proved to be(m-t)(n-I),andconsequently, if by way of distinguishing theseveral forms of I;}.we name thatone where £intheequation above cited is supposed to be divided into two parts, vandv,I;}",o,we have for all values of vandu, suchthatv+vis notgreaterthann,I;}",otoaconstant numerical factorpree identical with the (v+v)th simplified residueto(f,4»,sothatthe form of I;}",odepends only upon thevalue ofv+v. Art.22.Itmustbe well borne in mind thatthispermanency ofthe value of 1;}",'-11fordifferent values of vhasonly been established for the case where £canbe the degree of aresidueto / and ep,thatis tosay,when t islessthanthelesser of the two indices '11landn.When £does not satisfy thiscondition of inequality, the theorem ceases to be true. Itis clearthat whenm=nandv+v=m=n,I;}",o,whichalwaysremainsaconjunctive off and4>,canonly be anumerical linearfunction of / and 4>;and I have ascertained whenm=nongivingtovandIItherespective values succes­ sively (0, n),(1,n-I),(2,(n-2»...(II"0)that (n-1)(n-2) I;}o,n'=/jI;}I,n-l=(n-1)/+epj1;}2,,\-2=-~1.2- / +(n-1)ep;..• I;}n-I,l=f+(n-1)epjI;}..,o=cp. Thus, by way ofa simple example, let /=a;2+a.x+b=(IX-hi)(IX-h 2) , 4>=a;2+/XIX+f3=(IX-kl)(IX-k2) , *And more generally itadmitsof being demonstrated by precisely the same courseof reasoning, thatthenumberofarbitrary parameters in aconjunctive of the degree I,andofthe weight(m-I)(n-I)+fin the roots, cannot(abstraction beingsupposed to be made oCan arbitrary numerical multiplier) exceedthenumber f. 57J thatisoftwo Algebraical Functions. _sa;-hd_!_(X-kl)(hl:-kl)(hi -k,))} - hI- ~lkl-k,-(x-~)(h,.-~)(h..-kl) x-h=~h,._~(h,.-~)a;+[(lot+~)h,-(h,.h.+k1k.)]) =(x-hI)X+(x-hI)X-(kl+k,)a;+(h,.~+~k,) ={xl-(h,.+hi)a:+hlh,}+{xl-(kl+k,)x+k1k,} =(xl+ax+b)+(xI+a.x+ f3) =1+cfJ;469 so we find also ~"o=cfJ. Art.23.Theexpression ~".•, which isuniversally aconjunctive ofI andcfJ,continues algebraically interpretable so longastI+IIhas any value intermediate between 0 and m+n;whentI+II=0 wemustof course have V=0and11=0,and~o.obecomes the resultant ofIandcfJ;whentI+lI=m+n wemustalso have theuniquesolution v=m and II=n,and~m.nbecomes necessarilyIx<1',which we thusseestandsin asortofantithetical relation totheresultant ofIandcfJ,say(f,¢).Nor isitwithout interesttoremark thatfxcfJ=0impliesthata factor ofIor else of cfJiszero;and(f,cfJ)=0 impliesthatif a factor of theone ofthefunctions is zero, so also is a factor oftheother,thatisthata factor of each or of neitheris zero.As£increases from 0 to nordecreases fromm+ntom-I,thenumberofsolutions ofthe equation v+II=£intheone case, and thenumber ofadmissible solutions oftheequation tI+v=£intheothercase, which issubjecttothecondition thatIImustnotexceedn,continues toincrease by aunitateachstep; therebeingthusn+1different forms ~",.when tI+II=n,andthesame numberwhenv+II=m- 1.Forall values of £intermediate betweennand (m- 1)(bothtakenexclusively) itisveryremarkable that~".• willvanish, asIproceedtodemonstrate. 470 OnaTheoryojtheSyzygetic Relations [57 Art.24.Theweightofthecoefficient of thehighestpower of ~~." (v+vbeing equal to £)is(m-£)(n-£),andconsequently, when£isgreater thann,and less thanm,~~."wouldcontainfractional functions of theroots offand4>,iftherewere initapowerX',but~~."has been proved to be alwaysanintegerfunction oftheroots.Hencethecoefficient of afwill be zero,andso moregenerally thefirst power of Xin~~.",of which the coefficient is not zero, will be af-",subjecttothecondition (sinceevidently theweight of the several coefficients goes on increasing byunitsasthedegree of the termsinxdecreases by the same) that 0)benot lessthan(m-£)(£-n); letthen 0)=(m-£)(t-n),~~."becomes of theformAaf-"+Baf--1+&c., whereAis of zero dimensions; butthisis impossible if £-0)<n,forthen Aaf-"+&c. isaconjunctive of weight lower thanthelowest-weighted simplified residueofthedegree t-0).Hence 0)isnotgreaterthant-n, thatis(m-£)(,-n)is not greaterthan£-n,thatism-£cannotbegreater than1,thatis£whenintermediate between ·mandncannotbe lessthan m-1, otherwise ~~."will vanish identically. Moreover, when t=m-1, 0)=£-n,and,-0)=n,and accordingly ~~.m-l-~is not merely, aswemight know,apriorian algebraieal, butmore simply Nonumerical multiple ofepfor all values of v.The same isof course truealso,mbeinggreaterthann,for every form ~~,n-~,sincethisis always aconjunctive offand4>,ofwhich the former is of adegreehigherthanthe~inquestion, sothatthemultiplier offinthisconjunctive must bezero·. Art. 25. '1'0enterintoafurtheror more detailed examination ofthe valuesassumed by~v."for the most generalvalues of m, n, £.would be to transcend thelimitsI have proposed to myself in drawing upthepresent memoir. Whatwe have established is,thatto every form of ~,.•_~apper­ tainingtoavalue of ,between 0 andn,thereisasort ofconjugate form for which £liesbetween m+nandm;thatfor,=m-1ort;:::n,~v"-1Ibecomes anumerical multiplier of4>;andthatwhen £lies in the intermediate region betweennandm-1, ~~,,_~vanishes for all values of 11.Ipauseonly for amoment toputtogether forthepurpose of comparison the forms corre­ sponding to£and tom+n-c.By Art. 16, making £=V+u, ~.=~(x-hq)(x-hq,)•••(x-hq")x(:c-1/t)(:c-1/t,)'" (:c-1/t) [hq,.-;•••hq"]X[hqH1,<:hq_J 1/.,'1'...'1. '1"'It '1'x..0, Or 0!+1 r+I o.• [hq,• hq,•.. hq"J [1/E''It...'ItJ hqo+1,hqo+'J'..hq_x'1:•'It'•..'It" 0"+1 "+1 • •Itthusappearsthatif theindicesmandndo notdi1ferbyatleast8units, ~will have an actualquantitative existence forallvalues of •between 0 and m+n;orinotherwords,the failureinthequantitative existenoeof the forma ~.onlybeginstoshowitselfwhenthiBditJerenoe is8;thuaifm=n+8,~. exista,and~ft+texists,but~.+l=O. 57] oftwoAlgebraical Functions. 471 Theconjugate form for which ,'=m+n-,andm-v,n-JI,t1JItakethe places of n,JIand(m-v)(n-JI),will begotbytaking ~.,=~(e-hqIH-IHz-hq~)•••(z-hq",)x(z-1].Hz-1].) •.•(z-'7.)'.+1 '.~ '. X[hq"hq•'"hqwJ['7"'7....1].J'- Xn n "II' hqIH-l'hq~...i; 1]t.+1'1]t.+2···'7t._ whichitwill beperceived areidentical, termfor term, in thefractional constant factor,anddiffer only in thelinearfunctions ofe,which in ~.and in~.,arecomplementary to oneanother. Ourproperbusiness isonly with those forms for which,<n. Art. 26. Itwillpresently be seen to benecessary to ascertain the numerical relations between ~o.,and~•.owhen'<n,andthisnaturally brings underournoticetheinquiryintothenumerical relations whichexistbetween theentireseries of forms ~""_,,foragivenvalue of "corresponding to all values of vbetween 0and,inclusive. Inorderto avoid a somewhat oppressive complication of symbols, I shall takeaparticular numerical example, thatism=7,n=6,,=4,and compare thevalues of ~O.4; ~1.a; ~S.2; ~3.1; ~4.0'all of which we know tobeidentical [toanumerical factorpres]with one anotherand with thesecond simplified residuetofand4>,thatbeingofthefourthdegreeinz;ourobjectinthe subjoined investigation istodetermine thenumerical ratiosoftheseseveral forms of ~to oneanother. First,letv=0,JI=4.Theleadingcoefficient ~'.4is r L'75"1eJ I~hah,h4h5hGh., l::~:"Ia1]J which we know d,priori(itshould be observed) to be essentially anintegral function of thehandthe"Isystem. Inthis,thetermcontaining '7"willbe evidently (A) the"Isystemto which thelattersummation relatesbeingnowreduced to consist of '71>"la,1]1''74'1]5'Inthisexpression, again, thecoefficient of 1]5a isevidently 1.Hence,therefore, theleadingcoefficient in ~'.4contains the term'7"'7l. 472 OnaTheoryoftheSyzygetic Relations [57 (B)Secondly, let11=1,11=3.Theleadingcoefficient in ~IIIbecomes I[~"II'7IJx[~~::h5h8~1 l~hahi"lh8~Jxl"l4'15'70 . ~ '71"11'71 Inthis,thefactoraffecting "1lwill be ["II'71"11x'7~"II1 Ihi ~hah~hlh8h7 [hahah4h.h.~Jx[-'7~'71J' ~ "11"11'7. "18being now understood to beeliminated out ofthe'7systemincluded within theabovesummation. Again, in thislattersumthefactoraffecting '7e' willbe Il~'7I"1IJx[~:hah.h.hehJ lh,hlh4hlh.~Ix["14J' hi _"11"11'71 '71and'76being now both eliminated out of the '7system. Thislastsum can of course only represent anumerical quantity. So in like manner, again, if 11=2,II=2,thecoefficient of '781'711in~... willbesimilarly reducible totheform I[~:~:Jx[~Z:h8h8~J (C)rhah4h5h8~J ["I1"I4J• L~ha x~ha So,again,when 11=3,II=1,thecoefficient of "18'''151in~I.Iwill be (D) (E)["11'71"11"14J Ih5h.h7_ [h1heh7J' ~h..h,h~ out of all which sums it is to be remembered that"11and'78aresupposed excluded fromappearing. Alltheseseveral coefficients beingnumbers indisguise, we may determine thembygivingany values atpleasure tothetermsin thehand "1system. 57J oftwo Algebraical Functions. 473 Letnow'I'll=~,"11="",'II=ha,"14=h4,thenin (B) it willreadily be seen thatall the terms included within the sign of summation vanishidentically, exceptthefollowing, namely,- [i:"11"111 x[~hahahohoh,J [~h..hahoh.h,J x["14J' li, ."11'11'11 [i:"11"14] x[~""h4hohoh,J "[~h.zh4hohoh, x["11J' hi "11"11"14 [~:'71"14lx[~:hoh4hohohJ [~-hah4hohoh7Jx"11-J' i; 711'11"14 [~'11"141x[~hsh4hohoh,1 ""h8h4hohoh7J x[r;hI' ~ L"11"1s"14J Ineach of these expressions thefirst factor of thenumerator isidentical in value (by reason of theequations h}="11>""="11'hs="18'h;="14)with (-)Ixthe second factor of the denominator, and the second factor of the numerator with(-'txthefirst factor of the denominator jhence the coefficient of "10'"1osin~I.Iis - 4. Inlikemannertheonly effective terms of ~2.2will be [t~:x[~Z:hohoh,J [~~Jx[~i:hohoh,J ['''-thahohah,Jxl"18"14J'[hoh4hohoh,JX['11"1IJ-' hoh4 "11"11 ~ha 'Is"14 [~tJx[~~hohohJ [~:~:Jx[Z~Z:hohoh,J [~hohoho~Jx["12"141'[h2h4hohoh7J xl-"11"1IJ-, hah4 711"1s hlh8 "11"14 ~~Jx[~Z:hohoh,J [~i:Jx[~~h.hoh,J [~h4hohoh7Jx["11"14],lhah4hShoh,X['11"1.• hah.. "12'71 hlh. "11"14 Anyotherterm will necessarily containinthenumerator a factor, whose symbolical representation willcontainone of the quantities "11'"12''I.,"14'inthe upperline,andone of the quantities b«,"",ha,h4,havingthesamesubscript 474 OnaTheoryojthe Syzygetic Relations [57 index,inthelower line, and which will therefore vanish; thenumberof effective termsbeingevidently thenumber of ways in which four things can becombined 2and2together, andthevalue of each termisevidently (_)21(-1)21I,sothattheentirevalue of thecoefficient of 7]88"lS 'in~v is+6. Precisely inthesamemanner, weshallfindth~ttheleadingcoefficient in~8,1willcontaintheterm-4-rJl7]s3,the(-1)resulting fromtheoperation (-1)13(_)34,andin~~,otheterm+7]817]&"the+1resulting fromtheoperation (_1)43. Henceitappearsthat ~o,~i ~1,3i ~i,2; ~l,li ~4,oare to one another intheratiosof1;- 4i6j -4;1;and so in generalforanyvaluesof m, n,,(,beinglesethanmandlessthann)itwill be found that ~o," ~1,'-1l ~2,'-2 •••~.,o will be in theratiosofthenumbers ,-1 ,-1,-21;(_1)"'-1, i(-1)'("'-1) L-2-;(-1)3("'-3)'-2---3-'...i(_I)<lm-.I. Art.27.Themethod employed in thepreceding investigation will enableus to affix thepropersignandnumerical factor to ~o"or~"o,orin generalto~",'_'"inorderthatitmayrepresent theBezoutian secondary ofthedegree,inIX.Thislatterhas been already identified withthe simplified residueobtained byexpanding,j;undertheform of an improper continued fraction. Forthispurpose, itwill be sufficient tocompare a. singletermof anysuch~withthecorresponding one intheSymmorphic Bezoutian secondary. Letus firstsupposethatm=n,IandcI>beingof thesame degree. A glanceattheform oftheBezoutian squarewill show thatif we form theBezoutian secondary ofthedegree(n-i)inIX,the coefficient of its leadingtermwillcontaintheterm(-)(i-l)~(0,i)ii(0,i) asusualdenoting theproduct ofthecoefficient of a;"inIbythecoefficient ofil!'-iincI>,lesstheproduct ofthecoefficient of IX"incI>bythatofa;"-\ inI;andaswesuppose thefirst coefficients in IandcI>to be each I, if wetermtheothercoefficients lastspoken of Uianda,respectively, this saidcoefficient of theleading termoftheithBezoutian secondary will . (;-1)~. (i_I)' .contamtheterm(-) 2(a,-Ui)',andconsequently (-1) 2a;'and .Hl(-)'2Ui'. Now by thelikereasoning tothatemployed inthepreceding article, thecoefficient of theleadingtermin~m-i,o,thatis [hq1,hq,...hq'J "11'"l2•••7],,,I(IX-hqW-I)(IX-h q,+t)...(IX-hq".)h hh-J'q..q,'''q, hq'+I'hql+l•..i;.. 57J oftwo Algebraical Function». 475 willcontain thequantityI(hi~h,...hi)',andtherefore willcontain a tenn {I(hl~h, ...hi»)i,thatis(-)'"iai, whichisequalto(-)iai, since i-I(i-1)i isalwayseven.Hence ~1II_i.o=(_)ii xthecorresponding Bezoutian secondary. Art.28.Theaboveappliestothecasewherewehavesupposed m=n. Whenthisequality doesnotexistwe may proceed asfollows. Prefixto 4J:r.thefirstcoefficient of which is stillsupposed tobe1,atermEXm,where Eispositive andindefinitely small,andletepxsoaugmented becalled<I>(x). Thenif'TJ1ITJ2...TJnaretherootsofepa:,TJI''TJ2...'TJn,together withthe(m-n) ) valuesof(D"'-", will betherootsof<I>(x). Butithasalready beenprovedthatwhen(asheresupposed) thefirst coefficient offxis1,theBezoutian secondaries tofandepwill beidentical withthosetofand<I>respectively; atleastithasbeenprovedthatthese latter,when E=0,buttheform of <I>ispreserved, become identical withthe former,andconsequently thesameistruewhen Eistakenindefinitely small. Nowifwe call the(m-n)rootsof<I>whichdonotbelongtoep,'TJn+1I TJII-t2...TJm,andmake [hq"hq,...hq,] 'To ...(h)(h h) TJI,TJ2'••TJm ym-i,0=...x-ql+l::c-q/+2)'"(::c-qm[hh hJ'q"q,'" q, hql+1'hqW1' "hq", wehave 'To •_"'p(h h h )[hq"hq,...hq'JYm-I,o -~ ql'q,...q, , '1'>+1'TJnH'" 'TJm where Butsince'TJn+l,'TJn+!'"TJmareinfiniteinvalue, Hence and[hq" hq,'"hq'J_1(-)(-)(-))i(!)'-TJn+1 '1n+2'" '111I •'TJn+l,'TJn+!'"TJm E 1i 'l'm-i, 0=(;)IP(hq"hq,•••hq,) Butbywhathasbeenshownantecedently, takingaccount ofthefactofthe 476 OnaTheoryojthe Syzygetic Relations [57 leadingcoefficientof <I>being Ein place of 1,whichintroduces thefactor E'J we have, ''I'._(_)(i-l)2B.' E"'-1,0 - 'J whereB;'istheBezoutian secondary of the(m-i-1)thdegree in xtofand cp;butBo'hasbeen proved = B"theBezoutian secondary ofthesame .,-1 degreetofandcp;hence ~m-"o=(-)'-2B,. Art. 29.Ifnow we returnto thesyzygetic equation, Tf-tcp+~=0, ~may be treatedasknown,havingin fact been completely determined asafunction of the roots, aswell initsmostgeneralform,asalso soasto represent thesimplified residues tofandcpinthepreceding articles;it remains todetermine thevalues of Tandtasfunctions of therootscorre­ sponding to anyallowable form of ~,butI shall confine the investigation to thecasewhere ~isthelowest-weighted conjunctive or, which is thesame thing,asimplified residue to fandcpof any given degree in x;each value of iwillthenrepresent one oftheconvergents tojwhenexpanded underthe form ofacontinued fraction.If~be ofthe£th degree in e,Tis ofthe degree(n-£-1)andtofthedegree(m-£-1).Thisbeing supposed, and callingn-£-1=II,1/£-L-1 =1-',I saythattwill berepresented byGand TbyT,where [ltq,,i,....hq,.] G=(-)'!(x-hq,)(x-hq.) ...(x-hq,.)[h'II,~~'1hJ' 'I,''I.'" 'I,. hq,.. ,,hq~,...hq• andTis an analogous form I";hI.~...h",.,asheretofore, beingtheroots off,and'lJ1,'12'"'I,.ofcp.To fixtheideasand make the demonstration moreimmediately seizable, give mandnspecificvalues; thusletm=5, n=4,£= 2, sothatI-'= 5- 2- 1 = 2. Put~underthe form ~"O>sothat ~inthecasebefore us [hqJhq,hq,l =!(x-h)(x-h)_'TJI!l!~~ 'II 'I,[hhhJ. qJ'I, 'I, Itq,ltq, Now make x=~,thenf= 0,and~becomes 57J oftwo Algebraical Functions. 477 thatisI[~~I~JI;~~:~J [hI~hlJ ' h4h6 hIbeingkeptconstant in the above sum, butb«.ha,h4,h,beingpartitionable in allthesix possible ways into two groups, asintoh.,h6;ha,hain theterm above expressed. This sum is evidently identical with Again, t/>becomes IJHencet="ij,becomes But,whenx=hll thatis =(-l)·t. Thuswhenx=b-,t=G.In like manner, when x=h~,orhi,orh4,orh6, talways=G;huttandGare both functions of a:of the same, namely of only two, dimensions in a:Hencetisidentical withG.So ingeneral itmay be proved, thatwhenever x=hIor~orhi...orb«,tandG,which are each of only (m-1-t)dimensions in e,are equal. Hence universally t=G,aswasto be shown. To find Twe must avail ourselves of the sym­ morphic, or aswe maybettersay(itbeingatthe opposite extremity ofthe scaleof forms), theantimorphic, value ofIJrepresented by~o...takingcare to preserve ~strictlyidentical underboth forms of representation, inpoint of signaswellasquantity. Thatis to say, we mustmake 478 Ona Theory ofthe Syzygetic Relations [57 where fA)=t(m-t)+m(n-t), sothat (_)..=(-)""-,+mtl=(_)mn-,; andconsequently thesamereasoning as wasappliedtottoprovet=G.will serveto showthat-T=I',where or [hllh2..,h",] 'l'Jt''l'Jt••,'l'Jt 'T=(-)'"I(x-'lJt)(x-'l'Jt) ••,(x-'lJt) 1_2 • , 1 I •['l'Jtl''l'Jt.•••'l'Jt.] 'l'Jt.+I''l'Jt.+o•••'l'Jt. where fA)=mn.-1 -mil=mn-1- m(n-L-1) =mt+m-l. Art. 30. I have notsucceeded inthrowing tand'Tunderanyotherthan thesingle forms for each above given, and itisremarkable thatwhilst apparently tand'Tadmitonly ofthissinglerepresentation, ~admitsofthe varietyof forms included underthegeneralsymbol ~".,_"foragivenvalue ofL;anditoughtto beremarked thattheseforms,although themost perfectly symmetrical andexactlybalanced representations, and forthat reason possibly themost commodious for theascertainment oftheallotrious factorbelonging tothemrespectively, by no means exhaust thealmost infinite varietyof modes by which thesimplified residues, thatis,the hekistobarytic, orifwe like so to call them,theprimeconjunctives, admitof beingrepresented asfunctions oftheroots of thegivenfunctions; for ifin Art. 16,insteadofwriting 5i] we had madeoftwo Algebraical Functions. 479 P (hq"hq,•••hq"j'lEI''lEt'"'1E.)R= , [hq"hq,i;]X['lEI''lEI'" 1]E.] hq+ 'hq~hq'1., 1].•..1]."1"T' nI <r+1 <.+2 <m wherePrepresents anyfunction symmetrical inrespectofhq"hq,'"hq" , andalso inrespectof1]El'1]EI...1]E.'(theinterchanges, thatistosay,between onehandanotherh,orbetween one'1andanother 1],leavingPunaltered), itmightbe shown thatthevalueof~".•resulting fromtheintroduction ofthismoregeneralvalueofRwould (as for theparticular valueassumed) always be expressible asanintegral function of the roots jandconsequently, ifPbetakenof thesamedimensions intheroots as thenumerator ofR previously assumed, thatisvv,~".•wouldcontinue tobe(unlessindeedit vanish)identical (to some numerical factorpres)withthecorresponding simplified residue. If, on theotherhand,Pbetakenof lessthanvv dimensions inhand 1],we knowapriorithat~".•mustvanish,asotherwise weshouldhaveaconjunctive of aweightlessthantheminimum weight. WhenPis oftheproperamountofweightw,itis Ithinkprobable that another condition astothedistribution oftheweightwill be found to be necessary in order that~".• may not vanish, namely, thatthehighestpower of anysinglehinPshallnotexceedv,northehighestpower of any single 1]exceedv.ButasI havenothadleisuretoenterupontheinquiry,the verification or disproval of thissupposed law, and more generally theevolu­ tion oftheallotrious numerical factorintroduced into~".• byassigning any particular form to Psatisfying thenecessary conditions ofamount and distribution of weight, mustbe reserved, amongst otherpointsconnected withthetheoryoftheremarkable forms (19) Art. 15, as a subjectforfuture investiga tion, Art.:n.Aproperty ofcontinued fractions, which, if known, I have not metwithin anytreatiseonthesubject(butwhich has been alreadycursorily alludedto inthesepages), gives rise to a remarkable property ofreciprocity connecting Tandtseverally with~inthesyzygetic equation Tf-ttl>+~=o. Letthesuccessive convergents totheordinary continued fraction becalled1 1 1---ql+q2+q.+1 1 respectively; itis well known that 'lni-lZ.-"",1'-1=(_)l-11j 480 OnaTheoryoftheSyzygetw Relations [57 butI believe thatithasnot been observed thatthisis onlytheextreme caseofamuch more generalequation, namely 'lni-pli-mili_p=(-)i-pfJ-;-I-' wherefJ-J,/-'1...P-idenoterespectively thedenominators totheconvergents tothecontinued fractions formed with thequotients takeninareverseorder, thatis,the'continued fraction 1 1 1 1 1----- ...+---.qi+qi-I+qi--Jl+ q~+ql This iseasilyproved when p=1;f'ois ofcourse (asusual)to be considered 1. So more simply for theimproper continued fraction, li1 1 1 1-=--- ...---, '11liql-q2-qi-I-qi of which theconvergents are supposed to be II ~li-Ili 1nt''In..i'••m:'I'714' andthereversefraction 1 1 1 1- ---...--,qi-qi-Iq2-ql of which the convergents are supposed to be x,x,A.;-, fJ-J/-'1p.; we have the more simple equation li'TTL;-p-li-p'11li+fJ-~1=O. And it is well known, or atalleventseasilydemonstrable, that u;1 1 1 1t:=qi-qi-l-(i:~...q!' 1 1 1 'lni-I=..---- '1lL; Art. 32.Ifnow we use subscript indices to denote thedegree in a;ofthe quantities to which theyare affixed,we have thegeneralsyzygetic equation KT"_'_I/'" -Kt""""'_1l/>,.+K":!r,=0, whereK,aconstant (which I havegiventhemeans of determining inthe first section), being rightlyassumed, KT,._,_I,Ktm_<-Ibecomethenumerator anddenominator respectively of one of theconvergents toj,expressed as *SeeLondonandEdinburgh Philosophical Magazi,~, "On aFundamental Thllorem inthe TheoryofContinued Fractions," Vol.VI.,October, 1853. [See below.] 57] oftwoAl{]ebraical Functions. 481 animproper continued fraction,andn,becomesthedenominator to one of h t.-I hi hjh hi Tn-I•C I t econvergents to7,or,w1C ISt e same t 109,to~-. onverse y, itis obvious thatifweadoptasourprimitive functions ejmandt.-l' cbeingthevalue ofKwhen,=0, we shall obtainasthegeneralform of our syzygetic equation, bearingin mindthat(m-1) now replaces n, eK'Tn-,_ljm -K'~m--..-Itm-l +K't;=0; andsimilarly, if we adoptasourprimitive functions Tn-Iandq,n,weobtain for our general syzygetic equation, observing that(n-1)now replaces m, K'~n-'_ITn-l -eK'tm-._,c/Jn +K'T,=0; 80that(making abstraction oftheconstant factors and looking merely totheforms of theseveral functions which enterintotheequations) we see thatonthefirst hypothesis, namely of t,_1beingsubstituted forc/Jn,thecon­ junctives of each degree in xchangeplaces with thesecondconjunctive factors,thatis the original multipliers ofc/Jof the same degree in x,and viceversdjand in the second hypothesis, where Tn-Itakesthe place of jfit, theconjunctives of each degree in xchangeplaceswiththefirstconjunctive factors,thatistheoriginal multipliers offof the same degree in e,and viceveTsdjt.-landTn-Ibeingrespectively multipliers ofc/Jandj,suchthat thedifference of the respective products isindependent ofe.Theseresults oughtto be capable of being verified by aid of our general formulas fort,T,~, and88thisverification will serve to exhibitin aclearerlightthenature ofthereciprocity between theconjunctives andtheconjunctive factors, itmay be not uninteresting to setitout. Art. 33. Asusual, let ~,hs•••hmbe the roots of fe,and11"112'"'I,. theroots of cI>xjthelastconjunctive factor to c/J,which is of thedegree (m-1) ine,will berepresented, neglecting powers of (-),bytm-I>where [hg,,kg,...hgm-,] t.-I=I(x-hg.)(x-hg.)•..(x-hgm-I) [~::112"..l1nJ' kg"hg,...kgm-, Ifnow we for greatersimplicity maketm-I=t(x),and call the roots of t, ""1'"1'2'""I'm-I>any such quantity 88 [k,!""'J=t(hq,j=(hg",-hg,)(kg",-hg.)...(hg",-hg"...)111,"1I'""1m-I x c/J(kg.)c/J(kg,)..•c/J(kg",_,) (kg.-hg)(hg",-kg.)...(kg",-kg....l) =c/J(kg)c/J(kg.)...c/J(kg",,,,) 1 =Rc/J(kglft) ' *Since.isalwayssupposed less iliann(11being the degreeof the lower degreedof thetwo functionsjand </I).thefactof thelastquotient to'j-Ibeing wanting to "';1willnotaffectilia accurscy of thestatement in thetextabove, since this latterwillcontainas many quotients a8 canin anycaseberequired for expressing ~•• & 31 482 OnaThwryofthe Syzygetic Relations [57 Rdenoting aconstant independent oftherooth'lmselected, in factthe resultant of the two functions fxandepa:,thatistosay, Butby ourgeneralformuleethesimplified residuetofxandt(x)ofthe tthdegree in xwill berepresented by . (r h~,+t'h~'+1'"h?",1 } I L'71J'71···'TJm-l .~',0=I(x-hq,)(x-hq,)...(x-hq.)t[h h h ] 'ql'q.... q.. hq'+I'hq,n'..hq". therefore ~'-~(x-h)(x-h)(x-h) xjJlm-'If>(hq,+,)-l If>(hq'+I)-I...If>(hg".}-l}',0-~ q,q,.., 'q, [hh h ~q"q,'"q, hq,+t'hq,......h -R;rt-<-lI(x_h)(x_h)(x_h)If>(hg)If>(hq,)•••If>(hg) - q, g," • q,[hq,'hq,..,hq,] ' hq,+t'hq,...."hq". or therelationwhichwasto beobtained. So conversely, in precisely thesame manner, callingt'.theconjunctive factor of thedegree,inxtot(x)in thesyzygetic equation whichconnects}x andt(x)withacorresponding simplified residue, we have [hq,,-;'"hg,] I I I I~(h)(h)(h) 'TJI,'71'"'7_1t,=~x-s,x-q,•••x-q,lhhTt]q., q,'"q, hq,+!'hq,....••hq.,. theconjugate equation totheone previously obtained.". Andevidently thesamereasoning serves to establish thereciprocity, orratherreciprocal convertibility, between the~series and thel'series, when in lieu of theoriginal primitivesfeandepa:wetakeasour primitives l'(x)andIf>x,l'(x)beingthefunction which satisfies the equation l'(x)fx-t(x)epa:+~ =o. • M.Hermite, bya peculiar method, first discovered one of thesetwoconjugate relatious of reciprocity, applicable tothecaseofSturm'stheorem, where ~=f'z, and I am indebted tohim for bringing the subject undermynotice. - 57J oftwo Algebraical Functions. 483 "'lArt.34.Itmay be remarked thatifn=m- 1,thelastsyzygetic equation bein~thustm-l4>m-1 -T~m -~o-0, when ttn_1andfmaretaken astheprimitives, thecorresponding equation will be of the form t'm-ltm-l -T'm-a,/m+~'o=0; thesetwoequations musttherefore beidentical, andconsequently (m-I=4>m-1 (to anumerical factorpres),sothattm-IandcPm-1are reciprocal forms;this isalso obvious from theconsideration that(m-lmust, by thegenerallaw ofreciprocity (established above), be aresidueto(fm,4>7>1-1),whichthe latterfunction itselfmay be considered to be. Or thesamethingis obvious directly, bywriting , ()~(h)(h)(h)Ii>(hq,)cP(hg,)'"cP(hq"...)tm-I=t {C=..{C-q,1lJ--'q,.,.{C-q..-,(h-h)(h-h)(h-h)'q", g, q", g,'••~q"'-l andthenmaking . .,_~( _h) ( _h)(_h) t(hg,)t (hq,).,.t(hq"....,) ~m-l-.. {Cg,{Cq,'••{Cg.....1(h_h)(h_')(h_h)~g,~Itq, '••q", g.....l _Om-I~(h)(h)(h)P(hg,)-'...cP(hq.._,)-'-.n···..{C-"q,{C-q,,..{C-q"'-l('_')('_h) ftg".flq,'••ftq".q"._, -Dm-2~(-h)(-h) 4>(hg"J-.n'"..{Cg,'" {Cgill-I(h_h) (hF--_---.h~) , g..g."•q.. g".-l or finally, t'7>l-l=~cP, aswasto be shown. SECTIONIII. Ontheapplication oftheTheorems inthe preceding Section to theexpression intermsofthe roots ofanyprimitive function ofSturm's au:.ciliary functions, andthe other functions which connect these with the primitive function anditsfirstdifferential derivative. Art.35. The formuleeinthepreceding Sectionhad reference to thecase of twoabsolutely independent functions and theirrespective systems of roots: whenthefunctions become so relatedthattheroots of theone system becomeexplicitly orimplicitly functions of theroots of theothersystem, theformulse will become expressible intermsoftheselatteralone, and in somecasestheterms(of which thesum is always essentially integral) will becomeseparately andindividually representable underanintegral form. Such,asI shall proceed to show, is thecase.for two functions, of which one 31-2 484 Ona Theory oftheSyzygetie Relation» [57 isthedifferential derivative oftheother.Whenfand4>arethusrelated, sothat4>=ft:.callingasbeforehI.hi...hmtheroots off,and"l1'"l2..."lm-l theroots of 4>.weshall have in general [ltql+1]=(hq'+1-"l1)(hql+1-"l2).••(hql+1-"l,n-l) "l1>"ll..."l"'-I =f'h=[kgl+1]=[hgl+1]X[hg'+l ] • q'+1hq,.hg,...i,..hq,+l'"hg".hq,.hg....h91hgl+'J'hql+8...hg". Consequently [hgl+I'hg,+'J'"hg•] -[hgl+1J[h91+2 ] "l1>"l2..."lm-l-"111"11'""lm-Ix"l1>"ll'""lm-Ix ... x[hq". ] "l1."l2..."lm-I =[hgl+l ][hgl+1]hg,'hg,.•.hq,Xhgl+2'hql+l'"hq_1 x[~:::Ihq,...kJx[~::,i,......i;..J x . x[~:~hg....hJx[~::I'hg,.....hg..J. [hgl+!'hql+'Ji;] Hence "l1'"l2"l"'-1 [hql+I'hql+2' "-: hg"hg,ltg, =[~::,«;....kgJx[~::,hql+hq.Jx'"x[1::1,h9I+'J'"kq.J - ()ilm-i)(m-__l~J"(1.hh)- - .."'ql+l' gl....,'"g., thetdenoting theoperation oftakingtheproductofthesquaresofthe differences of thequantities whichthissymbol governs. Hence the Bezoutian secondary tofandf'ofthe(11t-i-I)thdegree in e,namely [hq"hg,•••hq,] '-1()'-~(h)(h) (h)"ll'"l2'..fJm-1 -2~a:-g,tla:-gl+.•••a:-q..[khhJ'g" g, g, hg..!,hg_hg., becomes ( -)'(\-I)!t(hq,,hq••••hq,)(a:-kql+l)(a:-kql+2)•••(a;-hg.) =!t(hg"hg••••hgJ(a:-hql+l)(a:-hql+2)...(a:-hq.), 57J oftwoAlgebraical Functions. 485 since(_)'1'-1)=1;thisgivesthewell-known formula- (enunciated - by me intheLondon and Edinburgh Philosophical Magazine for1839) for expressing M.Sturm's auxiliary functions in termsoftheroots of theprimitive, and which Ithereinstatedwereimmediately deducible fromthegeneralformulee (alsoenunciated inthesamepaper)applicable to any two functions. These moregeneral formulse appearto have completely escaped the notice of M.Sturmand others, who have used thespecialformulee applicable to thecaseof one function becoming the first differential derivative ofthe other. Art. 36. Inprecisely thesamemanner,if we form asusualtheordinary syzygetic equation tf''e:-TfIX+~=0, we may find thedifferent values of tgivenbythecomplementary formulee; and using t,todenotethemultiplier ofthedegreeiinIX,thatisappertaining totheresidueofthedegree(m-i-I)inIX,we have [hq"hq,...hq,J ~'TIl''TI'il'"'TIm-l(h)(h)(h)to=""-[-h h hJa;-q,a;-q,..•a;-q,91'q,...q, hqH 1,hq~...hq.. Art. 37. Thus, if we make i=m-1, .h'IX=tm-1=Inh91,hq,•••h9m-)(a;-hq.)(a;-hq,)•••(a;-kq.,...) . Itisevidentfromtheformoff/a;thatitpossesses relativetofa;,thesame property esf".»,I meantheproperty thatwhena;isindefinitely neartoareal root offx,and ispassingfrom the inferiortothesuperior side of such root,1:likej:willpassfrombeingnegative to being positive, or in other words,h'a;andf'IXhavealwaysthesame sign in theimmediate vicinity toareal root of fIX.Hence it followsthath'a;mightbe used instead off'IX,to produce, by theSturmian process of common measure, aseries ofauxiliary functions, which with fa;andh'a;would form arhizoristic series, thatis a series for determining (as inthemannerof M.Sturm's ordinary auxiliaries) thenumber of real roots of fa;comprised withingivenlimits. Therhizoristic seriesgenerated bythisprocess will, it is easily seen, be (to a constant factor pr~)thedenominators (reckoning+1asthedenominator in the zero place) of thesuccessive convergents toj:thrownundertheform [.p.45above.] 486 On aTheoryoftheSyzygetic Relations [57 1 1 1 1 ,ofacontinued fraction- -...----iM.Sturmsownrhizoristicql-q~-q..-l-q.. series, on thecontrary, will be(toaconstant factorp1'M)thedenominators oftheconvergents totheinversefraction1:'which will beoftheform K(_11..._1_!.);accordingly these two rhizoristie series will beq..-qn-l-q2-ql equivalent asregardsthenumber ofchanges and ofcombinations of sign (afforded by each) corresponding to anygivenvalue of a;,of which of course theq'sarelinearfunctions. Thisresultagreeswithwhathas been demon­ stratedbyme-by a more generalmethod(intheLondon and Edinburgh Philosophical Magazine, JuneandJuly 185:~),where it has been proved, by means of a very simple theorem ofdeterminants, thatthetwo series 1111111111 ql'ql:":"~'ql-q,-qs'...q--;:'"q2-is...q..' and111111 1111 q..' q..-qn-l'q..-q..-l-qn-s'q..-qn-l-qn-,...it' alwayscontain(for real values of qIJq"qsq..)thesamenumberof positive andnegative signs. Art. 38. Havingnowdetermined thegeneral values of ~andtin theequation tf?»-Tja;+~=0asexplicitintegral functions oftheroots offe,themore difficult taskremains to assign to Tits value similarly expressed. Thiscannotreadily be effected by means of substitutions inthe generalformulee, themethodweadopted for finding tand~;butallthe otherquantities except Tinthesyzygetic equation beingintegralfunctions oftheroots, it is evidentthatTalsomustbeanintegralfunction ofthe dbtaini he exnressi if'a;+~ same, an to0tamIt we may use t e expreSSIOn T=fe . Toobtainthegeneralform of Tbydirectcalculation fromthisformula would however be found tobeimpracticable ithe mode Iadopttherefore to discover thegeneralexpression forTcorresponding todifferent values of~,is toascertain itsvalue on thehypothesis ofparticular relations existing between theroots offe,andthenfromtheparticular values of T thusobtained to infer demonstratively itsgeneral form, as will be seen below.Thedemonstration ofTisunavoidably somewhat long, Tbeing in factrepresented by a double sum of partialsymmetrical functions. Usingthesubscript indices of each function as thesyzygetic equation todenoteitsdegreeina;,we have in general tm-i-lj'a; -Tm-H/a; +~,=0, [-Beebelowpp.616and621.] 57J oftwo Alqebralea; Functions. 487 where if we make ~-x=k1,h,-X=~...hm-x=km, sothat andtherefore nhSI'hS2..,hs;>=nkSl 1kS2' "ksp)' we have in effect found ~.=!kg,kgt."kg,~(kgl+ "k9l+2'"kg...) and tm-i-I=±!kg.kg,kgfA-4_,~(kg"kg,...kg__I) ; we have also /'(x)=t-)m-I~kg,kg, kg.._,. Letus commence with the casewherei=0;we havethen '~o=~(k"k2...km). tm-1=!kg,kg,...kgw&-!~(kg"kg,•..kg."....,); we havethus (_)mTm-'ilk,~'"s;=nkl>ks'"km) -!kg,kg,...kq."....,x!kg,kg....kg""""nkg"kg....kg."....,). Itmayeasilybe verified thatthenegative signinterposed between the two partsoftheright-hand memberof theequation hasbeen correctly taken,for ~(kt,k;...km)contains a term kl'(fIl-ll !c..'(m-'J) ...1f:4m-2kim-I> !kg,kg,...kq."....,contains atermkl!c.....km-skm-I, and !kg,kq.••.kq.-I~(kg"kq,...kg......,)contains a termkl2m- 'k,Im-O••.k'm--t1cJ.m-I, andthusthetermkI2(m- l)k2'('1l-2)•••1f:4m-'ilkim-I'which does not contain k,k,...k...,will(asitoughtto do)disappear from the right-hand side of theequation. Now suppose then and also nk9J,kg....kq,_,}=0, except when one or theotherofthetwodisjunctive equations q1Jq"q,qm-I=1, 3,4m, qllq"q,qfll-I=2, 3, 4 m, is satisfied (by adisjunctive equation, meaning anequation which affirms theequality of one set of quantities withanothersetthesameinnumber, eachwitheach,butin some unassigned order). 488 HenceOn aTheoryofthe Syzygetic Relations [57 !kg,kg,...k9__l.~(kg,.kg,kg_,) =2k1ka..,lem~(kl'kak",). Hence when k ,=k"(_)m7"",_, becomes 2i;!kg,kg,...kgfft-'~o;k•...km), thatis 2~(k"ka...km){k,Ikr,k r,...krm-l+2k.k•.. .Km}, the!referring tor"r•...rmsupposed to be disjunctively equal to 3,4...m. Now7"m-2is of(m-2)dimensions inIX.and whenever more thanone equality existsbetween thek's,~oandt...-,both vanish (infact every term . h .hI) d hf hi h ~o+tm_I/'1X1Deacvams es separatey,ant ere ore 7"m-" W IC - --_.--klk..,...km' will vanish. Hence(_)m7"m-amust be always of theform !~(hg" hg,...h9m-l)x'I"(kg"kilt...kq_l'kg",), 'I"denoting someintegralfunction of (m-2) dimensions in respectofthe system of quantities kg"kq,...kg",.Theresultaboveobtained enablesus toassignthe value of when k,=!c,.namely k,!(kra,kr••..krfl1l-1)+2k.k....k",. Now for amoment suppose. selecting(m-l)termsk,k,.k....le.,.out ofthemtermsof thekseries,that n(k"k"k....km,!c,)=k,m-,_!c,m-as,(kl,ka...km)+k,m-4S,(k ,•k....km) ±... =+=k,S_(kl,ka...le.,.)±2S_(k "ka...km). where S, means tha.tthequantities which it governs aretobesimplyadded together. S,denotesthattheirbinary,Sathattheirternary,and ingeneral S;thattheirr-aryproducts are to be added together. Whenk,=k..,nbecomes klm-2-klm-a{k,+S,(ka,k•...km)}+k,m-4{k,S,(ka•k•...km)+S,(ka,k•...km)} -k,""""{kIS,(ka,k•...lem)+S,(ka•k•...k",)}±... ±kl{k,Sm--4(ka.Ie•...km)+Sm-,(ka.k•.. .km)}±2S_(k"k•...km), whichevidently equals ±{2Sm-a(k.,k•.. .km)+klSm-a(k"k•.. .km)}, thatis ±{klI(kr.,k;•...krm_,)+2k.k•...k...[. 57J oftwo Algebraical Functions. 489 Hencewhen k 1=kl,'I'=n.and (_)m1'm--t=Inhq,.hq•...hq"'_t)xn(kq,. kq,...kq"'_t'kq",)j and so in like manner, when k 1is equal to anyone of the(m-l)quantities lc"kt••.km,theform of1'm- liabovewrittenwill have been correctly assumed. But1'~may betreatedasafunction of (m-2) dimensions in k.,and consequently any form of (m - 2) dimensions in k1•which fits it for (m-1) different values of k1,mustbe itsgeneralform, and accordingly we have universally, (-r1'm-2=It(hqtl i;•...hqm-I)x{(x_hq",)m-t -(x-hq",)m-tS 1(x-hq"x-hq,..•x-hqm-I) +(x-hqm)-Sli (x-hq"x-hq,x-hq",,..)±&c. =+=(x-kq",)Sm-I(x-kq..x-hq,x-hq,....J ±2Sm-I(x -hqpx-hq•'"x-hqm-I)}' Art. 39. Witha view to betterpavingour way to thegeneralform of l'forall values of i,let uspassoverthecaseofi=1 and go atonce tothe equation tm-sf'x-l'_fx+~2=0j and tobetterfix our ideas let m=7,sothattheequation becomes t.j/x- 1'./X+~lI=O; we have then,preserving the same relationasbefore,thatis, using hto denote any root of fx,andktodenoteh-x,theequation ±k.kllklkckDk,k.,1'1 =IkqlkqJ(kq,kq,kq,kq,kq.) -Ikqlkq,kq,kq.kq.kq, xI{kq,kq.kq,k q•nkq,kq,kq,kq.)} j now1'1will vanish whenever morethanthreerelations ofequality exist betweenthek's,fortheneachterminbothof the two sums in the right-hand memberof theequation abovewrittenwillseparately vanish;and of course threerelations ofequalitybetween the same are sufficient to make all the termsin the first of these sums vanish. This relationship between the different k'scorresponding to amultiplicity 3 may arise in different ways j themultiplicity 3 may be divided into 3 unitscorresponding to 3 pairs of equal roots, or into 2 and 1 corresponding one set of 3 equal roots, and a secondsetof 2 equal roots, or may be takenen bloc,which corresponds to thecaseof one set of 4equal roots. I shall make the first of these supposi­ tions, which will sufficiently well answer our purpose in the case before us. ThusI shall suppose then,asaboveremarked, 490 OnaTheoryofthe Syzygetic Relations [57 for all values of g"g"go,gug7'andtherefore "i.kq,kq,nkq,kq,kq,kq.kq,) =0j alsoIkq,kq,kq,kq,kq,kq. becomes k1k,k,{k1k2k,+u;(k1k.+~k,+kaka)}, and ~(kq,kq.kq,kq.) vanishes, exceptforthecases where gl,g2'g.,g,represent respectively, gltheindex 1 or 4, g,theindex 2 or 5, q,theindex 3 or 6, and g,theindex 7. Hence "i.kq,kq,kq,k q,~(kq,kq,kq,kq.) =2'k1k,k,k., n~k,kale.,), andconsequently 7",becomes ±8~(~k2k,le.,) x{~k2k,+2le.,(k1k,+k1lea+k,lea)}. Hence we are able to predictthatthegeneralexpression for our 7"inthe case before us will be T3=+Inkq,kq,kq,kq,} x{(kq,'+kq:+kq.')-(kq,·+kq,'+kq.·)(kq,+kq,+kq,+kg,) +(kq,+kq.+kq.)(kq,kq,+kq,kq,+kq,kq,+kq,kq,+kq,kq,+kq,kq,) -4(kq,kq,kq,+kq,kq,kq,+kq,kq,kq,+kq,kq,kq,)}. Forinthefirst place, thefactthatthe7"vanishes when more thanthree relations ofequality existbetween thek's, proves thatwe may assume 7". oftheform I~(kq,kq,kq,k q,)x4>{kq,kq,kq,k q,jkq,kq.kq.}, thesemicolon (j)separating thek'sinto two groups, in respectof each of whichseverally 4>is asymmetrical form.Butif intheexpression last abovewrittenfor7",we make k,=k"k.=k.,k,=k., itbecomes +8~(~kak,le.,) x{(~'+k2'+lea')-(kl'+kl+lea')(kl+k,+ka+le.,) +(kl+k,+k,)(~k,+k,lea+k,lea+k1le.,+kale.,+k,le.,) - 4(k1k,lea+k1k,le.,+k1k,le.,+k,k,le.,)}. Nowingeneralif Ur=G.tr+a.".r+al+...+at, and then Ur-Ur-ISI+UI'--'lS,±...±rS;=O. Consequently thesum ofthetermsconstituting thesecond factor inthe aboveexpression . =(3 -4)k1k,lea+(2-4)le.,(k1k,+~lea+k,lea). 57] oftwo Algebraical Functions. 491 Hencetheaboveexpression becomes ±8~(klksksk,) {klkska+2(~ks+~Ie,+ kaka)k7}. Thus,then,whenever klJks,Ie,arerespectively equaltoanythreeofthe quantities k4,ks,ka,k7,which may takeplace in twenty-four different ways (twenty-four beingthenumber ofpermutations of fourthings),ourTawill havebeencorrectly assumed jbutnkq,kq,kq,kq,) beingreplaceable by ~(hq,hq,hq,hq,), theTamay be treatedasacubicfunction inkl,ks•ka,and arranged according tothepowers of k).ka,kawillcontainonlytwentyterms; hence, since theassumed form is verified for more thantwenty,thatis, for twenty-four valuesof~.ha,ha,itfollowsthattheassumed form isuniversally identical withtheform of T,which was to be determined. Art.40.Now, again, in ordertofacilitate theconception ofthegeneral proof,letussupposejeto be of only five dimensions ine,istillremaining 3: it will no longerbe possible when we suppose amultiplicity threetoprevail amongtheroots, to conceive thismultiplicity to bedistributed intothree parts,forthatwouldrequiretheexistence ofthreepairsof roots, there beingonly five. Butwe may, if we please,make ~=ha=h"~andh,=h" or else ~=ha=ha=h4,or inanyothermode conceive themultiplicity to be divided intotwoparts,2and1respectively. or to be takencollectively en bloc. As a mode of proceeding themoreremotefromthatlastemployed, Ishall choose thelattersupposition. Thenweobtain(Tnowbecoming TlI-S-ll'thatisTl) ~lkale,k4k,Tl =±Ikq,kq,kq,k q•xIkq,kq,~(kq)kq,), andnkq,kq,) will vanish, exceptinthecasewhereqlrepresents theindices 1 or 2 or 3 or 4,andqatheindex5;also ~kq,kq,kq,kq. =kq)4+4~ak; Henceourequation becomes kl4k,T=±(kl4+4k)aks)4klk,nklk,), andTbecomes If,now, we assumeforthegeneralvalue of Tinthecasebefore us T=~nkq,kq.) {(kq,+kq,+kq.)-4(kq,+kq.)}. when~=ks=ka=k4,Tbecomes ±4nklks){3~-(4kl+k,)}, thatis ±4~(k)k,)(k) +4k,). Hencethenforthetwosystemsof values of ~,h"ha,namely ~=h4)(~=h, h,=h4oriha=h, h'J=h4lha=i; 492 Ona Theory ofthe Syzygetic Relations [57 theform of Twill have been correctly assumed. Butsincethederivedform is alinearfunction ofhlr~,ha,thisis notenoughtoidentifytheassumed withthegeneralform, since for such verification foursystemsof values must betaken,fourbeingthenumber oftermsinafunction ofthreevariables ofthefirst degree. If,however, we hadadopted aseparation ofthemulti­ plicitythreeintotwoparts,andhadstartedwithsupposing k1=k2=k., k4=kG,weshouldhave found thatTwould have become =6t'(k 1,kG)(~kl+3kG). Moreover, whentheseequalities subsist, ~~~~+~~~~+~~~~+~~~~+~~~~ becomes 2k1'kG+3k\'kl,andthecommon factor k1Sk 4disappears inthecourse oftheoperations forfinding T,andeventually we have to show (in orderto supporttheuniversality ofthepreviously assumed form for T)that kq,+kq,+k,.-4(kq,+kq,) becomes - 2kq,-3k.when kq,=kq,=kq,=k1, and kq,=kq,=kG, whichisevidently true.HencethenTwillhavebeencorrectly assumed for thefollowing cases, k,=k«=kG=k. k1=~=k;=k4; andalso forthecases .k1=k2=k.andkG=k41 k1=kG=k.andk2=1.."4 k2=k,=k.and1.:1=k4• k1=~=k4andk,=k.} k,=k,=k;andk.=k, , k.=k,=k,andk,=k; thatis, foreightcases in all, whereas four only would have sufficed. Hence, exabundalltid demonstrationis, theformassumed forTlisinthecasebefore usthegeneralform. Art.41. We may now easilywritedownthegeneral form which T assumes forallvaluesofiandproveitscorrectness. Iftheroots be i;s;h•...krn, and tm-i-Ij'x -Tm-i-sjX +~i=0, 57] oftwo A19ebraical Functions. 493 weshallhave ±'rm-i-'J=!{t(hg,hg,hg....hg.._+)x[O"m-i-s -um-._s81+O"m-H8s+&c. +(_)m-i-8 0"18m-i-s+(_).....-H(uo+1)Sm-i-ll)}, wherea;denotes in generalthe sum of the rthpowersofthe(i+l)quantities (x-hg",_,),(x-hg"H+I)'...(x-hg",), and8rdenotesingeneralthesum of the products of thecomplementary (m-i-1)quantities (x-hg.),(x-hg,)...(x-hg...~.J combined randrtogether. Itwill of course also be understood that 0",=i+1, sothat0"0+1=i+2. Art. 42. To prove thecorrectness of thisgeneraldetermination ofthe form of 'rm-i-ll'let us suppose in generalthati+1relations ofequality springup between themquantities kl,lea..•kmjwe shall theneasily obtain(Nrepresenting acertainnumerical multiplier) ±Q_NJ-(k1.k.)_!kg,kg,~:.~g"'_1 _- ~1,"'2•••m-t-l ,k,,",-Il~,..-1k""--I-I 1"'ll•••",-.-1 kl,ks•••km-i--Ibeing what theksystem becomes when repetitions are excluded, and being respectively supposed to occur JJ.1>~•••JJ.m-i--1times respectively, sothat !J.l+JJ.s+...+ JJ.m-i-1=m; thefractional partoftheright-hand member oftheequation immediately abovewrittenwill be readily seen to be equivalent to !JJ.fm.+-1k,.k,....ke.--s. Toestablish thecorrectness of the assumed form, we mustbe able, as in theparticular casespreviously selected, to prove two things;theone, and themore difficult thingto be proved is, thatwhen the series of distinct quantities k..,ks,ks•••kmbecomeconverted into!J.lgroups of kl;JJ.sgroupsof kl,•••f'm-i-Igroups of km-i--IJthenthat !f"1k..k"k"...ke-I-I' or inotherterms becomes identical with U''''''''_I- O"".....i-I81±&c.+(_I)m-H (0"0+1)8m--i-1' Theotherstepto be made, and with which I shall commence, consists in showing thatthenumberoftermsintheexpression last above written, considered as a function of (m-i -2)th degree of (i +1) variables, is never greaterthantheentirenumberofwaY8in which (i+1)quantities out ofm quantities may beequated to theremaining (m-i-I)quantities, namely each ofthefirst setrespectively to allthesame,or all different, or some the 494 OnaTheoryoftheSyzygetic Relations [57 sameandsomedifferent; inshort,inanymannereach ofthei+1quantities with some one or another(without restriction againstrepetitions) ofthe m-i-Iremaining quantities. Thislatternumber being in fact the number of ways in which (m-i-I)quantities may be combined (i+1) together withrepetitions admissible, by awell-known arithmetical theorem, . . )"+ d hfi berj(i+l)(i+2) ...(m-2) h" h . 1S(m-\-1\\an t e ratnum er 1S1.2 ...(m_ i _2) , W IC 1S always less thantheother.Itremains thenonly to prove theremaining stepof thedemonstration". Art. 43. To fix theideasletm=10,i=5, and consider the expression (kl+k.·+~.+kl+!ca'+klo·)-(k.·+k.·+~.+ks•+kg·+kIO')(kl+ka+k.+k,) + (k.+k,+~+ks+kg+klo)(klk.+klka+klk4+k.ka+k,.k,+kak4) - 7(~k.k.+k1k.k,+k1k.k,+k,.kak,). Now suppose thesixquantities k., k., ~,k..kg,klOto become respectively equal each tosome one or anotherofthefourquantities kitk,.,ka,k;asfor instance, I shall suppose k.=k.=~=kl k;=!ca=k,. Thenk10=k•. f/ot=4,,.,..=3,I.I.s=2,P-,=1, andtheformula of Art. 41becomes (3k1'+2k.'+k.')-(Ski·+2k.'+kl)(~+ka+ka+k,) +(3k 1+2k~+11'.)(k1kl+k1k.+k1k,+k,.lcs+k.k,+kak,) -7~k,.ka+~k,.~+~k,.~+k,.lcs~ =:3[{kl'-11'11(k.+lcs+11',)+k1]+11'1{(k,.k.+k.k,+lcsk,)+k,(k,.+ka+k,)j] +2{k.'-k.'(k1+k.+k,)+k.1+k.l(k1k.+k1k,+k.k,)+(k.~+lcs)+k,} +{k.'-k.·(kl+k.+k,)+lcs]+ka{(klk.+klk,+kak,)+k. (k1+k,.+k,)] -(k.k.k,+k1k.k4+klk.k,+klk.k.) = -klk.k.-2kJk,.k4-3klk.k,-4k,k.k, = _klk..k.k,{~+p..+~+~,}.k1k.k.k, *Ifthisfirst step of the demonstration appearunsatisfactory orsubjeoito doubt, it may be dispensed with,andtheresultobtained in the succeeding article(thedemonstration ofwhichis whollyunexceptionable) being assumed, it may beprovedthattheformula thereobtained on a partioular hypothesis mustbeuniversally true, in precisely the same way and by aid of the same Lemmain and by aid of which the formula obtained in theSupplement tothissection for the simplifled quotients to~:upon a like particular hypothesis is shown to beofuniversal applicanon, thatis, by showing thatotherwise afunction of2i- 1variables wouldcontaina function of 2i variables as a factor. 57J ojtwo..4lgebraical Functions. 495 Intheaboveinvestigation thequantities which with theirrepetitions makeupthele'ssystem,ares;lei'lcs,lei'appearing respectively 1, 2, 3, 4, times,thatistosayrepeated 0, 1, 2, 3 times;7 is one more thanthesum oftherepetitions 0+1+2+3, and the numbers 1, 2, 3,4arise from sub­ tractingfrom 7thesums 1+2+3;0+2+3;0+1+3j0+1+2jrespec­ tively,sothattheremainders 1, 2, 3, 4 denoterespectively one more than thenumberofrepetitions oflehlei'leltlea,thatis, arethenumberofappear­ ancesoflehlei'lea,lea;andthuswithaslightdegree of attention to the preceding process the readermay easily satisfy himselfthatthepreceding demonstration (although not soexpressed) is in essence universal, and the form of Tas anexplicitfunction of a;and oftheroots offa;isthuscom­ pletelyestablished for all values ofm and of i. Supplement toSECTION III. OntheQuotients resulting fromtheprocessofcontinuous divisionordinarily appliedtotwoAlgebraical Functions inordertodetermine theirgreatest Common Measure. Art.(a)·.We have now succeeded in exhibiting theforms of the numerators anddenominators ofj:developed into acontinued fractionin termsof the differences of theroots and factors of fx.Itremainstoexhibit thequotients themselves ofthiscontinued fractionunderasimilar form. LKMMA. Anequation beingsupposed ofanarbitrary degreen,there existsnofunctionofntuuiofless than 2i ofthecoejJicientst, whichvanishes forallvaluesofn whenever thenrootsreduce in any 'manner to idistinct groupsofe'J'U<Llroots;or in other words, any functionofn andthefirst2i- 1 coejJicients ofan equation ofthe nthdegree,whichvanishesforall values ofn inevery case wheretherootsretainonlyidistinct names, must be identically zero. Torenderthestatement of the proof more simple, let ibetakenequal to 3. And let theroots be supposed to reduce to prootsa,qrootsb,and •Thearticlesinthisandsubsequent sections towhichLatin,GreekandHebrewlettersare prefixed, although instrictconnexion withtheconted,aresupplementary inthesenseof havingbeensupplied sincethedatewhenthepaperwaspresented forreadingtotheRoyal 8oaie$y. Allthearticlesmarkedwithnumbers (from1to72),andtheIntroduction, appeared inthememoir lIBoriginally presented tothe Society, June16, 1858. tIntheproposition thusenunciated thecoefficient of the highestpower of zissupposed to beanumerical qU&Ilti$y. 496 OnaTheoryofthe Syzygetic Relations [57 and80ingeneralwe have for all values of e, 1, I, 1, 8, a,b,C,8H)rrootsc.Andlet8ringeneraldenotethesum oftherthpowers of the roots.Thenwe have evidently p+q+r=80, pa+qb+TC=8), pal+qbl+rei=821 pal+qbl+rei=8a, pa'+qb'+ro'=8,. &c.&c.,adinfinitum. Eliminating p, q,rbetween thefirst, second, thirdand fourth equations, weobtain 1, 1, 1,80 a,i,e,8)=0. alblci81 ,,, a·,bI,eI,8. Inlikemannereliminating ap,bq,orbetween thesecond,third,fourthand fifthequations, we have 1,1,1,8) a,b,c,~=0; albleI,B. , , aableI,8, ,, =0; ai,b'-,eI,8'H aI,bB,ca,8'+1 whenceitmayimmediately be deduced, that,uponthegivensupposition of therebeingonlythreegroupsofdistinctroots, we musthavethefollowing infinitesystem of coexisting equations satisfied, namely, 80t+S)U+SIV+8.W=0sayLo=0, 8)t+8aU+8IV+8,W=0 "L)=O, ~t+8.U+8,V+8GW=0 "La=0, 8.t+8,U+8.V+8eW=0 "L.=0, 8,t+SGU+8eV+87W=0 "L,=0, &c. &c.&c. &c.; 57J oftwoAlgebraical Function». and conversely, when thisinfinitesystemofequations is satisfied the roots mustreducethemselves tothreegroups of equal roots. LetnowtJ>be any function of 8o,81182•••which vanishes when thisis thecase.ThentJ>mustnecessarily containasafactor some deriveeofthe infinitesystem of equations abovewritten,thatis.some function of 80,81,82" , which vanishes when these equations are satisfied, thatis, someconjunctive ofthequantities Lo,LlIL2,La... ;butit is obviously impossible in any such conjunctive to exclude 8efromappearing, unless by introducing someother8 with an index higherthan6, andconsequently tJ>cannotbe merely a function of80,81,8.,8s,84,8D,norconsequently ofnand the first five coefficients; or if such,itisidentically zero. And so in generalany function of nand only 2i- 1 of thecoefficients which vanishes when the roots reduce to i groups of equal roots,mustbeidentically zero;aswas to be proved. Art.(b).Itoughtto be observed thatthepreceding reasoning depends essentially upon the circumstance ofnbeingleftarbitrary.Ifnwere given the proposition would no longer be true. Infact, onthatsupposition, the nrootsredncing to idistinct roots would imply theexistence ofn-i conditions between thenroots; and consequently n-iindependent equations wouldsubsistbetween thencoefficients,and functions could be formed of i only of thecoefficients, which would satisfytheprescribed condition of vanishing when the roots resolved themselves into i groups of distinct identities. Art.(c).LetDrllr•...r•beused ingeneralto denote thedeterminant thenthe simplified ithSturmian residueR;,may be expressed underthe form DI•••a....,Xn-i-1 -Ds,a...HI'xn-i-'J+Da•4....-t2,Xn-i-a."±Dn.....~I...n' which is easily identifiable with the known expression for such residue. Now obviously thenecessary and sufficient condition in order thatthen roots may consist of only repetitions ofidistinctroots is, thatRishall be identically zero.thatis tosay,wemusthave DI•2...i=O.Ds,a...i+l=0."Dn-i,n-i-I. ..n=O. Butthereasoning of thepreceding articleshowsthatalthough these equa­ tions are necessary and sufficient, theyarebuta selected system of equations of an infinite numberofsimilarequations which eubsist ",andthat,in fact, *Butqu.erewhetheranyotherruffidfflt systemcanbefound of equations so fewinnumber &8thissystem. & 32 498 OnaTheQrYofthe Syzygetic Relations [57 whatever bethevalue of n,we may takerllr,...r,perfectly arbitrary and asgreatas we please, andtheequation mustexistbyvirtueof theexistence ofthen-iequations lastabovewritten. Art.(d).I nowreturntothequestion ofexpressing the successive quotients of!Jiasfunctions ofthedifferences of theroots and factors; tha.t theymustbecapableofbeingsoexpressed is an obvious consequence of the factthatthenumerators anddenominators oftheconvergents havebeen putunderthatform, since, if Ni-'JNi_1s; Di---'J'Di-I'o.: are anythreeconsecutive convergents ofthecontinued fraction I I I QI-Q,-'"Qi' wemusthave Di-,Ni-Ni_,D,=Qi. Itwould not, however, be easy to perform themultiplications indicated in theaboveequation, so as to obtainQiunderitsreduced form as alinear function ofa:I proceed therefore tofindQiconstructively inthefollowing manner. LetR.-'l'Bi-I,Bibethreeconsecutive residues, fixcounting asthe residueinthezero place, thenQi=[li-'lR·.-11,i,and is of theformJ!.x+i,. i-I q q Now in generalif wedenotethenroots off»,wherethecoefficient ofIX"issupposed unity,byh,.,h.,....b«,andif we use Zitodenote Inhs"he....hsJ·,withtheconvention thatZI=n,Zo=I, we have, employ- ing(i)todenote ~{(_l)i+1), "titwillberemembered is the symbol of the operation oftakingtheproductof thesquares of the diJIerences of the quantities which it governs. 57J oftwoAlgebraical Functions. 499 ThepartofR..-Iwithinthesign ofsummation is Z,xn-i-~(h8>+l+h81~'+...+he.)~(h"8t,he,...he,)xn-'-l+&c., say z,xn-i-Z/xn-i-1+&c., andthepartofR.-2withinthe sign of summation is Zi-1xn-iH-Z'i-1Xn-i+&c., and HenceQ,=~Z2i_a~2,_~...Z21t)fZ2i-2~~' ~2(t1+I}-1 ZlZ2i_2Z2'-4'" Z2{'1+1lZ2i-1Z2i_a Z2(" X{Zi-1Z,X+(Zi-lZ;'-ZiZ'i-l)} =~2i-lZ"_aZ'H ...Z'{'Lt; Z?Z'i-2Z'i-4'" Z'(i)H T,denoting Zi_1ZiX+(Zi-1Z.;'-ZiZ'i-l)' Art.(e).Iftheprocess of obtaining thesuccessive quotients and residues beconsidered, it will easily be seen thateachstepintheprocess importstwo new coefficients intothequotients, thefirstquotient containing noliteralquotient inthepartmultiplying xandcontaining thefirstliteral coefficient in the otherpart,thesecondquotient containing twoliteral coefficients inthe one partandthreeintheother,and ingeneraltheith quotient containing 2i - 2 of the lettersIntheonepartand 2i- 1 of them in the other. HenceTibeingmadeequaltoLix+Mi,L,contains 2i- 2 andM,contains 2i - 1 of the literalcoefficients of fa: Moreover, we have Zioftheform T~~,=2-mPi ,Pi-I' where Pi.;="I.nh",h"...he:>"lel+1"l'l+I'""l,., Pi-'J="I.nh",he,.. ·h'i-I)"le,"lei+I"·"l'., andPi,whichistheithsimplified residue,vanishes whenthenroots in any mannerbecomereducedto only i distinctgroups. Iproceed to showthatif wemake Aix+Bi=Ui=Ati,1(x-hI)+A\2(x-hs)+...+A\n(x- h,.), where in general Ai,.represents ~'(h",he....h'i-I)(h.-h,)(h.-h,,)...(h.-h'i-I)' thenwill Ti=o; 32-2 500 On aTheoryofthe Syzygetic Relations [57 whereItwill be observed that.Ai,eisidentical with what thesimplified denominator ofthe(i-l)thconvergent becomes when we write hein place ofa;,and consequently, whenarranged according to thepowers of he'will be oftheform thhei-I+c.;.hei-s+...+Ci, whereth,Os•••Ciare functions of thecoefficients, butcontaining no more of themthanenterintoQi-l>thatis,containing only 2i- 2 of them. NowAiis made up ofterms, each consisting ofsomebinaryproductof combined with some termoftheseries "i,hti-'l,I,h2\-3•••'"iNi andanyoneofthislattersetoftermsexpressed asafunction of the coeffi­ cients of fa;containsatmost 2i - 2 of them. Hence only 2i- 2 of the coefficients enterinto.Ai,and in like manner only 2i - 1 of themintoBe: Thenumber ofletters,therefore, in.A,and inRiisthesameasinL, and inMi,namely 2i - 2 and 2i- 1 respectively. Nowlettheroots consist of only idistinctgroups of equalroots,80that PHT,becomes =ZtP-.'-I I shall show thatinwhatever waytheequal roots aresupposed to be groupeduponthissupposition, therewillresulttheequation Ti=Ui, T,=I~~("•.,"'•...",,)}9pPH ,'-I PH=~{"1""7'1+1•.."1...~("1'"""1'/-l)}' Pi-1=~{"'I+I"7'!+9 ..•"1'..~("1'"""7•.)}, and .A,meaning ~1("7,-"7••)("7e-"76.)...("7e-"76,.J ~("7••,"7....."7'i-I)}' and"1..meaning a;-h... Letthenfactors be constituted offfltfactors "71''1nsfactors"S··.mo factors"7i.Then where p.=fflt71l.s•••m.;, P'-I=p.'("1l1"1s···"11)"11m,-l"1sm.-1..•"r,-I, 57] and whereojtwoAlgebraical Functions. Pi-2=f£1~('112>'11....'11.)"11m.'11.m.-1...'11/m,-1 +f£2~('111.'112'"'11.)"1lm,-1'11.m,•••'11ime l +&c.&C.501 Hence m.=IL2"(.... .){'11ln'11••'11.'..'11/)+'112~('111> 7]....'11/)+ + '11i~(7]I.'112'""1i-I)}.L0r:~'II>'112...'110 .. • • illl "'-I 11Ii' Again. in U,the term containing '111will be 'llit'11dI ('111-'11.)('111-'11.)...('111-'11,)~('11.,'11.'"'11,)}1I ='llit'111x("'-1m,...mi)lIx('111-'1111)2(7]1-"11)2.:.("11-'11i)'(~('1111,'11,,,,'11.)}! f£1I= -"11xn'11l>'11.'"'11i)S('1111. '11....'11,).'llit 'Hence U.-'''( .){'111~('11.'11....'11,)+'111S('11I'11....'11.)+&c}=m,0-f£~'111''111'"'110 •.Lo'ml mll Hence. therefore. U,-T,vanishes whenever theroots ofIa;containonlyi distinctgroups of equal roots. and it hasbeen shown tbatU,andT,each containonly2i- 1 ofthecoefficients of Ia;.sothatU,-T.is a function only of 11and these 2i-1letters,and consequently, by virtueoftheLemma. in Art.(a),U,-T,isuniversally zero,thatis,U,isidentical withT"aswas to be proved. Inthesamemanner, asobserved in a preceding note [po494].theexpression givenin theantecedent articlesfor thenumerator of theithconvergents, havingbeen verified for thecaseof the roots consist­ ingofonlyidistinctgroups, could have been atonce inferred to be generally trueby aid of the Lemmaabove quoted. Art.(f).Sincethecoefficient of a;inT,isZi-IXZ"we deduce the unexpected relation I~(~,~...hi-I)XI~(hl'~...hi)=PI'+Pl+'"+Pnl, where P,=I{(h,-h,)(h,-h,,)...(h,-h'l-)~(h",h"...h'l-)}' Sothatevery simplified Sturmian quotient toj:,when the nroots offa; arereal,willbethesum of n~quares. Buttheequation is.otherwise kbl'hibi " h d fb fn(n-1) (n-i+2)remara e,10ex1itingt e pro uct 0t e sum 01.2(i-1) bh fn(n - 1)...(n-i+l) dthf fsquares yanoter sum 0 1.2...i squares un er e orm 0 thesum of n squares. 502 On aTheoryoftheSyzygetic Relations [57 we haveIfwe denote the ithsimplified denominator to theSturmian convergents toj.:byD,»,and if we call the ithsimplified quotient XiX,we have 1 XiX=I(Di-1h.)'(x-h.). " Ifweconstruct thenumerators anddenominators oftheconvergents to 1 1 1 1 Ql-Q,-Q,.,.Qi' according to thegeneralrule forcontinued fractions, as functions of QllQ"Q" &c.,sothatcalling the denominators ~1'~"~•••~i, ~1=Q,~~=QIQ'J-1...~i=Qi~i-l- ~i-'J' .Z2.Z2.Z,·A _'-2.-.'"/I-I)D. ~i-IX-Z2 Z2 Z2 .-1X, >-1i-I'" /11 d;-lXbeing in fact the multiplier off'xin theequation which connects fa; andj'xwiththe(i-1)thcomplete residue, and consequently, retaining Q(x) todesignate the complete ithquotient, we have Q.( )=Z'Ji_1Z\-sZ·i-5 ...Z~ ~{D.h.}2( _h) •X 17:JZ.Z.Z....-1. X•Li. >-2H...1.1+1 Zo.zs.zs.zs,=_'-1_'-3.~~I"_~ {~.h]'J(-h) Z2ZS ZS zs...-1.x., ii-'.lH'" (.1+1 whichequation gives the connexion between theform of any quotient and thatof theimmediately preceding convergent denominator of thecontinued fraction which expressesffi'x.x Art.(g).I have found thatthecoefficients of the nfactors of fxinthe expression above given for the quotients possesstheproperty thatthe sum oftheirsquare roots takenwith the proper signs is zero for each quotient except the first (thecoefficients for the first being all units),thatis Dih1+Dih,+'"Dih"=0 for all values of i except i=1.Moreover I find thatthedeterminant formed by the nsets ofthencoefficients of thefactors offein the complete set of nquotients isidentically zero,thatis,the determinant represented by the square matrix (D1h,)'J•..(D1hf&)~ (D,h,'f...(D'Jh,,)'J1, (D1h1)2, (D2~)2,1, 1, ], =0. 57J oftwo Algebraical Functions. 503 Art.(h).Itshould be observed thatU,is the form of the simplified quotients for all the quotients except the nth(thatis, the last), for which the simplified form is not Un,butUn+t;(hi'hi...hn),which arises from the circumstance of thelastdivisor. which is thefinalSturmian residue, not containing a;;it being evidently thecasethatthedivision of arational function of a;byanotherone degree lower, introduces into the integralpart ofthequotient thesquareof the leading coefficient of thedivisor,subject to theexception thatwhen the divisor is of thedegree zero, the simple power entersin lieu of thesquare. The generalformula gives for the reduced nth quotient the expression which equals Rejecting the first factor. we have It;(hi'h;'"hn)(a;-~). which is equal to the penultimate residue, which residueis(asitevidently 'oughtto be)identical withthesimplified last quotient. Art.(i).We have thussucceeded in givingaperfectrepresentation ffa;h . folx'tatIS, 0III--h+--r+...+-h-'a;- 1a;-,,'l a;-n undertheform of a continued fraction of theformIII ml(x-e,) -.rn";J(a;-e.)-......fn..(a;-en), wherem"m•...m..;e"el...enare alldeterminate and known functions of~,hi'"h.... We may by means of thisidentity. differentiating anynumberof times withrespecttoa;both sides of the equation, obtainanalogous expressions for the series 1 1 1 (a;-~)'+(a;-hl)t+...+(a;-h.,)t. Butto dothiswemustbe inpossession of a rule for thedifferentiation of continued fractions whose quotients arelinearfunctions of thevariable. I subjoin here the first step only toward such investigation. Letthedenominator of 1 1 1---,ql-ql-qn 504 On aTheoryoftheSyzygetic Relations [57 whereql>qt.··qnare anynarbitrary quantities, bedenotedby[ql>qt,q•...qn], sothattheentirefraction will be equal to [q,.q•...qn] [ql>qt,q•...qn]• Any such quantity as[qi,qiH...qn]may betermedaCumulant, of which q"qiH... qnmay beseverally termedtheelements or Components, andthe complete arrangement oftheelements may be termedtheType.The cumulant corresponding to any type remainsunaffected by theorder of the elements in thetypebeing reversed, as is evident fromanycumulant being in fact representable undertheform of a symmetrical determinant, thus,for example, the cumulant [ql'qi'q.,q,]may be represented bythe determinant s.:1, 0,°, 1,qt,1,°0,1,q.,1 0, 0,1,q, and[q"q.,ql>ql]will in like mannerberepresented by thedeterminant q"1, 0,°1,ql,1,0 0, 1,ql,1 0,0, 1,s. which is equal to theformer. Art.(j).Letit be proposed in generalto find the first differential coefficient in respectto{J;ofthefraction where each qis a function of one or more variables. I findthatthevariation ofF,may be expressed as follows: -8F,={8[ql' ql'"q,-t,qn]+8[qIJqt...qi-t,qn-I]qn' +8[qllqt,q•...qi-t,qn-t][qn,qn-I]'+ . +0[qllqt,q•...q'-I,qi-I][qn,qn-llqn-sq;]tJ +[ql'qt,q.'"qn]'. 57] oftwoAlgebraical Functions. 505 Art.(k).Supposei=2, andql=a1x+b.,qi=asx+b,...qn=anx+bfa, weshall have by virtueoftheaboveequation, d .d{l I II}J_FIIthatISd-- - - ... - UOli Xql-q2-q.q" I = - [ ]2{a,,12+an-lq"i+~[q",Q"-l]'+&c.ql'q2'"qn +~[q",q"-l,q,,-'J'"q,]'). f/>xIfwe callFi=Ixevery such quantity as[qn,qn-l...qdrepresents toa constant factorpresthe (i-I)thsimplified residue(4)xcounting asthefirst ofthem)toj:,andmakingcertainobviousbutsomewhat tediousreductions, andrejecting thecommon factor - (lX)2'weobtaintheexpression CoR12~2R,' Rn2 "-0+CC +CC +·..+-CC=4>xlx-f/> xlx, 1l' ,i 71-1" where ~,~...Rnrepresent 4>xandthesuccessive simplified residues tolx,4>x,whileC.means the coefficientof thehighestpower of xinR;,and Cothe first coefficient inlx·. Art.(l).Ifwetakegxofthesamedegreeasfa,and for greater simplicity make the first coefficients in Ixandgx,eachofthemunity, • Thisresultmaybeobtained directlyasfollows:- Letfx,t/>Zandilie(m-1)complete Sturmian residues becalledPo.Pl,PI...P.;letthe" complete quotients becalledql'q,...q.,andlet theallotrious factorstotherssiduesPI,PI...P. becalled~, 1A1J...p.;then ~=~Pl-P2.Pl=~PI-Pl,P2=~PI-h,&~; hence Pi&~-PO&Pl=Pi'&ql+(p,&Pl-Po&fl2) =Pl2& ql+p,i&qi+ ~&p,-P23p,) =&c. =Pli& ql+PI'3q,+Pa'3q,+...+P.'3q.; but wehave ingeneralp,=I/o;R" hence 3q,=1Z.=1~.::1&X, C,I/o; and but itmaybeeasilyseenthat /Io;-l/1o;=~. exoept when i=I,furwhichcase/10;-1/10;=1, C.-1 hence PI'3q,=c~RI'3X. wheni>I, and=~R1'&xwheni=I, ~1~ 1 which proves ilietheoreminthetext. 506 On a Theory ofthe Syzygetic Relations [57 tbesuccessive simplified residues to~:will beidentical withthesimplified residues to-I;:gx(including amongst themthequantity gx-Ixitself), and, since {Ix-gxJ1x-{fx-gx)'gx=g'xlx-I'xgx, tberight-band side oftheequation abovewritten,whentheresidues, instead ofreferring toIandep,are made to refertoIandg.takenofthesame degreeine,becomes equal to I'xgx-I.l:g'x;and if we now agree to considerIand.qashomogeneous functions each ofthenthdegreeinxand1, theequation becomes R,'Rsl ~I R"l-+~+-+ +--­C1C1C2CIC....Un-IGil d d=9(e,1)dxl(x,1) -I(x,1)ax9(x,1) =!(x~g+~ g)(_d_i\_!-:1+d:1\(~g)naxdldx}naxdl}ax .=~{~~i-g~}=~J(f,g), whereJindicates theJacobian ofthegivenfunctionsIand9inrespectto thevariables xand 1,meaning therebytbe so-called FUllJ::tional Determinant ofJacobito1and9inrespectofxand1,whichequation also obviously mustcontinue to hold good when we restoretothecoefficients of x"uifand 9theirgeneralvalues. Itmayhappenthatforparticular relations between thecoefficients of 1and9certainoftheresidues may be wanting, which will be. thecase when any of thesecondary Bezoutics havetheirfirstorsuccessive terms affected with the coefficient zero;theequation connecting theresidues withtheJacobian willthenchangeits form (assome of thequantities C1,O2, , ,C"will become zero);butI do not propose to enterfor thepresent intothetheoryofthesefailing, or asthey may more properly betermed, Singular casesin thetheoryofelimination. Art.(m).Theserieslastobtained forJ(f,g)leads to a resultof much interest inthetheory,and of which greatuse is made in the concluding section of thismemoir, namelytheidentification oftheJacobian (abstraction made of thenumerical factorn)withwhattheBezoutiant becomes when in place of the11variables in it,u1,Us...u",wewritex"-1,xn-'J...X,1.Thus supposeIand9to be each of thethirddegree, and let ..AxI+IIx+G, HxI+Bx +F, Gg;'J+F»+a, 57] oftwo Algebraical Functions. 507 be thethreeprimaryBezoutics; if wemake afJ=U,0;=v,1=w, thesemay bewrittenundertheform Au+Hv+Gw=L, Hu+Bv+Fw=M, Gu+Fv+Cw=N, and iftheBezoutiant be calledfl,we have dFTL=­du' Thesimplified residues tofand9areL, (L,M),(L, M,N),where(L,M) meanstheresultofeliminating ubetweenLand],f,and(L,],f,N)theresult ofeliminating uandvbetweenL,:AI,N;andbyatheorem (virtually implied in thedirectmethod- ofreducing aquadratic function totheform ofasum ofsquares), if we call theleadingcoefficients of these quantities Cl,Ct,C" we have .~2+(L,.:lf)~+(L,~~,N)2=fl.c.cc, UtO, Hence,whenn=3,iJ(f,g)=flwhen in fl,u, v,wareturnedintoafJ,o;,1; and so in generalforany values of n,theBezoutiant correspondingly modified, 1becomes - J(f,g),aswasto be showu['.n Art.(n).Theexpressions obtained forthequotients to.!J:may be generalized and extended tothequotients toj;,whereepo;andfeare two functions of 0;of anydegreesm andn,whoseroots are respectively, kl,~...km, andi;b«...s..Ifwesuppose epa;1 1 1 1 fo;=Q(0;)-qt(x)-q,(0;)-...qm+l(0;)' whereQ(0;)isofn-m dimensions, and q2(0;),q,(0;)...qm+l(0;)each of one dimension in0;,itmay be proved thatonwriting 1lIN, (0;) Q(0;)-qt{x)=···q.(0;)=15-;(x), •Namely, thatofM.Cauchy, adverted to in Section IV. Arts.44-45. [po511 below.] tCompare Jacobi, DeEliminatione, §2. The general expression for theallotrious faetor, I may here incidentally mention, is givenundertheheadTheorem 0,§16, which comes quite at theend ofthesamepaper. 508 OnaTheoryoftheSyzygetic Relations [57 weshallhave '~1{(N,k,)2:;/x-k,)}=Oq'+1(x), (A) '~1{(D,h,)2J~:(x-h,)}=O'qi+I(x), (B) where O±O'=O, (E) Oqi+I(x)beingthe(i+l)thsimplified quotient. WhenQ(x)is alinear function ofe,in finding qlxfromtheformula(B),wemusttakeDrfX=1.The proof of thistheorem beinggenerally true,may easily be shown to depend upon its beingtrueinthespecial case", when m=p.+i, andn=p.+i' (mbeingsupposed less thann),and~,~h..become [1'il...il"hI'h<J.•••h,.-, whileklJk2•••kmbecomei:I,i2•••l;kI,k; I.~,;andthetruthofthetheorem forthisspecial case (ifforinstance we wish to prove theformula (B» depends upontheexpression (~'~h.--I)(~'hi.,.h,.--I) kI,k«km-;-h;"h'-+I•••h.. (hI'~h;.)(h1,~...h..) XkI,k2•••km-;-h.".+1Jh.-H'"h,.. beingidentical withtheexpression {(~'~...h.--I) .(h1,~...h,.--I)(I.h)(hh)(h h)} k k k..,..h h x '..--1 ." - I•••...-'"-I 11I'"m,,-,h.-+ I..... (h.. )i;k2...kmxh ' (h~,~...s-:hH I...hJ asitmayreadilybe shown tobe. And theformula (A) may be verified inprecisely thesamemanner. Thereis no difficulty in finding thevalues of0and0',which are products of powers, some positiveand some negative, oftheleadingcoefficients in thesimplified residues, andrecognising that theysatisfytheequation (E);when<pxis of one degreebelowfxthisequation is oftheform0+0'=o. Art.(0).When<px=f''»,thisexpression forthe(i+l)thsimplified quotient becomes I(D,h)1(x-h),aspreviously found;thecorrelative ex­ pression will be -~(N·k\JA(x-k)..•I{"k ' •Byvirtueof theLemma,thatwhentfxrandIxare twoalgebraical functions, nofunction of the coefficients vanishing identioally whenirootsofIxcoincide with irootsof~respectively can be formed, in which thereare fewer of the coefficients of Iand",respectively thanappear in theleadingcoefficient of the (n-i+l)thresidue ofj. 57] oftwoA1gebraical Functions. 509 andthatkbeingBnyroot off'x=0, which is equaltotheformer expression. The generalexpressions above givenforthesimplified quantities are of course integral functions of handk,although givenundertheform ofthesums of fractions, by virtueofthewell-known theorem that~;~,where~is an integral function ofh,andthesummation comprises all theroots(h)of fh=0, is always integral. Art.(p).Itwillbe found thatfor all values of igreaterthanunity tilfie, ~(N;lc,) A,lk=0,'=1 .", " eph,s(D,h')f'h=0.'=1 , Thetheorem of Art.(n)is in effect atheorem ofcumulants oftheform wheretheelements are allindependent of oneanother, and fx=[QI(x),qt(x),q.(e)...qn(x)],epx=[qll(e),q.(e)...q"(x)], nbeinganynumber whatever greaterthani;thismakesthetheorem still moreremarkable. Theurgency ofthepressprecludes myinvestigating forthepresentthemoregeneraltheorem whichmustbepresumed to exist, whereby q\+1canbe connected with [ql' qt,q•...q,],or[qt,q•...q,].and with [qllqll'q•...qH.]and[qt,q•...q\+e],when each qrepresents afunction of an arbitrary degree in s:Thetheorem sogeneralized wouldcomprehend the complete theoryofthequotients arisingfromtheprocess of continued division, without exclusion of thesingular cases(atpresentsupposed to be excluded) where one or several consecutive principal coefficients in one or more oftheresidues, vanish. Art.(q).Thecomplete statement of two twin theorems suggested by andintimately connected withthebiformrepresentation ofthequotients ;:' given in thepreceding article,is tooremarkable to beomitted. f'xSuppose cf>x=f'x,andletthesuccessive convergents tofxbe called 1~xtn-'.lXtn-Ix Tlx'TsX'''Tn_Ix' T"x' wherethesubscript index totorTindicates thedegree in x.Thenif we calltheroots offe,h,.,~...h,.,thetheorem alreadycited in a preceding 510 Ona Theory oftheSyzygetic Relations [57 article,concerning thedenominators oftheconvergents, may be expressed as follows:- I(f'hl)', I\¢hl .(TIhl'f, (T,~'f,~ty...(p~:r (TI~)'•••(TIh,,)' (T,~'f '"(T.hn)1=0, (Tn_lhl)', (T n-Ih2) '...(Tn-Ih n)· whereitwill be observed thatthefirst line of termsconsists exclusively ofunits,sincef'x=¢Xbyhypothesis. Correlatively I haveascertained thatpreserving thesameassumption ¢'k f"kthat¢x=f''«,sothatconsequently ffmeansIk-,the following theorem obtains,namelythatifkl,k•...k"_1are the(n-1) roots of ¢X' (<1>'/,;)'fk,' (tl(I'I»)', {t.(I'I)}2,(¢'k)•...jk;:1 {t1(I's»),'"(tl(kn-IW (t.(I·.)}2...{t,(I'n_I»)2=0. (tn_.(k~I»)" {tn-I(k·2)}2•..{t'&-\l(kn-I))t I Itmayconsequently beconjectured, when¢andfareindependent functions ofxandrespectively ofthedegreen- 1 and n,andf<P~isexpanded under ;J; theform of a continued fraction, of which, as before, ~,i...tT."-1arethe II " successive convergents, thatwe shall have analogous determinants tothe twinforms above given, each separately vanishing, thesemoregeneral determinants differing only from theirmodel forms in respectof theupper­ most line of termsintheone ofthem,beingeachmultiplied bycertain functions ofhi,hs...hnrespectively (all of which become unitswhen¢X=f'x), and in the otherofthembycertainfunctions of k."I'k,...k". Theexactform, however, of such functions, and even thepossibility of such form beingfoundcapableofmakingthedeterminants vanish,remains open for furtherinquiry. 57J oftwo Algebraical Functions. SECTION IV.511 OnsomefurtherFormulce connected with M. Sturm'stheorem, and on the TheoryofIntercalations, whereof that theorem maybe treated as a corollary. Art. 44. As preparatory to someremarksaboutto be made on theformulas connected withM.Sturm'stheorem, itisnecessary topremise twotheorems ofgreatimportance concerning quadratic functions, one of which, notwith­ standing itsextreme simplicity, is as far as I know very little(ifatall) known, and theotherwas given in partmanyyears ago by M. Cauchy, but is also not generally known. Theformer of thesetwotheorems is asfollows. Ifaquadratic homogeneous function of any numberofvariables be (asitmay beinaninfinitevarietyof ways) transformed intoafunction of a new set of variables, linearlyconnected by real coefficients with the originalset,insuch a waythatonlypositiveandnegative squaresofthenewvariables appearin thetransformed expression, the numberof suchpositiveandnegative squares respectively will beconstant for agivenfunction whatever bethelinear transformations employed. Thisevidently amounts totheproposition, that if we have 2npositive andnegative squares of homogeneous real linear functions ofnvariables identically equaltozero,thenumber ofpositive squaresand ofnegative squares mustbeequalto oneanother, sothat forexample wecannothave 111'+11,2+...+ U,,2+U2n+1-u'n+2-U'nH-...-112m identically zero when nofthevariables arelinearfunctions oftheremaining n;andthisis obviously thecase, for if theequation could be identically satisfied wemightmake un+2=Ut,Un+3=U....1~=Un-1' andwe should thenbe able to find Un+1as a real numerical multiple ofUn, andconsequently should have theequation un2{I+k'}=0,which is obviously impossible;afortioriwe may prove thatintheidentical equation existing between thesum of an even number ofpositive and ofnegative ~quares of reallinearfunctions ofhalfthenumber ofindependent variables, there cannotbemorethana difference of two (as we have proved thattherecannot bethatdifference) between thenumber ofpositive andnegative squares. Hencetheremustbe as many of one as of theother;and as a consequence, thenumberofpositivesquaresor ofnegative squaresinthetransform of a givenquadratic function of any number ofvariables effected by any set of reallinearsubstitutions isconstant, beingin fact some unknown transcen­ dentalfunction of thecoefficients of thegivenfunction. I quote thislaw (which I have enunciated before,butof which I for thefirsttimepublish theproof)underthename ofthelaw ofinertiaforquadratic forms. 512 OnaTheoryoftheS1Jzygetic Relations [57 Art. 45. The othertheorem isthefollowing. Ifanyquadratic function berepresented intheumbralnotation-underthe form of (U1X1+~X2+...+ UTlX..)2, where U:1,~•.•Unaretheumbras of thecoefficients, and Q;1>X2...x"the variables, thenbywriting (a1X1+Uta;+...+Unx..}Jwill assume the form IU:1,~I I U:1,Ut,UtI I UhUtUn-l'UnI IalIY12+U:1 I'a,y,'+r0,'1y,'+... +0"0, y.', U:1 U1I U:1,~IU:1,UtUn-II U:1 U:1,Ut UhUt.••Un-l andconsequently thenumberof positive squaresinthereduced form of the given function will always be thenumberofcontinuations orpermanencies ofsign of theseries l'IU:1\.IU:1,UtI·IU:1,0.,.,Ut,.../U:1,0.,.11tl], ,U:1' U:1,0.,.' U:1,~,Ut U:1,UtUn theseveraltermsofthisprogression being in fact thedeterminants ofwhat thegiven function becomes when we obliterate successivelyall thevariables butone,thenallbutthatandanother,thenallbutthese two andathird. untilfinally, the last termis thedeterminant ofthegiven function with allthevariables retained. This comes to saying thatif we call the function (suppose offourvariables)/, andwrite down thematrix OJjdtjd'jdtj da:12'd~da:,' ~~'d3:-;dx: dtj d2fdtjdtj da:,da:l'da;2'd3:,dX2'da;da:,' dtj dtjdtjOJj da:,~' da:,da:,'dX,''da:,da:,' dtjdtjd'jd'j da:.da: 1'Cia:.d~' Cia:.dX.' da:l' •Foranexplanation of theumbralnotation, seeLondonandEdinburgh Philo,ophical Magazint, April 1861, or thereabouts [po243above]. 57J oftwoAl1Jebraical Functions. 513 (whereallthetermsare of course coefficients of thegivenfunction expressed asabove for greatersymmetry ofnotation), theinertiaoffwill bemeasured bythenumberofcontinuations of sign in theseriesformed of theBUCcessit16 principal minorcoa:.r:aldeterminants (inwritingwhich I.shalluseingeneral dtj)(r,8)todenoted:xrd:x.' I,(1, I).(I,2)I. (2,I),(2, 2)(1, 2), (2, 2), (3, 2),1,(1, 1), (1, 1), (2, 1), (3, 1), (4, 1),(1, 2), (2, 2), (3, 2), (4, 2),(1, 1). (2, 1), (3, 1), (I,3), (1, 4) (2, 3), (2, 4) (3, 3), (3, 4) , (4, 3), (4, 4)(1, 3) (2, 3) , (3;3) and in like manneringeneral.", Art.46.Reverting now tothesimplified Sturmian residues, since by thetheorysetoutinthefirstSection thesediffer from theunsimplified complete residues required bytheSturmian methodonly inthecircumstance oftheirbeingdivested of factors which are necessarily perfectsquaresand therefore essentially positive, thesesimplified Sturmians may of course be substituted forthecomplete Sturmians forthepurposeaof M.Sturm's theorem. Theleadingcoefficients in thesesimplified Sturmians, reckoning j'(e)asone ofthem,will be m!.~(l~,h.),Inh.,b«,h.)...~(h1Jh....h...), whichitiseasily seen, asremarked long ago by Mr Cayley, are thesuccessive principal minor coaxaldeterminants ofthematrix '0'Utn+., .•.o:~, •I have given adirectitpoIUriori demonstration in theLondonandEdinburgh PhilolophicaZ Magazi1U!, thatthenumberofcontinuations of sign in any seriesformed like the above from a symmetrical matrix,isunafJected byanypermutations of the lines andcolumns thereof,which leavesthesymmetry subsisting, thatistoBBy(using the umbra! noiation),if11.,112,11,.••11,are disjunctively equal,eachtoeach,inanyarbitrary orderto 1, 2,8...i,thenumberofcontinua­ tionsofsignin theseries 1,Ia••I,Ia••,aftI,Ia.paft,a.sI,..,Iaft,aft,a.s,.•a'lj,a.. aft,a'2a.l,a.t,a.,a••,aft,a.""a" isirrespective of the order of the naturalnumbers 1, 2,8...iin thearrangement 11.,11"11,•.•11" ~ ~ 514 OnaTheoryoftheSyzygetic Relation« [57 I{ n~,h,) .} (x-~)(i:::h,), I n~,h"h,) nJI.-"h,h".) . (x-~)(x-hi)(x-h,)'...(x-hl)(x-h,)(x-h.,.),where in general Ur=h{+h,r+..,+h".r,and of course Uo=m. M.Hermite hasimproved upon this remarkby observing, whatisimmediately obvious, thatifweusea;to denote, not thequantity abovewritten,but ~rh,r h".r--,+----,:-+...+------r-'x-II.-,X-'''t X-r.... thesuccessive coaxal determinants of the above matrixwill become re­ spectively I1x-hI' thatis tosay,these successive coaxal determinants, whenmultiplied up by (x,will become respectively I(x-h,)(x-h,)...(x-h".),Inhl,h,)[(x-h,)(x-h.)...(x-h".)},... ~~(hl'h,...h".), thatis tosay,willrepresent the simplified Sturmian series given by my general formulee, M.Hermite furtherremarks, thatthematrix: formed afterthisrule will evidently bethatwhichrepresents thedeterminant of thequadratic function (which may betreatedasagenerating function) 1I--h{~+~~+hl'Us+...+hlm-Ium}',e-« I in which, since only the squared differences of the termsin the(h)series finally remain in the successive coaxal determinants, we may write (x-hI)' (x-h,)...(x-h".)simultaneously inplaceof~,h,...h.,.withoutaffecting the result;consequently thegenerating function above may bereplaced by the generating function 1I--h{ul+(x-~)Us+(x-hI)'U1+...+(x-hl)m-Ium}',x- I thecorresponding matrixto which becomes I~h' 80,01'"Ofll-2'x-I 01,0"...Om, ...0__1, 57J ojtwo A19ebraical Functions. 515 1f'xwhere (J,denotes I(x-h)', andI--;:-=fi-' Hence every simplifiedx-'''l X residue is of theform j'xx(J1>(J,•••e: (J"(J,...(Jr+!+fxx0,(Jo,(Jl (Jr (Jo,(Jl (Jr+! Theresidueinquestion will be of the degree m-r-2ine,andconsequently we have, according to thenotation antecedently used for thesyzygetic equations 81>(J..••.(Jr (J"(J,•••(Jr+l 0,(Jo,(Jl (Jr (Jo,(Jl (Jr+! - Tr=(Jl (Jr+-. Elegant and valuable for certainpurposes asare these formulre fort,.+! andTr,theyare affected with the disadvantage ofbeingexpressed by means offormulas of a much higherdegree in the variablexthanreallyappertains tothem,theparadox (ifit may be termedsuch) being explained bythe circumstance of the coefficientsof all thepowers of xabovetherightdegree being made up of termswhichmutually destroyoneanother; uponthe face of the formulee, t,.+!andTrwhichare in fact only of the degrees r+1 andrrespectively inxwouldappeartobe ofthedegree 1+3+5+...+(2r-1), thatis ofthedegreerI. Art.47.Imay add the important remark,which does not appearto have occurred immediately to my friend M. Hermite when he communicated to me the above most interesting results,thatin fact, by virtueofthelaw ofinertiaforquadratic forms,we may dispense with any identification ofthe successive eoaxal determinants of thematrixtothegenerating function I~{Uj.+hlUt+~IUa+...+~m-lum}"P-'''l . with my formulse fortheSturmian functions, andprovea1Jinitiointhe most simple manner, thatthe successive ascending coaxaldeterminants 33-2 516 Ona Theory oftheSyzygetic Relations [57 (always of course supposed to be takenabouttheaxis ofsymmetry) ofthe matrixtotheform above written,or tothemoregeneralform (which I shall quoteas(0),namely) ~(p-h1'fllc/>l (~)~+4>1(}~)~+...+4>",(~)U",}I, (0) (where 4>1>4>1...4>",areabsolutely arbitrary integral forms of function with real coefficients), will formarhizoristic series in regardtofu;(thatis a series, thedifference between thenumber ofthecontinuations of signbetween thesuccessive termsof which corresponding to twodifferent values of pwill determine thenumberof real roots of u;lyingbetween such two assumed values), provided only thatqbeanodd positive or negative integer. Nothing can beeasierthanthedemonstration, forwhenever pisgreaterthanany one ofthereal roots as~:- Firstly,anypairofimaginary roots will give rise to two termsofthe form (l+m";-l)q(v+w..;-1'fand(l-m..;-l)q(v-w..;- 1)1, or more simply and(L+M";-I)(vI-w+2vw";-1) (L-M..;-1)(vi-W-2vw..;-1), wherevandware real linearfunctions of 1~,Us...Uno.Thesum of which couple will be 22{L(v2- w)-2Mwv}=L{(Lv-Mw)"'-(L2+M2)WI}=~-ql; 80thateachsuch couple combined will for every value of u;give rise to one positiveand onenegative square. Secondly, anyreal root of theseries ~,h,...h""whenpistakengreater thansuch root, will give rise to a positivesquareofareallinearfunction ofUt,l"-l.••u".. Thirdly,any real root of thesame series, when pisbeneath it invalue (qbeingodd),willgive rise to thenegative ofthesquareofareallinear function ofthesame.Hencethenumber of real roots between ptaken equal to one value (a),andptakenequal to any othervalue(b),will be denoted bytheloss of an equal number of positive squaresinthereduced form of the expression (0)whenpistakenaand when pistakenb; thatis byvirtueof Art.45will bedenotedbythedifference of thenumber ofpermanencies of sign in thesuccessive minor determinants ofthematrix corresponding tothequadratic form(0)-(which we have takenasour *Theinertiaof thequadratic form (G) is themeasure ofilienumberofrealroots offz comprised between coandp.andmaybeestimated inanymannerthatmaybefoundmost convenient. Ifpbemadeinfinity,andtf>.hbetakenequaltoh·-I,andtheinertia.ofthecorre­ sponding valueof (G)beestimated bymeansof the formulm in ordinary usebygeometers for 57J oftwo Algebraical Functions. 517 generating function) resulting fromthesubstitution respectively ofaand bin place of p,which gives a theorem equivalent tothatof M.Sturm, transformed by myformulee, when we choose toadopttheparticular suppositions q=-I,ep/t=I,ep/t=h,ep,h=hi,...ep,,.h=hfJ&-l. Thismethodofconstructing arhizoristic series toIxby adirectprocess isdeserving ofparticular attention, because it does not involve the. use of the notion of continuous variation, upon which all preceding proofs of Sturm's theorem proceed. Itcompletes the cycle of the Sturmian ideas.Happily this cycle wascommenced from the otherend, foritwould have been difficult tohavesuspected thattheroot-expressions for theterms in the rhizoristic series could be identified with the residues, hadthe former been the first tobediscovered, and much of the theory of algebraical common measure laid open by means of this identification would probably have remained unknown. Art.48.I proceed now to consider a theorem concerning therelative positions of thereal roots of two independent algebraical functions as indicated by the succession ofsigns presented bytheirBezoutian secondaries; this more generaltheoryofintercalations orrelativeinterpositions willbe seen to include withinit as a corollary thejustlycelebrated theorem of M.Sturm. Letthe real roots of Ixtakenindescending order of magnitude be hI'h«...hp,and the real roots of epxtakeninthelike order "II'"II'"1Jq, sothat Ix=(x-~)(x-~)(x-hp)H, epx=(x-1JI)(X-"II)(x-1Jq)K, HandKbeing functions of xincapable ofchanging theirsigns. Now, as in M.Sturm'smethod, let us inquirewhattakesplace inrespectto the sign of j~:~,which I shall call theIndicatrix, asxdescends thescaleof real magnitude from+coto -eo.Ifbetween+ocandhI'ireal roots of epxare contained, it is obvious thatasxtravelsfrom+X)to thesuperior brink ofhI'theIndicatrix willchangeits sign from +to - and from - to +alto­ getheritimes, so thatatthemoment whenxisabouttopassthroughhI'it detennining thenatureof asurfaceof theseconddegree, the criteriaof thenumberof realroots in1zwillbe,ormaybemadetobe,symmetrical inrespecttothetwoendsof theexpresaionfe. Thissystemofcriteria, however, is not 80good as thatgiven by theBezoutiant to the two differential coefficients off(z,1)takenwithregardtozand 1respectively, whichwillalso pollllll8Sthe like character ofsymmetrical indi1Jerence, andbe one less in numberthanthe former. 518 On aTheoryoftheSyzygetic Relations [57 will bepositiveifiis zero or even, andnegative ifiisodd;butthemoment afterIXhaspassedthrough thevalue ~,theindicatrix will benegative onthefirstsupposition, andpositive ontheothersupposition. Hence immediately afterthepassage ofxthrough hitheindicatrix will have been onceoftenernegative thanpositive ontheonesupposition, andasoften negative aspositive ontheother.Again, in likemanner asIXtraverses theinterval between hiandtheinferiorbrinkof~,ifno'"oraneven numberof",'soccupythisinterval, thesign which theindicatrix hadatthe beginning ofthisintervalwill have been reversed onceoftenerthanrestored; butiftherehe an odd number of",'s80interposed, thenumberofreversals andrestorations will have been identical; andso for each successive interval, reckoned fromavalue for IXimmediately subsequent to onerealroot oflx, down to avalueimmediately subsequent tothenextless real root of the same;and it is evidentthattheeffect upon thesignoftheindicatrix at theend ofeverysuchintervaldepends, notuponthenumber of",'sgrouped together in such interval, butupontheform of thegroupasregardsits beingmadeup of an odd or even numberofterms,thefirstintervalbeing of course understood toextendfrom+00toavalueimmediately inferior tohi,andthelastfromavalueimmediately inferiortohpto -00.Hence asregardstherelationofthesign oftheindicatrix atthebeginning tothe signattbeend ofeverysuchinterval, notbing willbealteredbytaking awayanyevennumber of",'sthatmay be found tberein.Ifwesuppose tbisto be done, we shalltbenbave in Borneoftheinterval!' one",occurring and intheotherintervals no"';thatis to say, some of theh'swill be separated by single ",'s,butotherh'swill come together. Again, by removing anyevennumber ofh'snotseparated by",'s(andthusremoving aneven numberofintervals), it is clear thatasmanychangesof sign of theindicatrix will have been done away witbfrom+to - as from - to +,and no effect upontheexcess of theonekindofchanges ofsignovertheotherkind of changes of sign will have been produced. Byremoving pairsofh'sinthis manner,itmayhappenthat",'swillagainbebrought together, anyeven numberof which, not separated byh's,mayagainberemoved andthenpairs ofh'snotseparated by",'sintheirturn,and socontinually totiesquotiesuntil atlengthwemustarriveatareduced systemofIt'sand",'s,where no two h'sandno two",'scometogether, or else all theh'sandall the",'swillhave disappeared. Letthescale of h'sand",'sthussimplified andreducedbe calledtheeffective scale of intercalations. Thenumberofh'sandthenumber of",'sin any such scalewill beequal,or willatmost differ from one another byaunit,sinceateachpartofthescale,exceptattheend, every his followed by an '"andevery",by anh.Ifthescalebeginsandendswithan h,therewill of course be one more hthann; ifitbeginand end with an"', therewill be one more",thanh ;ifitbeginwithanhor an'"and end with an'"orh,therewill be as manyoftheone as of theother. 57] ojtwoAlgebraical Functions. 519 Firstly,suppose theeffective intercalation scaletocommence withanh ; theninpassingfrom+00tojustbeyondthefirsththesign oftheindicatrix j;changesfrom+to-;itchangesagainfrom - to +asitpassesthefirst TJ,thenagainfrom+to -asitpassesthesecondh,and so on jthatis to say, therewill be a changealways in thesamedirection from+to -asxpasses frombeingjustgreaterthantobeingjustlessthananyhappearing inthe effective scale. Secondly, if theeffective scale beginwith"1,theindicatrix willconversely benegative afterpassingthefirst and every subsequent TJ,andchangefrom - to+intheactofpassingthrough thefirst and every subsequent h.Sothatoneithersupposition thechanges of sign for the effective scale always takeplace in thesamedirection, andthenumber ofh'sintheeffective scale will bemeasured bythenumberof such changes, andconsequently will bemeasured bythedifference between thenumber oftimesthattheindicatrixj:changes itssign from+to -asxpasses through each inturnofthereal roots of fe,and the numberof times tha.t inpassingthrough any such root it changes its sign from - to +jifthe formernumberbegreaterthanthelatter,theeffective scale of interpositions will begin with a root of fxjif it be less, the scale will beginwith aroot ofcf>:x.Ifinsteadofbeginning with+00andendingwith - 00we begin and end with any two limits, aandbrespectively (making abstraction ofallroots offxor ofcf>:xlyingoutsidetheselimits,and forming theeffective inter­ calation scale with therootscomprised withintheselimitsexclusively), we shall obviously obtainasimilarresult,butwiththecondition thatthe changes from+to - will be inexcessif an even number ofh'sand"1'S combined be cutoff bythesuperior limit, and theeffective scale begin with anh,orif an odd numberof h's and "1'scombined be so cutoff andthescale begin with an "1;andin defect if an odd number ofh'sand"1'Scombined be socutoff and the scale begin with an h,or an even numberbe socutoff andthescale begin with an "1.If,now,supposing fxto be of n,and4>x ofnotmorethann,saymdimensions, we form thesignaletic seriesfa,4>x, BI,B•...Bm(wheretheBI,B....BmaretheBezoutian secondaries or simplified successive residues corresponding toj:expanded undertheform of an improper continued fraction), it may be shown, in thesame way asfor Sturm's theorem, thatwheneverj;changes from+to - achangeof sign will begainedintheseries, and when from - to +achaugewill belost; andthatnochangecan begainedorlostexceptasxpassesthrough the successive real roots of fe,Hencethedifference between thenumber of changes of sign in theabovesignaletic serieswhenxistakena,andthe number ofthesame when xistakenb,willindicate thenumber of roots 520 OnaTheoryofthe Syzygetic Relations [57 offa;remaining intheeffective scale of interpositions formedbetween such oftheroots of fa;and of </>:xasliebetween aandb;callingtheone numberI (a)and theotherI (b),thesign ofI (b)-I (a)depends notonthe relative magnitudes ofaandb,butuponthemanner in which the effective scale commences; ifI(a)-I(b) is positive, theeffective scale formedbetween theaandbwill commence with a root of fa;jifnegative, itwill commence witha root of </>:X. Art. 49. Informingthescale of effective interpositions, it isevidently notnecessary to go on reducing thehseries and the'rJseriesseparately andalternately; thesameresultwill be effected more expeditiously by elidingsimultaneously any even numberofh'sthatcometogether without beingseparated by ann.and any even number of'rJ'sthatcometogether withoutbeingseparated byanIt,and,repeating thisprocess of simultaneous elision,asoften as may be required. untilno twoh'sor'rJ'scometogether. Tbus, for instance, denoting themagnitudes of theseriesof real roots of fand of¢bythedistances ofhand'rJpointstakenalong arightline from afixedpointtherein,andsupposing such series of roots between thelimits aandbto be hhh'rJ'rJ'rJh'rJ'rJh~'rJ'rJhh'rJh'rJhhhhh'r}'rJ~ our first reduction bringsthisscale totheform h'r}hh'T/'rJh'r} hhj thenextreduction bringsittotheform h'rJ~'rJh'rJj and athirdandfinalreduction bringsittotheform h'rJh'rJj andaccordingly we shall find for such an arrangement of thehand", system I(b)-I(a)= ±2. dfs:Art.50.Ifwe suppose </>:x=(],X,byawell-known theorem ofalgebra, any two consecutive roots offa;willcontainbetween theman oddnumber of roots of epa;,andthenumberof real roots of f'a;greaterthanthegreatest root offu;andthenumberof real roots of f'a;lessthantheleastroot offa; will each be even. Hence theeffective intercalation scalebetween any two limitsaandbwill be formed by merely reducing the'rJgroupstosingle units,andthenumberofh'sinthescale soformed will be thetotalnumber ofh'sbetween thelimitsaandb.Moreover, since such scale commences always with a root of fe,or with an even numberof roots off''»followed by 57] oftwo A1gebraical Functions 521 aroot offa;ifthenumberof h's and 'TJ'Scutoff be even, andwitharootof f'xor an even numberof roots of fxfollowed by aroot offe,ifthenumber 80cutoff beodd,itfollowsthatforthiscaseI(a)-I (b),abeingthe superior limit, will be always positive, andwillmeasure thetotalnumber ofrealroots offxlyingbetween aandb;this,then,isSturm's theorem, treatedasacorollary to theTheoryofIntercalations. Art. 51.Ifwewritedownthelastsyzygetic equation betweenfxof m and4>.xofndimensions, namely Tn-I(x)fx-tm-I(x)4>.x+~o=0, ithas been shown thatthesuccession of signs in theseries formed withfx, 4>xandtheirsuccessive Bezoutian secondaries willcontainthe same number ofcontinuations andvariations asthe series formed with fs;tm-I(a:),and theirsuccessive Bezoutian secondaries. Thisindicates thattheeffective scale of interpositions forfxand4>xwillcontainanequalnumberof roots offxwiththeeffective scale for fxandtm-I(x);thetwo scales however will notnecessarily beidentical, becausetheroots of 4>.xwillnotnecessarily beinthesameorderrelativetotheh'sintheone scale asthose oftm-1(x) relativetotheh'sintheotherscale. This equalityisperfectly wellexplained dposteriori by the form of tm-I(x),which by theformula in SectionII.will berepresented by ~(h)(h)(') q,hq,q,hq•..•q,hqm_•..a;-'I.a;-q....a;-ftq"'-I(h 9m_kg')(kq•_h'll)...(k==q-'-.----:-hq...---:I)' Now,whenever xisindefinitely neartoanyone oftheroots offx,ashq"" thissumreducestothesimpleexpression andconsequently intheimmediate neighbourhood of every real root of fa, 4>.xandtm-I(x)will have always thesame or always a contrary sign, according asq,kq,q,hq•...cfJhq",ispositiveornegative, which will dependupon therelative disposition ofthereal roots in fandq,;ineithercasethe effective scale of interpositions forfxwithq,xand for fa;withtm-1xmust containthesamenumber ofh's;butthedifference will be, thatif cfJhlq,~...cfJhmispositive anhwill occupy thefirst place in each scale, or thesecond place in each scale;butifnegative, thenin one scale anh will occupy thefirst place, andintheotherscalethesecond place. ,A.rt.52.Thesame process of common measure orresidues which serves tofurnisharhizoristic seriesforfeorasyrrhizoristic seriesforfxand4>.x, will serve also to furnish superior andinferiorlimitstothereal roots of a.ny proposed equation. Thussupposefxtobeanyrationalintegralfunction of 522 OnaTheoryoftheSyzygetic Relations [57 wherea;ofthedegreenandepa;anyotherfunction of(1;,which I shall begin withsupposing to be of thedegree(n-1),andletthesuccessive quotients resulting fromtheprocess of findingthegreatest common measure offe,ep.x continued untilthelastremainder isnotaconstant butzero, be supposed to be (as theymaygenerally betaken,butsubjectto cases of exception, which will hereafter bealludedto)nlinearfunctions q1,q2...q..;thenwe shallhave cf>a;1 1 1 1 fi--= - - ...-----J a;ql+q2+q..-1+q.. andtherefore cf>a;=KN, fa;=KD, whereNisthenumerator andDthedenominator ofthecontinued fraction andKis aconstant; thevalueofthisconstant isimmaterial butis in fact t;L2'Ll&±L12L3'Llc., Lo,L1,L.,L•.&c. being the leadingcoefficients of the last, thelastbutone, thelastbuttwo,&c.of theBezoutian secondaries tofa;andepa;.Accordingly, ifn=1,letD=q1=JIrI; ifn=2,letD=q2q1+1=JIrI{q,+~}=""lfLt; ifn=3,letD=q.{qllq1+ 1) +q1=""lILli{q.+~}=JlrIfLtf'a; ........................................................................... and ingenerallet D=JlrIf'2f's.••,.",., 1 1 1 ""1=ql',.",=qll+-,f's=q3+-,...,."..=q"+--. ""1 f'2 f'n-1 Now suppose a;to be sotakenthat q1doesnotliebetween+1and-1 q2"+2and-2 q." "+ 2and-2 q. +2and-2, (01) "., ................................................ qn-1" "2 and-2 q"" "1 and-1 whereitwill be observed thattheexcluded region lies between:+2and- 2 for alltheintermediate quotients, butbetween only+1 and - 1 for thefirst 57] ojtwo Algebraical Functions. 523 andlastquotients. Then J1'J.ispositively ornegatively greaterthanI, therefore.!.isapositive ornegative fraction; butq.ispositively ornega­ POI tivelygreaterthan2;therefore IJoiwill be of thesamesignasq2Jand also IJoi will bepositively ornegatively greaterthan1;therefore.!. will be a positive IJoi ornegative fraction; butq.ispositively ornegatively greater than2; therefore IJoiwill be of thesamesignasq«,and also IJoiwill bepositively ornegatively greaterthan1;andproceeding inthisway, we find thatall valuesof!Joi.fromi=1 toi=n-1,willbe ofthesamesignasqi,and positively ornegatively greaterthan1.Finally, _1_will beafraction, I'on-I andtherefore, sinceqnispositively ornegatively greaterthan1,p...=q..+1_ PO..-I willhavethesamesignasqn(butofcourseis notnecessarily greater than1.nor would thatcondition serveanypurpose wereitsatisfied). V-le inferconsequently, thatwhentheconditions (ro)aresatisfied, POl>1Joi, IJoi...po,.willrespectively havethesamesignsasql>q2...q..;andtherefore D=IJoIIJoiIJoi•••PO..hasthesamesignasqlg2qS...q...Nowsuppose ql=~x+bl,q.=a.zx+b•...q..=a"x+b.., andsolvethe2nequations ~x+bl=+CI,a..:x+h.=+C•.••aJl-Ix+bJl-I=C.._I,a"x+b..=c.., alx+bl=-CI•a..:x+h.= -c•...a.._lx+h"_1= -Cn-I.anx+b..= -Cn, where C1=1,c.=2,Cs=2...Cn-I=2,Cn=1. Whenever inanyone ofthenpairsofequations abovewrittenthecoefficient ofxispositive, theupperequation ofthepairwillbringoutthegreater valueofx;butwhenthecoefficient isnegative thelowerequation willgive thegreatervalue.Takethepair a.x+hi=Ci, a.x+bi=-Ci. IfltiispositiveaiX+hiwillalwaysbepositive, andgreaterthanCi.between x=coandx=thegreaterofthetwovaluesofx;ifltiisnegative a.x+hi willalwaysbenegative, andless(thatisnearerto -co)than-Ci,forall values of xbetween thesamelimitsasbefore. So againitwill beseen in likemanner,thatwhether a·ibepositiveornegative, between a:= -coand x=thelesserofthetwovaluesofxcorresponding totheabovepairof equations, a.x+hiwillalwaysretainthesamesign,andwill begreaterthan +Ci,or lessthan-Ci,according asa.isnegative orpositive. If,then,we 524 OnaTheoryoftheSyzygetic Relations [57 takethegreatest ofthegreaters ofthenpairsof values of x,thatisthe absolute greatest of the2nvalues, and the leastofthelessers,thatisthe absolute leastofthesame,sayLand A,thenbetweenLandA,gilqa...q«will each always retainaninvariable sign,andwillthenfallwithoutthelimits ±Cl,±o,•...±c,,-1I±C",sothatbetween+00andLandbetween Aand - 00 , P-llJ-t•••P-",thatisaconstant multiple off(x),willretainthesamesign as qlg,...q",thatis willneverchangeits sign from thebeginning totheend of oneinterval, nor from thebeginning totheend oftheother;and con­ sequently LandAwill beasuperior andinferiorlimitrespectively tothe real roots of fa:Itwill of course be observed thatitisindifferent forthe purposes oftheforegoing theorem, whetherj:beexpanded undertheform of a proper or an improper fraction,thatiswhether we employ theordinary ortheSturmian process of successive division; forchanging the signs of the residues will only have theeffect of changing qiinto(±)qi,andthepair ofequations (±)qi=±c.remains thesamewhether the+orthe- sign be prefixed toqi.Theresultis,thatif we form the211quantities ±1 -hI±2 -b,±2-ba±2 - b"_l±1 -btl ~~- al Cl..l al a"_l Un thegreatest ofthemwill be a superior, andtheleastof them aninferior limittotheroots offx·. Itmay beremarked thatifthesuccessive dividends inthecourseof theprocessbemultiplied respectively by~,k,...k",j;willtakethe form klk,klk" ql+ql+ql+...q"; and if we write ~x+bl=±Cl,arc+b~=±<;•••anx+btl=±C" and make Cl=1,Ca=1+kt,Ca=1+ka...c"=1+k", thesamereasoning as above will show thatthegreatest andleastofthe2n quantities ±(!_+k,)-ba±(1+k,,)-bn-l±1-btl Ut a"-l an will beasuperior and inferiorlimittotheroots offe. Forgreatersimplicity, again,consider kll~...k"tobeall equal to unity; we may makethisaddition tothetheorem as above stated.namelycalling •Forageneralization andimproved form ofstatement of thistheorem IlOOSupplement to thepresentSection. 57] oftwoAlge1Jraical Functions. 525 ~,A,;Lt,At...L",A"thegreatest andleastvalues of thetermscontained respectively intheseriesmarkedbelow1,2,3...n,namely- ±I-b,±2-b ll±2-bt±2-b_,±I-b" ~ ~ a, a_I Un ±I-bll±2-bs±2-bfl,-'±I-b" a, a, UfI,-l Un ±1-bt±2-b_,±I-b" a, a-l Un ±1 -b_,±1-b" a-l Un ±I-b" Un(1) (2) (3) (n-1) (n) ~,A,;Lt,At..,L",A"will berespectively superior andinferiorlimitsto/IX, ~andtheirsuccessive residues. As a corollary, we see, of course, thatL andA,thesuperior andinferiorlimitstotheroots of thegivenfunction/IX, mustalwaysliebetween+00andthegreatest root,andbetween -00and theleastroot, ofthearbitrarily assumed function ~ Art.53.Letus nowassumesomewhat moregenerally that4xcisany numberofdegrees0,inIXlowerthan/IX,which will cause thefirstquotient q.,to be of thedegree01in:r;;andletusfurthersupposethatc/Ja:standsin such arelation to/IXthatthefollowing quotients, q't' q't ... q.p'are ofthe degrees Oil'Ot...OpinIX(Oil'0....Opbeingsupposed notnecessarily units, astheywouldgenerally be,butanypositive integers whatever. asmay happeninconsequence of one or more of theleadingcoefficients in any residuevanishing); then c/Ja:__1__1__1_+~ f:r;-q.,+q't+q't+ ...q.p' where0,+Ot+O.+ ... +Op=n;andconsequently fIXwill beequaltothe denominator ofthelastconvergent abovewritten,multiplied byaconstant, 80thatwe have now cfo:=m,lI~.,.fflp,where 1 1m1=q'l'~=q.+- ...mp=q'P-l+--. t,n, mp-I Andasinthecasepreviously considered, so longas qe,(:rI ),qet(:r2 ),q't (:r2 ),... <-1 <-2 <-2 fIXwillhavethesamesignasqe,q.....q.p' 526 OnaTheoryoftheSyzygetic Relations [57 Letnow g'l=±CIlq.,=±Ct•••g.p=±Cp, where ~=1,Ct=2...cp-1=2,cp=1. Consider anypairoftheaboveequations asg••'-c.?=O. Firstly,suppose all theroots of thisequation areimpossible jg.,1-ct mustbepositivefor allvaluesofe,andg'lcanneverliebetween+Caand -Ca;moreover, since upon thehypothesis made,g..+Caandg'l-c,always retainthesamesign,namely,thatofthecoefficient of thehighestpowerof g,"itfollowsthatg'lmustalsoalwaysretainthesamesign;for if we con­ strnctthetwo curves y=g'l+Caandy=g'l-Ci.thesewillbothlie onthe sameside oftheaxis ofa,andnevercuttheaxis,consequently thecurve y=g,"which lies between them,mustalso lie on thesameside aseitherof them,andnevercuttheaxis. Hence,then,iftheroots of theequation are all impossible, g'Iwill alwaysretainthesame sign, and will neverfallwithintheregionbounded on two sides by +Caand -Ca. Secondly, suppose theequation to have one or more possible roots, and Iito bethegreatest, andx...theleast(which of course, if thereisbutone possible root, will be identical). Iftheleadingcoefficient of q'lispositive, thegreatest root(I)oftheequation g'l-Ci=0 will exceed thegreatest root (I')oftheequation g'l+Ci=0;forbetween x=aoandx=I',q'lmustgo through all values intermediate between aoand -Ci;hencetheremustbe a qualityIintermediate between l'and+ac,which will make q..=Ca.In likemanner,iftheleadingcoefficient of g,"isnegative, itwill be seen that thegreatest root ofg..+Ci=0 will exceed thatofg.,-e,=O.Moreover, intheone case g'lwill be always positive andgreaterthanCi,andinthe otheralwaysnegative and less thanCi.Ineverycase,therefore, between +aoandli,q••retainsthesame sign, and does not fall withintheregion bounded by+Ciand - Ci;thesamethingmay be shown to be truefor all values of xbetween -aoaridA;.Hence,then,bythesamereasoning as thatemployed in thepreceding article,we areenabledto affirm, thatif we form the equation (g"t-I)(g.,t_4)(g••I-4).,.(q\r-I-4)(q'p'-I) =0,("") itsgreatest root will be a superior limit,and itsleastroot aninferiorlimit totheroots of theequationfx=0,whatever bethevalue of theassumed function cf>xjand if the above equation ("")has no real root, all theroots of fxwill beimaginary. Art. 54. Inthepreceding twoarticlesithas been supposed thatallthe quotients aretakenintegral functions ofx;buttheprocesaofsuccessive division may be80conducted as to give rise to quotients oftheform ,b-'-l d lax+ar+...+c+-+...+-;;.xx' 57J oftwoAlgel»-aical Functions. 527 Suppose thenthatwe have in general epx=_1__1_+! fxql+qi+...q..' where ql' qi...q..are each of thegeneralform above written(butof course iandi'being not necessarily the same for any two of thequotients), and supposethatthesum of the degrees in xofql,qi".q..isn+t,wheretis essentially (asitmust be) positive. Thenwe shall find, asin the last article, thatLandAbeingcalled the greatestand least roots of (q1t-I)(qit-4)...(ql.._1-4)(q..t_I), D,thedenominator ofthelastconvergent to thecontinued fraction above written,willneverchange its sign between+00andL,nor between A and -<Xl;buthere we shall have fx=Kaf xD. Hencea;lDwill beinvariable in signwithineach of these two intervals. Firstly,lettbeeven;thenfxwill beinvariable in sign, whatever L and A may be for each such interval. Secondly, let tbeodd;thenifLis>0 and A<0,fxcannotchange itssign ineitherinterval; butifLis<0 or A>0,fxwill change its sign as xpassesthrough zero,butwill beinvariable for each of thethreeregions contained between+coandL,Land0, or 0 and A (asthecasemay be), and A and - co;sothatuniversally Land A will be a superior and inferior limitto the roots of fx,makingabstraction oftheroots(ifany such therebe infx)whose value iszero. Art. 55. I shall close thissection with offering (for what itis worth) abaresuggestion asto the mode in which thetheoryofIntercalations may hereafter be found to admitof being extended from a system of two general functions ofe,to a system of threegeneralfunctions of e,y,fourgeneral functions of e,y,z,and ingeneralto a system of Egeneralfunctions of E-1 variables, or which is thesamething,ofI:homogeneous functions of I: variables. Inthecaseof two functions of e,fxandcf>x,fx=0 andepx=0 maybeconsidered to represent twosystems ofpointsin arightlinejand thetheoryrelatesinthiscaseto the relative positions of these two "Kenothemes " orpointsystems; and of course using xandyto denote the distances of anypointin a line from two fixed points thereinrespectively, insteadoffxand~,we may employ two homogeneous functions of xandy, asf(x,y)andcf>(e,y),to denote these two systemsof points. So, similarly, ifwehavethreefunctions of two variables, (x,y),9(x,y),h(x,y),which Ishallsuppose to be of thesame degree, we may consider the mutual relations oftheMonothemes, thatisto say,thethreeplane curves, denoted 528 OnaTheoryoftheSyzygetic Relations [57 bytheequations f(x,y)=0,9(x,y)=0,h(x,y)=O.Now every two of thesewillintersect oneanother in a system of points, which we may call (f,g)fortheintersections offandg,(g,h)for those of 9andh,and(h,f) for those of handfIfwetakeany two of these systems of intersections, as(f,g)and(g,h),theywill both lie upon one of thegiven curves (g). And by readingoffthetwosystems ofpoints(f,g)and(g,h),arranged according totheorderupon which they aredisposed upon thecurveg.we may, by following thecourse of such curve, form a scale of effective inter­ calations forthesetwo systems, and in like manner forthetwosystems (g,h)and(h,f);(h,f)and(f,g).Now I believe thatitwill be found that whenf,g,hrepresent anyalgebraical curvesconsisting ofasinglecontinuous line,eitherextending toinfinityin both directions. orreturning toitself (and I have fully satisfied myself of thetruthofthisfurthecase of ellipses), eacheffective scale of intercalation willcontainthesamenumberof pairsof points;if, however, thecurves consist of more thanone branch, as if hyper­ bolae be considered, such is no longer necessarily thecase;fromthesefacts, conjoined with thelightthrownuponthesubjectby itsrelation tothe theoryofcombinants explained in thesucceeding section, I am induced to infer the probability of thetruthofthefollowing law (which, for avoidance offurtheruncertainty, I confine to thecase of functions of the same degree). namely,thatiff,g,hbethreehomogeneous functions of x.y,andzofthe same degree. and if U, V, W be anythreelinearfunctions of f,g,h,andif U=0,V=0,W=0 betreatedastheequations tothreecones. and if we form an effective scale of theintercalations ofthelines ofintersection ofU andW,andVandW,according to theorder in which theyare disposed uponW(which seems to requirethatthelines shall be continuous, in order toadmitof a fixed order of readingofftheintersections of any two of them uponthethird);then,whatever value may have been giventothecoeffi­ cientsinthelinearfunctions, thenumberofelements remaining in anysuch scale will (as I conjecture) beconstant, and some theory(to be discovered) forthreefunctions, analogous to thatofBezoutian residues for two functions, will serve to determine thenumberoftheelements soremaining. And so, in likemanner,butwithadifficulty increasing ateachstep(asatthenext stepwe should have to pass into quasi-space of four dimensions), a theory of intercalations may be conjectured toexistfor anyngeneral functions of any(n-1) variables. Development ofthe method ofassigning a superior andinferior limit to the roots ofany algebraical equation. Art.(a).Since the articlesinthepreceding partofthissection on the methodof discovering limitstotheroots of an algebraical equation were written,themethodof which thegermisthereincontained haspresented 57] oftwoAlge1Jraical Functions. 529 itselfinamuch more fully developed form, which I proceed to exhibit: for greatersimplicity I shall suppose 4>a:tobeofn-1, andfa:tobe ofn dimensions ine,andthatby means of theordinary process for common measure (exceptthatas inSturm'stheorem thesigns of all the remainders arechanged)j;hasbeenthrownundertheform of the improper continued fraction 1 1 1 1------...-,ql-q2-qa-q" whereql'qa...qnare allrestricted to signify simple linearfunctions of s: Supposetheseriesql'ql'qa..'q"to be resolved into thedistinctsequences glq,···q.,q'+lq'+i'" q..,q'"..I···q...,...,q('l+1...q", insuch amannerthatin each sequence, asq'+I'q'H...q,",the coefficients ofa:have all thesame sign, butthatin any two adjoining sequences the coefficientsof a:have opposite signs, so thatforinstance inq.andq'+lthe coefficients of a:areunlike,as also in qeandg'"+I;therewill of course be nothingtopreclude any of these sequences becoming reduced toa single term. The first theorem is,thatthegreatest and least roots of theproductof thecumulants [po504above] [qlq,...q;]x[q'+lq'H'" qe]...X[qhHlq(,Ha ...q,,] aresuperior and inferior limitsto the roots of fa:.To prove thistheorem I begin with premising thetwo following lemmas, one virtually andthe otherexpressly contained in thePhilosophical Magazine forthemonths of September and October of the presentyear-[p,641 below]. •Eachofthesetwolemmata flowsreadilyfromthefacultypreviously adverted to engaged byeverycumnlant of being representable underthe form of a determinant. As to the second lemma,itbecomes apparent immediately when the cumulant is80represented. byseparating the matrixintotworectangles andexpressing theentiredeterminant according toa well-known rule for thedecomposition ofdeterminants as afunction ofthedeterminants belonging tothesetwo rectangles takenseparately. As to the first lemma, by reasonof thecumulant ["'I"'a'""'/-1"'i"'/+lJ being80representable. we know thatwhen["'I"'a"''''I-l'''/]=O. [CoIt"'a"''''I-tl and[CoIt"'S"''''I+lJ musthaveopposite signs. Suppose. now, thatthetheorem istruewhen the numberofelements in thetypedoes not exceed ijthentherootsof[CoIt"'a''''''H]' say of"'H'being called hi'hs...h,-I>and of ["'I"'S'"""-1""]' sayof"'I.being called k1•ka...k,.these may be arranged in thefollowing order of magnitude kl•hI'ka.h.."ka...k'-I' h'-I' k,;andiftheroots oq"'I"'I..."'i-l"""'Htl. sayof"'HI'becalled~, ~...I'H'from the factof theleadingcoefficients in"'Hand"'HIexpanded according tothe powers of zhavingthesamesign. it follows that whenZ=oo.""-1and"'HIhave the samesign.butthey have contrary signs when z=k1;but +'-1doesnotchange its sign between z=00andz= kl•hence"'HIdoeschangeitssignbetween z=00andz=k1•andtherefore a root of "'1+1lies between 00andk1;in likemannerpreciselyit may be shown thata root of "'HIliee between -00andk,jand since "'/-1changes itssign betweenk1andka.between ksandka...and between k'-1andkl•"'/+lmustlikewise change itssign between one and the otherextremity ofeachoftheseintervals, and hence theroots ~,z,...IH1areintercalated between 00,~.ka...k,.-00,or which is the samething.k1,ks...k, arerespectively intercalated between ~.~...IH1;consequently, if thetheorem istrueuptoi, itistruefori+1.andtherefore trueuniversally jbutismanifestly true when i= 2. forthen z=2000makes["'I"'J.thatis."'1"'s-1positive; but"'1=0makesitnegative, which proves thetheorem oont«ined inLemmaA. ~ M 530 OnaTheoryojthe Syzygetic Relations [57 LEMMAA. The roots of thecumulant [qIqll...qd,in which each element isalinearfunction of x,andwherein thecoefficient of xfor each element hasthelike sign, are all real, and between every two of such roots iscon­ tainedaroot ofthecumulant [qIqll'"qi-I],andexconverso aroot ofthe cumulant [qaqa...q;];and (as an evidentcorollary) for all values of pandpi intermediate between 1 and ithegreatest root of[qlqll...qi-Iq,]will be greater,andtheleast root of thesame will beless,thanthegreatest and leastrootsrespectively of[qpqP+l...qp'-Iqp-]. LEMMAB.For all values of the elements qIqa'"qR,thecumulant [qlqa...q..-lq..q..+lqw+2"·qn]=[qIq2'" q.. -Iq..]X[q..+!q..-r2'"qn] - [qIqll'" q..-I] X[q"+2...qn]. Thusforexample thecumulant [abed],thatis abed-ab-cd-ad+1=[ab]x[cd]-[a]x[d]=(ab-1)(ed-1) -ad, and[abcde],thatis abode-abc-abe-ade-cde+a+c+e=[abc][de]-[ab][e], thatis =(abc-a-cHde-1)-(ab-1)e. Art.(fJ).Also suppose thatqlqll'"q..q..+1...qnare alllinearfunctions of a,andthatthecoefficients of xhave all one (saythepositive) signin ql,qa'" q..,and allthecontrary signs in q ..+1•.•qn,and letLbe not less tha.n thegreatest root of[qlq2'" q..]or of [q..+I'"qn],and also letA benot greaterthattheleast root of each of thesesametwocumulants; thenby LemmaA,Land A will also be respectively greaterthanthegreatest, and lessthantheleastroots of [qIqll•..q..-I]and of[q"+2...qn].Nowthecoeffi­ cientofthehighestpower of a:inboth[qIqll'"q..]and in[qlqa...q-tlis positive,butas to[q..+!...qn]and[q"+2...qn]is ofcontrary signs inthetwo, namely,negative inthatone of those cumulants whichcontains an odd,and positiveinthatone ofthetwo which contains an even number ofelements. Hence by virtueofLemmaB,Land anyquantity greaterthanLsubstituted forxwillmake[qIqa'"qn]to have always thesame sign, and in like manner itmay be shown thatA and any quantity lessthanAsubstituted forxwill also cause [qIq2..•qn]toretainalwaysthesame sign. HenceLand A are superior andinferiorlimitsto[qlq2'"qn];and the samereasoning would evidently apply if we had supposed thesignsof the coefficients of xinthe firstpartialseries of elements to have been negative, and intheotherseries ofelements to have been positive. Thegreatest and least roots of [q,q2...q..]x[q..+!•..qn]evidently satisfy thecondition to whichLand A are subject,and may be takenin place of L and A respectively. They will accordingly be superior and inferior limitsto thecumulant 57J oftwoAlgebraical Functions. 531 Again, by virtueofLemma B itmay readily be shown that [g,g,...g."9'"l+l9...+2•••q...,9...+1'"qft] =[q,98'"g.,]X[g••+lg.,+I'" g...JX[g+1•••q..] -[g,gl···9.,-1] X[g"'+1'"q..]X[g+1...qft] - [g,q, q.,]X[g..,+1g_I] X[g+1'"g..] +[q,g,g..,_I]X[g.,+,q_I]X[q+I'"q..]j and hence if g"ql'"g..arealllinearfunctions of xin which thecoefficients ofxhaveallthesame algebraieal sign in anyone(takenper88)ofthethree senes g,g,...g••,g.,+I'"q..., q..+I'"q.., but sothatthis sign changes in passing from one series to another. itis easily seen, by thesame reasoning asinthepreceding case,thatthetwo positiveandtwonegative products ontheright-hand side oftheequation allgivethesamesignto the coefficient of thehighestpower of e,and consequently thatifLand A be superiorand inferior limitsto [g,...g..,),[g.,+I'"g.,].[g..+I'"g..]. andconsequently byLemmaA, to [g,gl'"q..,-I), [q.. ,+I'"g••],[g..,+I'"q...-I],[q.,+I'"q..-I]. and to [g...+1•••q..], Lor Asubstituted forxwillcause[g,ql...q..]toretainalwaysthesame sign. and will consequently besuperior and inferior limits thereto jand so in general jwhenceitfollows,returning tothetheorem to bedemonstrated, thatthegreatestandleastroots of [q,ql'"q,]X[qi+lqi+1...q,.]x...X[q(i)+I...qft], will besuperior and inferior limits to thecumulant [q,qll'"qft].thatistocIx·,and therefore tofe,aswasto be proved. Art.(ry).The second theorem isthefollowing: if q"q,...g..belinear functions of x,saya,x+~,alx+bl...a..x+b..,in which thecoefficientsof x •Irj;expanded as acontinued fraction bymeans of thecommon measure processgives risetothequotients q"ql'"q..,andifL,•L,...L.._I,L..be theleadingcoefficients of the successive simplified residues, (L..being, in fact, thefinal simplified residue, thatis.the resultant to¢a,fx),wemusthave¢a=C[q"q•...q..],fx=C[q"ql...q,.],where(supposing ¢a tobeofn-1,andfxofndimensions inx). C=~lL..2L2..-IL1.._4&c.! L..LII .._ILII.._.LI.._1&c.\. 34:-2 532 On aTheoryofthe Syzygetic Relations [57 have all thesame sign, andif wetakethequantities Ji'J"#1'"/l-n-J.,all havingthesamesignaB~,Cl.t•••a..,butotherwise arbitrary, andmake 1 1 1 1lei=Ji'J"leg=#1+-,lea=#1+-...kn-l=P-n-l+-,Ie..=-, Ji'J, #1 ~ P-n.-l thenthegreatest ofthequantities k,.-blk,-b gk..-b.. ~agan sayL,is asuperior limit,andtheleastofthequantities -k,.-b l-k,-b a-k..-b.. ~ Cl.t a.. sayA, is aninferiorlimittotheroots offa: LandanyvaluegreaterthanLsubstituted fora;willevidently make ql-k,.,qa-lea...q..-Ie..,all ofthempositive. Hence,whena;=or>L,ql ispositiveand>Ji'J,',and lk1Ilh,···d qg-->g-->f£a+- --,tatIS,ISpositive, an>#1.ql Ji'J, Ji'J,P-I 11,.1 11h" " dq.- - ->"'3- ->#1+- --,tat18,ISpositive, an>P-"qa- ql f£a f£af£a "and 11111 hati.,an qn---- ... ->----,tatIS, 18positive,qn-l-q..-lI(]IP-1I-1P-n-l andconsequently thecumulant [qlqaq....q..],which =qlx(qg_-!.) x(q.-~!)x &c.,s. qa-ql remains of aconstant sign when Land any quantity greaterthanLis substituted fors:HenceLis~superior limit.Inlikemanner Aandany quantity lessthanAwillevidently make ql+lei.q.+leg...q..+lenall ofthem negative, sothat,whena:=or <A,qlisnegative, and< -Ji'J" 1Ie1, . dqa--<a--IsnegatIve,an <-f£a,ql Ji'J, 1k1. .dq.--<a--ISnegative, an< -P-.,qa f£a and1 1 1 1 1. .q..----...-< - --ISnegative,qll-l-qn-llqlP-n-lP-n-l 57] oftwoAlgebraical Functions. 533 HenceSothat[ql'q....qn]for all values of xlessthanAwillpreserve aninvariable sign, and consequently Ais aninferiorlimittofe. Art.(8).Itmayberemarked thatthequantities I I I I I Il-J,Pi+-,p.,+-,...P-n-2+- •P-n-l+--, fJ-J JkJ """__ P-n-tP-n-l may be derived successively from one another, according tothesamelaw, fromwhichever end oftheseries we begin. Ifwetakeanytwoconsecutive termsas I I 1-',+-,Poi+l+- ,f'i-l f'i theeffect of diminishing f'iis todecrease thefirst ofthesetwoterms,and protanto,totendto reduce thelimitjbutontheotherhand,!being f'iincreased, thereisbrought intoplay an opposite tendency, whichoperates protantoto increase thevalue ofthelimit. Art. (e).Itis ofimportance toremark,thatby arightselection of the system of quantities 1-'1'Pi...P-n-l'whichenterintothecomposition of lelllc,...le.".,Lmay bemadeto coincide with thegreatestroot of[qllq....q,,]j and80in like manner byarightselection ofanother system of these quantities, whereby to form leI'lc,...len,Amaybemadeto coincide withthe leastroot ofthesame.ThusletfJ-J,/I"l...P-n-lbeso chosen, that ql-let=0,q,-lei=0...qn-len=0, are allsatisfied bythesamevalue of e, I I IThen ql=fJ-J,q.=p.,+-,q.=p.,+-...'I»=-, fJ-J p.."""-1existsimultaneously. I I I Ip..==qll--,p.,=q.- -==q.- - - ,ql p.,q.-ql I I If'n-l==qn-l---- -...-,qn-t-qn-a ql I I I"""=---- ...-,qn-l-qn-t-ql which is satisfiedbymaking [qn,qn-l,qn-....ql]=o. Itremainsthenonly to show thatthegreatest root ofxinthisequation substituted forxinq10q,...qnwill make 1-'10p.,...P-n-lall of one sign, and thattheleastroot ofxsimilarly substituted, will also make themall of one, butacontrary sign, which may be proved asfollows. 534 OnaTheoryojtheSyzygetic Relations [57 We have JI-J.=guJLt=[gl'gs]+-s..P-a=[glgaga]+-[gl'g,]&c. p..,.....1=[glgs...gn-I]+-[glg,...gn-!]; andbyLemmaBthesuperior limitto[glga...gn]will beasuperior limit also to[glgi...gn-!],and to [gIg,].[gIgSgS]...,[gIg,...gn-I]' Consequently thissuperior limitwill make JI-J..p..z•.•p..,.....1have all thesame signasthatofthecoefficients of a;ingl'gi...gn.And in like manner,the inferiorlimitto[gIgs...g,,]will cause JI-J.,JLt•••p..,.....1to have all thecontrary sign tothatofthese coefficients. Thusthenwe seethatwhenthecoefficients of a;inthepartialquotients toj:expressed asanimproper continued fraction form asingle series of continuations of signs,by arightchoice of thearbitrary constants P-t,JLt.••f'n-I thesuperior or inferior limitgiven by thisnew method may severally and separately be made to coincide with the greatest andleastreal root, or each inturnwiththesole real root of fe,iftherebebutone. Art.(t").Thegeneralmethodof enclosing theroots of fa;withinlimits is founded upon thecombination ofthetwotheorems abovedemonstrated. Anarbitrary function ~,onedegree in a;belowfx,being assumed, and by aid oftheauxiliary function ~a;.fxbeing thrown undertheform C[glgs... q.,q/qs'...g'..,gt...(qMg)s...,(g)('l]' in which thecoefficient of a;is supposed to change sign in thepassagefrom g.togI"fromq',.,toqt,&c.,asuperior limitis found to eachofthe cumulsnts [qIgS'"q.],[q/gs/... q'..]...[(qh(q)s· ..(q)('l], takenseparately, by means of thesecond theorem, and thenbyvirtueofthe firsttheoremthegreatest of these superior limitsisasuperior limittothe cumulant [glgi...g•...(g)1...(g)('1], andconsequently tofe,andsomutatismutandis theleastoftheinferior limitsofthesamepartialcumulants isaninferiorlimittothetotalcumulant [gIgs'"q....(g)1(g)s...(g)(\1]. Art.('1).Whenall the roots of fxarereal, if~be soassumed thatall itsrootsareintercalated between those offx,thepartialquotients toj; will form butone single series. Inorderthat~may fulfil thiscondition, itis necessary thatthecoefficientsof ~shall besubjecttocertainconditions 57J 0-/twoAlgebrawal Functions. 535 ofinequality, not necessary to be investigated here;butno conditions of equality, thatis, noequations between thecoefficients of f/xx,areintroduced bythiscondition; or inotherwords, the coefficients "off/xx,theauxiliary function, areindependent andarbitrary withinlimitsjand we have shown thatinthiscasetheauxiliary constants f1'J.,fl-l•••p.,1r-lmay be so determined thatthelimits may be made to come separately and respectively into contactwiththetwoextreme roots. When all theroots offxarenot real, thequotients (however 4>xis chosen) can no longer be made to form a single series.Itstill however remains true,that,by a due choice of theauxiliary function followed by a due choice of theauxiliary constants, thiscoincidence may bebroughtabout,so long as thereis a single real root infx. Itisratherimportant todemonstmte thisuniversal possibility of effecting a coincidence of thelimits to theroots with theextreme roots themselves, because it is themoststriking featurewhichdistinguishes the method of limitation here developed from all otherspreviously brought to light. Art.(0).Beforeentering uponthisdemonstration I may make the passingremark,thateverymethodofroot-limitation isimplicitly amethod ofroot-approximation. Forinstance, letebe any given quantity between which and+Xlit is knownthata root offo:lies.Thenif we write x=e+~,and form they equationynf(e+~)=0, and find Lasuperior limittoy,itis clearthat e+~will liebetweeneandtheroot'offo:sayE,nextsuperior toe.Again, makingx=e+~+},and finding a superior limitL'toy',we shall have e+~+bstillnearertoEthane+~was; and so we may proceed advanc­ ingnearerand nearer, and always from thesame side towards Eateach step, and finally obtainEundertheforme+~+l,+~,+&c.And in like mannercallingE,therootnextbelow e,we may find Art.(L).Inestablishing thetheorem of coincidence above adverted to, the following notation will be found very advantageous. Letndenote a Type of any numberofElements, asq"q2'"qi-I>q.,and letn'denotethis •Itneedscarcely bestatedthat!,zisthesimplest CormoCtf>:e,whichsatisfiesthecondition inquestion. 536 Ona Theory oftheSyzygetic Relations [57 sametypewhen the lastelement, and'.0.thesametype when thefirstelement iscutoff,and'.0.'thesametypewhen both extremes arecutoff,80thatthe apocopated typeH'will mean ql' q2'"q'_I;theapocopated type'.0.will meanqsqs'"q.,andthedoubly apocopated type'.0.'will mean qi'q•...q'_I' Ifnow atype0be made up of the types ~,0,...n.putinapposition, and if we use ingeneral[.0.]to denote the cumulant corresponding tothe type.0.,therewillbea very simple law·connecting [.0.]with [.0.1],[.0.,],[.0.,][ll'-i],[Oi-I],[ll.], [ll'I],[ll'i],[n'.][O"-ll],[.0.'i-I], ['ll2],rna]['lli-a],['!la-I],[,.0..], ['.0.'2],['ll'a]['ll'i-a],['n'i-I]. This law will be seen to be obviously deducible by successive steps of expansion from the fundamental theorem given in LemmaB,Art.(a),for thecaseof.0.=nlll2,and willbebestunderstood by showing itsoperation inafewsimple cases. ThusletII=~ll2t.Then [ll]=[.0.1]x[.0.2]-[ll'l]x['ns]. LetII=lll.o.,Oa. Then [ll]=[nl]xEns]x[.0.,]-[.0.'1]x['ni]x[.0..]-[.0.1]x[ll'2]x['.0.2] +[ll'l]x['ll'i]x['na]. [llJ=[lll]x[lla]xrna]x[0.] -[.0.'1]x['lls]x [lla] x[ll.]-[.0.1]x[.0.'2]X['.0.,]X[ll.J -[.0.1]x[Oi]x[n'a]x['n.H+[0'1] X['n'i]x['ll.]x[ll.] +[.0.'1]x ['ll2] x [ll'a] x ['ll.]+[01]X[ll'i]x ['ll'a] x ['ll.] -[ll'l]x['ll'2]x['n's]x['ll.J, *Thecumnlant corresponding toanyportionorfragment ofatypemaybesaidto be apartialcumnlant totheentiretype,anda type whose elements areconstituted outofthe elements of two or more types placed in juxtaposition maybesaidtobethelIggreg&te ofthese types;the law given in the textabove may then be saidto have for its object the expansion of thecomplete cumulant toanytype intermsofcomplete andpartialcumulants tothetypes of which the given type is the aggregate. tThe sign of equalityis employed here to denote the relationbetween a concrete wholeand theaggregate of itsparts. :::Thenumberofdistinctfactorsentering intotheseproducts, takencollectively, is evidently i+2(i-1)+(i -2),thatis4(i-1). 57] oftwoAlgebraical Functions. 537 2\-1_~(1+./5)1+1+~(1-...I5)'+!. ./52 ./52andso ingeneralifn=n10•...n.,[n]may beexpanded undertheform ofthesum of2\-1products separable intoialternately positive and negative groupscontaining respectively 1, (i-1),1(i-1)(i - 2),...(i-I),1products. Art. (~).Ineveryone oftheabovegroupsformingaproducttheaccents enterin pairs andbetween contiguous factors,itbeingacondition thatif anynhave an accentontherightthenextnmusthave one on theleft, andifithave one on thelefttheprecedingnmusthave an accentonthe right,andthenumber ofpairsofaccentsgoes on increasing ineachgroup from 0 toi-I.Thisrule serves completely to define thedevelopment in question", Forgreaterbrevitylet[n,], [n',], ['ne],['n'e]bedenoted respectively by0)"0)'"'0)"'0)'"thenwhenthetypeneconsists of asingleelement, OJ'e=l, 'OJe= I,'OJ',=O. Itshouldbe observed thatthetwoequations OJ,=0,OJ',=0cannotexist simultaneously, forifnerepresent ql'q,...q., sothatifOJ,=0andOJ',=0, we have OJ"e=0,OJ"'e=0,&c.,andthus,finally, - 1=0, which is absurd. Now,ifwe suppose n1,n,...Deto betypeseveryelement ineachof which is alinearfunction of e,thecoefficients of xintheseelements being positivein~,negative innt,and so on alternately, andnistheaggregate ofDlJfi.'"fie.it may easily be madeoutthateachterminthedevelopment ofOJintermsofOJ1,OJ'I,'OJlJ'OJ'I;OJ"OJ'"'OJ"'OJ'"&c. will have thesame sign when we give to xa value which is a superior limit,oraninferiorlimitto •Wheneachpartialtype0oonsists of&singleelement, every doubly accented 0willvanish, andeverysinglyaccented 0will become unity;hence we mayderivetherulefortheexpa.nsion oUhecumn1&nt [lltasaa...a,lintermsoflit,Ut...a"whichwillacoordingly consistof 1 1litasUa...a,-2:-- (litasoo.a.)+2: (a1a,.ooa.) T&c.,a.ao+l a.aO+1xa,a'+1 theindiceseandI,e+1andI,&c.beingunderstood tobe&11dietinct integers (whichagrees withtheknownrulefor theexpression of thedenominator ofacontinued fraction intermsof thequotients). Thenumberoftermsinthisexpansion, inconsequence of thevanishing of the quantities affected with&doubleaccent,reducesfrom2\-1down to theithtermin theseries commencing with1, 2,8,&c.defined by the equation11'+1=1£,+1£'-1' th&t is ~(1+./5)1+1_.!.-(1-./~)I+l. ./52 ./52' thenumber, therefore, ofproducts inwhichdoubleaceenteoccurinthegeneralexpansion of ["'l"',.•."'i!is 538 OnaTheoryojthe Syzygetic Relations [57 the roots of eachofthecumulants ro1,""•••"'"andconsequently tothose of thecumulants ""I'",',...",',;''''1>'''''•.•'",,;'''''I''",',...'",',;theproducts affected with positive signs beingall positive or negative in themselves, and those affected with negative signs being reversely all negative, or all positive. Thus, for example, if O=~O,' andthesign of the leadingcoefficient in ' ""will bethecontrary ofthat inw,.but"'1andro'lhave both the same positive sign;so againif 0=01020" wheretheleadingcoefficients in "'2and'''''havecontrary signs, as have also those in "'2and",',while"'2and'",',havethesamesign;and of course the leading coefficients in "'1>"'3'""1>'""have all the same sign, theybeing all positive, and so in general. Butthesuperior limitto the roots of any integralalgebraical function of (Xsubstituted in place of (Xcausesthesigns of theresulting values of thefunctions to coincide with thesigns of the leadingcoefficients, so thatin the example last above given, Lasuperior limitto all the factors in the several products in theequation substituted for(Xwill make "'1"'2"'"-W'/"'ll"'"-"'1",'2'""."'1'",'2'"" to have all the same sign. The like will be trueof A the inferior limit;for if~,0"0,contain respectively n1,n"naelements, thevalues of thefourproducts lastabove written, when (X=-co,will be to the values of the same when (X=+coin therespective ratios of (_)m1+m.+m.: I,(_)m.+m.+"'r2: I,(_)m1+m.+mo-l: I,(_)mt+mo+m.-4: 1, and so in general. Hence wededuce the theorem, thatif thetotaltype0 represent theaggregate inapposition of thepartialorders ~,n,...0,(the elements beingunderstood to be linear functions of ai,whicharesubject to the law of alternation inthesigns of the coefficients of (Xinpassingfrom onepartialtypetoanother), nosuperior limittoro1,"',•••"',can make ro vanish unless each separate product intheexpansion of roin terms of "'1>""••."',andtheappurtenant apocopated cumulants vanishseparately. Art.(X).Fromtheabovetheorem we may deduce thefollowing law, namely,thatiftheroots of"'11ta•••"',be supposed to be arranged inorder ofmagnitude, andXto bethatone ofthemwhich is nearestto+coor to -co,thenif6is evenitis impossible for Xto be a root of ro.Thussuppose e=2,andconsequently", ="'1"'ll-ro'l''''lljifXbe a root of ro1and one of the twoextremes of the roots of "'1'""putin order of magnitude, Xcannotbe a root of ''''2'for the roots of ''''2areconfined between theroots of "'2;but 57J oftwoAlgel»-aical Functions. 539 ifAmakeI»and1»1each vanish, we musthave1»'1'1»2= 0,hence1»/=° aswellas1»1= 0,which is impossible. Inlikemannerifaroot of 1»2were theextreme root,thesameimpossibility could be in like mannerestablished. Again, suppose 6=4,sothat LetAcontinue to denote one or theotherextreme oftheroots of 1iI1Ii1,I»,I»..IfAmakesI»=°we have 1»11»,1»,1».=0, 1»\'IiI,I»,I».= 0,1»II»','IiI,I». =0,1»11i1,I»','IiI.=0, 1iI'/I»','I»,1».= 0,1»'/I»IIiI'a'I».=0,1»11»'2'1»','1».=0,1»'/1»','1»','1». = 0. Now suppose thatAis a root of 1»1>thentheequations remaining to be satisfied are Since 1»1and1»\cannotboth be zero together, Acannot make 1»\or'1»1 zero; and because Ais anextreme to the roots of 1»2,1»" 1».,Acannotmake 1»',or'I»Ior1»,or'I»,or1»'.or'I».zero, sothatin fact when X=Anone ofthe singlyaccented quantities I»can be zero. Asregardsthedoublyaccented quantities 1»,thesamethingcannotbe affirmed, because ifanyncontains only one elementthecorresponding value of I»withadouble accent vanishes spontaneously. Again,any oftheunaccented quantities I»may vanish, because we may suppose any of theseto have an extreme rootA.Conse­ quentlythefirst,second and fourth of the equations remaining tobe satisfied, mightbe satisfied on makingthe necessary suppositions asto the form of the quantities I»andthevalues of the extreme roots;butthethirdremaining equation 1»'/(1'11»','1». =0,in which only singlyaccented quantities I»occur, remainsincapable ofbeingsatisfied on anysupposition whatever. Andthe samethingwould be trueif we suppose Ato be a root ofany otherI»instead of1»1'Hence ~cannotmakeI»=°whene=4. Inlikemanner,ifebe any even number2e,therewill beanequation tobe satisfied by thatvalue(ifitexist)ofxwhich, besides beinganextreme (oneitherside) of the roots of 1»1'1»2'"1»2rarranged in order of magnitude, also makes I»=0.Butassuchequation cannotbe satisfied, neitherextreme root oftheroots of 1»1'1»,...I»Ifcanbearoot of 1»,aswasto be proved. Consequently, unless4>xissoassumed thatthenumberofchangesof sign in the coefficients of xin thequotients resulting fromj:expanded asan 540 OnaTheoryojtheSyzygetic Relations [57 improper continued fraction is even (for if thechangesfrom sequence to sequence areoddthenumberofsequences themselves is even). themethod oflimitation inthetextcannotgive the means of drawing eitherlimit indefinitely nearto one or theotherextremerootsofJ:c. Art.(1-').Itnowremainsto prove the converse, andto show, first. that when the numberof changes is even, thatis.thenumberofsequences odd, this coincidence can always be effected jand secondly, thatitisalways possible when J:chasone or more real roots, so toassumetP:ethatthe numberof sequences shall be odd. The first partof theproposition is easily proved. Thus suppose e=3, sothat Ifwe suppose A.,eitherextreme of the scale formed by writinginorder ofmagnitude the roots of OJI,OJI,OJa•tobearoot common to OJIandtoOJ.,and if'OJ'a=0, which lastequation may be satisfied by supposing thetypefit to consist ofa single element, theseparate equations will all be satisfied; and so in generalitmaybeshownwithoutdifficulty thatife..2e+1, and if A.be a root common to and if OJIIOJ••••OJitbeallsimplelinearJunctions ofe,sothatconsequently 'OJ'.=0,'OJ'.=°...'OJ''M=0, each separate terminthedevelopment ofQI will vanish singly and separately, andconsequently A.willbearoot of Q): for since A.makes QlI=0,OJa=0...612<+!=0, everyproductin thedeveloped form OJ,in which 61lJOJ....OJ'M+!do not each bearatleast one accent, will vanish; and if we consider anyproduct in which 611.OJ....61'M+!areall accented, if in any two of these immediately following one aftertheother asWILt-It OJILt+!.an accent falls to the rightofthefirst. and to the left of the second, the intervening term OJItwillbeara double accent, and will therefore vanish, since OJILtis supposed to be a linearfunction of :c;butitis impossible when every OJisaccented topreventtwo accents of contiguous oddterms in any such product. from falling to the rightof the left, and to theleft oftheright,termof the two, since thecontrary would imply thatallthe accentswould fall to theright,or all to theleft, which, as above remarked. is impossible, on account ofthetwoextreme termsbeing only simply accentable, thatis,OJIonly totheright,andOJ'M+Ionly to the left. Hence, when:csubstituted forA.makes OJI,OJa'"OJk+!all vanish. and when OJa,OJ....OJtk arealllinearfunctions of e,:c=A.will bearoot of OJ. 57J ojtwoAlgebrairol Functions. 541 Art.(II).I believe thattheremaining partof the proposition may be rigorously demonstrated, namelythatwhen any of theroots offxare real, and the numberof oddintegers not exceeding the index of the degree of fxism,andthenumberofimaginary pairs of roots in fxisp.,epa;maybe80 assumedthatthequotients toj:expanded underthe form of an improper continued fraction, may be made totaketheformnIln"n.,n•...ntH-II wheren"!l.,..n,.arelinearfunctions of e,andiis anynumberassumed atwill, not less thanp.,and of course not greaterthanm;and where "'II"'...."'1i+1will have in common a root }..,which may be made atwillthe greatestor the least root of "'I"',"',...rotH-I;theinvestigation, however,accord­ ing to the presentlightwhich I possess on thesubject,appearscomplicated and tedious, and therefore, in order thatthepress, which is waitingforthe completion of these supplemental articles, may not bekeptstanding, must beadjourned to some futureoccasion. ForthepresentIcontentmyself with showing the truthof the law for the simple casewherefxis a cubic function of e. Firstly.If7%gives rise to asingle sequence of quotientsn,we know, fromthetheory of intercalations, thatit isnecessary thatall the roots of fx shall be real, and in order thatwhenthisisthecasethequotients may form asingle sequence n,it is only necessary 80to assume epa;,thatits roots may beintermediate between thoseoffx. Secondly, Iftheroots offxare not all real, or if theyare all real, but do not comprise the roots of 4>xintercalated between them, and if for greater brevityofratiocination westipulate thatepa;shall have its leading coefficients of the same sign as thatoftheleadingcoefficientof f»,theleadingcoefficients of thethreequotients willeitherbeartherespective signs++-,orthe respective signs+ -+,ortherespective signs+ --;in the first and last of these cases therewould be two sequences, and therefore, by what has been shown above, the methodoflimitation ofthetextcould not give a limit coincident with a root. Letusthenlook to the remaining case,andinquire whether, and how, 4>xmay beassumed sothatfxshall become representable toaconstant factorpresby thecumulant [p(x-a), -q(x-fJ), r(x-a)]. wherep, q,rare all positive, and ais a rootoffx. Letthiscumulant be called hfo: Nothinginpointofgenerality will be lost if we suppose theleading coefficientof hfxto be-1.Wethenhave hfx=[p(x-a), -q(x-{3), r(x-a)] = -pqr(x-a)'(x-b)-(p+r)(x-a) 542 On aTheoryojthe Syzygetic Relations [57 and andhenceandwriting-hfx=x'+B»+0andmakingx=a,we find from theabovex-a identitythat p+r=at+Ba+0,thatis,p=at+Ba+0-r, pqr(x-fJ)=x+a+B, fJ+a+B=0,thatis,fJ=-B-a, 1 1pqr=1, andtherefore qr= - = tB 0 .pa+a+-r Hence if 4>xbe soassumed thatthequotients totarep(x-a),-q(x-fJ), r(x-a),we have hcf>x=[-q(x-fJ),rex-a)]= -qr(x+B+a)(x-a)-1 1= -qr(x'+Bx-at-aB)-1= - -{x'+Bx-at-aB+p).p Hence 4>(x)isoftheform m{.xt+Bx-at_aB+(at+aB+0-r)}=m(x'+Bx+0-r). Ifwe callthethreerootsoffx,a,b,crespectively, we have 1 1 q=r (at+Ba+C-r)=r {(a-b)(a-=-c)-r}; and since qandrare both to be positive, we see thatamustbetakenthe greatest or least of thethreeroots iftheyare all real, so thatat+Ba+0 may be positive, whichit will of course necessarily be if bandeareimaginary; wemustalso have at+Ba+C-rpositive, so thattheform of 4>xis m{(x'-at)+B(x-a)-t}, tbeingnecessarily positive, butotherwise arbitrary, a formcontaining twoarbitrary constants, one of which issubject to satisfy a certaincondition ofinequality; whereas when fxisof such a form as to admit,and4>xis supposed to be 80assumed as to causeitto cometopassthatthequotients toj:form a single sequence, thenthethree coefficients in cf>xremainexemptfrom all conditions ofequality butare subjectto twoconditions ofinequality. And so in generalwhenthedegree offo:isa:andthenumberofsequences 2i+1, it is to beinferredthatthe ncoefficients of cf>xwill besubjecttosatisfyn-i-Iconditions ofinequality andiconditions of equality. Art. (~).The theory of thedetermination ofthemimmum interval between eitherlimitdeterminable bythismethodandthenearestroot, orbetween thetwolimits 80determinable when4>xis80assumed thatj: givesriseto a defined even number of sequences (which will includethe 57J ojtwoAlgebraical Functions. 543 theory of the casewhere all the roots of fa;areimaginary), mustbedeferred toanopportunity more favourable for leisurely contemplation. Asregards theapplication ofthetheorytotheveryinteresting caseofalltherootsbeing imaginary, theprincipal pointremaining to be cleared up isthedetermination oftheleastvaluethatcanbeassigned to thegreatest, and the greatest valuethatcan be assigned to theleastroot ofthealgebraical product X1XiX•...XZ1UwhereXI'Xi...Xmare all of themreal linear functions ofe,subjectto the condition thatthecumulant [XI'XI'X•...X2n]shall (toanumerical factor pres)be equal to agiven function of thedegree2n ina;incapable of changing its sign, which condition implies, asanecessary consequence, thatthe coefficients of a;in each of the terms XI'XI...Xom must be affected with the same algebraical sign. Art.(0).Itshould be observed thatintheapplication of the above method, thedivision of theseries of quotients intodistinct sequences governed by the signs of thecoefficients of a;isintroduced for the purpose of drawing thelimits closer to the roots, butis notnecessary forthemere object of assigning limits. Thus, for instance, if therebe two sequences so that [qlqi'"q"q'Hq'H'" qi+i']' thegreatestandleast rootsof a;deduced from these equations willbesuperior andinferior limits respectively to the roots of fe;from which it is clear that if leaving all theotherequations unaltered, except those which contain respectively q?andqii+lJwe writeinplace of these q?=(p+~)I, f'i-I q1i+1=(~+III)i the roots of thesystem of i+i'equations thusmodified willafortioribe limits to the roots of fe,butthenthequantities 1 1 1 1 1 1 ,."",f't+-...f'i-I+-,P+--,III+-,IIi+- ... ,."" f'i-I /Li-I PIII11;'-1 formthesame single series aswould correspond to the two sequences 544 OnaTheoryojthe Syzygetic Relations [57 treatedas asinglesequence, andthesameisobviously thecase for any numberofsequences ". Art.(7T).Ifweconsider asinglesequence as'I»q,...q..,andwrite ql=aJ(a:-c;),q3=as(a:-es)...q..=an(a:-c..), wherea,.,0"1'"a..aresupposed tohaveallthesamesign,andwrite a,.3(a:-c;)i=J.£J.',a,l(a:-es)'=(JLt+~y...a..3(a:-c..r=(~J, itseemsnotunlikelythattheintervalbetween thegreatest andleastofthe roots of theaboveequations will be a minimum whentheinterval between anypairisthesameforeachpair,thatis, when 1 1 111-,+- JLt+- P-1 P-1 JLt JI-n-l ~=a:;-=--a;-=...=an. Ifweassumetheseequations, andwriteP-1=a,.E,theequation fordetermining Ewillbe Ifn=2thisequation becomes al~~- 1=O. Ifn=3,rejecting thefactorE,itbecomes al~asr-(a,.+as)=O. Ifn=4 itbecomes a~asasa4E4 -(a,.as+asa4+a,.04)r+1=O. Ifn=5,rejecting thefactorE,it becomes al~asa4alE4 -(a1a3as+a,.~al+a,.u4al+asa4al)r+(a,.+as+as)=0, •Itfollows from this,thatifql'q•...q..bealllinearfunctions ofe,andif Q=(ql'-i<t·){q,'-(I;+~n {q3'-(I;+~Y} ......(q..._~.~-J, norootofQcanlie between theextremeroots ofthe funotion X, usedtodenotetheenmulant [.JqJ',-.Jq3',./q3·'......,~Jq...J, thesquarerootsbeingunderstood to betakenso as to make the sign of thecoeffioiente of x all ofthempositive; andfromapreceding artiolewe know thateitherextreme rootofQcan bemadetocoincide with 0.corresponding extreme root of X. Hencewe have anitpriori solution ofthefollowing question, namely, ..Todetermine the(n-1)positive quantities i<t,I""J.•••~-l'so as to make the greatest root ofQaminimum anditsleastrootamaximum; " forthegreatestroot of X will be theminimum greatest root ofQ,andtheleastrootofKthe maximum leastroot of Q.Callingtheserespectively Iand",the two systems ofvaluesoC 1'1'I;...~-lrequired will beobtained bysubstituting respectively Iand>.forxintheequations 1 1 1i<t=.JqJ',I;=-.Jql--, 1;=+.Jq3·--· ..·..p.,.-1=~.Jq' ..-I--·i<t I's p.,.-, 57] oftwo Algebraical Function». 545 and80in general, theequation in~being always of adegree measured by theintegernearestto and not exceeding i';and itiseasy to be seen that for all values of n,thesecond coefficient divided by the first will be an inferior limit to~(of course actually coinciding with itforthecasesof n=2 andn=3).Hencewe have the following valuable practical rule for finding asuperior and inferior limittothecumulant [a1(x-c1),as(x-ci)...Un(x-cn)], where ~,a."•••Unhave the same sign, namely if Cbe thegreatest, and Kbetheleast of the quantities C1,C1•••Cn,C+~will be a superior, and K-A an inferior limit, A being takenequaltothe positive value of I(]+~+~+...+_1_);V~as~asaaa. Un-IUn anditmay be noticed thatCandKarethequantities which would them­ selves be the superiorand inferior limitsto the given cumulant iftheseries ofterms ~,al...Un,insteadofpresenting only a sequence of continuations orpermanencies, presented only a sequence ofchanges or variations ofsign. SECTION V. OntheTheoryofIntercalations asapplica1Jle totwofunctions ofthe same degree,and011theformalproperties oftheBezoutiant withreference to themethodofInvariants. Art. 56. Iffxandrf>:.cbe any twogiven functions of xofthesame degree m, we may form a system of m Bezoutics to fandcf>(asshown in thefirst section), thecoefficients of thepowers of xm-t, X".....I...xl,:Ifin which will compose a squarematrixofm linesofmterms each, whichwillbe symmetrical inrespecttothediagonal which passes through the first coefficient of the first Bezoutic and thelastcoefficient of thelastBezoutic; and we may construct aquadratic homogeneous function of m new variables, such thatits determinantive matrixshall coincide with theBezoutic square so formed. Thisquadratic form may be considered in thelightofagenerating function. Allitscoefficients will be formed of quantities obtained bytakingany two coefficients in one of thegiven functions, and two corresponding coefficients intheothergiven function, multiplying themin cross order, and taking thedifference: each coefficient of the generating function in question will consist of one or more such differences,and will thusbe of two dimensions altogether, beinglinearinrespect to thecoefficients of f,and also linear in respect to thecoefficients of cf>.Thisgenerating function I termthe & 35 546 Ona Theory ofthe Syzygetic Relations [57 Bezoutiant, anditmay bedenoted bythesymbolB(f,4»:thedeterminant ofBis of course theresultant tof,cf>,andthematrixtoBistheBezoutic squaretoj,4>.Now we have seen thatthedecrease in thenumber of continuations of sign in theseriesI, B1(x).Bs(x)...Bm(x)(where B 1(e), Bs(x)...Bm(x)arethe7nBezoutics tof,4»,asxchanges fromatob, measures thenumberof roots of fxretained intheeffective scale of inter­ calations takenbetween thelimitsaandb.Ifwe take theentirescale between+00and - 00thetotalnumber of effective intercalations will be thesame,whether reckoned bythenumberof roots offor ofcf>remaining; forthesetwonumbers canneverdifferexceptbyaunit,since no two of either can ever come together jbutthenumberof eachremaining intheeffective scale will be m - 2i and m-2i'respectively, ibeingthenumberofpairs ofimaginary roots and pairsofunseparated real roots of f,andi'beingthe similarnumberfor4>;sothatwemusthavei=i', Now obviously thisnumber becomes measured bythenumber of con­ tinuations of sign in thesignaletic series I, (B 1) ,(Bs)...(Bm),where in general (Bi)denotestheprincipal coefficient in B,(x). But(B1) ,(Bs)...(Bm)arethesuccessive ascending coaxal minor deter­ minantsabouttheaxisofsymmetry totheBezoutic square; andaccordingly thenumberofcontinuations justspokenof,measures thenumberofpositive termsintheBezoutiant whenlinearly transformed, 80astocontainonly positiveandnegative squares, or in otherwords,measures theinertiaofthe Bezoutiant, theconstant integerwhichadherestoitunderall its real linear transformations. Art. 57. Thisinertiaisthesamenumberas,inthecaseof ahomogeneous quadratic function ofthreevariables usedtoexpressaconicreferred to trilinear coordinates, servestodetermine whether such conic belongs to the impossible classor tothepossibleclassof conics, being 3 or 0 in the former case,and 1 or 2 in thelatter;orasinthecaseofahomogeneous quadratic function of fourvariables used todenote II.surfacereferredtoqusdriplanar ortetrahedral coordinates, serves to determine whether such surface belongs to theimpossible class or to theclassconsisting oftheelIipsoid and thehyper­ boloid of two sheets(which are descriptively indistinguishable), or tothe hyperboloid of onesheet,being0 or4inthefirstcase,1 or 3 in thesecond, and2inthethird.The most symmetrical (butleastexpeditious) method of finding theinertiaof anyquadratic form isthatwhichcorresponds tothe method oforthogonal transformations, and is, in fact, theusualmethod employed in geometrical treatises on lines and surfaces ofthesecond degree. Ifwe apply this methodtotheBezoutiant Bconsidered asahomogeneous quadratic function ofthemarbitrarily namedvariables UI>Us,U;.•.11m inordertomeasure itsinertia,thatis to say, the number ofeffective 57J ojtwo.A19ebraical Functions. 547 interpositions between the two systems of roots, we must construct the determinant litB dlB litB dlBd"+X,d~d'Ut' duldu•...d~dUm Ul litB dlB daB dlB d1-'1d~'crs+X,duad1-'1••.d'Utdu", UI D(X)= dlB litB dlB dlB du".dul'du.mdUs' du".du,"· du",a+X Alltheroots ofD(X)=O.asis well known, are real;theinertiaofB,being measured by thenumberof positive roots of D(-X),will be equal to the numberofcontinuations of sign in D(X)expressed asa functionofXofthe mth degree. IfinfIXandepa:we reverse theorder of thecoefficients, and fIXandepa: sotransformed becomeAIXandcf>lIX,it is obvious thattheroots ofAand cf>lbeing the reciprocals oftheroots offandcf>respectively, thenumber of effective intercalations toAandcf>,.mustbethesameasforfandcf>. Accordingly we findthattheform of the Bezoutiant tofandcf>is the same asthatoftheBezoutiant ofAandcf>,.,thesole difference (one only of names) beingthatB(ul,Us•..Um-llum)fortheone becomes B(u""Una-l•••Us,~) fortheother. The equation D(X),whichdetermines theinertiaofB, remains precisely the same, asitoughtto do, for eitherofthetwo systems fandcf>orAandcf>,.. Art.58. The theoryinthepreceding articlesof this section may be made toembrace thecaseinvolved in Sturm'stheorem; for if fIX='loIX'"+lZta;"'-l+... + G.m-la;"'-l+amlX"', f'IX=maoa;"'-l+(m-1)lZta;"'-t+...+ am-l, and AIX=mfIX-fIX =ala;"'-l+2a.a;"'-1+...+ma"" theBezoutian secondaries,or which is thesamething,the simplified Sturmian residues to fIXandf'IX,willevidently be the same 118those to AIXandf'IX. Accordingly, if we form the signaletic series whereBllB•...Bm-lare the Bezoutian secondaries to AIXandj'IX,the numberofvariati(i1l8 of signbetween consecutive terms in thisseries, when 35-2 548 Ona Theory oftheSyzygetic Relations [57 Xis made+co, will measure thenumberof pairs of imaginary roots inj8:; andjxandj'xforming always a continuation, andthehighestcoefficient of j'xbeing supposed positive, we see thatthetermsoftherhizoristic series will be 1, (BI) ,(B2)•••(Bm_I) ,consisting of positive unityand the successive ascending eoaxaldeterminants of the Bezoutio.n matrixtoj'xandAx. Hence thentheform of the Bezoutiant toj'xandAxwill serve to determine the number ofpairsofimaginary, andconsequently also the number ofreal roots to fo:Itshould be remarked thatthe form of theBezoutiant toj'x andAx,considered as aquadratic function of ~,u.:.•••Um-land ofthe coefficients in jx,will remain unaltered when forjxwe write Ax,forthis will change thesignsthroughout ofjxandAx;andconsequently the coefficients in theBezoutiant, whichcontains in every term one coefficient fromj'x,and one from Ax,willremainunaltered in sign. Art.59.Itappearsthenfromthepreceding article,thatfor every function of xof the degree m,thereexistsahomogeneous quadratic function of (m -1)variables, theinertiaof which augmented byunitywillrepresent thenumber ofrealroots in thegiven function. Now thisinertiaitself maybe measured by the number of positive roots of a certainequation inAformed from the quadratic function (in factthewell-known equation forthesecularinequalities oftheplanets), all whose roots will be real. Hencethenwe are led tothefollowing remarkable statement. "AnaJ,ge­ braical equation ojany degree beinggiven, an equation whosedegree isoneunit lowermaybeformed,alltherootsofwhichshallhereal,andofwhichthe numberojpositiveroots shall beonelessthan thetotalnumberofrealroots ofthegiven equation." Let us suppose jxwritteninitsmostgeneralform,thefirst and lastas wellasall the intermediate coefficientsbeing anything whatever: byreversing theorder ofthecoefficients j'xwill become jlxandAxwill become fx;the Bezoutiant toAxandj'x(which wemay term the Bezoutoid tojx)willremain unaltered exceptin sign,and the equation ofthe(m - 1)thdegree in ~formed fromtheBezoutoid remainunchanged; consequently theequation inAenables us tosubstitute, for the purpose of calculating thetotalnumberofrealroots injx,in lieu of Sturm's auxiliary functions to.jx,anotherset of functions which remain unaltered whentheorderofthecoefficients is completely reversed, thatis in effect, when we consider thenumberofrealroots of j(~)in lieu of those of j(x).And of course more generally theequation ofthemthdegree in ~formed from theBezoutiant to any two functions jxand¢Xof the mth degree eachinx,supplies &set of functions for determining thetotalnumberof effective intercalations between the roots ofjxand¢X'which do not alterwhen we consider in lieu of thesethe 57J ojtwoAlgebraical Functions. 549 rootsoff(~)and~(~).Thissubstitution offunctions symmetrically formed inrespecttothetwo ends of an equation forthepurpose of assigning the totalnumber of real roots in lieu of theunsymmetrical onesfurnished bytheordinary methodofM.Sturm,had been long felt by me to be a desideratum, and as an objecttheaccomplishment of which was indispensable totheulteriordevelopment ofthetheory, and it is certainthatI did not inanticipation exaggerate theimportance oftheresultto beattained. Art. 60. ItmayhappenthattheBezoutiant tofand~(each of the mthdegree)maybecome a quadratic function of lessthanmindependent variables, ortheBezoutoid to f(afunction ina;ofthemthdegree)of less than(m-1)independent variables. This will takeplacewheneverfand4> haverootsin common, or whenever fhas equal roots. The numberof independent relations ofequality between theroots offand4>,andthe amount ofmultiplicity, however distributed, amongtherootsoff,will beindicated bythenumberofordersthusdisappearing outofthegeneral form of theBezoutiant and Bezoutoid in therespective cases".Inwhat particular modetheform of each would be affected according to themanner ofthedistribution oftheequalities andthemultiplicity requires a specific discussion, which I mustreserve for some futureoccasion. Art.61,I shall devote theremainder ofthismemoir to a consideration oftheproperties and affinities of Bezoutiants or Bezoutoids, regarded from thepointof view of theCalculus ofInvariants..Forthispurposeitwill be moreconvenient hereafter toconvertallthefunctions which we areconcerned withintohomogeneous forms, and I shall accordingly for thefutureuse fand4>todenotefunctions eachofa;andy,which I shall writeunder theform f=aoa:'"+~a:"'-ly+1m(m-1)asa;"'-ty+ + a",ym, ~".ber'£'"+n~a:"'-ly+1m(m-1)b2a;m-,y+ +bmy"'. Inwhatfollows a knowledge of thegeneralprinciples oftheMethod of Invariants ispresupposed, butaperusalof my two paperson the Calculus ofFormstintheCambridge and Dublin Mathematical Journal, February and May, 1852, will furnish nearlyalltheinformation thatisstrictlynecessary torthepresentpurpose. The first pointto beestablished is,thatB,the • I have elsewbere defined bow iliisword order, as bere employed, i~tobeunderstood. IfF,abomogeneous fonction ofxl>x,...x"'can beexpressed as afonction of~,",...",,_, (alllinearfonctions ofXl'X,•••r,,),Fissaidto be afunction ofn-iorders,or to have lost iof theordersbelonging tothecomplete form. [tBeepp.284,828,411 above.] 550 Ona Theory ojthe Syzygetic Relations Bezoutiant ofIxandtf>x,isaCovariant to the system f,4>;the variables inBbeing in compound relationof cogredience with thecombinations of powers of xandy, Thatis to say, I propose to show thatifJ,g,h,kbeanyfourquantities, takenforgreatersimplicity subjectto therelationfk:-gh=1,and if on substituting Ix+BYforxandh»+leyfory,f(x,y)becomes Aox~+mA1xtJHy+im(m-1)Aix"'-lyl+AmY"',sayG(e,y), and4>(e,y)becomes Box"'+mB1xm-1y+im(rn-1)B i,x"'-2yl+Bmy"',sayT(x,y), andifB'(u/,~'...um')be theBezoutiant toGandT, B(u1,~...urn)being t~attoIand4>,then, on making Ul>U1•••u,,,,thesamelinearfunctions of t'-1',ul'•••u,,,'as (Ix+By)m-l,(/x+gy)m--'J(hx+ky)...(/x+BY)(hx+ky)m-'l, (hx+ky)"'-l, are respectively of Bwill become identical withB'.Iwasled to suspect thehighprobability of thetruthofthisproposition concerning theinvariance oftheBezoutiant fromthefollowing considerations: Firstly,thatfortheparticular case whereIand4>arethedifferential derivatives in respect to xandyre­ spectively ofthesame function F(x,y),theBezoutiant ofIand4>,which thenbecomes the Bezoutoid of F,determines thenumber of real factors inF,which obviously remains thesame for all linear transformations ofF. Secondly, thattakingIand4>intheirmost general form, the invariant to theirBezoutiant, thatisthedeterminant oftheirBezoutiant, is aninvariant ofIand4>,being in fact the resultant of these two functions; nowasevery concomitant (aninvariantive form of the most general kind)to aconcomitant isitselfaconcomitant to theprimitive, soitappeared to me, and is I believe true(although awaiting strictproof),thatany form satisfying certain necessary andtolerably obvious conditions of homogeneity and isobarism, aconcomitant to which is also aconcomitant toagiven form, willbeitself aconcomitant tosuch form; thisprinciple, if admitted, would be of course at'once conclusive asto theBezoutiant beinganinvariantive concomitant tothefunctions from which it isderived. Art.61-.Sincethepublication of the two papersabovereferredto on theCalculus of Forms, I have made the important observation thatevery species of concomitant, however complex, to a given system of functions, may betreatedasa simple invariant of a system including thegiven system 57J oftwo Algebraical Functions. 551 together with anappropriate superadded systemofabsolute functions; thus anordinary covariant involving only one systemof variables, asu, v,w... cogredient withx,y,z...thevariables of a system S,is in fact an invariant ofthesystemScombined with thesystemuy-vx,vz-wy,uia:-uz,&c., u, v,w...beingtreatedasconstants; soagaina.simplecontravariant ofS is aninvariant ofScombined with theformux+vy+wz+&c.j so again, tomeetthecase before us,a covariant tothebinarysystemIand4>expressed asafunction of Ul>'U.J...Um,where ~,~...ltmarecogredient withxm-I, xfJt-Sy...ym-l,may be regarded asaninvariant oftheternary system f,4>,n,where n=1llym-1 -muiyfJt-2x+~m(m-1)Uaym-s:x2.••+(_)m-Iumx"'-\ (Ul>U"J'"Urnbeinghere to be treatedasconstants); and accordingly the differential equations which serve to define in themostgeneralandabsolute mannersuchcovariant off,4>,orinvariant tof,cp,n,sayI,willtake theform . (c,d~+i.d~)+2(UId~+bld~)+3(aid~+i.d~J+"'1 Id d )+m~am-Idam+bm_1dbmI=0, (d . d •d d)j - U ld-+2~d··-+3usd~+...+(m-1)Um.-Id--- 1~ Us .U. 14n ((amda~n-I+i;db~)+2 (am-IdaC::2+bm_1db~.J i+3(a..-..00:-'+6..-.db:J+...+m(a,.:"+i.,;,lI~0 (d"d d d)-Urnd- - +_um-Id--+31~d~- -+...+(m-l)~d'- Um_1 Um-s Um-s UI Theseequations may beprovedto besatisfied whenIistaken=B,the Bezoutiant tof,cp,andthusBmay be proved to be a covariant tof,cp, butthedemonstration is long and tedious. An admirable suggestion, well worthy of its keen-witted author,for which I am indebted to Mr Cayley, will enable us to prove theinvariantive character ofBby a much more expeditious method. Art. 62. Forgreatersimplicity begin with considering functions of a single variable xjand inorderto fixtheideas, snppose mto betaken 5, and writeIx=a:I!+b~+ex'+dW+ex+l, 4>x=a.:rf+f3~+'Yx'+ow+E$+A, andlet~=lx4>:x'--(;'cf>:x;thisis of course an integralfunction of xandx',x- 552 On aTheoryoftheSyzygetic Relations [57 sincethenumerator vanishes when x==x';and we have by performing the actualoperations, ~=(af3-ba)re'x'4+(ary+ea)wa;'>(x+x')+(as-da):r;IX'I(tIP+u+X'I) +(aE-ea)xx'(w+tlPx'+XX'I+afS)+(aX-la)(re'+wx'+xix's+ul+af4) +(bry-ef3)WX'I+(bS-df3)tIPx's(x+x')+(bE-ef3)xx'(tIP+xx'+X'I) +(bX-lf3)(w+xlaf+XX'I+X'I) +(eS-dry)tlPx's+(CE-ery)xx'(x+x')+(eX-try)(tIP+xx'+x") +(dE-eS)xx'+(dX-lS)(x+x') +(sA-lE); and if we arrange ~undertheform A4,4re'x'4+A4,Sa:'a;'1+A4,Sre'X'1+...14, 1re'x'+A4,ore' +AI,4WX'4+...11,1xix'S+AS,IWX"+Aa,lwa;'+Aa,ow +Aa,4x1x'4 +As,lxlx's +Aa,sxlx's +Aa,lxla!+Aa,ow +AI,4XX'4+AI,sxx's+AI"xa;'s+...11, 1xx'+AI,ox +Ao,4x'4+Ao,lx's +Ao,lx's +...10,1x'+...10,0 it will rea.dily beperceived thatthematrixformedbythetwenty-five coefficients, namely A4t4J...14,1,...14,1'A4,I'.A4,O' ...11,4,AI,s,Aa,I'...11,11Aa,o, Aa,4'Aa,I'Aa,IJAa,1IAs,o, ...11,4,...11,1,...11,1,...11,1'...11,0, ...10,41...10,1,...10, 11Ao,1I.Ao,o, will be symmetrical aboutitsdexterdiagonal (thatone, namely, which passesthrough A4,4and...10,0) ,and will be identical withtheBezoutian squarecorresponding tothesystemf,4>;in fact, usingthenotation previously employed in thefirst section, it becomes (0, 1) (0, 2) (0, 3) (0, 4)(0, 5) (0, 2)ro+3)}ro+4)}rO+5)} (1, 5) (1, 2) (1,3)(1,4) (0, 3)(0,4)}C..G)IrI+ 5)1.(2, 5) (a) 1(1:3)(1,4)>- (2:3»)(2, 3)J (0, 4)rO+5)} rI+5)} r2+5)} (3, 5) (1,4)(2, 3) (3,4) (0, 5) (1, 5) (2, 5) (3,5)(4, 5), 57] oftwoAlgehraical Functions. 553 ~=/(x,y)cf>(x',y'~-/,(x',y)cP(x,y), a;y-xy we seewithoutdifficulty that ~=IA.."{rym-l-f'x"y'm-l-'}, whereA.."isthetermintherthline and sthcolumn of theBezoutiant matrixto / and cf>.Thisistheidentification, theideaof which, as before observed, is due to Mr Cayley.(r,8)beingused in generaltodenotethedifference between thecross products ofthecoefficients of arrandr'in / and cf>.Restoring now to mitsgeneralvalue, and taking/ and cf>homogeneous functions of xandy, andmaking Art.63. If,now, weconsider thesystem of functions /(x,y)=ag:crn+'TTULta;m-ly+ +Umym, cf>(x,y)=bo:crn+mb1a;m-ly+ +bmym, n(x,y)=u...:ym-l_ (m-1)Um-lym-t X±...+(_)m-1OUta;m-l, evidently/(x,y)cf>(ai,y)-/(aI,y')cf>(z,y)is acovariant with / and cf>,and therefore (which is ameretruism)withtheentiresystem,f, cf>,n.So also is:cy'-x'y,andtherefore ~,thequotient ofthesetwo, is a covariant tothe system. Hence, therefore, byvirtueof ageneraltheorem givenin my Calculus of Forms, n(~-~)~dy'dx is acovariant tothesystem; and, again, therefore, n(d~'-:x')n(;y'-f.x)~ isacovariant thereto. Now~is of(m-1)dimensions in x,yand also of the same in x',y.Consequently thislatterform will containonlythe quantities Ut,~...Um_1>andthecoefficients of/ and cf>,sothatthe powers ofx,y;x',y'will notappearin it. Nowo0 ~=IIA.."(x..ym-l-rx"y'm-l ....}, ",-1",-1 (d d) (d)m-l (d)m-td(_~-ln dy'-d:c=umdo;+(m-l)Um-l do;dy+.., (d)m-l...+u1dy, (dd) (d)m-l (d)m-td ( -)Jll-lndy"-do;'=Umdx'+(m-1)Um_1\dx'dy+... (dr...+OUtdy, 554 thereforeOn a Theory ojtheSyzygetic Relations 1(dd)(dd)j1.2.3...(m_l»)·n dy"~da!ndy'-dxIJ o 0 0 =I(A"."UlI"+l)+2II(A".u,,+lUa+l),.-1 .-1".-1[57 randsbeingexcluded inthelattersumfrombeingmadeequal;butthis latterexpression istheBezoutiant tof,</>.HencetheBezoutiant off,</> is aninvariant tof,</>,fl,thatis acovariant tothesystemf,</>'as was to be proved. Themode of obtaining thecovariant IJ,used inthisandthepre­ cedingarticle,is veryremarkable. Ibelievethatthetruesuggestive view oftheprocess for findingit,is toconsider j(x,y)</>(x',y')-j(x',y')cf>(e,y) as aconcomitant capable ofbeingexpressed undertheform of a function ofIJandQ),Q)standing fortheuniversal covariant xy'-x'y;IJisthento be considered, notproperly as aquotient. butratherasaninvariant oftheform IJQ),afunction ofQ)ofthefirstdegree,whereIJistreatedasconstant. Art. 64. Bisnotanordinary covariant ofjand</>'itbelongs to that specialand most important family of invariants toasystemtowhich I have giventhenameofCombinants ",namely Invariants, which, besides the ordinary character ofinvariance whenlinearsubstitutions areimpressed uponthevariables, possessthesamecharacter ofinvariance whenlinear substitutions areimpressed uponthefunctions themselves containing the variables; combinants being, as itwere,invariants to asystemoffuuctions intheircorporate combined capacity qudsystem. ThattheBezoutiant possesses thisproperty isevident; forifinsteadofjand</>wewritekf+iep andk'f+i'</>,any such quantity asa"b.-a,br(ar•brbeingcoefficients in f, anda"b.thecorresponding ones in </»becomes (ka;+ibr)(k'a.+i'b.)-(ka,+w.)(k'ar+i'br). thatis (kt-k'i)(arb,-a.br). sothatB,theBezoutiant, becomes increased intheratioof(!ci'-k't)"', thatisremainsalwaysunaltered inpointof formandabsolutely immutable, provided that1.-i'-k'ibetaken,as we may always suppose to be thecase, equalto1. Wederiveimmediately fromthisobservation, thesomewhat remarkable geometrical proposition, thattheintersections withtheaxis ofxmade by anytwocurvesofthefamily of curves U=V(x)+p-</>(x),(jand</>being functions ofxofthesamedegree)give rise to aconstant numberof effective intercalations, whatever values be given to ~orp-forthetwocurvesso selected. •Forsomeremarks on the Classification ofCombinants, seeCambridge andDublin Mathematical Journal. November, 1858 [po411 above]. 57J ojtwoAlge1Jraical Functions. 555 dfdcf>dfdcf> =d:tdy-dyd:t.Art. 65. B(Ut, 'lJ..,...Um)beingacovariant ofthesystemfandcf>,and Ut,tit•.•u".cogredient withItm-I,Itm--'.ly...ym-I,itfollowsfromageneral principle in the theory of invariants, thatonmaking Ut,'Ut...Umrespectively equal to thequantities with which theyarecogredient, Bwill become anordinary covariant tofandcf>.Bythistransformation Bbecomes a function of Itandyofthedegree 2 (m-1)inItandyconjointly, and linear inrespect tothe coefficients of j,and alsoinrespectto those of cf>.The onlycovariant capable of answering thisdescription is what I am in the habitofcallingtheJacobian (afterthename of thelatebutever-illustrious Jacobi), atermcapable of application to anynumber of homogeneous functions of asmany variables. Inthecase before us, where we have two functions of two variables, theJacobian dfdcf> da:'dx J(j,cf»= dfde/> dy'dy We have thentheinteresting proposition", thattheBezoutiant to two functions, when the variables in the former are replaced by the combinations ofthevariables in thelatter,with which they are cogredient, becomes the Jacobian']', So in the caseof a single functionFof the degree m,the Bezoutoid, thatistheBezoutiant to~,~:'onmakingthe(m-1)variables which it contains identical with·1tm-2,xlTlr-3y...ym--'.lrespectively, becomes identical withtheJacobian toa-,:;,~:'thatistheHessian of F,namely d2Fd2F du;2'dltdy d'F d'F da:dy'dy2 Asanexample of thisproperty oftheBezoutiant, suppose f=a;cS+bry+ca:y2+dya, cf>=a;c8+fJry+ryxya+oys. TheBezoutiant matrixbecomes af1-ba,ary-ca, 0.0-da, ary-ca,(~;d)bry-cfJ, bry-cf1 0.0-da, bry-cf1'co-dry. *I havesubsequently foundthatthisproposition iscontained underanother mode of statement, at the end of Section2 of the memoir of Jacobi, "DeEliminatione," above referred to. tForastrictproof ofthisproposition seeSupplement toThirdSection of thismemoir. 556 OnaTheoryofthe Syzygetic Relations [57 TheBezoutiant accordingly willbethequadratic function (a{3-ba)u1'+{(as-da)+(bry-c{3)}1It'+(cS-d"t)ut' +2(a"t-ca)U1'U:z+2(as-da)us~+2(bry- c8)'U:z~, which on making becomes La::'+Mry+Nw!l+Px!l+Q'!!, ({3) whereL, M, N, P, Qrespectively will be the sum of thetermslyinginthe successive bands drawn parallel to thesinisterdiagonal of theBezoutiant matrix,thatis L=a{3-ba, M=2(~-ca), N=3(as-dr:z)+(b"f-c{3), P=2(b"f-c{3), Q=cS-~. Thebiquadratic function in a:andy,(ft),abovewritten,willbefoundon. computation to beidentical inpointof form with the Jacobian tof,~, namely ~~+~+~~+~+~-~+2~+~~+~+~ thislatterbeing in fact 3La::'+3Mwy+3Nwy~+3Pxy'+3Q'!!. Theremarkis notwithoutsomeinterest. thatin facttheBezoutiant, which is capable (ashasbeen shown already) of being mechanically constructed, givesthebest and readiestmeans of calculating theJacobian jforinsumming thesinisterbandstransverse totheaxis ofsymmetry the only numerical operation to be performed is thatofaddition of positive integers, whereas thedirectmethodinvolves thenecessity of numerical subtractions aswell asadditions, inasmuch asthesametermswill berepeated with.different signs.Thusif f=azI+ba::'y+cry2+dWy'+eX'!!+lJl, ~=ru!'+{3a::'y+"try'+Swy'+fIX'!!+XJI, using(r,8)intheordinary sensethathasbeen considered throughout, we obtain by takingthesum ofthesinisterbands in (a)-for the value of B when we write a::',ry,w~,xys,y'in place of u1,Ut,~,U"Uu (0,I)xl+ 2(0, 2) x7y+(3(0, 3)+(I,2)}xly2+(4(0, 4) + 2 (I,3)1xay' +{5(0,5)+3(1,4)+(2, 3)}a::',!!+(4(1, .j)+2(2,4)}wyl +{3(2, 5) + (3, 4»)a,.2y1+ 2(3, 5) xy7+(4,5)!t. •VideArt.62[po552above]. 57J oftwoAlgelYraical Functions. 557 Thedirectprocess requires the calculation of ~+~+3~+2~+~~+2~+3~+~+~ -(5ax'+4{3a;'y+3rywyS+2oxy'+Ey4)(ba:4+2cWy+3daflys+4ex!l+51y), each coefficient of which will containthenumerical factor5;80thatto reducetheJacobian to itssimplest formeachcoefficient will necessitate theemployment ofadditions, subtractions, and a division, insteadofadditions merely,aswhentheBezoutic square is employed. Forinstance, to findthe coefficientof :c'yfromtheabove expression (ex)we have to calculate i{25(0,5)+16(1, 4)+9(2, 3)+4(3, 2)+(4,1)J, thatis !{25(0, 5)+ (16- 1)(1, 4) + (9 -4)(2, 3)}, which is 5(0, 5)+3(1,4)+(2, 3),agreeing with what hasbeen found above forthevalue of such coefficient,by a simple process of counting. The same remarkwill, of course, also apply to the computation of the Hessian of F by means of its Bezoutoid. Art. 66. This relationbetween theBezoutiant and theJacobian led me toinquirewhether, as would atfirstsightappearprobable, the Bezoutiant were the only lineo-linear quadratic function of mvariables covariantive tofand~(thewordlineo-linear being used to denote theform of coefficients, suchasthose in the Bezoutiant, linearin respect of thecoefficients in f andthecoefficients of t/».If50,thentherewould have existeda method ofperforming the inverse process of recovering theBezoutiant fromthe Jacobian, almostassimple as thatofderiving theJacobian from the Bezoutiant. Oninvestigating thematter,however, I found thatsuch is by no means thecase",butthatthereexistsa whole family of independent •Thismillhthavebeenconcluded immediately fromthefollowing observation. LetJ, theJacobian offand"',beexpreBBed underthe form .doz2m-s+(2m-2).dla;2m- llI+!(2111-2)(2m-8).dsztm-Sys+ ...+.d_-tV---', thenweknow [po282 above] from theCalculus ofForms,that,Dbeingtakentorepresent the persymmetrical Determinant . .do..dl•.<12,......,.<1..._1, .dl,A2,.d"......,A", As,.d".du......,.d"'+1 D=Oistheoondition tobesatisfied inorderthatJmayberepresentable underthe form of the sum of powers of (m-1)linearfunctions ofzandy,andDitselfisaninvariant toJ.and consequently aninvariant and(asis obvious from itsform)aoombinanUve invariant tofand",. Moreover, whichismoreimmediately to thepoint,weknow thatthequadratio formQ {,((m -1)(m-2))} ,.<Iouls+2.<1dudm-1)u,}+.d, {(m-1)u,} +2ul 2 u,+&o·+.&""-s",,,' 558 OnaTheoryofthe Syzygetic Relations [57 lineo-linear quadratic covarisnts of m variables to every two homogeneous functions of a;andyofthemthdegree. I have, moreover, I believe, succeeded in determining thenumberof such Iineo-linear quadratic forms for any value of m, of which all the rest, in whatever mannerobtained, may be expressed as linearfunctions, thecoefficients of the linearrelations moreover beingabstract numbers; inotherwords, I have succeeded in forming the fundamental orconstituent scaleoflineo-linear quadratic forms of mvariables covariantive tofandt/>;aresultoftoogreatinterest, asexhibiting theaffinities oftheBezoutiant to its cognate forms, to be altogether passed over in silence. Supposing thenumberof linearly inde­ pendent forms of the kind tobev,thenspeakingaprioriany oftheforms takenatrandommightseem to be equally eligible to form one of the II included in thefundamental scale,combined with any (v-1)othersinde­ pendentinterse,and of which the selected one isalsoindependent. Infact, however, thisisnotso; for it will always be more satisfactory tocontemplate thefundamental scaleof forms asgenerated successively or simultaneously by a uniform process; and in thecase before us, the process which I have hitupon, and which I believe is the simplest thatcan be employed for generating thefundamental scale, will be found not to include directlythe Bezoutiant among the number. Therewillthus arise two subjects of inquiry; firstly, the mode of forming thefundamental scale, and proving itsfundamental character; secondly, determining thenumerical relations willbeaninvariant toj,.pand()(thislastquantity ()being defined asin p.[551]),andacom­ binantive covariant tofand.pinthesamesenseprecisely as the Bezoutiant isacovariant tothesame,andlike the Bezoutiant isIineo-Iinear inrespectofthecoefficients ofjandtf>. Ifweoperatewith the symbol E,whereErepresents !ld"-d(!l2d.. !ld tildao+"vIV!ldill+Vs+VIV,)dA~+",c.+v..dA2llI_!l' uponKanyinvariant 01fand.p,weshallobtainE K,aquadratic funotion ofvi'tl,'"tI"" which by the rules of the Calculus ofFormswe know will be a contravariant tofandtf>. andthematrixoorresponding towhichmustevidently bepersymmetrical. Itis aninteresting subjectofinquiry,whichIreservefor some futureoccasion, todetermine theCo-bezoutiant, theDiscriminant ofwhichmustbeemployed forK,sothatwhenthisdiscriminant isoperated uponbyE,thematrixcorresponding toEKmay become identical (termfor term) with the matrixwhich is the inversetotheBezoutisnt matrix,whichinverse,asJacobihassosimply andbeautifully demonstrated, possesses thispersymmetrical oharacter. Videthe"DeElimina­ tione,"Section5. The investigation of thearithmetical connexion between the Qofthisnote andthefundamental Cc-bezoutiants mustbe alsosimilarly reserved. Ibelieve it to be generally true,andhave verified the fact for the case of two cubic functions, thatEQgivesaquadratic form such thatthecorresponding matrixis theinverseto thematrixofQ.Thecalculations necessary forextending theverification ofthisremarkable proposition forfunctions ofz,y exceeding thethirddegree(notwithstanding thatthey are muchabbreviated by theapplication of the rules of the calculus) stillremainexcessively laborious. Theabbreviation alludedto consists in confining the verification inquestion tothecomparison ofeitherone of the two unreiterated termsat opposite cornersof thematrixtoEQwith the corresponding term inthe inversematrixofQ;ifthesecoincide, it is easy to prove thateveryotherpairofcorresponding termsin the two matrices mustalso coincide respectively with one another. 57J oftwoAlgebraical Functione. 559 which connect thatveryimportant form,perhaps ofallits kind the most important, withtheforms comprised in thefundamental orconstituent scale. These questions I propose toconsider more fully atafuture period. For the presentI shallcontentmyself with giving a methodof forming theconstituent scale(without, however, seeking theproof of all the forms extrato such assumed scale being linearfunctions of those comprised withinit), and withdetermining thenumerical relations between theforms in thisscale andtheBezoutiant for alimitednumber of values of m.Alltheforms which we are seeking, besides being lineo-linear quadratics, mustalsobe combinantive invariants tofand¢,remaining (asforms)unaltered for any linearsubstitutions impressed eitheruponthevariables or uponthefunctions containing thevariables. Art. 67. I musthere premise thatiftherebe any two forms of the same degree (and thatdegree odd) in xandy,acombinant may be formed fromthem,which will be linearinrespectto each set of coefflciente ",Thus callingthe two functions aoa;l"+l+(2n+1)~a;I"y+t(2n+1)2na,a;tn-lYS+ +Cl.mHy"'+l aownH+(2n+1)a1wny+t(2n+1)2na..a;I"-lyl+ + ~+lyllr&+l, thelineo-linear combinant inquestion will be T==aorJ.mH- (2n+1)~rJ.m+t(2n+1)2naslZm-l (2n+1)(2n)(2n-l) &&+ 1.2.3 aslltn-t c.- c., which, using our customary notation, will be of the form (2n+l)2n(0, 2n+1)-(2n+1)(1,2n)+1.2 (2,2n-1)±&c. (_)"(2n+1)(2n)(2n-1)...(n+2)( 1)+ 1.2.3...n n,n + . Asacorollary to thisproposition (which, aswellasthe proposition itself, will be needed for thepurposes of theensuingdetermination), takingany function of an even degree in e,y,F(x,y),therewill exist a combinant to dFddFdo;andy'byvirtueof what hasbeenstatedabove, which will be .. Imayaddhereincidentally (although notwantedfor ourpresentpurposes) thatas a com­ binantin which eachsetof coeffioients enterslinearlyoanalwaysbe formed toasystemof functions twoinnumberof as many variables and of any odd degree, so reciprocally can a com­ binantin which eachsetof coefficients enterslinearlybealwaysformed to asystemof funotions eaoh of the degree 2,of whioh and of thevariables oontained in them, the numberisany odd integer[cf. p.606below]. 560 Ona Theory oftheSyzygetic Relations [57 Mr Cayley's well-known quadrinvariant toF;namely, if F=aoxl"+2~xm-Iy+ ... +amy'lA, this will be 22n(2n-1) 1(),,2n(2n-1) ...(n+1) ,aoam-~~I+--2-- a,a.m-,+...+ -2- 1 2 a,......n The proposition itselfiseasilyproved; first,theexpression Tbeing expressed entirelyintermsofquantities of the form (r,8)remainsunaltered forlinearsubstitutions impressed upon theformsfandtI>;itremainsthen onlytoshowthatTsatisfiesthedifferential equations toTtreatedasa.mere invariant, namely id d d d}aod~+2~clat+30"da,+...+(2n+1)amdam+1T=0 d d d d'+«0d-+2ald-+3a,-d-+...+(2n+1)cr".-d-al CIs IX. CZz.t1 and d d diamHdam+2amdam-I+ .., +(2n+ 1)aldaO} d d dT=O. +cr...+1dam+2a",dtx.-I+...+(2n+1)aldao Fromthehemihedral symmetry ofT,which only changes its sign when the order ofthecoefficientsin fandtI>issimultaneously reversed, it is obvious thatone of these equations cannotbesatisfied withouttheotherbeing80too. Looking thenexclusively atthe first of them, we see thatthisis satisfied by virtueof theequations { ao~+(2n+1)~~JT=O, {2~~ +2na...-Id~J T=O, {(2n+1)am~+1 +aod~J T=O. Hencethenthedifferential equations toTbeing satisfied proves thatitis aninvariant, and,asabove observed, its form shows upon its face thatitis acombinant. Precisely in thesame way it may be demonstrated, thatto twofunctions each of the same even degree 2mas :c""2a;tmI2m (2m-1)..--., ....ao+~-y+2 0'2"'-- -Y+...+a....y, 57J oftwoAlgebraical Functions. 561 Make2m(2m-1)and aoW'"+2ma1a;2""'-ly+2f4.X--2y'+.,.+rx-.y'''''', therewill be a quantity 2m(2m-1)G-11otx,.,.-21nattx,.,.-1+211..Ia--,±&c.-2m21asm-1+aolZsm, which,although notacombinant, willsatisfythedifferential equations necessary toproveittobe anordinary invariant tothetwogivenfunctions. Art.68.Nowletusconsider thethreeforms,f,4>andthesubsidiary form0,where 1=aox'"+1natX7n-Iy+ +amym, cp=boxm+mb1xm-1y+ +bmym, 0=u".ym-I-(m-1)~ym-lx±&c.+(-)7n-lUma:"'-I, where ~,~...Umaretobetreatedasconstants. 1(dd)2\+l E2i+d-m(m-1)...(m_2i)fda:+11dyJ, 1(dd)'HI E"+l4>=m(m-l) ...(m-2i)fda:+11dycp, ibeinganyintegersuchthat2i+1 doesnotexceedm,andnowconsider EIi+J.Eli+!4>88twofunctions ofthedegree2i+1 inf,'7(xandybeing regarded asconstants); andbyvirtueoftheformula inthelastarticle. formT"thelineo-linear combinant ofE2i+dandE"+lIf>;T,willthenbe lineo-linear inrespecttothecoefficients in fand4>,and ofthedegree 2{m-(2i+1)J inrespecttoxandy.Again,let E.O- 1(f!!+!!)ria,.-m(m-1) ...(m-2i+1) da:11dy. E"ntreatedas afunction offandfJofthedegree2iwillfurnishaquadrin­ variantQ,ofthedegree2(m-1-2~)inrespectofxandy,andquadratic inrespectofthesystemu".,U,...Um•We have thustwo forms, T,andQ" each ofthesameevendegree2{m-(2i+1)J inrespectofx,y.Forming between thesethelineo-linear invariant G"G,will be a function lineo-linear inrespectofthecoefficients of fand4>,andquadratic inrespectofthe system f.tx,~•••u"..Moreover, Giwill (bythegeneralprinciple ofsuccessive concomitance) beaninvariant inrespecttothesystemf,4>,il,and combi­ nantiveinrespecttoIandcp.ThusthenG,for alladmissible valuesof iwillbelongtothefamilyof forms to which theBezoutiant is to be referred. Itrequirestobenoticed,thatwheniistakenzero, sothatT.andG, are ofthedegree2(m-1),E"forthiscasemustbetakenequal toa',which L M 562 OnaTheoryoftheSyzygetic Relations [57 evidently fulfilstherequired conditions of being of thedegree2(m- 1) in (x,y),andquadratic inrespectofthecoefficients of n.If,now,mbe even, we may takefor 2i+1successively all theoddnumbers from1to (m-1) inclusively, and therewill betmformsGi;when 1nis odd we may take for 2i+1 successively all the odd numbers from 1 to 'Tn,andthenumber of forms of G,will bet('Tn+1).Itshould be observed, thatwhenmis odd and 2i+1=m,T,will become identical withthelineo-linear combinant tofand1/>,andQiwiththequadrinvariant ton;and no power of xorywill enterintoeither,sothatGmwill become simply TmxQm'Iamnow able toenunciate theproposition, thatGo,GI•••Gm,whenmis even, and--12 Go,GI'"Gm-l,whenmis odd, form theconstituent scale of forms, of which 3 theBezoutiant and allotherlineo-linear quadratic functions of m variables, which are combinants of thesystem/. 1/>,will benumerically-linear functions. I propose to termthemembers ofthisscaleCo-bezoutiants. .ABregardsthepresentmemoir, I shall contentmyself with exhibiting apartialverification ofthislaw as regards theconnection oftheBezoutiant withtheGscale of Co-bezoutiants, and acomplete determination ofthe. numerical multipliers which express thisconnection for thecasescomprised between m=2 and m=6takeninclusively., Itis impossible to predict forwhatulteriorpurposes inthedevelopment oftheCalculus ofInvariants thesenumbers mayormay not be required, and it seems tomedesirable thatacommencement of atablecontaining themshouldbemade and placed on record. Theremaining pages of thismemoirwill accordingly be devoted totheascertainment of them. ThetheoryoftheBezoutoid being included withinthatoftheBezoutiant, need not hereafter callforany special attention; I may merely notice that theBezoutoid to a function of thedegreemwill be a numerico-linear function ofHm-3) of the G'sifmbe odd, and t(m-4)oftheG'sifm beeven. Itwillbemoreconvenient hereafter to denote theG'sasGI>Ga.G. respectively, in lieu of Go,GI,G«,&c., and to continue atthesame time to give totheTsand Q'sthesamesubscripts asthecorresponding G's. Art. 69. Firstly. Suppose m=2,.f=a:x"+2b,xy+Cyi, I/>=cu.i+2{3xy+"'fyi, n=UIY-u,x. 57J Then andthereforeoftwoAlgebraical Functions. Ed=(ax+by)E+(b:.c+cy)"1, E,4J=(c:w;+fly)E+(fl:.c+'YY)"1, T1=(ax+by)(f3:.c+"tY)-(b:.c+cy)(arc+fly) =(afl-ba)[L.J+(wy-ca):.cy+(b"f-c(3)y'J, QI=il2=~y- 2~Ut:.cy+Ut2:.c2,563 01=(af3-ba)U12+(wy-ca)~Ut+(b"t-c(3)Uti. Letus now form in theusualmannertheBezoutiant tof,4J;this isthe quadratic function which corresponds to thematrix (2afl-2b2),(wy-ca)L (u"t-ea),(2b"f-2cfl») thatis tB=(afl-ba)~I+(wy-ce)~Ut+(b"f-c(3)Uti=01orB=201, Secondly. Suppose m=3. f=ax'+3ba;ly+3c:.cy'+dys, 4J=a:.c'J+3fla;ly+3"t:.cys+oy, il=~yl-2u,y:.c+Ut:.c'. We have then Ed=(ax'+2b:.cy+cyl)E+(b:.c2+2c:.cy+dy')"1, E,4J=(a:.c2+2f3:.cy+"tyl)E+(fl:.c'+2"t:JJY+oyl)"1, T,=(ax'+2b:.cy+cy')(f3:.c2+2"ttry+oy')-(b:.c'J+2ca;y+dy')(a:.c'J+2f3a:y+"ty') =(afl-ba):r:'+2(u"t-ca):.c2y+{3(b"f-cfl)+(aO-da)}:.c'y2 +2(bo-d(3):.cy'+(co-dry)y4, Ql=ill=u1"!t-4u,usy':.c+(4u'.l'+2~u.)'Y':.c'-4u.u.y:.c2+u,1a:'. Supplying for facility of computation thereciprocals of thebinomial coefficients to theindex 4, namely 1,-t,-t,-t,1, weobtain 01=(afl-ba)Ux'+2(wy- ca)ulUt+{2(b"f-cfl)+Hao-da)}Ut2 +{(b"f-cfl)+·!Cao-da)}~Us+2(bS-dfl)UsUs+(co-dry)u,I. Itwill here and henceforth be more useful to employ [1',8]to denote, not the difference of the cross products ofthe(r+l)thand(8+l)thentire coefficients in Iand4J,butthedifference of the cross products of these 36-2 564 On aTheoryojthe Syzygetic Relations [57 coefficients divided eachbyitsappropriate binomial coefficient. We may thenwrite GI=[0, 1]Uti+2[0,2]UtUs+([1,2]+HO,3])UtUI+(2[1, 2]+HO,3])Usl +2[I,3]UsUs+[2,3]Usi• Again, GI={(aO-da)-3(bry-c,8)}(UtUs -~I)=([0, 3] -3[1, 2])UtUs - ([0, 3]- 3 [I,2])ul. Hence GI-iGI=[0,1]Uti+2[0,2]UtUs+2[1, 2] UtUs+([0,3]+[1, 2])u.1 +2 [1, 3] UsUs+[2, 3]u.1• But,again,theBezoutiant off,4>corresponds to thematrix 3[0, 1], 3[0,2]. [0,3],3[0,2], [0, 3]+9[I,2]. 3[1,3],[0,3], 3[1,3]. [3,4]. Hencesumming thesinisterbands to form thecoefficients,we have B=3[0, 1] UII+6[0, 2]UtIUs+(3[0, 3]+9[I,2])Usl+6[1, 3] UsUs +[2, 3] u,1=3GI-Ga. Thirdly. Suppose m=4, f=ax'+4b:ry+6e:c'yl-+4dzy'+ey', 4>=au:'+4,8wy+6ry:r:'y'+4oxy'+ey', n=Uty'-3u.yl:c+3ulya;l-u,w. Then E./=(aa;+by)EI+3(ba:+cy),s"+3(e:c+dy)",11+(d:c+ey)"}3, therefore T._{(aa;+by)(&;+ey)}_3{(ba:+cy)(ryx+oy)}.--(a.a;+,8y)(d:c+ey) -(,8a;+ryy)(e:c+ dy) =([0, 3] - 3[1,2]):r;I+([0,4]-2[I,3])a;y+([1,4]- 3[2, 3]) y' and Q.=(Uty-u.:c) (u.y-u.:c)-(u.y-u.:c'f =(UtUs-'Us1)yl-(Utu,-UsUs):cy+(Usu,-UsI)w. Hencesupplying thebinomial reciprocals 1,-l,1, wehave G.=([0,3]-3[1,2](UtUs-u.1)+i([0,4]- 2[1,3])(UtU,-!Lau.) +([1,4]- 3[2,3])(Usu,-u.1) . 57J oftwoA19ebraical Functions. 565 Again, TI=(ax'+3b:i'y+3CX!f+dy')({3w+3'Ywy+3&cy'+fy') -(ax'+3{3wy+3'YXY'+oy')(ba:-'+3c:r:'y+3d.xtl+ey') ~[0, 1]~+ 3 [0, 2] a!y+ (3[0, 3]+6[I,2])~y'+ ([0,4]+8[I,3])wy' +(3[1,4]+6[2,3]):r:y+3[2,4]xy"+[3,4]y', and QI=.Q' ='-'1'y'-6'll1~y'a;+(9u,'+6~'Us)y4w- (2~u,+18'Usu,)wy' +(9u,'+6'U,u,) ys~-6u,u,y~+u.'~. Hence,supplying the reciprocal binomial coefficients, I,-!'+-h,-n,-h,-i,1, we find GI=[0,1]~'+ 3[0, 2] ~U2+(![0, 3] +i[I, 2])(9Ut'+6~u,) +(-h[0,4]+ttr[I,3])(~u,+9u,Ua) +<![1,4]+1[2,3])(9u.2+6Ug'll,)+ 3[2,4]u,'ll,+ [3,4]u,'. NowtheBezoutic square, takingaccount of the binomial factors in fandt/J, may bewrittenundertheform 4[0, 1], 6[0,2], 4[0,3], [0,4], 6[0,2],[4[0,3]][[0,4]]4[1,4],+24[1,2],+16[I,3] , 4[0,3], [[0,4]J[[1,4]]6[2, 4],+16[I,3] ,+24[2,3], [0,4], 4[I,4], 6[2,4], [3,4]. Hence the Bezoutiant Bbecomes 4[0,1]~'+ 12 [0, 2] ~'Us+(4[0,3]+24[I,2])~'+ 2[0,4]~u, + (2[0,4]+32 [1,3])U,'Us+ 8[1,4]'Us'll,+([I,4]+ 24[2,3])us' +12 [2, 4]u.u,+[3, 4]u,'. And we oughtto haveB=cGI+eGs.to satisfy which equation we must manifestly have c=4;to find e, compare thecoefficientsof u,',thisgives 4[0, 3]+24[1, 2]=¥[0,3]+.y.[1, 2]+e(3[I,2]- [0,3]); accordingly we oughtto be able to satisfy thetwoequations ¥-e=4,.y.+36=24, each of which accordingly we find is satisfied by the equalitye=1/. Substituting intheequation forBabovewritten,wethusobtain B=40GI+1/Ga, which will be found to be identically true. 566 On aTheoryoftheStJzygetic Relations [57 Art.70. We maynow see our way to amore concise mode of obtaining thenumerical coefficients, by which theymay in fact be computed and verified with comparatively littlelabour,connecting theBezoutiant with the Co-bezoutiant forms of theconstituent scale.Itwill not fail to have been remarked,that throughout thepreceding determinations I havepresumed thetruthof the formula, which admits'of animmediate verification, thatfor all values of mandI»w~have the identical equation (E::x+."ty)"'{Coa!"+mc"a;m-lY+lm(m-1)C~y,+ ...+mCm-la-ym-l+CmY"'} =m(m-1)...(m-I»+1){LoE"+I»L1E'"-I"7+jl»(I»-1)LtE-V+ ...+L...".}, where m-I»-1Lo=coa;m_+(m-I»)Cta;m--ty+(m-l») 2C;,Xm--,y'+ ...+c",_y'" L"'-( ) "'-_I ( ) m-I»-1m--t.,2 '" .=c.,a;+m-l»c ..+J'xy+m-I» 2C;,X:/+...+cmy. Letus now proceed to determine by anabridged method the linearrelations corresponding to thecasesofm=5,m=6,and first for m=5. Let f=agfl+5ba:'y+10c.xBy'+10a.rya+5e:T:!t+h'!l, ~=a.afl+5{Ja:'y+1Oty:i'y'+10ory'+5e'x~+."ye, .n=uJY'-414:!I,X+6u,y'r-4u.yr+Ua.x4. InformingGs,Gs,G1,let us confine our attention totheterms ~',~!.ta,~Uf• Acomparison of the coefficients of these with those in theBezoutiant (B) willbe sufficient for assigning thethreenumerical quantities which connect BwithG1>G.,Gs•I omit Ut'Ut,becauseG1is the only one of theG'sfor any value of mwhichcontains Ut'orU1Ut,and inG1thetermscontaining Ut'andUt'Utare [0, 1]ut'+(m-1)[0,2]U1U" andthecorresponding partof theBezoutiant is m[O, 1]u l'+m(m-1)[0, 2]UtUti sothatif we write B=C1G1+caGa+csGs+&c., thetwotermsUt'andUt'Utwill only enable us to form one equation withthe c'a,namely, Cl=m:Again,insteadofconsidering theentirecoefficients 57J oftwo A1ge1Yraical Functions. 667 ofU1'll,andU1U4'it will be sufficient to takeasingleargument ofeither ofthesecoefficients (intheforms to be compared), asforinstance [0, 3]and [1, 3].Thenc;beingknown,c.,c"will bedetermined; butforthepurposes ofverification Ishallfurthermore compute thewhole of thecoefficient of ~'ll,. Accordingly, calculating theGsysteminreverseorder, we have Gs={EO,5]- 5[1,4]+10 [2, 3]} {~u"-4~u,+3ul} ={EO,5]- 5[1,4]+10[2, 3]} ~u"+..., Ed=(aa;I+2bxy+cy')f·+3(br+2cxy+di)E", +3(er+2d:r:y+ei)~'+(rk.t+2ery+fy')'1]' E.4J=&c.&c.; therefore T,={(aa;I+2bxy+cyt)(or+2EXy+"Iyl)-(ax'+2fJ:xy+",(yl)(dw+2exy+hyt)} - 3{(bxJ+2ca:y+dy2)(-yw+2Sxy+Ey')-(fJw+2"'(xy+oy2)(c:rfJ+2dxy+eyt)} =([0,3]-3[1,2])x4+ (2[0,4]+...)ry+{EO,5J+[1,4]-8[2,3]}ri+&c. Thenumber- 8resultsfromthecalculation 1 - 3(4 - 1) = -8. Again, Eln=(~'!/-2~yx+u.w)~'-2(u,yl-2u,yx+u,w)~ +('ll,y'-2u,yx+u"w)'1]1, therefore Q.=(~yl-2l4.Yx+u,x')(u,y2-2u,yx+usw)-(u,y'-2UaYx+u,r)1 =~Uly'-2u1u,ylx+~u"y'w+&c., allthetermsandpartsoftermsunexpressed beingfree of ~,andtherefore notnecessary forourpurpose. Hencesupplying thereciprocal factors 1,-i,!,..., wehave G.=[0, 3J ~'ll,+([0,4]+)~u,+*{EO,5] +[1,4J+[2, 3]} ~u,+&C. Again,expressingEJandE1epintheusualway, we obtain T1=(a$'+4bry+6eryl+4d:xy1+ey')(/3x4+4"'(ry+6&c'Jyt+4EryI+'1]y') -(cz.x4+4fJry+6"'(wy'+4Sry'+Ey')(ba,.4+4ery+6dx'y'+4exy.+hy') =[0,l]r+4[0,2]rriy+(6[0,3]+)a;8yt+(4[0,4]+)a;Dy. + ([0, 5]+ 15 [1, 4] + 20 [2, 3]) x4y'+&c. (whereitmay beobserved thatthenumbers 15 and20inthecoefficient of x4y'arise from thequantities 41-1, 61-42) . 568 Ona Theory oftheSyzygetic Relations [57 Again. Ql=fit=u12r+8u1Ut:CY+12ulu,ryt -BUt.u.a:I!I+2Ut~a:ty'+&c. Hencesupplying themultipliers -11-11 1.S'28'56'+70'&c. we have G1=[0, 1]Uti+4[0, 2]UtUt+J,}i[0,3]u1u.+t[0,4]UtU. +n([0,5]+15[1,4]+20[2,3]) U1Ua· Again.theBezoutiant B=5[0, 1]Uti+2.10[0,2]UtUs+2.10[0,3] U1Ua +2.5[0, 4]UtU.+2[0, 5JUlUa+&c. Accordingly, if we write B=C1G1+c.G.+c.G••we have, asaboveremarked, Ct=5;and todetermine Cs•c.,we have, by comparing thecoefficients of U1Us.Utu.inB,Gl>Gs,G., 20=¥+es 10=¥-+Ca. These two equations, then, asitturnsout, are not independent, butare satisfiedsimultaneously by es=¥. Finally,equating thecoefficientsof the several arguments inUI~'we have 0=5xn+¥xi+c.fromtheargument [0,5], 0=5 xit+¥xi+5c.from the argument [I,4], o=5 xH+¥xi+10c.from the argument [2, 3]. The first of which equations gives c.=2-t-H=H=i; the second gives andthethirdgives c.=H+f=J· We have thusabundantly verified the accuracy of thecalculation, and there resultstherelation Lastly, let m=6, f=a:J!l+6bx'y+15ca:ty'+'!.OdWyt+15eafy'+6hxy'+l!!, cf>=a:r;S+6{3x'y+15rya:ty'+20oa-'!I+15Eafy'+6'T]xyI+")I,:!!, fi=Ut!l-5Uty'x+10u,ysaf-lOu.ytr+5u.ya:t-u.x'. 57J oftwoA19ebraical Functions. 569 I shall here confinemyself to thedetermination of a single argument in each ofthetermsult,UtUs,UtUa,UtU4,ulua,UtUaithis will be ample for the purpose of verification, astheequation to beassigned is oftheform B=CtGI+caG.+caGa. Thearguments which I select asthe most simple, will be those expressed by thesymbols (0, I), (0,2),(0,3), (0,4), (0, 5), (0, 6) respectively; thenwehave Ta=(~+by)(7]X+AY)+&c.-(h:c+ly)(ax+(3y) =([0,5]+...)r+([0,6]+...)xy+ (...)yt, Qa=(Uty-~x)(Uay-Uax)+&c. -(Utua+...)yt-(UtUe+...)yx+ (...)r. Hencesupplying thebinomial reciprocals 1,-t,1, Ga=([0,5]+...)'1.£1'1.£.+t<[O,6]+...)UlUa+&c. Again, T.=(a:r;I+...)(Bw+3ery+37]xy'+Ayt)+&c. -(da;I+3e:c'y+3ha:yt+lye)(ax'+...) =([0,3]+...)~+(3[0,4]+...)~y+(3[0,5]+ )flfyt + ([0, 6]+ ) zly'+&c. Q.=(Uty'+&c.)(u.,yt+3u4Y'+3ua7r-u.r)-&c. =(UtUs+...)y'-(3UIU4+...)yex+(3'UtUa+...)rr-('1.£1'1.£.+...)yaw+&c., andthe reciprocal binomial multipliers will be -1+1-1 1"6'IT'20'&c. Hence G.=[0, 3] '1.£1Us+i[0,4]UtU4+t[0, 5]UlUa+n[0, 6]UtUa&c.&C. Finally, TI=(~+&c.)({3:c1+5ryflfy+1O&xay'+10ery'+5"1xy4+Aye)-&C. =([0,1]+...):cI0+ 5([0,'2]+...)aJly+ (10[0,3]+...)wyt +(10[0,4]+...)x'y'+(5[0,5]+...):c'y4+ ([0, 6]+ ...)~ye+&c. QI=Ot=1l-tll.to+(10u1Ut+...)y'x+(20UtUa+...)'!Iw+(20Utu4+...)y':cI +(lOUtua+...)y':c4+(2UtUs+...)ye:cl+&C.i 570 Ona Theory ofthe Syzygetic Relations [57 andsupplying thenumerical series we haveI.1 1-11-1 - 10'45'120·210'252'&c., G)=[0, 1]U]2+5 [0, 2] ~'U..I+-¥[0, 3]~u.+HO,4l~u, +It-[0,5]~u,+rh[0, 6]U]Ue+&c. Again,theBezoutiant =6[0, 1] ~2+30 [0, 2] ~'U..I+40 [0, 3] ~u,+30 [0,4]~U, +12[0, 5] ~u,+2[0.6]U]Ull+&c.&C.=B. Hencemaking B=C:1G)+C,GI+C,G" from~2and~U2weobtainrespectively 5c]=30; hence from U]UIand~U4weobtainrespectively ~+ea=40}¥+tea=30 or CI=.y; hence from ~Ueand~U8weobtainrespectively 6 xfr+-¥i+c,=12,thatisc,=12- 8 -J,f=J,f-. 6 xm+.yn+tc,=2,thatistC,=2-i-~=,; hence andtheequation soughtfor is B=6G]+.yG.+J,f-G,. Art. 71. The following tableexhibits therelations between the Bezoutiant andthecorrespondent system of Co-bezoutiants for allvalues of mbetween 1 and 6 underasynoptical form. m=1,B=Gil m=2,B=2GIl m=3,B=3G]-GI• m=4,B=4G]+J.fG., m=5,B=5G]+¥G.+iG" m=6,B=6G]+JfG,+J,f-G,. 57J oftwo Algebraical Functions. . 571 These series could if wantedbe easily extended, andthecalculation ofthe coefficientsreduced toa mere mechanical procedure. Ifwe suppose mto be 2i or 2i -1,we have the equation B=C:1G1+OsGs+...+t;&-1Gai--l, and itappearsfromtheforegoing instances thatthecomparison of the coefficients, eitherofu1a,or ofUtUsonthetwo sides of theequation, will serve togive01(mbeingknown), Osmay be found by acomparison of the coefficients eitherofUtus,or ofu1u"and so on for Ca..•C;;-I; allthecoefficientsin theequation forBabove given, thusadmitting ofbeing foundseparately and successively andin two modes, so thatthereisa check at eachstepupon the correctness ofthecomputations: the only exception tothislastremarkis (when misodd) for the lastcoefficient of which the above condensed method affords only a singledetermination. I need hardly addtheremark,thatinsubstituting zm-t,~y,..•f1Jym-t,ym-lin place of Ut,Us•••um-1,U'"respectively, all theG'sbecome (to anumerical factorpres) identical with one anotherandwith the Jacobian tothesystem(f,rjJ). Art. 72. The foregoing theory took its origin (aswillhave been readily imagined) inmeditations growing out of the celebrated theorem of M. Sturm. Thereappearto be several directions in which a development orextension of thesubjectmatterofthattheorem maybe soughtfor.Thusatheorymay beconstructed relativetoasingle function of one or more variables, viewed in all cases asrepresenting ageometrical locus. In the limitingcase,when thislocus becomes a system of points in arightline, we have thetheorem ofSturm;generally thetheorywill bethatofcontours. Or, again, a theory may be formed in which the numberof functions is always keptequal tothat of the variables. We have thenatheoryofdiscrete pointscorresponding to roots, the numberof realonesof which comprised within given limits itis the object of such theory todetermine. M.Hermite, inamemoir recentlypresented totheFrenchInstitute, appearstohavemade a valuable addition totheSturmian theoryextended inthisdirection, to which the beautiful researches of M. Cauchy andthejointlabours of MM.Liouville and Sturm,with reference tothedisposition of theimaginary roots of equations appeartohave led theway.Finally,thenumberof variablesmay besupposed to bearbitrarily increased, butmade always inferior by aunittothenumber of the functions in which theyare contained, or which comes to the same thing,we may construct thetheoryof a system of homogeneous functions equal in numberto the variables in them, which in its simplest case becomes thetheoryofIntercalations whichhasbeen here partially considered, and which(ashasbeen shown) embraces (notasaparticular case,butasan impliedconsequence and easilyextricated result)thetheorem ofM.Sturm. 572 On aTheoryofthe Syzygetic Relations General and Concluding Supplement.[57 Art.(N).Theexpressions giveninArt.(n)[po507 above] for thepartial quotients ofthecontinued fractionrepresented byj;.arerestricted tothe supposition of allthesepartialquotients (exceptthefirst)beinglinearinx; whenthefirstpartialquotient islinearthe formula (B)ofthatarticlecontinues applicable onreplacing (Dih,)by1.Iwasforciblystruckbythepeculiarity oftheseformuha notceasingtobetrueinconsequence ofthefirstpartial quotient beingsupposed non-linear; andreBecting upon this, IwasBOOnled toperceive thatallthepartialquotients mightbesupposed tobearbitrary integral functions ofe,andtheformulee wouldstillcontinue toapplyto any such of themasmighthappentobelinear,although, asitwere,imbedded amongagroupofothernon-linear partialquotients. Fromthisitwasbut aneasysteptoperceivethattheformulre(A) and(B)mustadmitofextension totherepresentation ofpartialquotients of any form,and thatthedimorphism oftherepresentation ofthelinearpartialquotients could only be aconsequence oftheequation inintegers u+v=1havingtwosolutions u=0,v=1and u=1,v=O.Inow proceed to enunciate theveryremarkable general theorem (orasit mayperhaps notinappropriately betermedAlgebraical Porism), by virtueof which any partialquotient ofagivendegreeinx belonging to aninfinitecontinued fraction, all of whose partialquotients are algebraical functions ofai,may be expressed toaconstant factorpres,by means of thenumerator anddenominator (or if we pleaseeitherone ofthese) oftheconvergent immediately antecedent to and of thenumerator and denominator ofanyconvergent notantecedent tothepartialquotient which is to bedetermined. Art.(:1).Theorem. LetQI'Q'J...Qi.Qi+I...Qn,&c.each of an arbitrary .degreeine,be thenfirstpartialquotients ofanalgebraical continued fraction; letQi+lbethepartialquotient to bedetermined and ofthegiven degree (c)i+l;let and1 1 I 1 4>i(X) QI-Q~~Q.-'"Qi=}.(x)' 1 1 I I 1 1 <t>(x) QI-QI-Q.-Qi-Qi+I'"Q:=F(x); letuandvbe any couple of integers ofthe(c)i+l+1 couples which satisfythe equation v+u=(c)i+l;then,asusual,denoting theproductof thedifferences ofeachof one set of termsfromeachofanotherset, bywritingtheformer underthelatter,andcalling '11>"11'""1,.the,.,.roots of<t>(x),andhI'hs...h.". 57J oftwo.A19ebraical Functions. 573 themrootsofF(x),(<I>andFbeingsupposed respectively ofII-andm dimensions inx),andformingthedisjunctive equations 81,tJi,tJa•••tJ,..=I, 2,31-£, tl,~,ta•..t".=1,2, 3m, wehavethefollowing equation, whereinq,andfarewrittenfortP.andji, Qi+l=Ku,.x~{(tPfJ'ltP".....4nJ••Jx(fhe.Jht.·.·fhc.J ['7'1''7.....'7.][he1,hta•••he:J he.+!,he.+I•••~X'7'.+1''7'7. Xl'7'I ''7.....'7'.X[hel 'he,he. '7'0+1''7'0+1'"'7.he.+1,he•.."he. x{(x-fJ,,)(x-'71,)•••(x-'7••)!{(x-hc,)(x-ht.)...(x-hc.)}}, andmoreover thedifferent valuesofKu,.depending uponthedifferent modes ofbreaking up0)i+1intotwopartsuandIIareall (to a numerical factorpru) equaltooneanother. Thusthenthetheorem pointedatinArt.(p)is discovered, andtheway laid open (by anunexpected channel) for acomplete discussion ofthetheoryofthesingular caseswhich may occur in the expansion ofanyrationalalgebraical fractionundertheform of a continued fraction. Art.(.'I).Intheaboveexpression, ifwesuppose 0).+1=1, we have u=1 andII=0, oru=0 and II=1,andremembering that [h ] ...<l>hand['7 ] ...F'7, '71''7""'7,.. i;~...h", [he,]=F'heand['7'1 ]=4>''7",ht.,he,...he", 1'71"'7es'"'7'''', Qi+lbecomes by virtueofthegeneralformula representable undereither. oftheequivalent forms Ko,II'{(tP'7.J:~.(x-'7,)}andtc.,It{UheJ;~:(x-he)}, KO•landKl•Obeingeitherequal,ordiffering only inthesign,agreeably to theformulse (A) and(B)[p,508 above]. Art.(').Itmay be worthwhile to notice, that,although (ofcourse) these formulas andthegeneralformuleeof Art. (~),whensupposed converted intofunctions ofxandofthecoefficients of Fandof<I>bythereduction, integration andsummation ofthesymmetrical functions oftherootswhich enterintothemremainuniversally valid,andsubjecttoDOcasesofexception, 574 Ona Theory oftheSyzygetic Relations [57 yetantecedently totheseprocesses being performed theformulse asthey standmay become illusorywhen any relations ofequality existbetween the roots of 4>interse,orbetween theroots-ofFinterse.Thusin thecasebefore us, if<I>have equal roots theformulacommencing withKo,Iis illusory, and ifFhave eqnal roots theotherofthetwoformulee becomes illusory. Letustakethesecond of theseand suppose thatF(x)has kirootsCt,~rootsc;...kprootscp, we may pass to theactualcase from any case where theroots are infinitesi­ mallyneartotheactualroots ofF(x),and allinfinitesimally different from oneanother. Moreover thechoice of theinfinitesimal variations being arbitrary, letthek,rootsCtbereplaced byagroupofroots CI+S,Ct+SPI'Ct+SPI'.•. CI+Sp/'I-t, wherePIisaprimeroot oftheequation pl"=0, andSis aninfinitesimal quantity, and suppose eachoftheothergroupsto be varied in an analogous manner. Thenitmay easily be shown from thisthatthesecond of the formulas inquestion will become p(ir-1 {(fCt)l(<I>Ct)(x -c,)] s;t~lkc(d)k '--Fc,de,_ andsimilarly, thetwinformula becomes ft(~Y-l{(t/Y't.)I(Fry.)(x-ry.)] KOI.~/,' (d)« -.- - -<1>"(. dry. Corresponding modifications will admitof being made by aid of alike method in thegeneral formulse of Art.(:1)uponasimilarsupposition asto equalities springing upbetween theroots offxperseand of ~(x)perse, orbetween theroots offo:andtf>xinterse. =(d).. {1{1(e+")+p1{l(e+pa)+p'1{I(e+p'&)+ ...+p"-I 1{1(e+ p..-Ian ikfe&",-I (:cr--I t/lewa..-I(~r-I ~e =(~rfea"'-I =111(~rfe •• For in generalifpisaprime root of theequation /,,"=1,andiffzhaveIIIroots all equal toeand1{Izis anyotherfunction ofzand if&isaninfinitesimal quantity. thenrejecting all powersof&higherthanthe(III-l)thdegree, 1{I(eH) 1{I(e+pJ) 1{I(e+p'&) 1{I(e+p°l-l&)r(e+&)+F(e+p&)+f'(e+frl&)+ .-,+J'(C+PW-' &) 1 57J oftwo.A1gebraical Functions. 575 Art.(n).Ifin Art.(.:1)wetakei=0,theformula for Qi+1will become ["1'1'"111t "1••1X[hel'ht,heu] .he...l,ht...,heJ "1......."1.....1"1'''' QI=s;.[ ] [l I.h] •"1.1'"1"•.•"1..Xlei''''tt•••eu "1'''''1'"1•.-+!•••"1''''ht...1•he...,•..he.. uandIIbeingany two integers whose sum is Cell>which is identical (as it oughtto be) with theexpression virtually contained intheformulas of Section II. forthesyzygetic multiplier of<I>(a:)inthesyzygetic equation connecting Fa:and<l>a:withtheirfirstresiduewhen<l>a:is supposed to be Cell dimensions ina:lowerthanFa:identical, videlicet, inotherwords, with the integerpartofthealgebraical fraction~~~~. Art.0).When<I>(a:)=F'(a:), <I>(hh,.),<I>t)·..<I>~h-u.,~becomes identical with(_)i(";+1-1)01;+1~(~,hs...h...+1)' [I' .., "'<+IJ ~+oo<+..ht+"l+1..,h". and we may consequently (usinganextreme termintheforms in the polymorphic scaleof forms representing Qi+I),write Qi+l=(_)i("HI-I) "'>+1KO."'I+1In~,hs...h"'l+l)(fi~),(fihs)1... (fihOll+I)1 (a:-hi)(a:-hs)...(a:-h0ll+1) ' Art.(T).Thefollowing observations will serve to complete thetheoryof thesingular casesintheexpansion of an algebraical continued fraction. Preserving thenotation of Art.(:1),let ui=m-(Cell +~+...+Celi-l+1). Then(callingtheroots of Fa:,h«,hs...hm)the(i)thsimplified residue to ~:' in accordance with thegeneralformulee for theresidues inthesecond section (for greatersimplicity selecting anextreme termofthepolymorphic scale), will be represented by ~<l>h1,<l>hs,<l>h,<l>h<l'(h)(h)(1.)(h) ~[hi'b«,hihal]a:-Ia:-~a:-'''J•••a:-<I" hH<I"hs+<t"ha+<tl'"s; which will be of theformLitt·;-..;+I+&c.,allthetermscontaining higher powers of a;vanishing bythecoefficients becoming zero.Ifintheabove expression we should use ulin lieu of Ui,whereu(isu.diminished by any integerinferiortoci"we should getotherforms of thesameresidue,but 576 OnaTheoryofthe Syzygetic Relations [57 thesewill all be of higherdimensions intheroots or coefficients than theonejustgiven, and in fact theformsthusobtained corresponding tothe valuesas,tT,-IJtT,-2...tT,-""+1substituted fortTlin succession, would, by aid of the relations ofcondition between thecoefficients of <I>xandFx impliedin the value of ""Jadmitofbeingexhibited asa scale in which each form would bean exact algebraical productoftheform which precedes it, multiplied by a function of thecoefficients, and did space permitthereof itwould be perfectly easy to give theforms of these multiplicators. But Ipasson totherepresentation of what is more material, namely, theform ofthecomplete residueinthecasesupposed, merely observing (asan obiterdictum)thattheexistence of eachsingular partialquotient (meaning thereby aquotient non-linear inx)only affects the form of thesingle simplified residueinimmediate connexion with itself, and not atalltheform of theotherresiduesantecedent orsubsequent tothatone. Art.(M).Lettheithsimplified residuebecalledRoandthecorrespond­ ing complete residue[Ro].thenapplying amethodsimilartothemethod given in SectionL, we shall find that L.....,+IL-+l&c (_)'[R.]=i-I (-4 •Ro •L~I+1 L~I-o+l&c'.-1.-3. L,representing theleadingcoefficient in theithsimplified residue, and the sign ofinterrogation (7)denoting some function of "'11"'I'"""(possibly a constant) remaining tobedetermined. Andreverting to Art. (:1),the quantity thatwould be called K o,..according to the notation employed in theformulas expressing Qi+1inthatarticle.will(abstraction being made of thealgebraical sign and using for greaterbrevity(t),(, -1),&c.toexpress 1+""J1+"'i-1I&c.)come to be represented by L(' - I )L4(.-3)L4(' - 6)&i-Ii-3i-6c. L\')L!('IIL4(.- 4)&c'• <-2 (-4 • asimilarconvention being supposed to be made respecting thenumerator anddenominator of eachconvergent aswas made respecting them in the particular casetreatedof inArt.(f).page [502]. Art. (~).I will merely add avery few words in generalization ofthe methodoflimiting the roots of fxgiven in theSupplement tothefourth Section [po528 above]. As an inferior limittofxisidentical witha superior limit tof(-x)Jwe may confine our attention tosuperior limits alone. Suppose thenthat ¢X1 1 1 1 1 1 1 1 1 fx=QI-QI-...Q,-Q/- QI'-'"Q'f...(Q)I-(Q)I-'" (Q)(t'), 57J oj twoAlgebraical Functions. 577 wherethepartialquotientsQare each of any arbitrary degree in e,and have all onealgebraical sign in the coefficients of the highestpowers of a:fromQI toQ"and allthesame sign (contrary totheformer). in the coefficients of thehighestpowers of xfromQ,'toQ',-,and so on alternately, thenfirstly asuperior limitto thesuperior limits of the cumulants [QI'Qg...QiJ. [Q/,Qg'...Q'e],...[(Q)l'(Qh..·(Q)(,)]will beasuperior limittofe,sothatit remains only to give a rule for finding asuperior limitto acumulant [QI'Qg,Qs...Q,],which, secondly, isto be found by making QI-M1=O.Qa-Ma=0,Qs-Ms=0...Q,-M,=0, 1 1 1where M1=/l-J"Mg=/'1+-,Ms=/'1+- ...M,= - , IJ-J. /I"J /-'i-I IJ-J.,/I"J.../-'i-Ibeing any quantities entirelyindependent andarbitrary except inregardtotheirbeing all of thesamesignastheleadingcoefficientsin the elements QI'Qa...Q,. We may thenfindL1,Lg•••L,anysuperior limitstotheroots of xin theseiequations respectively; L,thegreatest of these, will be asuperior limitto the proposed eumulant [QI'Qa'"Q,]janditmay be observed that M1,Mg•••M,arethegeneralvalues which satisfytheequationIIIM1-M,-xr_•..M,.=0,g-lUs , subjecttothecondition thatfor all values of e 1 1 1 1 M.-Me-I-M&-S-...M1 shall have a given invariable sign. The first partoftheprocess,asjust shown, consists in separating thetypeof thetotalcumulant whichrepresents Ixintopartialtypes,thepointfor each fracture of the total type being markedbyachange of sign in theelements ofthetype for the value x=+00;itiseasilyseentherefore from this, thatif;;isthegeneratrix ofthecumulant in question, the number of such fractures (thatis,the numberone lessthanthenumberofpartialcumulants) will be the number of changes of algebraical sign in the signaletic series,consisting of the leadingcoefficients in Fxand in each of theodd-placed complete residues respectively, together with the numberof changes of sign in the signaletic series,consisting oftheleadingcoefficients in $xand in each of theeven­ placed complete residues respectively. The syzygetic theoryof two algebraical functions, and thealliedtheory ofalgebraical continued fractions with theirprincipal applications, may, Ithink,now besaidto be completely made out, aswell for thesingular cases88forthegeneralhypothesis. ~ 37 578 OnaTheoryoftheSyzygetic Rekuion« [57 Art.(').I willconclude withobserving thatthetheorywithindeveloped givesthemeans of transforming (explicitly andwithout theaid of sym­ metrical functions) intoanalgebraical continued fraction, anygivensum of algebraica.l fractions oftheform ~ ~ ~ ~ -r+----,:+------,;-+...+------,;-,q;-,"t q;-,"Iq;-,.. q;-,.. whereeach candhissupposed known. Forlettheabove sum be called ~:'thenifh"o.beusedtodenoteanypairofcorresponding termsofthe hseriesandthe0series, we have;t=c.,asiswell known andeasily proved. Again,ifD.q;represent thesimplified denominator oftheithcon­ vergenttothecontinued fractionequalto~;whichisto be found, say I I 1 (-;-"A-;-t-q;-+---CBt)-(A,q;+B,)-...(A"q;+-B~), we have [po476above] D.q;=sI~:'~~~.\(x-h1)(x-hs)...(x-ho) "-o+t,ho+sb« _~(_)'('-1)n~,hs...ho)<l>ht,<l>hs...<I>~( _I.)( _I.)(_h.) -~ I F/~F'hs...F'ho x'''Ix'''1''' q;, ,.-1 =(-) II{~o,...O('(ht,h, ...ho)(q;-~)(x-hs) ...(q;-hi)}' Therefore (D,~)'={I(~Ca•..O(+t)t'(~,h,"-o+t)(ht-hs)(~-h,)...(~-ko+t)}' ={I(GaGa...Ci+t),.(~,hsko+t),.(~,h,...ho+t)}I; andthesimplified (i+l)thquotient, thatis,thevalue of Ai+tx+ Bi+1Jwhen divested oftheallotrious factor,hasbeenproved [cf.p. 508 above] to be equalto <I>~ ~(D.~)' F/~(x-~); itistherefore now known asarationalandintegralfunction ofq;;~,hs...h,.; Ct,Os...0".Theallotrious factoritselfis made up of theproduct ofsquares ofquantities all ofthesame form astheleadingcoefficient in D.x,which, fromwhathasbeenshown above, isseento beequalto ,-I (_f2I{(~CI'"c,)t'(~,~...hi)}' Henceeachterminthecontinued fraction I I 1 (Atx+Bt)-(A,x+B,)- ...(A"q;+B,,), 57J oftwoAlgelnaical Functimul. 579 which is to bemade equal to Ot C1I Cn----z:+-------,:+...+-h-,re-,"I re-,~ re-n iscompletely assigned intermsofreand the given quantities 0andh. Art.(':').Thenumberof effective intercalations between the roots of <I>oX,Freis easily seen to be equal to the excess of thenumberof positive realnumerators over the numberof negative real numerators inthepartial fractions of which ~:is the sum, and hence we see apriori,asan obvious consequence of asimpleextension of thereasoning in Art. 47 [p. 515 above], thattheinertiaof thequadratic function I{c,(~+h,u."+hiu.+...+h,n-1u..'f re~h,}, wherec,=;~"willrepresent the value of the index in question. Sotoo we may see thattheforrriuhe given for theresidues toIre,{'oXin Art.46 continue to apply to the residues FoX,<l>re.Thatis tosay,theseresidues when divided out by Ftrwill be respectively represented bythesuccessive principal coaxal determinants to thematrix So,8,..S.,.S_lI SIIS2'S.s; S2'S8'S,Sm+ll where in generalSm-IIs:Sm+l'"s",....1J S-~h,.r+~h.{+ +~h rr-oX-h,. re-h.,. ...a;-h,.n, and using the same matrixasabovewrittenwithS'substituted forS,where ingeneral Sr'=Ot(x-h,.)h{+c..(x-ha)h{+...+en(re-h,.)hw,r, thesuccessive principal coaxaldeterminants ofthenewmatrixrepresent the successive denominators totheconvergents of thecontinued fraction which <l>xexpresses Fre' The expression for the numerators totheconvergents may also, thereis no doubt, be obtained by some simple modification (dependent onintro­ ducingthequantities Ot,~•..cn)of the formula in Art. 41, p. [492]. I annex, more with thehope of suggesting than(in allinstances) of conveying a full conception of the force of the definitions, a Glossary, or ratheraRepertory oftheprincipal termsofartemployed in thepreceding pages,whichmightotherwise beapttooccasion some difficulty to persons unfamiliar with the subject. 37-2 5~O OnaTheoryoftheSyzygetic Relations [57 GLOSSARY OF NEW OR UNUSUAL TERMS, OR OF TERMS USEDIN A NEW ORUNUSUAL SE~SE. INTHEPRECEDING MEMOIR. Allotrious.-The allotrious factortoaresidueorquotient intheprocessof common measure appliedtotwoalgebra.ical functions istheconstant factorof which such residueorquotient mustbedivested inorderto become anintegral andirreducible function. Apocopated.-Applied toatypeintheTheoryofCumulants, denotesatypethe final orinitialelement ofwhichhasbeentakenaway.IfbotharetakenaWIlY, thetypeissaidtobedoublyapocopated, Bllzoutic.-For definition ofPrimary andSecondary Bezoutics see firstSection. Bewutiant to twofunctions, eachofdegreen,isahomogeneous quadratic invarian­ tivefunction ofnvariables, theform of whichservestoa.ssigntheindexofthe scale oftheeffective intercalations oftherealrootsofthetwogivenfunctions. Bllzoutoid.-The Bezoutiant totwohomogeneous functions obtained by dif­ ferentiation from one homogeneous function of twovariables. TheBezoutoid toa givenfunction ofmdimensions inthevariables isaccordingly aquadratic function of (m- I)variables, theform of which issufficient fordetermining thenumber ofrealroots inthegivenfunction. Characteristic.-The employment ofthiswordhasbeenavoided inthepre­ cedingmemoir; butasitcontains anideaofcapitalimportance inanalysis, and especially in allinquiries of thekindheretreatedof, Isubjointhedefinition of itsmeaning. Thecharacteristic ofasimplecondition ofanykindistherational integral function (initslowestterms)whoseevanescence necessarily anduni­ versally impliesandisimpliedbythesatisfaction of such condition. Asimple condition hasalwaysasinglecharacteristic, abstraction beingmadeofthealg~ braicalsign, which remains indeterminate. Inlikemanner, amulriple condition, orasystemofconditions, willhaveforitscharacteristic aplexusofrational integra.l functions, whoseevanescence necessarily anduniversally impliesandis implied bythesatisfaction of such multiple condition orsystemofconditions. Thenumber offunctions inthecharacteristic plexuswillhowever ingenera.l greatlyexceedtheindexofthemultiplicity oftheconditions, and need notalways beauniquesystem. Therearehowever exceptions tothis:thustheduplex condition, thatabiquadratic function of:x:shallcontainacubicfactor,ortha.t&. curveofthethirddegreeshallhaveacusp. will eachbedefinitely characterized byaplexusof twofunctions, andno more. Thespiritofthehigheranalysis resides,andistobesoughtfor, inthelogic ofcharacteristics. Co-bllzoutiant.-Any homogeneous quadratic function similarin form andin itsproperty ofinvarianoe totheBezoutiant. 57J oftwo Algebraical Functions. 581 Cogredient andContragredient.-A systemofvariables iscogredient toanother systemwhenitissubjecttoundergo simultaneously therewith linearsubstitutions ofalikekind,andcontragredient when it is subjecttoundergo linearsubstitutions simultaneously therewith butofacontrary kind. Combinant.-A function ofthequantities appearing inagivensetoffunctions whichremains unaltered aswellforlinearsubstitutions impressed uponthe variables asforlinearcombinations ofthefunctions themselves. Concomitmd.-Nomen generalissimum foraforminvarientively connected with agivenform orsystemof forms. Conjunctive.-A syzygetic function ofagivenset offunctions. Anyfunction whichuniversally, andsubjecttonocasesofexception, vanishes when acertain number ofotherfunctions allvanishtogether mustbeaconjunctive (thatis asyzygetic function), orarootofaconjunctive of such functions. Butif its vanishing issubjecttocasesofexception, thenallthatcanbepredicated ofit isthatit issyzyy"tiMlly relatedto suchfunctions, hutitmay,andusuallydoes happen,thatit will be syzygetically relatedto them in more thanone way. Contrava1iant.--A function whichstandsinthesamerelationtotheprimitive function from which itisderivedasanyofitslineartransforms to aninversely derivedtransform of itsprimitive. C01JarW.nt.-A function whichstandsinthesamerelation totheprimitive function from which it is derivedasanyofitslineartransforms toasimilarly derivedtransform of itsprimitive. Cumulant.- Thedenominator of the simple algebraical fractionwhichexpresses thevalue of animproper continued fraction. SeeType,infra. Determinant.-This word is used throughout inthesingle sense, afterwhich itdenotes thealternate orhemihedral function thevanishing of which is the condition ofthepossibility ofthecoexistence ofasystemofacertainnumber of homogeneous linearequations of as many variables. Dialytic.-If therebeasystem of functions containing ineachtermdifferent combinations ofthepowers of the variables innumberequaltothenumberofthe functions, aresultant maybe formed from thesefunctions by,asit were,dissolving therelations whichconnect together thedifferent combinations ofthepowers ofthevariables, andtreating themassimpleindependent quantities linearly involved inthefunctions. Theresultant 80formed is calledtheDialytic Resultant ofthefunctions supposed; and any methodby which theelimination between two or more equations canbemadetodependontheformation of sucharesultant iscalledadialyticmethodofelimination. Insuch method accordingly theprocess ofelimination between equations ofa.higherdegreethanthefirst isalwaysreduced toaquestion ofelimination between equations whichareofthefirst degree only. Discriminant.-The resultant ofthendifferential coefficients of ahomogeneous function ofnvariables. SeeResultant, infra. 582 OnaTheoryoftheSyzygetic Belatione [57 Diajunctive.-A disjunctive equation isarelation between twoBetsofquantities suchthateachone ofeithersetisequalaccording tosomeunspecified orderof connexion withsome one of theotherset. Effective scaleofintercalations istheseriesoftherealroots of twofunctions of:I:writteninorderofmagnitude afterrepeated prooesses ofremoving pairsof rootsbelonging toeitherthesamefunction (whennotseparated byrootsofthe otherfunction): therootsofthetwofunctions followeachotheralternately. EjJluent.-From everyhomogeneous function ofanynumberiofvariables of thedegreemm',wherem, m'areanytwointegers, may be formed (asshown in the Oalculus ofForms,SectionII.)acovariantive function ofthedegreemandofp. variables, where",·isthenumberofpermutations thatcanbeobtained bydividing m'intoiparts(zerosadmissible), in which all thecoefficients arenumerical multiples ofthegivencoefficients; covariants BOformed may betermedeffluents oftheirprimitive. Anexample ofthisoccursinthefootnote toSection V., [po557],wherethequantity therecalledQisaquadratic effluentoftheJacobian. Ekment.-A simplecomponent ofthetypetoacumulant. SeeCumulant; supra. Emanant.-The resultofoperating anynumberoftimes(supposeitimes)upon agivenhomogeneous function ofanynumber ofvariables :1:,y,Z•••twiththe operative symbol (, d,d,d,d) :I:dz+Ydy+ZdZ+...+t dt ' iscalledtheithemanant ofthefunction operated upon.Everyemanant is& covariant toitsprimitive, thenewvariables :1:',y',Z'...t'beingcogredient with thevariables :1:,y,z...twithwhichtheyarerespectively associated. E2&+J'. E~Hep,page[561],areemanants offandep.Theprocessofemanation is one of incessa.nt occurrence inthetheoryofinvariants. Whentheorderoftheemanant isthesameasthedegreeofthefunction (supposed to berational andintegral) fromwhichtheemanant proceeds, theform of theoriginal function isrepro­ ducedinthefinalemanant, thenamesonly ofthevariables beingchanged. E'I'llbJscopic, Ezo,copic.- Whenthecoefficients of thefunctions concerned in anyinvestigation areregarded asintegral indecomposable monads, themethod iscalledeX06COpic, andendoscopic whenthecoefficients aretreatedwithreference totheirinternal constitution ascomposed of rootsorotherelements. Inaddition totheexamples inthefootnote toSectionI..,thesewordshavea markedandmoatimportant application inthetheoryofInvariants, especially oftwovariables. Form.-Any function may be regarded asanoptUloperatum jthematter operated uponbeingthevariables, andthesubstance oftheoperations beingthe form, which residesinthefunction asthesoul inthebody. A form is always common to aninfinityoffunctions, butforgreaterbrevitymay beandfrequently iscalledbythenameof some specified function inwhichitiscontained. [.. p.431above.] 57] oftwoAlgebraical FU'IlCtions. 583 Fundamental.-The fundamentalsca.le ofasystem of Invariants orConcomitants is•setofthesame,whereof every otherisaRational Integral Function. Hessian orHeeseon, namedafterDrOttoHesse, of Konigsberg (theworthy pupil of his illustrious master,Jacobi,butwho, tothescandalof themathematical world,remains stillwithoutaCha.irintheUniversity which he adornswithhis presence andhis name), is theJacobian tothedifferential coefficients of ahomo­ geneousfunction ofanynumberofvariables.ItistoaJacobian whataBezoutoid istoaBezoutiant, oraDiscriminant toaResultant. H~inant8.-See Memoir ofMrCayley, Cambridge andDublin Mathematical Journal, May 1845, and Crelle'sJournal ofaboutthesamedate. Improper continued fraction isacontinued fraction differing only from an ordinary one inthecircumstance ofnegative signs being substituted forpositive signs toconnecttheterms. Inertia.-The unchangeable numberofintegers in theexcessofpositiveover negative signs which adherestoaquadratic formexpressed asthesum ofpositive andnegative squares, notwithstanding anyreallineartransformations impressed upon such form. Intercalations.-The theoryofintercalations isthetheoryoftherelative distribution oftherealroots,orpoint-roots, of two or more equations, butinthis theorythenumberofrootsmutually interposed is to betakenonlywithreference tothenumber2asamodulus. Invariance.-The property (underprescribed or implied conditions) ofre­ maining invariable. Invariant.-A function ofthecoefficients of one or more forms which remains unaltered when these undergo suitablelineartransformations. Inverse.-The inversetoagivensquarematrixis formed by selectinginits tumeachcomponent ofthegivenmatrix,substituting unityinitsplace,making alltheothercomponents inthesamelineandcolumntherewith zero,andfinally writingthevalueofthedeterminant corresponding tothematrixthusmodified inlieu oftheselected component. Ifthedeterminant tothematrixbe equal tounity,its second inverse, thatistheinversetoitsinverse, will be identical, term forterm,withtheoriginalmatrix. Jacobian.-The Jacobian tonhomogeneous functions ofnvariables isthe determinant represented bythesymmetrical collocation in asquareofthen differential coefficientsof eachofthenfunctions. KenotMme.-A finite system of discrete pointsdefined by one or more homo­ geneous equations innumber onelessthanthenumberofvariables contained therein. Limiting Series.-One setofquantities whoseextreme values are exteriortothe extreme values of asecondsetis set to limitthelatter. Matri:r:.-A squareorrectangular arrangement oftermsin linesandcolumns. 584 OnaTheoryofthe Syzygetic Relations [57 MinorDeterlllinant.-Any determinant retained represented by asquaregroup oftermsarbitrarily chosenoutofa.matrixis aminordeterminant thereto. The simpletermsofthematrixaretheIBBtminors,andof courseifthematrixis asquare,.itwillitselfinitstotalityrepresent asinglecomplete determinant. Morwtheme.-A line, or finite systemof lines, defined by one or more homo­ geneousequations two lessinnumberthanthenumber ofthevariables contained therein. Order.-The ordersof a homogeneous function arethelinearfunctions ofthe variables theleastinnumber by aid of which thefunction admitsofbeing expressed. PersymTTIRtrical.-A symmetrical matrix,in which all the termsinthediagonal bandstransverse totheaxis ofsymmetry areidentical, issaid to be persymmetricaL Example. Anaddition table. Quadrinvariant.-An invariant of which thetermsarequadratic functions of the coefficients of theprimitive. Relation (simpleandcompound). Vide Substitution, infra. Resultant.-The resultant of n homogeneous generalfunctions ofnvariables isthatfunction oftheircoefficients which, equated to zero, expresses inthe simplest termsthecondition ofthepossibility oftheircoexistence: Rhizoristic.-A rhizoristic series is aseries of disconnected functions which servetofixthenumber ofrealroots of agivenfunction lyingbetween any assigned limits. Signaletic.-A signaletic orSemaphoretic series is asequence of disjunctive terms,considered solelywithreference tothealgebraical signs of plusandminus whichtheyrespectively carry. Singtdar.-A properalgebraical function of a given degree,n,in onevariable in its most generalform, will, in respecttothatvariable, be ofthenthdegree inthedenominator andthe(n-l)thdegreeinthenumerator, and will admit ofbeingrepresented byacontinued algebraical fraction ofnterms,all ofthem linear. Butforparticular valuesof, orrelations among, the coefficients entering into thegivenfraction this mode of representation fails,andthecontinued fraction, insteadofconsisting oflineartermsn innumber, willconsistof terms, some of thematleaat,non-linear, andfewerthann innumber. Thesethenarethe singular cases(orcasesofsingularity) inthetheoryofthedevelopment of an algebraical fraction underthecontinued fraction form;anditwillbe seenthat according tothisdefinition thecaseof thedevelopment ofanyproperalgebraical fractionin which thedegreeofthenumerator is morethanoneunitbelowthatof thedenominator, belongs(strictly speaking) totheclassofsingular cases;and thisview ofthecasesupposed isperfectly correctandconformable totheanalogies ofthesubject. . 67J oftwoAl{]ebraical Functions. 585 Substitution (linear,similarorcontrary).-A linearsubstitution issaidtobe impressed uponasystemofvariables wheneachvariable isreplaced byalinear conjunctive ofallthevariables. Thematrixformedbythecoefficients of sub­ stitution arranged inregularorderiscalledtheMatrixofSubstitution, andis of courseasquare. Whentwosubstitutions (impressed on twosystems ofvariables) havethesamematrix,theyaresaidtobesimilar,andcontrary whentheirmatrices arecontrary, thatismutually inversetoeachother.Whentwosystems of variables aresupposed tobesubjecttothecondition thattheirsubstitutions arealwayssimilaroralwayscontrary, theyaresaidtoberelatedor in simple relation, therelationbeingofcogredience intheonecaseand ofcontragredience intheother. Whenalinearsubstitution isimpressed uponasystemofindependent variables, acorresponding linearsubstitution isnecessarily impressed atthesametime upon everycomplete systemofhomogeneous combinations (thatis,products andpowers andproducts ofpowers)ofthesevariables, thematrixtowhichlattersubstitution willconsistoftermswhich will be functions (depending uponthedegreeofthe homogeneous combinations) ofthetermsofthematrixtotheprimitive substitution. Thismatrixmaybetermedacompound matrix, having theprimitive matrix _foritsbase, If, now, two systems ofindependent variables aresubjecttobesynchronously impressed withsubstitutions, thematrices towhich (not beingbothofthemsimple matrices) have for theirbasesmatrices whichareeithersimilarorcontrary, these twosystemswill be said tobeincompound relationofcogredience intheonecase, andofcontragredience intheother. Syrrhwristic.-A syrrhizoristic series is aseriesofdisconnected functions whichservetodetermine theeffective intercalations oftherealrootsoftwo functions lyingbetween anyaasigned limits, Syzygetic.-A syzygetic function orconjunctive ofanumberofgivenrational integral functions isthesum oftheseaffected respectively witharbitrary functional multipliers, whicharetermedthesyzygetic multipliers. Whenasyzygetic function ofagivensetoffunctions canbemadetovanish,theyaresaidtobesyzygetically related. Trans/oTm.-Equivalent totheFrenchnounsubstantive "trans/O'f"TTIie." Type.-The typeofa.cumulsnt istheseries of thesimpleelements (orquotients), arranged inafixedorder,ofwhichthecumulant iscomposed. Vmbral.-The umbral notation isa.notation according towhichsimple quantities aredenoted bysyllables, insteadof bysingleletters(thecomposition ofthesesyllables beinggoverned bythemode in which thequantities whichthey expressareobtained); andthesinglelettersof suchsyllables aretermedumbral quantities orum!JrOJ. Weight.-In thismemoir(throughout theearliersections) theweightofany quantity composed oftheproduct ofthecoefficients of anygivenfunction or 586Syzygetic Relations oftwoAlgebraWal Functions. [57 functions ofxis used to denotethenumberofrootsofxappertaining tothegiven function orfunctions whichmustbe employed toexpresssuchquantity. More generally, whendealingwithasystem of homogeneous functions, theweigl'" ofaquantity may be defined withrespecttoany'electedmriahle therein 88the sum of the weightsinrespecttosuchvariableoftheseveralcoefficients of which thequantity iscomposed (theweightofeachseveralcoefficient meaning theindex ofthepower of theselectedvariable inthattermofthegivenfunction orfunctions which is affected withsuch coefficient). These two definitions ofweightmay be perfectly well reconciled with eachotherbyunderstanding theweightofaquantity formed from thecoefficients of afunction or system of functions ofxtomeanthe weight, in respecttounity,of suchquantity whenthegivenfunctions aretreated 88homogeneous functions ofxand1. Zeta.-The symbol' (preceding arow ofbracketed terms) is usedtodenote theproductof thesquareddifferences of thetermswhichitaffects. [J.Abracketofthisform, when enclosing asuperior andaninferiorrow oftermsmandninnumber respectively, indicates themnproducts ofthe differences obtained bysubtracting eachterminthesecondrow from eachterm inthefirstrow;when enclosing an arrangement oftermsinasingle line, itis usedtodenotethecumulant of which such anarrangement isthetype. 58. ONTHECONDITIONS NECESSARY ANDSUFFICIENT TO BE SATISFIED IN ORDER THATAFUNCTION OF ANY NUM­ BER OF VARIABLES MAY BE LI~EARLY EQUIVALENT TO AFUNCTION OF ANY LESS NUMBER OF VARIABLES. [Philosophical Magazine, v,(185:~),pp.1Hl-126.] INtheCambridge and Dublin Mathematical Journal forNovember 1850·, I defined anorderassignifying anylinearfunction ofagivensetofvariables, andspoke of agenemlfunction ofnvariables as losing rorders when the relationbetween itscoefficients is such thatitiscapableof being expressed asafunction of (11-r)orders only. Itwill behighlyconvenient topreserve thesamenomenclature forthepurposes of thepresentinvestigation. DrOttoHesse,in a long memoir in Crelle:«Journal, thecontents of which have been described tomet,butwhich I havenotyetbeen able to procure, hasgivenarule for determining theanalytical conditions forthe 1088of one order. I propose to give amore simple and comprehensive scheme ofconditions thanProfessor Hesseappearsto have discovered, applicable not tothiscase only, buttothatof the loss of anynumber whatever of orders, and shall moreover show in whatrelationthesubstituted ordersstandtothe givenvariables. DrHesse'srulehadbeenpreviously statedby me in the4thsection of myCalculus ofForms(Cambridge and Dublin Mathematical Journal, May 1852Dasapplicable tothecaseof ageneralfunction of the3rddegree [.p. 171above.] tAdistinguished mathematical friend in Pariscommunicated tome with greatadmiration Professor Helllle's resultovernight. Iventured toaffirmthat,to oneconversant withthe calculus of forms, the problem oouldollernomannerof diffioulty. Anhour'squietreflection in bedthe following morning, ormorning af\el,sufficedtodi8010eetomethetrueprinciple ofthe solution. [Cf.Noether, Math.Annal. L.(1898) p. 188. ED.] :I:VUhVol.VII.p. 187[po 8Mabove]. .. WhenUrepresents apencilofthreeraysmeeting in apoint,~=0,~=O,&0.,andalsotherefore T=O"(8andTbeing the two Aronholdian In­ nriants ofU,anda,b,e,&0.theooetllcients orU);..also inplaceofthissystemmaybe subetituted thesystemobtained bytakingaUthectN.fficitmu oftluHemanzero." 588 OnPolynomial Functions which [58 ofthreevariables becoming therepresentative ofthreerightlinesdiverging fromthesame point, which is thecase of a cubic function ofthreevariables becoming afunction of twolinearfunctions of thesevariables, thatis tosay, losing one order:this,perhaps, mighthavebeennoticedintheProfessor's memoir. I gave also anotherruleforthesamecase;butthetruefundamental schemeofconditions aboutto be set forthwill be seen to embrace as mere corollaries all such and such-like rules, which in fact supplymore or less arbitrary combinations oftheconditions, ratherthanthenakedconditions themselves intheirsimpleformandabsolute totality. I shall call thefunction to bedealtwithU,andshallconsiderUtobea homoqeneous" rationalfunction of 11Ldimensions inrespectofXI>Xt••.x..,and shallinquirewhat are theconditions whichmustobtainwhenUiscapable of being expressed asafunction of only (n - r)orders, say ll'l,... l.._r,each of which is of course a homogeneous linearfunction ofthegivennvariables. Letthetermderivative ofUbeunderstood to mean any resultobtained bydifferentiating Uanynumberoftimeswithrespectto one or more of the variables Xl'Xli...X...The first derivatives will be of (m - 1) dimensions, thesecondderivatives of(m-2)dimensions, andso onjandfinally,the (m-l)thderivatives will be homogeneous linearfunctions ofXl>~•••a;.. SupposeUtobeexpressible asafunction ofll'l,...lfl-r'Itisimmediately obviousthatthederivatives fromthe1sttothe(m-1)thinclusive willbe allexpressible ashomogeneous functions ofll>lll'"In-r,and vanish when thesevanish. Butthisstatement is insubstance pleonastic; for by means ofEuler'swell-known law,anyderivative ofU,sayK,maybeexpressed (toanumerical factorprM)undertheform of andconsequently, whenever thelinearderivatives ofUvanish, all theupper derivatives ofU,including Uitself,mustvanishatthesametime.The number oftheselinearderivatives, say",will hethenumber oftermsin ahomogeneous function ofnvariables of(m-1)dimensions, thatis tosay, n(n-1)(n -m+2) 1.2(m-1) Again, if all the"linearderivatives vanishwhenthe(n-r)equations ~=0,III=0...In-r=0aresatisfied, rbeinggreaterthanzero,thiscanonly happenbyvirtueofthese"derivatives beinglinearfunctions of(n-r) •Itisacommon errortoregardhomogeneity ofexpreallion WImerely a means for satisfying ihedesireforsymmetry; the ground of its application andutilityinanalysislies,infact,much deeper;itisessentially amdlwdandapouur, 58Jadmitofreduction. inthenumberofVariables. 589 ofthem.Now, conversely, I shall prove, thatifitbetruethatallthelinear derivatives ofUarelinearfunctions (n-r)ofthem,thenUmay beexpressed asafunction ofthese(n-r)jandthisrule,aswill beimmediately made apparent, will give the necessary andsufficient conditions forthe1088of rordersinthemostsimpleandcomplete form by which theyadmitofbeing expressed. Fortheproof of therule, only one additional remarkhasto be madeinaddition tothatalreadymade,ofthevanishing ofthelinearderiva­ tivesnecessarily implying thesimultaneous evanescence of alltheother derivatives; thisadditional remarkbeing,thatifthederivatives of any class, linearorotherwise, q'l.ldonesetofvariables, becomeallzero,thederivatives ofthesameclass,q'I.Idanyothersetofvariables linearfunctions ofthefirst setandthesameinnumber, will also become zero, for theyareevidently expressible aslinearfunctions ofthefirst set. Now let d1,dt•••dfl-rbe any(n-r)linearderivatives ofU,of which alltheotheroftheIIderivatives ofthisclass are linearfunctions, sothat theyvanishwhenthese(n-r)vanish, and letUbeexpressed asafunction of(~,~...dn-f'; ICl>Il:t.•.ICr).Thenwe may write U=4>...,0+4>'-1,1+4>"0-2,2+...+4>1,"'-1+4>0,..., where in general 4>...-.,-denotesa function homogeneous andofm-€dimen­ sions in respect tod1,d'l....dn-r>andhomogeneous andof€dimensions in respectto11;,IC2...ICr•Nowthelinearderivatives ofUall vanish when d1=0, ~=0...dn-r=0 for all values of ICl>a;...ICr•HenceU=0 onthesame supposition, and hence 4>0,...issimilarly zero. Also thefirstderivatives ofU, q'l.ldd1,d'l'"dn-f"mustvanish on thesamesupposition. Hence4>1,'-1 isidentically zero;and so by takingthe2nd, 3rd ...up tothe(m-l)th orlinearderivatives ofUinrespectto~,dt...dn-"we find successively t/>,,7IO-'l'4>"m-3...4>'-1,1eachidentically zero,andconsequently aswasto be proved. To expressthefactoftheIIderivatives beinglinear functions of(n-r)ofthem,formarectangular matrixwiththecoefficients oftheIIlinearderivatives. Thismatrixwill bentermsinbreadthandII termsindepth.Letr=1:itisadirectconsequence oftherulewhichhas beenestablished, thatevery full determinant consisting ofasquarenterms byntermsthatcanbe formed outofthisrectangular matrixmustbezero: again,letr=2;allthefirst minors, thatis to say, all thedeterminants composed of squares(n-1)termsby(n-1)terms,mustbe zero, and so in generalaloss ofrorderswillrequirethatthe(r-l)thminorsshall all vanish;ifr=n,the(n-1)thminors,thatisthesimpletermsofthematrix which are all coefficients of U,mustvanish,or inotherwords, when the function is of zero orderallthecoefficients vanish(anobvious truism), ",.. ·590 OnPolynomial Functions which [58 Thus,then,we seethatthetrueruleforthe1088of one order in apolynomial ofanydegreeisprecisely thesameasthe well-known rule for the 1088of one orderinaquadratic function jthespeciality in thelattercaseconsisting merelyinthefactthatIIbeing equal to n,therectangular matrixbecomes a square, andthereis only one full determinant. Moreover. for anyother value of rthe above rule coincides with thatgiven by me some timeback in thePhilosophical Magazine for thecaseofquadratic functions. Professor Hesse's rule for finding conditions applicable totheloss of one orderis,asI havealreadystated,aconsequence of themore simple scheme of conditions above given. Itconsists in forming thedeterminant d'Ud'Uetr du;l"dxldo;...~dlx~ d'Uet: d'U dx.dxl'dx,dx.···dx.dx. eoet: dJU dxndxl'clxndx.··· dXndxn andequating thecoefficients of thisdeterminant fully developed separately to zero",Theattachment of the Professorto thisparticular formofcovariant (Iuse thelanguage ofthecalculus of forms) is readily intelligible, seeing theadmirable application which he hasmade of it to the canonization of the cubic function of threevariables, butit is really foreign to the natureofthe presentquestion; the coefficientsof thiscovariant mayeasilybe shown to be merely the full determinants of thenxIIrectangular matrixabovedescribed, orlinearfunctions of these said determinants withnumerical coefficients. Hencetheground ofits applicability. Returning to the rule of the matrix,if wesuppose the numberofvariables to be two, and call thecoefficients of U au,n~,in(n-1)a....an, ourrectangle becomes •Aform capable of being so derived I have elsewhere termed(incompliment toM.Hesse) theHessian of the function to which it appertains. This is the trivialnamewhich is much needed on accountof thefrequent occurrence of the form, and hasbeenadopted byMrSalmon ~8Jadmitofreduction 1.UtheuUrrWerofVariables. 591 andthe conditions become aoav-all=0, ~~-avl=0, lln-aan-an-Ian-l=0,&c., all of which equations are obviously true(whenthefunction loses an order, thatis tosay,becomes aperfect power) and are satisfied (special cases excepted) when any (n-1)independent equations out oftheentirenumber obtain;sothatthenumber of conditions implied in the property to be represented is inexactconformity with thenumberofindependent equations derivedfrom the matrix,thatisequations which, when satisfied, will in generalcause all the rest to be satisfied. This conformity manifests itself also in the caseof aquadratic function of n variables. Butexceptin these twolimiting (and, in anoccult sense, reciprocsl ")casesofafunction of two variables of thenthdegree, or of thedegree 2 and nvariables, this con­ formity in measure asthedegree or number of variables rises, although it mustsubstantially continue to exist, becomes, and in anaccelerated degree, lessand lessapparent. Thus,takethesimplecaseof a cubic function of threevariables, and let us confine ourselves to theconsideration oftheconditions which mustbe satisfied when itlosesasingle order. LetUbewrittenoutatlength, aW+by'+cr+3kyzl+3ua;!+3ja:'!f+3k'ylz+3i'zig;+3frely+6rrucyz. inhisadmirable treatiseonthehigherplanecurves.Insystematic nomenclature itwouldbe termedthediscriminant ofthequadratic emanant, or more briefly,thequadremanative dis­ criminant. Ihavediscovered quiterecentlythatthe long soughtforsymmetrical, andbyfarilie mosteasypractical processfordiscovering themunber of therealrootsof anequation, is contained in,andmay bededuced immediately from,acertaintransformation ofitsHessianI •Therearefrequent casesoccurring inthecalculus of forms of interchange betweenilie degreeofafunction andthenumberofvariables whichitoontains. Thus,toselect astriking example (although one where theinterchange isnotexact),thetheoryoftherealandimaginary rootsorfactorsofahomogeneous function of twovariables andof thenthdegreemaybe shown to beimmediately dependent uponiliedetermination ofthespecificnatureofaconcomitant homogeneous function ofthe2nd degree andof (n-1)variables. Forinstanoe,ifanyordinary algebraical equation of the6thdegreebe given, ahomogeneous quadratic function of fourvariables maybeconstructed, representing, consequently, asurfaceofthe2nddegree[theooefficients ofwhich(asindeedistruewhatever bethedegreeoftheequation) will bequadratic functions of the coefficients of thegivenequation]; andsuchthat,according asthesurfacesorepresented belongstotheclassof (I),impossible surfaces; (2),theellipsoid orhyperboloid of twosheets; (3),thehyperboloid of onesheet;thegivenequation will have 6, 3,oronly1realrootIMoreover, anequality between twooftherootsof theequation will bedenoted by the loss of one order in theassociated quadratic function; andsomanyordersaltogether will belostasthereare independent equalities existing between theroots.Anentirely newlightisthusthrownon M.Sturm'stheorem; andthenumberof realandimaginary rootsinanequation isfor thefirst timemadetodependuponthesigns of functions symmetrically construoted inrespectto the two endsoftheequation, whiohbaslong been feltasadesideratum. 592 OnPolynomial Functions which [58 Thematrixformed out of thecoefficients of thelinearderivatives becomes a,r.i J,b,h' '1h,C ~, m,h',h t.m,i' f,J,m Now by thehomaloidal law, if thetermsinthisrectangle were all unlike, thenumberof fulldeterminants (3termsby3terms)whoseevanescence (exceptfor special values) determines theevanescence of all therest,should be (6 - 3+1)(3- 3+1),thatis4<jbutintheactualcase,sincethe evanescence of allthefulldeterminants isanecessary consequence of the function becoming acubicfunction of twoorders(thatis,breaking upintothe productofthreelinearfunctions ofe,y,z),andasthisdecomposability, asis well known, impliesonlytheexistence ofthreeaffirmative conditions, thefour fulldeterminants a,i'.ia,j',ia,j',ia,i'.i'. j,b,h'J,b,h'J,b,h'J,b,h' c,h,cm,h',hi,m,i'i'.j,m •Thatistosay,asyzygetic relationmustconnectthesefourdeterminants. I may as well hererepeat,thatwhenthevanishing ofasetof,rational integral functions necessarily, andwithoutcasesofexception, impliesthevanishing ofanother rational integralfunction, thenthisfunction istermedasyzygetic function oftheothers;andsome power of itmustbe expressible undertheform of asum ofibinaryproducts ofrational integral functions, one factorof each of which products mustbeone ofthe,givenfunctions. When the vanishing of allbutone ofasetoffunctions ingeneralnece88arily impliesthevanishing ofiliatone,but subjecttocasesof exception for specific values of thevariables, thenitcanonly beaffirmed thatthefunctions of thesetare insyzygy; thatistosay,thatthesum of the products of each ofthemrespectively by some rational integral function will be zero:theequation expressing thisrelationistermeda syzygetic equation. Thus,ifwetakethethreefulldeterminants thatcanbeformedoutof thematrix a,Q., b,fJ, thatise,oy, a{J-bu,boy-cfJ,ell-a')', theseare in syzygy, for we canformtheeqnation e(afJ-bll)+a(boy-cfJ)+b(c..-a')')=0. This,however, is not the only equation of the kind thatcanbeformed, for ')'(afJ-bII)+II(b-y-cfJ)+fJ(ell-a')')=O isalsoidentically true. We see in thiscasethattheevanescence of anytwoofthethreefunctions 58]admitofreduction inthe number ofVariables. 593 which in the generalcase would be entirelyindependent, inthiscase cease to beso;and thevanishing ofthreeofthemmustdraw along with it by necessary implication (exceptfor special values) theevanescence of the4th, forthusonly can thenecessary conformity between thenumberof affirmative conditions andthenumberofunimplicated equations come to takeeffect. The clear and directputtingin evidence of thispeculiarspeciesof implication demands and deserves to be minutely considered; and as it mustinpart borrowitsexplanation from the very littleyetknown of syzygetic relations, so itmustalso throw new lightonthatgreatandimportant, butasyet unformed and scarcelymorethannascenttheory. Inconclusion, itisapparent from the demonstration above given, that whenU,afunction of nvariables, becomes expressible asa function of (n -r)orders,theseorders may be takenrespectively anyindependent linear functions of thelinearderivatives ofU,whichremarkcompletes thetheory of functions subjecttotheloss of one or more orders. Itis obvious (and I amindebted to myesteemed friend Mr Cayley for the remark), thatthe conditions furnished asabove by the(m-1)th,thatislinearderivatives, areidentical withand may bemoreelegantly replaced by those involved intheassertion oftheexistence oflinearrelations between the1stor (m-1)th degreed derivatives, and we have thenthisvery simple rule; ifIf>,afunctionofXI'X,•.•Xn,isexpressible asafunctionofn-rlinear functions ofXl>x...,a;"itisnecessary andsufficient that rindependent linear relations shallexistbetween dlf>dlf>dcf> da:I'dx."·dxn• afJ-bel;boy-cfJ;CII-a')'will ingeneralimplythethird,subject, however, tospecialcasescf exception. Thus,ifthe1stand2ndvanish,the3rdmustvanishunlessbandfJbothvanish; ifthe2ndand3rdvanish,the1stmustvanishunlesscand')'bothvanish; if the Srd and lBtvanish,the second will vanishunlessaandIIbothvanish.Itwillthusbeseenthata peculiar species of cutricud syzygyobtainsbetweenthethreeproposed functions, whichenablesus toaffirmthatingeneral,andexceptunderextraspecialconditions, allthreemustvanishsimnl­ taneously. Iftwooutofthethreevanish,andthe3rd does notvanish,itisnotmerely(asmight atthe first blush of the theoryof syzygy be conjectured) becausesome one otherfunction vanishes initsplace,butnecessarily becauseaplurality ofentirelyindependent functions (twosimpleletters asithappens here)eachseparately vanish. Thuswe see how allbutone ofasetoffunctions XI'X2,.,X..mayingeneral,andyetnotuniversally, necessarily vanishwhenalltherestvanish: tosaythatonesyzygetio equation suchas XIXI'+X.X.'+ '"+x..x..'=O obtains,isnotenoughtoexplainthecircumstances ofthecase;the fact is, thatseveraldistinct systems ofvaluesofXI"X,'...X..'will be found capableofsatisfying theequation, sothateach ofthefunctions XI'X•.,.x..will have a'y,tnnof syzygetio factorsattaohed toit,andthese unrelated, inthewidesensethat,if we take X..', X..", any two of the syzygetio factorsattached toX..,theywillnotbeinsyzygywithXI'X2...X,,-I;sothatwhenthese(n-1)functions vanish, thevanishing ofx.:andX.."represents twodistinctandoompletely independent oonditions. Thus,in fine,themutualimplication offunctions will ingeneraldenotethepossibility offorming asme,ofsyzygetic equations betweenthem,-a remark, this,of nominorimportance. a 38 594 OnPolynomial Functions. [58 This rule itselfalso,itisevident, is capable of an independent and immediate demonstration bymeans of integrating thepartialdifferential equation orequations by which it admitsofbeingexpressed. Theabove theorymay readily be extended to functions of several systemsofvariables, Thus, for instance, thedeterminant a, b, eI,b'e'Ia,, a",b"e", vanishing willbeindicative ofthefunction {axu+ba:v+eXW} +a'yu+ b'yv+e'yw, +a"zu+b"zv+e"zw beinglinearlyequivalent toafunction oftheform {Ax'u'+Btrlv'} + Cy'u'+Dy'v'' thatis losing anorder inrespectofeachofthetwosystemsx,y,z; u, v,w; and so in general. 59. ONMRCAYLEY'S IMPROMPTU DEMONSTRATION OFTHE RULEFORDETERMINING ATSIGHT THEDEGREE OF ANYSYMMETRICAL FUNCTION OFTHEROOTS OF AN EQUATION EXPRESSED INTERMS OF THECOEFFICIENTS. [Philosophical Magazine, v.(1853),pp.199-202.] FORaconsiderable timepast,among the few cultivators of thehigher algebra, aproposition relativetothetheoryof thesymmetrical functions of the roots of anequation hasbeen in privatecirculation, which, to say nothingoftheimportant applications of which it has been found susceptible to the calculus of forms, merits(by reason of its extremesimplicity), although, strangetosay,ithas,I believe, not yetobtained, aplaceinelementary treatises on algebra. The proposition alludedto I have reasontothink first came tobe observed in connexion with my well-known formulae for Sturm's auxiliary functions in termsof the roots given in this Magazine. Thetheorem is briefly asfollows.Ifa,b,c,&c.betheroots ofanequation a;A+Pig;A-I+PIa;A-J+&C.=0, anysymmetric function such asIa&fiJc"...,wherea,fl,'Y'"are positive integersarranged according to the order of theirmagnitudes inadescending (or, to speak more strictly,non-ascending) order, when expressed asafunction of the coefficients,will be made up of termsoftheformPl"lJI"pl" ...P"''',such that81+81+81+...+8/rwill be equal to afor some terms, butwill for no term exceed a; abeing,asabove described, thatone of the indices a,fl,'Y••• which is not less thananyoftheothers. I hadprepared. andindeeddespatched, asomewhat elaborate proof ofthistheorem fortheOamhridge andDublinMathematical Journal;but ou proceeding to explainmymethodto MrCayley, elicitedfromthatsagacious analystthe following excellent impromptu, which Ithinktoo valuable tobe lost;andasit is now atwelvemonth or two since our conversation on the subjecttook place, andtheauthorhasnot cared to putiton record, I feel 38-2 596 On an Impromptu Demonstration ojMr Oayley. [59 myselfunderanobligation so to do, themore soasitentirelysupersedes the comparatively inelegant demonstration of my own which I had previously intended topublish. Themethodrestsessentially onthefollowing well-known theorem given byEulerrelative tothepartition ofnumbers jto wit,thatthenumber of ways of breaking upanumbernintopartsisthesame,whether weimpose thecondition thatthenumberofpartsinanypartitionment shallnotexceed m,orthatthemagnitude ofanyone ofthepartsshallnot exceed m. Ofthisrulemorehereafter-for thepresenttoitsapplication tothematter in hand. Sincea,b,C..•aretheroots of a;"+Pla;"-l+...,we have Pt=a+b+c+ ... P2=ab+ac+bc+ ... p,=abc+abd+ acd+'" Leta+fJ+'Y+...='11,none of thequantities a,fJ,'Y'"beinggreater thanm,buta,fJ,'Y".beingotherwise arbitrary andcapable ofbecoming equaltoanyextentinter86.AlsoletA+I'"+II+...='11,thenumber of quantities X,1"',II,&c.beingnevergreaterthanm,butthequantities themselves beingotherwise arbitrary, andbeingcapableofbecoming equal to anyextentinter86.ByEuler'srulethenumber ofsystemsa,fJ,'Y...is thesameasofthesystems A,1"',II•••,sayPfor each. Foranysystem A,/-"II•• , ,weshallhavePAp,.p•...,byvirtueoftheequations abovewritten, expressible asthesum oftermsoftheformIa-bI'cY•••;itmay easily be madeostensible, thatallthecombinations ofex,fJ,'Y'"subjecttotheabove prescribed conditions mustcomeintoevidence bygivingA,1"',II•••allthe variations of which theyadmit;butthisis alsoimmediately obviousindirectly fromtheconsideration, thatwere itotherwise, linearrelations wouldsubsist between thedifferent values of PAPvP•...,which is obviously absurd.Hence, then,weshallbe able to expressthePquantities oftheformPAP,....by meansoflinearfunctions ofthePquantities Ia-bI'cY•••;andconversely, by solving thelinearequations thusarising,thePquantities Ia-bI'cY. may beexpressed intermsofthequantities PAP,....jconsequently Iambl'c Y, where m is greateror not less thananyofthequantities fJ,'Y...,will be expressible bymeansofcombinations PAP,.... ,wherethenumber of co­ efficients PAP,....(anynumber of which may become identical) is forsome ofthecombinations asgreatas,butfor none of thecombinations greater thanm,aswastobe proved. Itwill of course beseenthat,forthe purposes ofthedemonstration above given, itwould have been sufficient 59JOn anImpromptu Demonstration ofMrCagley. 597 to have been able to assumethatthenumberofpartitions, when the greatest partis not allowed to exceed m,is notgreaterthanthenumberofpartitions when the numberofpartsinanyone partitionment doesnotexceedm. Theequalityof these two numbers wouldthenevinceitselfinthecourse of thedemonstration asaconsequence of thisassumption. A word now as to Euler'sbeautiful law upon which the above demon­ strationis based. A corollary from it, obtained bysubtracting theequation which it gives whenthelimiting numberistaken(m- 1) from theequation which it gives when the limiting numberism,will bethefollowing proposition. The numberof modes of partitioning nintompartsis equal to thenumber of modes of partitioning nintoparts,one of which is always m, and the othersmor less thanm. This proposition wasmentioned to me by MrN.M.Ferrera", whosedemonstration of it(probably notdifferent from thatofEuler'sfortheotherproposition, of which it may be viewed asa corollary) is so simple and instructive, thatI amsureevery logician will be delighted tomeetwith it here or elsewhere. Itaffordsamostadmirable example ofthatratheruncommon kind of reasoning whereby two abstract integers areproved to be equal indirectly, by showing thatneithercanbe greaterthantheother. Iftherebe agroupofA'sand agroupofB's,and every Acan be shown to produce a B,and every Bcan be shown to produce an A,nomatter whether theAproducing aBis the same as,or different from, theA produced by thatB,it is obvious thatthenumberofA's'cannotexceedthat of theB's,nor oftheB'sthatof theA's,and the two numbers willtherefore be equal. Takeany such grouping as 3, 3, 2, 1, sa.,...4..This may be writtenas 1, 1, 1 1, 1, 1 1, 1, 1, and byreadingoffthe columns as lines, may be transformed into thegroup 1, 1, 1, 1 1, 1, 1 1, 1 thatis4,3, 2,sayB. *IlearnfromMrFerrerathatthistheorem waBbrought underhiscognizance through a Cambridge examination paperBetby Mr AdamBof Neptune notability. 598OnanImpromptu Demonstration ofMrCayley. [59 In.Athenumberofpartsis+.InBthegreatestpartis4;theothers mightbe(although theyhappennot inthisparticular instance to be)+,but cannotbegreaterthan4. And so every .Ain which the number ofparts is4will give rise to aBin which+is one of theparts,andeveryotherpart is4or less, and evidently (although, asaboveremarked, thisisimmaterial to thedemonstration) every such Bgives reciprocally thesame.Afrom which itisitselfderived; hencethenumberofA.'sand B's is equal. This is the theorem which, for thesake ofdistinction, I have called theCorollary to Euler's. Euler'sown is proved by the same diagram; for if we define .A asagrouping where the numberofpartsdoesnotexceed+,we getadefinition ofBasagrouping where the greatest partdoes not exceed +.and soin general. We see thatthistheorem may be varied also by affirming thatthe numberof ways in which nmay be brokenup,80thatthereshallnever be lessthanmparts,is the same asthenumberof ways in which it may bebrokenup intoparts,thegreatest of which in anyone way is not less thanm.So, again, asimilardiagram makes it apparent, thatifwe break up each of inumbers intopartssothatthe sum of thegreatest partsshall not exceed (or be lessthan)m,thenumberof ways in which thiscanbe done will be the same asthenumberof ways in which these inumbers canbe simultaneously partitioned sothatthetotalnumberofpartsin anysimul­ taneous partitioument shall never exceed (or never be less than)m;and doubtless anextensive rangeof analogous general theorems relativeto the partitioning ofnumbers may be struckout by aid of thesamediagram, by no means easily demonstrable unlessthissimple mode of conversion happen to bethought of,butinthateventbecoming intuitively apparent. This mode of conversion is precisely that(onlyapplied toamoregeneralstate ofthings)whereby, in elementary arithmetic, itisestablished thatmtimes nisthesame as ntimesm.Aconsideration of the process by which themindsatisfies itselfoftheuniversality ofthislaw, has been always sufficient to convince me of theabsurdity ofascribing to aninductive process thecapacity ofthehumanmind for forming general ideasconcerning necessary relations. 60. A PROOF THAT ALL THEINVARIAN'rS· TOACUBIC TERNARY FORM ARE RATIO~AL FUNCTIONS OF ARON­ HOLD'S INVARIANTS AND OF ACOGNATE THEOREM FORBIQUADRATIC BINARY FORMS. [Philosophical Magazine, v.(1853),pp.299-303, 367-372.] ALTHOUGH contrary totheorder of exposition indicated inthetitleto thispaper, I shall, as thesimplercase,beginwithestablishing thetheorem for abiquadratic form,sayFine,y.Let F=aa;4+4bx'y+6c:rfly2+4d:x!l+e'!t, s=ae-4bd+3&, t=ace-ad2-c'-ble+2bcd, sandtarethe two well-known invariants ofF.I propose toprovethat therecan exist no otherinvariants toFexcept such as are explicitrational functions of sandt. LetF,by means of thesubstitution ofjx+gyforx,andf".»+g/yfory, bemade to take the form h=rc'+'!t+6rrt:rfJy2. Then by the characteristic property ofinvariants, ifI(a,b,c,d,e)be anyinvariant toFofthedegree q,we must have 1(1,0,m,0, 1)=(fg/-j/g)2IJI(a,b,c,d,e); anditwillbesufficient to prove that1(1,0,m,0, 1), or saymore simply I(m),can only have thetwo radically distinct forms corresponding to 8andt,thatis (8)=1-311£2and(t)=11£-m', anyotheradmissible form ofIbeing arationalexplicitfunction of these two. •AOomtant inanalysis isanyquantity which in itsownnature,or bytheexplicitconditions towhichitissubjected, isincapable ofchange. AnIllllariant isanexpre8llion apparently liable tochange,butwhich,owingtocertaincompensations inthemodifying tendencies imprelllled upon it,remains &8awholeunaltered. The former maybecompared toafixedpointorsystemin mechanioa; thelattertoapointorsystemCreetomove,butkeptatrestunderthecombined operation ofcontending forces. 600 OnAronhoUls Invariants. [60 Itmaybeshown-thattheparameter minItwill have sixdifferent values and no more. In thefirst place, if we write LXforxinIt(Lmeaning ";-1),it is obvious thatmbecomes - m.Again, let x+''!Jandx-L'!Jbe substituted in place of xand'!Jrespectively; thencalling(/)thevalue assumed by/1'whenthissubstitution ismade, (f)=(x+''!I)4+(x-''!I)4+ 6m(x!+'!I)t =(2 +6m)(x4+yf)+(-12+ 12m)xtyi =(2+6m){x4+y4+6~~~=x!'!I}. Henceif we write I , (2 +6m)!x+ (2 +6m)!'!Iforx, and I , (2+6m)ix- (2+tim)!yfor'!I, and call what Itbecomes afterthesesubstitutions h, h=x4+y4+6'Y(m)x!yt, -1+m'Y(m)denoting----.1+3m Inlikemanner,bywritinginIt 1 , 12+t5'Y(m)}1 x+ {2+6'Y(m)}l'!Iforx, and weobtain whereI , {2+&y(m)}!x-{2+6'Y(m)}!yfor'!I, .r.=x4+y4+6r(m)x''!I, -2-2m -1-m - 2+6m= -I+3mj-1+-I+m r(m)= I+3m 1+3-I+m 1+311£ 'Y(m)isaperiodic function of mofthethirdorder, for we find -(1+3m)-(-I+m) r(m)=rh(m)} =-(I+3m)+3(-1 +m)=m. Itwillof course be observed, also, that r(m)=-'Y(-m) and'Y(m)=-r(-m). •SeeA.ddendum [po607 below]. 60J HenceOnAronhold's Invariants. 601 IS I (m)=(1-3m)qI(m+1).-2 I-3m(-'Y)(-'Y)(m)=-"'I(-m)=m,(-"'1)(-"'I)(m)=-r(-m)=m. Sothat,in fact,thesix values of the parameter are m,'Y(m),r(m), -m,-'Y(m),-r(m), forming two cycles, havingtheremarkable property thatthetermsinthe samecycle are periodic functions of thethirdorderof oneanother, andeach termin one cycle isa periodic function of thesecondorderof every term in theothercycle. The modulus of substitution forpassingfrom};.toI..thatis thesquare ofthedeterminant [(2 +1 am)l'f2+'am)l], 1 - , (2 +6m)l'(2 + 6m)l (-2,)1 -2---or--2+6m'1+3m· SothatifI(m)bethevalue of any invariant ofthedegreeq,corresponding totheform};.,andconsequently IG~+-3~) the same for A,wemusthave I(m)=(1+3m)q I(m-l).-2 1+3m In like manner, by meansofAit may be shown thatwemusthavethe furtherequation Theseequations are easily verified for thevalues of (8)and(t). Thus (1+3m)' {(m-1)Il(8)=1+3m'=~1+33m+1J =(1 -3m)3{I+3(~+1)I}, 4 I-3m (1+3m)'{m-1(m-1)I}(t)=m-m' =------83m+13m+1 = _(1 -3m)'{m+1 _(m+1)'). 8 1- 3m1-3mJ' 602 OnAronhold's Invariants. [60 anditismoreover obvious, thatthe values of (8)and(t)mighthave been foundaprioriby means of these functional equations. Theessential pointofinference for my presentpurpose from theequations above, which are of theform I(m)=Hx 1(~~~)=KX I(t_~~), isthis,thatifI(m)containany power of m,saym',itmustalsocontain (m-1)'and(m+1)';inaword, (mS-m)',which,by,theway.itmay be noticed, is (t)·.Now, if possible, let therebe anyinvariant Iq(m)of the qth degree in mwhich is not a rational function of (8)and(t).Ifwe make 2x+3y=q.asmanyintegersolutions asexist of this equation (in which zero valuesof:r;andyareadmissible), so many functions of the form (a'f(t'1may be formed of thedegreeqin m, and all of themof course invariantive functions. Asregardsthegeneralnatureof anyinvariantive function in m,since thechangeof:r;into-:r;in:r;t+'!I+6m:r;2y2introduces no change into the invariant ifqbe even,butchangesthesign ifqbe odd, it followsthatIq(m) is oftheform4>(m2)whenqiseven, and of theformm4>(mS)whenqis odd. Letp.be thenumber ofsolutions oftheequation inintegers above written. Then, by linearlycombining all the different values of (8'f(t)Ywith Iq(m),it is obvious thatwe may form a new invariant, sayI'q,in which the ,."firstoccurring powers of mwill bewanting, thatis in which theindices 0,2,4...(2p.-2) will be wanting whenqis even, and 1, 3, 5 .., (2,."- 1) when qis odd. Hence in theformercasethe newinvariant willcontainm"'",and inthelattercasem","+l;and therefore, by virtueof what hasbeen shown already,I'qwillcontain(ma_m)2I&intheone case and (ma-m)2I&+1inthe other. Firstly,letq=6i,or 6i+2, or6i+4;then,."=i+1jandtherefore (ma_m,,+I,which is of thedegree 6i+6 inm,iscontained asa factor inIwhichisof the degree qonly, aquantity lessthan6i+6, which is absurd. Again, secondly, letq=6i+1,thenp.=i;and(ma_m)2I&+1is ofthe degree 6i+3 in m, and is contained asafactor in I,which is of the degree 6i+1, which isagainabsurd. Finally,ifq=6i+1, or 6i+3,,."=i+1; andthefactor (m a_m)2I&+lisof the degree 6i +9,thatis, in each case,greaterthanq,whichisabsurd, andthusthetheorem is completely demonstrated Itmay foramoment be objected, thatwe have been dealingonly with aparticular formg;I+6m:r;2y2+'!I,insteadofthegeneralform a:r;4+4b:c'y+6c:x'y'+4d:cy'+e'!/j 60] OnAronhold's Invariants. 603 butthelatteris always reducible to the former by means of adefinitelinear substitution jand if we callthe modulus of the substitution, thatisthe square of thedeterminant formed by thecoefficients of substitution, M, toeveryg-eneralinvariant 1qoftheqth degree, to thelattercorresponds apartialform(1q)ofinvariant totheformer, such that 11q=M/1q)j and consequently, since every (1)isarationalfunction of (8)and(t),80must every 1 be thesame of 8andtjunless,indeed, it were possible to have 1"=~"(1q),q/being different from and greaterthanq:butifthiswere the case, sincei,=~q(1q),apower of Mthe modulus would necessarily be an invariant jbutin passing from :If'+!t+6rnafJy2to:If'+v:+6ry(m)q;lyl,1+ 3m becomes themodulus, which we know is not an invariant. Hencethe proposition is completely established for thecaseof thebiquadratic function (z,y)4_. Now let us proceed to Aronhold's famous SandT,theinvariants tothe generalcubic function (z,y,Z)3,formsequallydeartotheanalystand geometer. (VideMrSalmon's HigherPlaneCurvespassim.) The method will be precisely thesame 8..<>thatappliedto8andtt. We commence with the canonical form q;I+'!I+zI+6m:cyz. Onsubstituting z+y+z,a:+py+piz,lX+ply+p.fore,y,e,wherepis the cube root of unity,theabovequantity takesthe form (3 +6m)l.xl+'!t+zI+6,8(m):cyz), where I8-I8m I-m ,8(m)=6(3+6m) =1+2m' aperiodic function in m of thesecond order only, for I+2m-I+m ps(m)=1+2m+ 2 _ 2m=m. • I have madeataci~assumption throughout theforegoing demonstration (which is, however, capable ofaneasyproof),namelythatifanyfractional function ofthecoefficients ofany formbeinvariantive, thenumerator anddenominator mustbeseparately invariantll. tThe,isAirCayley's property, ~hetbelongs to ProfllllllOr Boole,havingbeenbyhimimparted, in theinfancyofthetheory,toAirCayley, by whom it was first given to the world, atleastinits character &IanInvariant. 604 OnAronhold's Invariants. [60 Butif wewriteforxintheoriginalformpx,itbecomes :r;'+y'+zS+6pmxye; and if for xwewrite p~x,itbecomes :r;'+!I+zS+6p'mxyz. Hencewe can by liuearsubstitutions obtainfrom:r+y+zS+6rnxyzthe threeadditional forms xl+y+zS+6f3(m):eyz, xl+y'+zS+6'Y(m):eyz, xl+y'+zS+60(m):eyz, where I-m I-pm p'-m f3(m)= 1+~m''Y(m)=p\+2pm= 1+2pm , I-p'l.mp-mo(m)=p = .1+2p~m1+2p'l.m In all,therewill be twelve values of m forming threeremarkable compound cycles, m,f3(m), pm,pf3(m), p'l.m,p'f3(m),'Y(m), P'Y(m), p'l.,y(m),o(m), po(m), p'o(m). Itwould be beside my presentobjectto seek to develope fully the functional relations in which the several termsofthesecyclesstandtoone another: theinteresting relations fJS(m)=r(m)=OS(m)=m, f3'Y(m)='Yfl(m)=0(m), 'YO(m)=0'Y(m)=fl(m), of3(m)=flo(m)='Y(m), have been already"statedby me in another place(Oambridge and Dublin Mathematical Journal, March1851t). The(8)ofthecanonical formcorresponding totheSofthegeneralform is m -m·;andthe(T)corresponding totheTofthegeneralform is 1-20m'-8m!.(See my Calculus ofFormat, Oambridge and Dublin Mathe­ meticalJournal, February 1852.)Itis my object toshowthatanyother invariant (I)tothecanonical form mustbe arationalfunction of SandT. Inthefirst place, I observe thateveryinvariant to anyfunction ofan odddegreeiof any odd numberpofvariables mustbe of even dimensions; for ifthedegreeofthedimensions beq,andDthedeterminant ofthe [*p.192 above.] tVid~Addendum [po607below]. [:::p.811 above.] 60J OnAronhold's Invariants. 605 coefficients of substitution, theinvariant tothetransform becomes theoriginal iq . invariant affectedwitha factorDp,where ":.-q,mustbe an even integer,sincep otherwise thesign ofthismultiplier would be equivocal andindeterminable i hence when iandparebothodd,qmustbe even. Thus,then,I(m)inthe case before us mustbe aneveu-degreed function ofm.Moreover, since the changeoftxintoptxconverts mintopm,andIq(m)intopq]q(m), forDbecomes pwhentx,y,zbecome ptx,y,z, Iq(m)mustbe oftheformcf>(m'),mit/>(ml) , met>(rn'),according astheindexqis oftheform6i,6i +2,6i +4. Byprecisely thesamereasoning aswasappliedtothepreceding caseof (8)and(t),we seethatanyinvariant of m which contains mCmustalso contain(1-m)C,(1-pm)C,(1 -ptm)c,thatismustcontain(m-m4)C,which in fact is (8)c.If,now, we consider anyinvariant oftheqthdegreein m,[(m),andsupposeitto beotherthanarationalfunction of(8)and(T), andifwetake1J.todenotethenumber ofthesolutions of4o:x+6y=q, itwill follow thatwe may form an invariantI'(m),which, when qis ofthe form12ior12i+6, willcontainm,andconsequently (m-m4)"'+1asafactor; and in like manner whenqis oftheformI2i+2 or12i+8,willcontain (m -m4)"'+1asa factoriandwhenqis oftheform12i+4or12i+10will contain(m-m4)"'+lasa factor. Now when q=12i, p.=i+l, q=12i+6,.p.=i+l; when q=12i+ 2, JI.=i, q=12i+8,p.=i+1i when q=12i+ 10,p.= i + 1, q=12i+ 4,p.=i+l. HencethefactorsdividingIqintheseseveralcaseswill be of therespective degrees 12i+I2,12i+12; 12i+8,12i+12j 12i+16,12i+I6; corresponding toq,beingoftheseveralvalues 12i, 12i+6iI2i+ 2, 12i+8i 12i + 10, 12i+4; which is clearlyimpossible. Thisprovesthetheorem inquestion (the passagebeingmade from thecanonical tothegeneralform,asintheformer partofthisinvestigation), towit,that8andTformwhatI haveelsewhere termedafundamental scaleofinvariants tothecubicternaryform,entering astheexclusive ingredients intoeveryotherinvariant thatcanbederived from such form. 606 OnAronhold« Invariants. [60 A word of warning isnecessary before I lay down my pen:thattherecan be only two algebraically independent invariants to(.x,y)4or (z,y,z't,is an immediate consequence of thecanonical form of each havingbutonepara­ meter;so ingeneraltherecanbeat mostbut(n-2)absolutely independent invariants of(.x,y)";butthepointestablished inthepreceding investigation goes to show thattherecanexistnootherinvariants thansuch as are rational functions of8andtintheone case, and SandTintheother. I shalltakesomeotheroccasion to establish asimilarconclusion for theforms (.x,y)6and(.x,y)'. I have shown thatthereexistthreeinvariants to the one of the degrees 4, 8, 12, and four totheotherofthedegrees2, 4, 6,10;and Ishalldemon­ stratethatanyotherinvariant toeitherformmustbe arational function of those above stated. Forthecubic form (z;Y'lwe know thatthereisbutone invariant, namelyitsdiscriminant. Thus,then,forn=3,n=4,n=5,n=6 thenumber ofabsolutely independent invariants isn-2, andthenumber oflinearlyindependent invariants is nogreater. Butthisresultis by no meansgenerally true.Itmaybeprovedby means of a greatlaw of reciprocity· which I myselforiginated, butunfortunately threwaside,and which M. Hermite has since demonstrated, thatthereare more thanfive linearlyindependent invariants to(.x,y)7,and more thanten,infact twelve atleast,to(.x,y)1J;thatis to say, itis impossible in thelattercaseto findtenof which all therestshall be ratioual functions, although an algehraical equation connects any 11. So, again,if wetakeasystemof two cubicequations, thereare only five absolutely independent invariants; but thereare not less thansevenlinearlyindependent fundamental invariants, • Thetheorem ofreciprocity alludedtoin thetextis the following :-Iltoanyfunction (x,y)1tthereexistsaninvariant oftheorderm intheooeffiaients, thento(x,y)'" there enm aninvariant oftheordern inthecoefficients; or more generally, which is M.Hermite's addition, iftoanysystemoffunctions (x,y)It"(x,y)'\...(z,y)It,thereexistsaninvariant ofthe severaldimensions mlJ~... m, in the respective setsofcoefficients, thenconversely toasystem (z,y)"'"(x,y)"'•...(x,y)m,thereexistsaninvariant ofthedimensions ~,n,..._n,intherespective setsofcoefficients. Ihlldpreviously shown in thi8 Magazine [p,279 above], thatMrCayley's formnlal for finding thenumberofbiquadratic invariants toanyfunction (x,y)It, given in thatremarkable paper of his on lineartransformations [Cayley's Collected Paper»,Vol.I.,p.1M],wherefirstdawned upon the world the clearandfull-formed ideaofinvariants (themostoriginalandimportant in­ fusedintoanalysis since the discovery of fluxions), could be espreesed bymeansofthenumberof solutions of theequation inintegers2.x+3y=n, thesquareof thequadratic invariant (which only existsfor even values of n) counting for one in the fundamental biquadratic scale;thisis ofcourse adirectconsequence, throughthelaw ofreciprocity, ofthefundamental scaleto(x,y)'eonsisting of aquadratic andacubicinvariant. Mydiscovery ofthefundamental scaleofinvariants to (x,y)Oand(x,y)'nowenablesus,throughthe same law of reciprocity, to expresa the numberof distinctQuinticandSextlcinvariants to(x,y)lt,namely a8 being thenumberofintegersolutions of:t:+2y+S~=iin the one ease, andof:t:+2y+SZ+5t=~ intheother. 60] OnAronhoU£s Invariants. 607 of which any otherinvariant must be a rational function. Infact, if we takefor our two cubies U=a:c'+3bry+3c,xyt+d'!l, V=a:r;I+3fJry+3"(xy2+o'!l, the fivecoefficientsof the powers of )..in thediscriminant ofU+)..V,eachof which is of four dimensions in the two setsof coefficientscombined, are all invariants ofthesystem; buttherewillbebesides two more, one of which isaCombinant of six dimensions, being the resultant ofUandV;theother is aCombinant of two dimensions only, namely ao-3b-y+3cfJ-da.These seventogether form the fundamental constituent scale. The twolast-mentioned maybe expressed algebraically (bytheintroduction ofsquareroots)asfunctions of theotherfive,butof course not asrational functions of thesame. My attention was more particularly called to the search ofa proof of thecompleteness of theAronholdian system of invariants, by aninquiryasto thepossibility of rigidly demonstrating thattherecould existnoothersnot made up of these, addressed to me in the springof last yearby one of the most gifted geometers of this or any othercountry. A morning or twoaftertheinquiryreached me, in a walk before breakfast by the side of the ornamental water inStJames'sPark(atimeand place by no means, according to my experience, unfavourable to theinspirations of the analytic muse), I had the satisfaction of falling upon the ratherpiquant demonstration above given, which essentially rests upon a principle, requiring noharderexercise of faith thanthe concession of theimpossibility ofa greaterbeingcontained in or proceeding outof a less. ADDENDUM. Onthenatureofth6threeOyclesoffourtermseachwhichcontainth6twelve valuesoftheparameter tothecanonical formofacubicfunctionofthree variable8. Theequations given in thetext[po604above] show thateach term in anyone cycle is a periodic function of the secondorder of each otherterm inthesame cycle. Moreover, itmay be shown thateachterminanyone cycle is a periodic function of thethirdorder cf every termineitherof the othertwo cycles; a sort of relation between thecycles taken perse,and with oneanother,precisely the inverse of what obtains(asalreadyshown) for the two cycles of threetermscontaining the six values of theparameter tothe biquadratic function of two variables. For as regardsthatcase,itwas shown 608 OnAronhold:s Invariants. [60 inthefirstpartofthispaperthatthetermsinthesame cycle are periodic functions ofthethirdorderof oneanother, andofthesecondorderof each ofthosenotinthesame cycle with themselves. lfwemake m=,A,I-m pi-m p-m 1+2m=B, =C1+2plm=D, 1+2pm , p,A=.A', pB=B', pC=C', pD=D', pl.A=.A", plB=B", plC= C", p'D=D". Thefollowing tablewillexhibitalltheternaryperiodsthatcanbe formedbetween thetermsoftheseveral cycles s-« (1)AB'IY', (4)B.A'C", (7)CA'D", (10)tus: (2)sc:s: (5)BO'IY', (8)Off.A",(11)DB'C", (3).AD'O", (6)BIYA", (9)OIYB", (12)DC'.A". Forinstance, asanexample ofthemeaning ofthetable,takeline (8), namelyCff.A".Thisindicates that.A"is formed from B'andCfrom.A" inthesamewayasB'from0,and of course .A"fromCinthesame way asCfromB'andB'fromA",&c. By means of thistableitwilleasily beseenthatatermineachof two cycles beinggiven,theterminthethird which forms withthegiventwoaternaryperiod may immediately be assigned. Theremarks which I have to add on thenatureoftheequations for findingtheparameter" m,as well for (x,vras for(x,y,z)',will begiven hereafter. 61. ON A REMARKABLE MODIFICATION OF STURM'S THEOREM. [Philosophical Magazine, v.(1853),pp.446-456.] LETme be allowed to use thetermimproper continued fraction to denoteafraction differing from an ordinary continued fraction, in the solecircumstance ofthenumerators beingallnegative unitsinsteadof positive units, 88thus: .!.1ql--1q.-q.-&c. Thesuccessrve convergents of such afraction 88thatwrittenabove willbe 1ql s.'qsql-1' Ifwe calltheserespectively NIN.N,&c DI'D1'D,'. we havethegeneralscaleof formation N.=q.N'-l-N.-'/., D.=q.D'-l-D.~. Moreover,we shall have universally N.D'-l-N'-lD.equal to+1, insteadofalternating between+1 and - 1, 88isthecasemcontinued fractions of theordinary kind. Again, let me be allowed to use the termsignaletic series to denote a series of disconnected terms, designed to exhibitacertainsuccession of algebraical signs +and-,and to speak of two series beingsignaletically equivalent whenthenumberofcontinuations of signs and of variations of & 39 610 On a remarkable Modification ofSturm'sTheorem. [61 signsbetween the several termsand those thatareimmediately contiguous tothemisthesame for thetwoseries;acondition which evidently may be satisfied without theorderof such changes and continuations being identical. I am now able to enunciate thefollowing remarkable theorem ofsignaletio equivalence between twodistinctseries of terms,eachgenerated from the sameimproper continued fraction. ButfirstI must beg to introduce yetanothernew term in addition to those already employed, namely reverse CO'nvergents, to denote theconvergents generated from a given continued fraction by readingthequotients inareverseorder, or if we like so to say, theconvergents corresponding to thegivencontinued fractionreversed. The two forms and 1 are obviously reciprocal; and ifthetwolastconvergents of eitherone of them be respectively Nfl-IN" Dfl-I'D,,' Dn:1will serve to generate theother.Fortheclearersndmoresimple enunciation ofthetheoremaboutto be given, itwillbebettertotakeasour firstconvergent ~,sothat1willbetreatedasthedenominator of thefirst convergent in every case;andcallingDosuchdenominator, we shallalways understand thatDo=1.LetnowDo,DI>D2•••D"bethe(n+1)denominators of anyimproper continued fraction of nquotients, and([0'([I>([t•..([" thecorresponding denominator series for the samefractionreversed; then, Isay,thatthesetwoseriesaresignaletically equivalent. I do not here propose to demonstrate thisproposition, towhich I was ledunconsciously by researches connected with thetheory of elimination, whichafforda complete and generalbutsomewhat indirectandcircuitous proof. Doubtless some simple and directproof cannot fail ere long to be discovered". ForthepresentI shallcontentmyself with showing aposteriori thetruthofthetheorem for aparticular case.Letn=3. The two series whichareto be proved to besignaletically equivalent maybewritten 1,A,BA-I,CBA-C-.A, 1, C,BC-l, ABC-A-C. •BeePOB~8Clrip~ [p,616below]. 61JOn a remarkable Modification ofSturm'sTheorem. 611 Calltheserespectively 8and(8).In8 we may substitute inthethirdterm, in place of BA-1,OAwithout affecting thesignaletic valueoftheseriesj for ifthesecondandfourthtermshavedifferent signs,thethirdtermmay betakenanything whatever, sincethesequence ofthesecond,third,and fourthtermswill give one continuation and one change,whatever themiddle one may be. Suppose, then,thatthesecondandfourthtermshavethe samesign, and let OBA-C-A=miA, therefore C(BA-l)=(m l+I)A, therefore (BA-1)AO=(ml+1)AI. HenceBA-1 andACwill have thesamesign;hence 8 is signaletically equivalent to8',where8'denotestheseries 1,A,CA,CBA-C-A. Now, again, if OAisnegative, we may putinstead ofAanything whatever, andtherefore, if weplease,C,without affecting signaletically the valueofS'.ButifCAispositive,AandCwill have thesame sign, and therefore onthissupposition alsoCmaybesubstituted forA.Hence always8'issignaletically equivalent to8",where8"denotes 1,C,CA,CBA-C-A. Again,ifCandCBA-0-Ahavedifferent signa,thevalue of the intermediate termisimmaterial jbutifCandCBA-0-Ahavethesame sign,let CBA-C-A=mlOj then A(CB-I)=(I+ml)O, and AI(CB-I)=(I+ml)AOj andconsequently CB-1andAChavethesame Sign. Ineverycase, therefore, S"issignaletically equivalent to 1, C,OB-I,ACB-A-C; thatis 8 issignaletically equivalent to8',andtherefore to8",andtherefore to (8),aswastobeproved. Theapplication oftheforegoing theorytoSturm's processfor finding thenumber ofrealroots of anequation isapparent jforaverylittlecon- sideration will serve to show, thatif weexpand ~:'febeingofthenth degreeinx,algebraically undertheform ofacontinued fraction 39-2 612Ona remarkable Modification ofSturm'sTheorem. [61 whereQI>Q2'Q....Qnmay be supposed linearfunctions of a;(although, in fact,thisrestriction, aswill behereafter noticed, is unnecessary), the denominators ofthereverse convergents o1Qn-l Qn-IQ1>-IQI-&c.I'Qn'QnQn-I-1...QnQn-l Ql- &C.' will besignaletically equivalent with tbe Sturmian series of functions for determining thenumberof real roots of fa;withiu given limits;in fact, 1,Qn,QnQn-l-1,... ,QnQn-l...QI-&c. will betheSturmian functions themselves, divided out by thenegative oftbelastorconstant residuewbich arises in theapplication oftheprocess ofcontinued division, according toSturm'srule;andaswehave shown that theseries of tbedenominators to theconvergents of anycontinued fraction, andtheseries of thedenominators totheconvergents ofthesame fraction reversed, are signaletically equivalent, we have tbissurprisingly new, interesting, andsuggestive mode of statingSturm's theorem, namely, the denominators to theconvergents of thecontinued fractionwhichrepresentsj:constitute aRbizoristic series for fa;,thatis asignaletic series which serves to determine thenumberof rootsofja;comprised within any prescribed limits. Moreover, in applying thistheorem it is by no means necessary that, inthecontinued fraction which representsj:,all or any of thequotients should be takenlinear functions of e.A verylittleconsideration ofthe principles upon which thedemonstration ofSturm'stheorem is founded will serve to show thattheconvergent denominators to anycontinued fraction whatever whichrepresentsj: 'whetherthequotients belinearornon-linear, integral or fractional, or mixed functions of e,andwhatever thenumber ofquotients, which,itmay be observed, cannotbe lessthan,butmay be made to any extentgreaterthantheexponent ofthedegree of fa;,will equally well furnisb aRhizoristic series for fixing theposition of tberoots, provided only thatthelastdivisorin the process of expanding-j: underthe form ofan improper continued fraction be aconstant quantity or anyfunction ofa;incapable ofchanging itssign. Letus, however, for thepresentconfine our attention totheordinary Sturmian form, where all the quotients arelinearfunctions of e.Letthese quotients berespectively CZta;+bl,a,a;+bll,a,a;+b•...ana;+b«. Inorder to determine tbetotalnumber of real and imaginary roots offe,we must count the loss of continuations of sign in theRhizoristic 61]Onaremarkable Modijication ofSturm'sTheorem. 613 series in passingfroma:=+00tox= -00•Whenxisinfinitely great,it is clearthat,whether positive or negative, thepartsbl,bs•••btlmaybeneglected, and only thehighestpowers of xneed be attended toinwritingdownthe signaletic seriescorresponding to these two values of e.Accordingly for x=±00thesignaletic series becomes I,~x,a)a.;r:2,...,ala....~f1, andconsequently thenumberof pairs of imaginary roots of/xisthenumber of changes ofsign in the series 1,~,ala.,...,~~•..an, thatis, is the numberof'negative quantities intheseries Hencewe have the curious andhithertostrangely overlooked theorem, that inapplying Sturm'sprocess of successive division to /xand/'x,thenumber ofnegative coefficients of xin the successive quotients givesthenumber of pairs of imaginary rootsof/x;as a corollary,welearn the somewhat curious factthatnever more thanhalfofthesecoefficients canbenegative jand in generalit wouldappearthatthebetterpractical method of applying Sturm's theorem would be not to deal with the Residues, which have hithertobeen the sole thingsconsidered, butratherwiththelinearquotients which have beentreatedas merely incidental to the formation of the Residues. To find the value of the Rhizoristic series corresponding to a given value ofe,thebettermethodwould accordingly seem to be to commence with findingthearithmetical values of thenquotients alx+bl,¥+bs...~+bfl' Wethusobtainnnumbers ILl'1L'J.../Ln,and have onlyto form a progression according tothe well-known law 1,NlJ!{s...N.., where N I=p.,andingeneralN,=jl,N'_1-N,-'j' Thenumberofarithmetical operations required bythismethod(afterthe divisionpartoftheprocess which is common to the two methods has been performed) will be"2nmultiplications and2nadditions orsubtractions; whereas if we deal with the residues directly, the numberofmultiplications will be thatisn+(n-I)+ ...+I, n(n+1) 2 (besides havingtoraisexto thenthpower),andthesamenumber of additions. Thepractical advantage, however, of thismethodovertheold [*joomote, p. 622 below.] 614On a remarkable Modification. ofSturm'sTheorem. [61 methodis notquitesogreatas it may atfirstsightappear, in consequence ofthequantities operated with on applying itbeinglargernumbers than those which have to be used in theold method. Ifwe were to employ, insteadofthedirectseries, 1, Nt,N,Nt-I, &c., thesignaletically equivalent reverse series 1, Nn•N_tNn-I, &c., thearithmetical difficulty would be much increased in consequence of the quotients becoming rapidlymore complex asthedivision proceeds. Itwere much to bedesired thatsomeperson practically conversant with theapplication ofSturm's method, such asthatexcellent andexperienced mathematician, myesteemed friend Professor J.R.Young, would perpend and give his opinion upon therelative practical advantages ofthetwo methods of substitution; theonethatwheretheresidues areemployed, theotherthat where the quotients. Iambound to state,thatbutforavaluable hintfurnished to me by my friend,thatmost profound mathematician, M.Hermite, who discovered atheorem virtually involving thetransformation ofSturm's theorem here presented, butfounded upon entirelydifferent and less general considerations, and in the origin of which hint, asarisingout of myownprevious speculations upon which I wasin correspondence with M. Hermite, I mayperhapsmyself claim a share, this theorywould probably not have come to light. Itis of course not confined to Sturm's theorem. which deals only with thespecial caseof two functions, whereofone is the first derivative oftheother. Thereisalarger theory, to which M.Sturm's isacorollary. which contemplates the relations of theroots of any two functions whatever. This is what I term thetheoryofinterpositions, upon which I do not propose here to enter,butwhich will be fully developed in a memoir nearly completed, and which I shortlypropose to present- to the Royal Society, wherein will be found combined and flowinginto one currentvariousstreams ofthoughtbearinguponthissubjectwhich had previously existeddisunited, andappearing tofolloweachaseparate course. Remark. Iamnot aware thatanyonehasobserved what the effect would be ofomitting to change thesigns ofthesuccessive residues in theapplication ofSturm's method,thatis, of employing aproper in lieu of an improper continued fraction to expressj:. [.pp.429-586 above.] 61]Ona remarkable Modification 0/Sturm'sTheorem. 615 Although easily made out, it iswell worthy of beingremarked. Suppose f=-!. 1fQI-Q_-.! 1IQI 1 -Q..' and ingeneral(Pbeinganyletter)usePto denote - P.Now we may write 1=Qllf>-PI> If>=QtPI-PI' PI=QIPI-PI> p.=Q4P.-P4' p.=QoP4-Pu P4=QoPs-p.. &c.=&c. Thisgives 1=Qllf>+PI' If>=Q,pl+PI> PI=QaPl+PI' PlI=~+P4' P.=QoP4+ps, &c.=&c. The law evidently beingthatthequotients changetheirsignalternately, thatisinthe2nd, 4th, 6th, &c.places, and remainunaltered inthe1st, 3rd, 5th, &c.places;whereas the residues or excesses changetheirsigns in the Istand 2nd, 5th and 6th, 9th and 10th, &c.,and remain unaltered inthe3rd and4th,7th and 8th, 11thand 12th, &c.places. The effect is, thatif,in applying Sturm'smethod, we omit to change the signaoftheremainders, and take as our signaletic series la;,j'e,~,R"R,...R...-h ~,R"R"&c.beingthesuccessive unaltered residues, the signaletic index corresponding to any value of a;insteadof beingthenumberofcontinuations intheabove series, willbecomethenumberofcontinuations in going from a term in an odd place to a term in an even place plusthenumberof variations in going from a terminanodd place to a termin an even place. Ifweadoptthequotient method,therule will besimply to change the sign ofthealternate quotients (beginning withthesecond)informingthe signaletic series. 616Ona remarkable Modification ojSturm'sTheorem. [61 As anartistdelights inrecalling theparticular timeandatmospheric effectsunderwhich he hascomposed afavourite sketch, 80I hope to be excusedputtingupon record thatitwasinlistening toone of themagnificent choruses in the'IsraelinEgypt'that,unsought and unsolicited, like aray oflight,silentlystole into my mind theidea(simple, butpreviously un­ perceived) oftheequivalence oftheSturmian residues tothedenominator series formed by thereverse convergents. Theideawasjustwhatwas wanting,-the key-note tothedue and perfectevolution of the theory. Postscript. Immediately afterleavingtheforegoing matterinthehands of the printer, amost simple and complete proofhasoccurred to me of thetheorem left undemonstrated inthetext[po610]. Suppose thatwe have any series of terms'It,.,Ut,Ua...Un,where 'It,.=ill,Ua==illilll-1,Ua=illillill-ill-ill,&c. and ingeneral U,=il,U.-I-U.-lI' thenUx,~,Ua...Unwill bethesuccessive principal eoaxaldeterminants ofasymmetrical matrix. Thussupposen=i5;if we write down thematrix ill>1,0,0,0, 1,ill>1, 0, 0, 0, 1,ill,1, 0, 0, 0, I,il.,1, 0,0,0, 1,il., (themode of formation of which is self-apparent), thesesuccessive coaxa.l determinants will be 1IillIIAI,1IAI,I,°AI'1,0,°All1, 0, 0,°1,illl1,AI>11,illll1,°1,AI,1, 0,°0,1,ill0,i:AI,10, 1,AI,1,°, 0, 0, 1, il.0, 0, 1, il.,1 0, 0, 0, 1, A. thatis 1,AI,Alilll-1,AIAsAl- ill-ill,AlilsAlil. -AIAlI-Alil.-Alil.+1, ilIAsA..A.il. -AlilllA.-Alil..A.-A..A.A.-illAsAl+A.+ill+AI' Itispropertointroduce theunitbecauseitis, in fact, thevalue of adeter­ minantof zero places, asI haveobserved elsewhere. Now I have demon- 61JOnaremarkable Modification ofSturm'sThem-em. 617 strateddirectlyinthisveryMagazine (August 1852)-,undercover of the umbralnotation, thatthesignaletic value of aregularly ascending series ofprincipal coaxaldeterminants formed from any symmetrical matrixis unaffected by anysuchtransposition whatever of the lines andcolumns of thematrixasdoes not destroy the symmetry abouttheprincipal axis. Hence, then, beginning fromthelowerextremity oftheaxis.A.,andreading offthe ascending series of coaxal minors from thatpoint, we obtain the reverse series, I,AI,.A.A.-1,..A.A..A..-AI-A.,.A.A.A•.AlI-..A.A.-.A.A.-A.A.+1, A.A.A•.A..A,-A..A.A,-A..A...A.-A•.A~,-A..A.A.+Al+A.+A•. Hence we see thatthedenominators totheconvergeuts of -.!.1 Al-All--.!..-1A.--1 A.-..AI' beginning with I, form a series signaletically equivalent tothatsimilarly formed from thefraction -.!.1 ..AI-A--.!.1 •..A.--1..A.-..AI; andthe reasoning is of course general, andestablishes thetheorem in question. Itseems only properandnaturalthatI should not leave unstated here thesignaletic properties of the series of numerators totheconvergents toj:expanded undertheform ofacontinued fraction. Letthenumberof changes of sign in thedenominator series for any given value aofa:be called D(a),and for thenumerator seriesN(a). ThenN(a)-N(b)may be equal to, or atmost can only differ by apositive or negative unitfromD(a)-D(b).The relation between these differences dependson thenatureoftheintervalbetween the greaterofthetwo limits aandb,and the root of f(a:)nextlessthanthatlimit, and of the interval between the less of thetwo limits aandb,andtheroot offa:nextgreater thansuch limit. Ifarootoff'a:iscontained ineachsuchinterval, N(a)-N(b)=D(a)-D(b)+ 1i ifaroot off'a:iscontained withinoneinterval, butno root within the other, N(a)-N(b)=D(a) -D(b); ifno rootoff'a:iscontained withineitherinterval, N(a)-N (b)=D(a)-D(b)-I. [-p.S80 above.] 618Onaremarkable Modification ofSturm's Theorem. [61 Imayconclude with noticingthatthedeterminantive form ofexhibiting thesuccessive convergents to animproper continued fraction affords an instantaneous demonstration oftheequation which connects any two con­ secutive such convergent8 as N'_IandN, D,_ID,' namely N,D'_I-N'_ID,=1. Forif weconstruct thematrix,which for greatersimplicity Ilimitto five linesandcolumns, A,1,0, 0,° 1,B,1, 0,° 0, 1,0,1,°(M) 0, 0, 1, D,1 0, 0, 0, 1, E andrepresent umbrally as (~'~,~,a..as) bllb21i;b"ba' and if, by way of example, we takethefourth and fifth convergents, these will be in theumbralnotation represented by (~'as,a,) (~'~,a"as)i;i;i,dbi,i;i;ba -~----"----'----- an , (aI,~,aa,a,) (~'~'aa,a"as) ~,i;ba,b,i;s,ba,i,b, respectively. Hence NaD,-N~a=(b~'asb,ab"asb)x(b~'aab,ab4,~\ i'.."a i'a."bJ (~'aa,a,)x(~'as,a"as,~) - ba,ba,b,bl>ba,i,ba,~, =(as,as,a"aa\x(~'~,aa,a,\i;i;b., bJi;i;i.bJ' thatis 1, B,1,°1, 0,0,°=1x1=1, 0,1,0,1B,I,0,°x 0, 0,1, D 1, 0,1,° 0, 0, 0, 1 0,1, D,1 61JOna remarkable Modification ojSturm's Theorem. 619 aswasto be proved. And thedemonstration isevidently generalinits nature. We may treatapropercontinued fraction in precisely thesame manner, substituting throughout V(-1) in place of 1 in the generating matrix,and we shall thus, by thesame process ashasbeenappliedto improper continued fractions, obtain N'+lD,-N,D'+l={v(-l)j'x{v(-l)j' =(-I)'. I believe thattheintroduction ofthemethodofdeterminants intothe algorithm ofcontinued fractions cannotfail to have an important bearing uponthefuturetreatment anddevelopment of thetheoryofNumbera". •Ifin~heabovematrix(M)wewri~ ~hroughout"; (-1)inplaoeof 1,we haTearepreaen~tion of~enumera~re anddenominators oftheconvergente to apropercontinued fraction, andsuch repreeen~tion gives animmediate andvisible proof of the simple and elegantrule(nots~tedin theordinary treatises onthesubject, nor 80well known asitdeserves to be)for forming any such numerators ordenominators by means of the principal termsineach;therule, I mean, aocording ~which the t~hdenominator may be formed from qlqsll,q,... q,(ql'qs...q,being the sUOO98sive quotiente), and the tthnumerator fromqslla...q"by leaving outfrom the above products respectively anypairor anynumberofpairsofconsecutive quotiente asq,qP+l' Forins~oe, fromqlqslltl,qo, by leaving out qlqS'qslla'qaq,andq,qo'weobtain qil,qo+qlq,qo+q,qsllo+qlqslla; andbyleavingoutqlqSxqil"qlqsxq,qo, qsll,xq,qo' weobtainqO+qS+ql; 80thatthe~Ul denominator beoomes qlqsllaq,qo+qaq,qo+qlq,qO+qlqsllo+q,qslls+ql+qa+qo; and in like mannerthenumerator ofthesameconvergent is qtlil,qo{I+~+-.!..+~+__1_}, qsllaqil,q,qoqsqri,qo tha~is qtlil,qo+q,qo+qsllo+qslla+1. Themostcursoryinspection of the form of the generating matrixwill show atoncethertB80D of~hisrule.Itmayfurthermore be observed, thateveryprogression oftermsconstructed in conformity withtheequation u"=G.u,,-l-b.u,,-s+c.u..-a=&c., may berepresented as anascending series of principal coauldeterminante ~a common matrix. Thusifeachtermin such progreasion istobemadealinearlanction oftheU1reepreceding terms,it will be representable bymeansof thematrix A,B,Oil,0,0 1,A'B",0'''0 , , 0,I,A"Bill0"", , 0, 0, 1,.A"'B"", 0, 0, 0, I,,A'III indefinHely continued, whichgiTestheterms I,A,AA'-B;'AA'A"-BA"-AB"+C", &0. 62. NOTE ON A REMARKABLE MODIFICATION OFSTURM'S THEOREM, AND ON ANEWRULE FORFINDING SUPERIOR ANDINFERIOR LIMITS TOTHEROOTS OF ANEQUATION. [Philosophical Magazine, VI.(1853), pp. 14-20.] INmypaper[po609above] on this subjectin thepreceding Number ofthe Magazine, I showed how by means of the quotients ~X+bl'¥+b....a.,x+b.., obtained bythrowing-j: undertheform ofacontinued fraction, the process for finding the signaletic index for anygiven value of xin the series for deter­ miningthenumberof real roots of fxwithingiven limits wasreduced to performing two sets of nmultiplications andasmanyadditions orsubtractions. Butbymeans of avery simple observation, I can now show thatthe second and more laborious set of multiplications may be dispensed with and replaced bythesimpleoperation of finding reciprocals, which can be done by mere inspection by means of Barlow's or similartables, which arefamiliartoall computers. Ifwe call the quotients ~x+bI,¥+bl•••ana:+b.., we must, asexplained in thepreceding article, find thennumerical values ~,Jl1...JLnwhich these quotients assume for any assigned value of x.This beingdone,thesignaletic index corresponding tosuch value of e,thatis thenumberofcontinuations ofsign in thesignaletic series 1,/lrl,J.'1Jl1-1,~-fIoa-1'-1'&c., isevidently thenumberofpositivetermsin the series ...1 1 ~!1fIoa-- 1fIoa-­ ~1"'"-­JLn-l1'-- 62) On a remarkable'Modification ojSturm'sTheorem. 621 Thesetermsmaybefound with theutmostfacility in succession from oneanother; for ifM.beoneofthem,thenextwill be (}J-.+I-M.)-l.Thus, then.thenecessity forthemore operose set of multiplications isdone away with, and the actuallabour of computation reduced much more than50 per cent. below thatrequired bythemethodindicated inthepreceding article onthesubject. I needhardlyadd,thattheoldmethodofSturmwould admitofasimilarabbreviation; butin using itwe should besubjected to thegreatpractical disadvantage ofhavingto begin with themore heavy andcomplicated quotients Jl-n.Jl-n-lt&c.insteadofJl-l,Jl-t.&c.,which would verygreatlyenhancethelabour of computation. I will conclude by aremark of someinterestunderanalgebraical pointof view. Ithasbeenstatedthatthedenominators ofthesuccessive convergents to 1 ql areequivalent (toaconstant factorpres)withtheSturmian functions. and thereadermay becuriousto knowsomething ofthenatureofthesignaleti­ callyequivalent series formed by thedenominators of theconvergents to thedirectfraction 1 1ql-- 1qi--&c.q. :1q.: Thesedenominators are(abstracting fromaconstant factor not affecting' thesigns) the Sturmian residues resulting fromperforming the process of common measure betweenf'a:andha:;f1a:beingrelatedinaremark­ ablemannerinpointof form to f'e.Call the roots of fa:~,as...an;we knowthatj'a:is I{(a:-as)(a:- as)...(a:-an)}. and I am able to statethatha:is (toaconstant factorpres)equal to I[t(as.as...an){(a:-~)(a:-as)...(a:-an))]. t(1b.l.as...an)denoting theproductof thesquaresof the differences between the(n-1)quantities as.a....an'Accordingly itwill be seen thatwhenever a:isindefinitely near.whether ontheside of excess or defect, to a real root offa:.j'a:andha:will have the same sign;which serves to show. upon anindependent and specific algebraieal ground,why the two seriesof residues. corresponding toj:and1:are(asbyadeduction fromageneralprinciple theyhave been previously shown to be) rhizoristically equivalent. 622Ona remarkable Modification ojSturm'sTheorem. [62 Observation. Incomparing therelativemeritsoftheold and new methods ofsubsti­ tutionforthepurposes of Sturm's theorem, theeffect of theintroduction of positive multipliers intothedividends in order to keep all thenumerical quantities integraloughtnot to be disregarded. Ifwe callthequotients corresponding to thismodification of thedividends QhQ'JI>Q.,Q.,&c.,and thefactorsthusintroduced m,.,11I.tI,ma,m.,&c.,thetruequotients will be ~,::a;m,.r:.Qa,:::Q.,&c.; andit will be found thatwe may employ asourrhizoristic indexeitherthe numberofcontinuations ofsign in the series thelaw of formation of the successive termsUa,u,.,'Ut,&c.being or thenumberof positive signs in theseries thelaw of formation of the successive terms VhVIIVa,&c.being V,=Q,-m,. V,_I Theremay therefore, in fact,be ineachcase(n-1)moremultiplications thanhave been takenaccount ofin the textabove. Ifintegernumbers be usedthroughout (sothataccordingly theuseries isthatmadeuse of),thetotalnumberofmultiplications will ingeneral ben+2(n-1)"or3n-2jthe old method, aspreviously stated,would requirein(n+1)multiplications jfor if we callanyone oftheSturmian functions AoZ'+A11lf-1+A~+...+A" we shall, using themostabbreviated methodofcomputation, have to calculate successively •Ifall theextraneous factors are units,thenumberofmultiplioations (likethatof the additions) would be 2n-l,andnot2n,asinadvertently statedinthe preceding numberof the Magcm.M. 62JOn a remarkable Modification ofSturm},Theorem. 623 givingriseto£operations (but,itmustbeadmitted, withthepractical advantage oftheuse ofaconstant multiplier) jandas£maytakeall values from nto 1,thetotalnumberof suchoperations will beIn(n+l). Whenn=4, in(n+1)=3n-2. Consequently (ifitbethought necessary toadheretointegersthroughout), forvaluesofnnot exceeding 4, theoldmethodwould be probably the moreexpeditious. ADDENDUM. Onamethodoj finding Superior andInferior LimitstotherealRoots ofany.A1gebraical Equation. Thetheoryabove considered has incidentally led me to thediscovery ofanew and very remarkable methodfor finding superior and inferior limitstothereal roots of anyalgebraical equation. Suppose in general thatNIl 1 1-----+-'D- ql+q,+q.+...q..' thenitiseasilyseen tha.t where Ingeneralletanynumerical quantity withinbrackets be used to denote itspositi",enumerical valuejsothat,forinstance, whetherq=±3,(q)will equallydenote+3. And now suppose thatneitherqlnorq..,thefirstorlastofthequotients, liesbetween+1 and - 1, and thatno one of the intermediate quotients q"q•...qn-lliesbetween+2 and - 2 j80that,inotherwords, (ql)>1,(q.)>2,(q.)>2...(qn-l)>2,(q,,)>1j then,I say,thatM1,M., M•...M..will have thesamesignsasql,q"q....q.. respectively; for therefore but(M1)>1j 1M,=q,+ MI' therefore624Onaremarkable Modification ojSturm',Theorem. [62 (Mt)=(qt)±(~J>2±1j therefore Mthasthesame sign asqt,andal,o(Mt)>1j therefore in likemanner, (MI)hasthesamesignasql'andalso(M.)>1j therefore in like manner, (M.)hasthesamesignasq.,andalso(M.)>1j andso onuntilwe come toM_IIand we shall find M_Iofthesamesignasq.....11andalso(Mn-I)>1. Finally, 1Mn=qn±M-,_I where(qn)>1and(M~J<1,therefore Mnhasthesamesignasqnj butwecannotsay(noristhereanyoccasiontosay)that(Mn)>1;therefore D=MIM.M•...Mnhasthesamesignasqlq.q....qn' Now let fa;beanygivenfunction ofa;ofthenthdegree,and~any assumed function whatever ofa;of the(11-1)thdegree,andlet ~1 1 1 1 fa;=ql+qt+q.+...q,., whereqllq.,q•...qnarenow supposed to be linearfunctions of e,which, exceptforspecialrelations betweenfand<1',willalwaysexist,andcanbe found by theordinary process of successive division. Writedownthe11pairs ofequations, u,.=ql+1=0,Ua=q.+2=0,Ua=q.+2=0 '"Un=qn+1=0, u'l=ql-l=O, u'.=q.-2=O, u'.=q.-2=O ...u'n=qn-1=O. Ifthegreatestof the values of a;determined from these 2nequations be calledL,andtheleastof these values be called A, itmayeasilybe made out thatbetween+coandL,each of the quantities qllqllq•...qnwill remain unaltered in sign jandbetween - 00and A also the sameinvariability of signobtains; and, moreover, between+00andL,andbetween A and-00, (ql),(qt)...(q_I),(qn)will berespectively greaterthanI, 2...2,1.Con­ sequently, byvirtueofthepreceding theorem, between+00andL,and between A and - 00,Dwill always retainthesamesignasqlq.qt...qn, 62JOn a remarkable Modification ojSturm'sTheorem. 625 andtherefore no root of fxwill becontained withineithersuchinterval. Andhencefx, which is manifestly identical withD(thedenominator of the continued fractionlastabovewritten), affected with acertainconstant factor, willretainaninvariable signwithineach such intervalrespectively. Hence, then,thefollowing rule. Callingql'qt,qs...qflrespectively a.x-bl,a"x-ba,aax-bl'"UnX-bn, if we form the2nquantities bl±1bt±2 b l±2bfl-I±2i;±1 -a;-(i;' alUn-I an thegreatestof these will be a superiorlimit, and the leastoftheman inferior limittotheroots offx. The values of these fractions will depend upon the form of the assumed subsidiary function cf'.Hence, then, arisesamost curious question for futurediscussion-to wit,to discover whether in anycasethesubsidiary function canbe soassumed asthatthesuperior limitcanbebrought to coincide with the greatest, ortheinferiorlimitwith the leastrealroot, supposing thatthereare any real roots. I believe thatit will be found that thisis always impossible to be done. Then, again, if all the roots are imaginary, caninconsistent limits(evincing thisimaginariness) beobtained by giving different forms to thesubsidiary function, which would be thecase if we could find thatthesuperior limitbrought out by one form were less thanthe inferior limitbroughtoutbyanother,ortheinferiorlimitbrought out by one form greaterthanthesuperior brought out byanother? If,asI suspect,thisalsocannever be done, thenthegeneralquestion remainsto determine for allcasestheform to be given to thesubsidiary function, which will make the interval between eitherlimitand itsnearestroot, or between thetwolimitsthemselves, aminimum. Thus,itappearstome,afine field ofresearch is thrownopento those whoare interested inthetheoryofmaxima minimorum, and minimamaxirnorum, and one likely to leadtounexpected andimportant discoveries [cf.p. 533 above,and theAuthor's footnote,p. 495J. Itmay beaskedhow is the above rule to be appliedif any of the leading coefficients in ~,or of the successive residues offxandcf'xvanish jin which case, insteadofthecoefficients beinglinear,some ofthemwill be,asin factallmightbe, polynomial functions of x.The rule, itmay be proved, willstillsubsist. Equating thefirst and lastquotients each ofthemto+1andto-1, andtheintermediate onesto+2 and to - 2, the greatest root of all the equations 80formedcontinues to beasuperior, andtheleastroot aninferior a ~ 626 On aremarkable Modification ojSturm'sTheorem. [62 limitto the roots of fe:Nor is it ever necessary, even in these special cases, actuallytosolveany oftheseequations; forevidently it willbesufficient to findasuperior limit and aninferior limit to eachof them, and adoptthe greatest of thesuperior andtheleastoftheinferior limits asthesuperior and inferior limits to the roots of the given equation. Thus, then,we should have torepeatuponthequotients increased and diminished by 1 or 2 (asthe case may be), thesame process asis supposed to be originally applied to fe, andthusbyacontinued process of tritumtion (since every new function so to be operated upon is of a lower degree thanthe original function) we must finally descend to linear equations exclusively. Itisinteresting thusto seethatthereare no failing casesinthe application of the rule, and thatasolution of equations of ahigherdegree thanthe first is never necessary. Butasamatterof fact, the chances areinfinitely improbable (if rf>:eis chosen atrandom), of any of thequotients afterthefirst ceasing to be linear;andthefirstis of course linear, provided thatthe degree of rf>:eistakenonly one unitbelowthatofjre. Inworking with Sturm'stheorem, asystem of quotients issupplied ready tohand;and these quotients, byvirtueoftherule given above,may be used to assign asuperior and inferior limitinthefirst instance, before setting abouttodetermine thedistribution of the roots between theselimits by aid eitherof these samequotients or oftheresidues. For the change of sign of the residues required by theSturmian process will only affect thesigns, and nottheforms of thequotients; butintheapplication oftheaboverule for finding thelimits,thesign ofany quotient isevidently immaterial. 63. ON THE NEW RULE FOR FINDING SUPERIOR ANDINFERIOR LIMITS TO THEREAL ROOTS OF ANY ALGEBRAICAL EQUATION. [Philosophical Magazine, VI.(1853), pp. 138-140.] THElemmaaccessory tothedemonstration oftherulefor finding limits totheroots of anequation, givenintheaddendum [po623above] to my paperintheMagazine forthismonth,admitsof two successive and large stepsofgeneralization, in which thescopeoftheprincipal theorem will participate in anequaldegree. 1.Whatever thesigns may beofqloqs,qa.••q"thedenominator ofthe continued fraction 1 1 1 1---ql+qs+qaq, will have thesamesignasqlqsqa'"qr,provided that 1 1[ql]>PI,[qa]>J.'i+-,[qa]>J.'i+- ...PI J.'i 1 1...[qr-l]>f/or-l+-,[qr]>-, f/or-l I-'r-I where /ki,J.'i...I-'r-lsignifyanypositive quantities whatsoever; inthe particular casewherePI-J.'i=J.'i=...=P.r-I=1,we fallbackuponthelemma asoriginally stated. 2.Butthelemmaadmitsofanothermodification, which will in general imposefarlessstringent limitsuponthearithmetical values of theseries ofq's.. Letallthepossiblesequences of"Sbetakenwhichpresentonlyvariations ofsign;forexample iftheentireseries be ql>qa,qa,q..andthecorresponding algebraical signsare+- - +, weshallhavethetwosequences ql'qa;qa,q•. Iftheentireseriesbeql>qa,qa'"qUIandthesigns be ---+-++++ -++++-, 40-2 628Onthe NewRuleforfindingSuperior andInferior [63 thenthe sequences to be takenwill be and so in general. Suppose,now, thatq,..H,q~...qp+iare thetermsofanyonesuch sequence. Then,provided that 1 1[qP+l]>1'-10[q~]>/102+-...q,.·H-1>JIo\-I+-, Jl'J I'-i--t 1and qP+i>-, JIoi-l (itbeingunderstood thatthevalues of /Iol,/102...JIoi-lare perfectly arbitrary, exceptbeingsubjecttothecondition of being all positive, and thatthere areasmanydistinctandindependent systems of such values 88thereare sequences of.variations of sign),itwillcontinue to betrue(and capable of beingdemonstrated tobe so by precisely thesamereasoning aswasapplied tothedemonstration ofthelemma in its original form) thatthedenominator of~_1_....!willhave the same sign 88theproductqlq2q•.•.qr'Itwillql+qi+qr beobserved that,asregardstheresidual quotients not comprised in any sequence, theirvaluesareabsolutely unaffected by any condition whatever. Asadirectconsequence from thislemma, we derive thefollowing ~tly improved Theorem forthediscovery of thelimits. Let, as before, 1m=0 be any given algebraical equation; 4Jxany assumed arbitrary function of IXof an inferior degree tothatofIx; and let 4Jx_1 1 1 1 . ].:V-'-'X1+XII+X.+X/ let the leading coefficients of Xl'XII'X K; beql'gi'q,....gr,andlet thislatterseries be divided into sequences of variations andresidualterms not comprised in any such sequence, 88explained above.LettheX's corresponding to theresidualtermsbe called and letthesuccessive sets of X'scorresponding tothesequences be called respectively VIIVII...VI" V/,VII''''V'p', Vt,Vt... V""", 63JLimitstothereal Roots ojanyAlgebraical Equation. 629 And let X=PIP, ...p.. X(V1ll_cl)(V,'-~2)...(Vp2-cpt) X(VI'li-~")(Vll"-c,'2)...(V'l-c'l) &c. &c. where, in general, any system ofvalues ~,c"C,...Cp-I'Cp• represents 1 1 1 ""1'JI-ll+- ..."",-1+-,-Jl'l ""p-IJ. /lop-I Then the largestroot ofX-0 isasuperior limit, and thesmallestroot of X=0 is an inferior limitto the real rootsof/:x=0;and ifX=0hasno real roots,neitherwill/:x-Ohave any. For thecomplete demonstration and somefurtherdevelopments of thistheorem seetheforthcoming numberof Terquem's Nouvelles Annalesfor thepresentmonth", [- p.428andp,424above.] 64. NOTE ON THENEWRULEOF LIMITS. [Philosophical Magazine, VI.(1853), pp. 210-213.] ITmayappearlikeharping too long on thesamestringto add any furtherremarks ontherulerelatingto so simple and elementary amatter asthatofassigning limitstotheroots of agivenalgebraical equation; butit will be remembered thatsome of thegreatest mastersof analysis, including thehonoured names of Newton and Cauchy, have not disdained totreat,and to give to theworldtheircomparatively imperfect results on this very subject. I hope, therefore, to standexcused of anyundue egotism in addingsome observations which may tendtopresent, undera cleareraspect and more finished form, thenew and beautifully flexible rule laid before thereaders of this Magazine inthetwo preceding Numbers. Firstly,I observe thatanysuccession of signs may be considered as made up of, and decomposable into, sequences of changes exclusively, if we agree to consider, where necessary, a single isolated sign +or-asa sequence of zero changes. Thus, for instance, +- - ++++- +++- +- - may be treatedasmade up of thevariation sequences +-,-+,+,+,+-+,+,+-+-,_•. Secondly, I observe thatifXl>Xi'"Xibe alllinearfunctions of e, andthesigns of thecoefficients of xin these functions constitute a single unbroken series ofvariations, thedenominator ofthecontinued fraction 1 1 1 1 Xl+Xt+X1+'"X, (reduced totheform of an ordinary algebraical fraction) will have all its roots real. *The rule is, th&tthe given series of signs is tobeseparated intodistinotsequences of vlU'iations, 80thatthefin&!term of one sequence andtheiniti&1term of the nextshallform a continuation, thatis we must have variation sequences connected together bycontinuations at theirjoinings. 64J NoteontheNewRuleofLimits. 631 Thirdly, suppose, for greatersimplicity, that4>rcis of one degree in x lowerthanfe,andthatbytheordinary process of common measure we obtain 4>rc1 1 1 1 fx=X--;+Xs+XI+'"x.., whereXl.Xs,XI'"Xnare all of themlinearfunctions of e; LetXl'XI'"Xnbe divided into distinctandunblending sequences, XIX S'"Xi,Xi+IXi+,'" Xi',X"+l'"X,,-,...,X1il+IX(ilH ...Xn; sothatineachsequence thesigns of thecoefficients of xpresenta single unbroken series of variations, which by virtueofobservation (1), may be considered to be always capableof being done, and let cf>lx1 1 1 1 Itx=Xl+XI+XI+...Xi' cf>,x_ 1 1 1 fiC-Xi+l+XiH......Xf' (cf»x_ 1 1 . (f)x-X(il+l+ Xn' then,according toobservation (2),theequations Itx=O, faX=0...(f)x=0, haveeachofthemalltheirrootsreal;and theobservation nowtobemade is,thatthehighestofthehighestroots and the lowest of thelowest roots of these equations furnishrespectively asuperior and inferior limittothe rootsoffx=0·. "Thistheorem may be more concisely statedasfollows:-"IfUwith any subscript be nnderstood tomeanalinearfunction ofit:in which the sign of thecoeffioient of it:isconstant, thenthefinite roots of the equatiou1111111 111 UI-Us-UI- '"U,+Uj+l-U/+2-'"U,-+'"UW+l-UI;)+2-'" U,,=co lie between thegreatestandleastfinite roots of theequations . 1 1 1 UI-U,-'"-u.=co, 1 1 1 U'+l-U'+2-'" U(=CO, 1 1 1 .. U(j)-UlIl+l-'" U,,=co. Thetheorem underthisformsuggests amuch more generalonerelatingtopara-symmetrical determinants, thatisdeterminants partlynormalandpartlygaucllt,which willbe given hereafter; oneexample among the manyoonfirming the importance of the viewfirst statedinthisMagaziM bytheauthorofthispaper, whereby continued fractions areincorporated with the doctrine of determinants. 632 Note on the New RuleojLimits. [64 N.B. The single root of anyoneor more of these which may be of the first degree in xis to be treated,inapplying thepreceding observation, asbeingatthesametime the highestand the lowest root of such equation or equations. Fourthly and lastly, theproblem of assigning limits to the roots offx=0 reducesitselftothatoffinding limitsto j;x=O, f.jJJ=O...(j)x=O; forthegreatest andleastofthesecollectively will evidently, dfortiori, byvirtueofthepreceding observation, be limitstotheroots offx=O.Of any such of these asare linear, theroot or roots themselves may betreated asknown; leaving these out of consideration, thefunctional partofany otherof them, such asj;x,isthedenominator ofacontinued fraction of the form 1 1 1 1 (~x+bl)+(~+b2)+(aax+bl)+...(Clix+b.)' in which aI,Ut,as...u,presentasingle sequence of variations of sign,and thelimitstotheroots ofj;x=0 maybefoundasfollows. 1 fI-t 1 P1j£s+-~I a,;r;+bl=a.;c+~=1-P1- - fI-t a.x+bl=1j£s+- P-JFormthetwo systems of equations (in which fI-t,P1...1Joi-1arenumerical quantities having all thesamealgebraicsl sign,butareotherwise arbitrary andindependent), ~x+bl= (_)H a.-Ia:+bi-I=(_)HIJoi-I+-­ P-H (-Y-I then(supposing fI-tto have the same sign as~)thehighestofthevalues ofa:obtained fromthefirstsystem,and the lowest of thevalues of a:found from the second system of these equations, willbeasuperior andinferior limitrespectively to theroots ofj;x=0;and so for all the rest of the equations f2(e)=0,.fa(x)=0...(f)x=0, excluding those ofthefirst degree. Itwill be seen thatthetheorems contained intheobservations (3)and (4) combined (which presuppose thestatements made in observations (1) 64J Note on the NewRuleojLimits. 633 and(2»,containbetween themthetheorem given in thelastNumber of the Magazine [po627 above], butrendered in one or two particulars more simple and precise, and,asit were, reduced to its lowest terms. In thewhole course of my experience I neverremember atheorywhichhasundergone so many successive transformations in mymindasthisvery simple one, since theday when I first unexpectedly discovered the germ of it in results obtained forquiteadifferent purpose. Infact, itneverenteredinto my thoughts thatin sobeatenatrack,andin80hackneyed asubjectasthat of finding numerical limitsto the roots of an equation, therewasleft any­ thingto be discovered; and my sole merit, if any, in bringing thenew role to light, consists in havingbeen able to detectthepresence and appreciate the value of atrothwhich fortune or providence hadputintomy hands. [~+a, Xl=~+a, X2= b»+{3,65. THEALGEBRAICAL THEORY OFTHESECULAR-INEQUALITY DETERMINANTIVE EQUATION GE~ERALIZED. [Philosophical Magazine, VI.(1853), pp. 214-216.] ART.1.Let [~+a.b»+{3.d:c+S] ~:J' X.=bx+{3,c.x+ry,ex+E&c., d:c+S,ex+E,jx+q, and letthefirst coefficients of Xl'X2,XI'&c.have all the samesign;then Isaythattheroots of any such function asXiwill be all real, and will lie respectively intheintervals comprised between+00 ,the successive descending roots ofXi-Iand-00.Whena=l,c=l,j=l, &c.,andb=O,d=O, e=0,&c.,X,=°becomes the well-known secular-inequality equation. Demonstration. Forgreatersimplicity, let allthefirst coefficients be takenpositive, and suppose thetheorem proved up to i, it will be true fori+1.Forby a well-known property ofsymmetrical determinants, when Xi=0,Xi-IandXH Iwill have contrary signs.Lettheroots ofXi-Ibe andtheroots ofX"i;k2...~l'ki• Whenx=klJwhich is greaterthan ~,thegreatest root ofX'-lwill be positive; whenx=~,which lies between thefirstandsecond roots of Xi-lJXi-Iwill benegative jand so on, X'-lalternately becoming positive andnegative aswepassfrom root to root of Xi. HenceXi+l,which is positive when X=00,becomes negative whenx=klJ positive again when x=~,and80alternately; being finally, when x=k" positive or negative, and when x= -00 ,negative or positive, according as iis even or odd. Hence XH I•which changes sign i+1timesbetween+00 and - 00,musthave all its roots real, and lying severally intheintervals included between+00,thesuccessive roots of X,and - 00.Henceif the theorem betruefori-Iandi,it istruefor allnumbers abovei;butif wetake d[ax+a,b»+{3Jax+aan b»+{3,ce+ry 65JSecular-inequality Determinantive Equation. 635 thelatteris(ax+a)(ex+"I)-(bx+{:J)2,whichispositive for x=00,negative forQ.11;+a=0, and positive for x= -00.Hence the theorem istrueforXl andX2,andtherefore universally. Inthe above demonstration itwassupposed thattheleading coefficients are allpositive; butthedemonstration will be precisely thesame,mutatis mutandis, iftheyare all negative. Art. 2. And muchmoregenerally it may be shown, in like manner,that ifthesuccessions of signs, in theseriesconsisting of the sign +followed by the signsof the principal coefficientsin Xl'Xi...X...-tn,consist of m variations andncontinuations, thenumberof real roots of theequation Xm+n=0 will beatleastasgreatas the positive value of thedifference between m andn. Thistheorem, moreover, remains trueifXl'Xi'X..&c. be formed from asymmetrical matrix,in which theterms,insteadof being linearfunctions ofx,are any odd-degreed rational integral functions of e,or fractional functions of which thenumerators (whenrendered primetotheirdenomi­ nators)areodd-degreed functions of x.My friend M. Borchardt, whohasso beautifully effected thedecomposition of my formulee for the Sturmian criteriaofrealityintothesums of squares for the secular-inequality form of theequation, may now, if he pleases, tax his ingenuity to effect a similar decomposition for thegeneralcasesupposed in Art. 1-. Art. 3. Itisobvious that,inapplying thetheorem contained in Arts. 1 and 2, it isindifferent whetherwe look to thesigns of the successive determinants a;I~:~I;&c.,or to those of ajI~,~I;&c.;or,more ~enerally, to those of a+aejI:~;~,::~:Ij&c.,ebeing any arbitrary butreal quantity. Conversely we obtaintheremarkable theorem, thatwhen any homogeneous quadratic function, whose coefficients are linear functions ofe, • So,too, my own more simple methodforproving theomni-reality of the roots of the secular-inequality equation, August 1852, [po364 above], oughttobe capable of being extended to the general form in Art.1,thatis weought to be able toprovethattheequation whose roots are thesquaresof therootsofX,=0 will have all its coefficients alternately negative and positive. Ifwe take for examplei=2.theequation tothesquaresof the roots becomes (tU:_1Jll)2%2-{(aoy+CIZ-2bfJjJ+2(1Jll-tU:){aoy-(P)}%+(aoy-{p)2=Oi andwe have to prove thatthe ooefficient of - %inthisequation isessentially positive when tU:-b'ispositive: thismay be shown by various modes of decomposition; amongst others, bywritingthe coefficient in question nnderthe form 1c:{(cia+oyb'-2bcfJ)'+r(tU:-bl)'+ 2(boy-cfJ)2(tU:-b2)}. In general, ifLisessentially positive when LlJL2...L,are positive, then,discarding all artifices of calculation, thismustbe capable of being proved by virtueof&IIidentity of the form 636 Secular-inequality Determinantive Equation. [65 islinearly concerted by real substitu.tions into a sum ofpositiveand negative squares, thegreatest difference forany value of8betweenthenumberof positive and thenumberofnegative squares has foritslimitthe number of real roots of8in theDiscriminant (otherwise called the Determinant) ofthe givenfunction. Thetheorem actuallydemonstrated aboveteachesonlythis much, namely thatthemaximum difference in number between thetwo species of squares(whichdepends only on thevalue given to 8) cannot exceedthenumber of real roots in thediscriminant; itadmits,however, of aneasyproofthatthismaximum difference is equal to thenumberofreal roots, so thatthe onenumber is•.inthestrictsense of theword, an e.xact limittotheother. Art. 4. I wasled tothetheorem, asgiven in Art. I, by havingto consider thefollowing curious andimportant question. "Givenilinearfunctions ofx,sayXI'X,...X"tofindthei-Ipositive quantities, sayJl.J.,111•••~l>whichshall give theleastvaluetothegreatestroot, orthegreatestvaluetotheleastroot,oftheequation Thetheorem in Art. 1 enables me easilytodemonstrate, thatif wetake X/,X,',XI'...X/identical with ";1.s;";1.X,...";1.X" thesign ofthesquarerootbeingselectedin eachcasesothatthecoefficients ofxinXI"X,/...X/shall have all thesamesign,thentheleastvalue of thegreatest root, and thegreatest value of theleastroot, of thegiven equation will be respectively thegreatest andleastfinite roots of the equation thetwosystemsof values of JJrJ,111•••p.o_1required beingthetwo systems ofvaluesof X'X'1X'1 1 X' 11 1I' I-XI" 1-X,'-X/'" i-I-X"-S-X/i-I'''X/' corresponding respectively tothesetwo values of e. And it is by means of thissolution thatthestatement oftherule for findingthesuperior andinferiorlimitstothereal roots of an algebraical equation made in thelastAugustNumber of theMagazine, iscapableof beingconverted intothestatement contained inthethirdobservation on thesame rule in thepresentNumber [po631 above]. • Thefiniteroots ofthisare the same as those of Xi'-x,!-x~...xl,=o.i-I-i-I- I 66. ON THE EXPLICIT VALUES OF STURM'S QUOTIENTS. [Philosophical Magazine, VI.(1853), pp. 293-296.] BySturm'squotients is of course meantto beunderstood thequotients whichresultfromapplying theprocess for thediscovery of thegreatest common measure between j:r;(analgebraical function ofthenthdegree inx;and whose first coefficient is unity)andj':r;its firstderivative, as in Sturm's theorem jor which is thesamethingin effect, supposingj: to berepresented by 1 1 1 1 Ql=.Qs- Q.-...Q"' (whereQ1JQs... Q"are alllinearfunctions of :r;),thequotients inquestion areQl'Q2'"Q".Beforeproceeding to discuss these quotients, it will be well tostatetheformunderwhichtheotherquantities whichappearinthe course of theapplication of theSturmian processadmitof beingrepresented. First,then,it will be remembered thattheresidues withthesignschanged areall oftheform R.=M.'i.{~(~,hs...hi)(:r;-hi-u)(:r;-hi+a)...(:r;-h,,)}, where ~(~,hs...h.)indicates thesquared differences between every two ofthequantities ~,hi'"hi,and~,hi'"11."are supposed to be thenroots ofj:r;;andwhere, using ~.todenote !~(~, h~...h.),withtheconvention that ~o=1,~l=n,andunderstanding by (i),t{I+(-)i), M,.-rHri-4 rlil+l 0-r;-lri-ario). Hereitwill be observed thattheonlyquantities appearing arethefactors andthedifferences of theroots offe;and since theselatterarethesame 88thedifferences between thecorresponding factors, for (:r;-11.)-(:r;-h')=11.'-11., 638OntheexplicitValuesofSturm'sQuotients. [66 theentirequantity whichexpresses anyresidueR.maJ'beconsidered as afunctionofthefactorsoffxexclusively. Again, if we solve thesyzygetic equation N;jx+D;j'x=R" I havepublished manyyearsago inthisMagazine thevalue ofD.,andsubse­ quentlyinapaperreadbeforetheRoyal Society on the16thofJunelast[po429 above]thevalue ofN.,bothwhich values arealso functions of thefactors of f:r; exclusively. ~:'itis easily seen, represents thesuccessive convergents to thecontinued fraction by which .r;:is supposed to be expressed, andB.(to aconstant factorpres)isthedenominator ofthereverseconvergents ofthe samecontinued fraction. To thecompletion ofthispartofthetheoryit evidently therefore becomes necessary toexpresstheqnotients QI'QIlQs... Qn-l'Qn(of which thefirst(n-1) are those which appearinSturm'sprocess, andthelastissimplythepenultimate Sturmian residuedividedby the ultimate residue) underasimilarform,thatisasfunctions exclusively ofthefactors of f»,or, which comes to thesamething,ofthefactors and the differences of theroots.Guidedbyaninstinctive sense of thebeautiful and fitting,inahappymoment I have succeeded in grasping thismuch wished forrepresentation, withwhich I propose now andfor ever to takemy farewell ofthislonganddeeplyexcogitated theorem. Ifwewrite[cf.p.499above, and theAuthor's footnote, p. 495] B.-I=Mi-l{A._l,xfl-'+l- Bi-lxn-·+&c.}, and we have T.=Ai-lA.x+(A._IBi-A.Bi_l), T.may berepresented bythedouble sumAi-I=I~(hl'~14-1),Bi-l=I.(14+hot-l+ +hn)nh,.,~h.-I), Ai=I.nhl,h2h.), B.=I(lli+l+hiH++hn)nhl,~14), andtheithquotient isevidently M'-lA._IA,x+(Ai-lB.-A.Bi_I).M. Al- , andthisisthequantity (unpromising enoughinaspect)to betransformed inthemannerprescribed. Mi-llM.,andAiarealreadygivenunderthatform, and I find that, putting 66JOntheexplicitValuesofSturm' 8Quotients. 639 This ofcourseimpliesthetruthoftheidentity ~{Ink'l'he,...h'I-l)(~-h'l)(hl -he,)...(~-k'HW=Ai-lA,=~i-I~" initselfatrulyremarkable equation, which it will be seen isof 2(i-1)1 dimensions in respect of theroots", Wheni=1, and when i =2, thatisT2=~{[~(~-h,)]11(a:-~)}, =~{[(n-1)~-(~+h,+ ... +hn)]1l(a:-~)}. Wheni=n,Tnbecomes asitevidently oughtto do.Substituting forTi-l>T,and.A"theirvalues, we have asthe complete general expression of theithSturmian quotient thefollowing expression, in which, agreeable toanotation which I have previously used and explained, [hhlh h ]means(~-h,.)(hi -h,,)...(h~-h"_l)''1'"...'I-. namely Itoughtnot to be passed over in silence, thatif we write 1 1 1 1 N.(a:) QI-<J2-Q.- ...Q.=D.(a:); and if we suppose N,(a:)andD.(a:)to be expressed integrally, and to be algebraically prime to one another,then Di-I(a:)=s{nh'l'h'Il'"h'l-l)[hiT.hh]}''I'"...'t-l •Thusifn=4andi=2 and wehave 4{(hI -h,jll+ (~_1~)11+ (hi-h4)11+ (hll-ha)1l+ (hll-hJIl+(ha-h4)2} =(3~-1~-h.-h4)2+(3hll-hi-h:J-h4)2+(3ha-~-h2-h,Jll+(3h4-"1-I~-h,jll, andso in general t'-Ito.which is the product of two sums of variable numbers ofsquares. is expresaible rationally asthesum ofaconstant number(n)ofsquaresforallvalues of i, t(i)denotesit(_1)'+1}. 640Onthe explicit Values ofSturm'sQuotients. [66 HenceQiiscontained asa factor in (Di-Ih,.'f(x-'h.,)+(Di-I~'f(x-~)...+(Di-1hn'f (x-hn). Itmay be observed also, thatfor all values of ibetween 1 and n inclusively, Dih,+Di~+Dih,+...+Dihn=0, and alsothatthedeterminant 1, 1, 1, 1 (DI~)2 (D,hn)' (D2~)'J (D,hn)' (Dfl-Ih,.)', (Dfl-I~'f, (Dfl-lh,)2...(Dn-1hn)' is always zero [cf.p. 502 above]. To complete the theory, I subjoin thevalue ofNi•the simplified numerator oftheithconvergent toj:,expressed asanimproper continued fraction. Letthesum oftheproducts ofx-h.x-k...x-lcombined iandi together be denoted by 8i(h, k ... l), andthesum oftheithpowers of the samebyu.(h,k...l),thenNiis equal to ~nhephis'"he,)x{ui-l(hel•h"...h,;)-ui...Jl(hel,hes•..he.)81(h'H-I'"he.) +Ui-3(hephe,...he,)82(heH-1...he.)+&c. ...±(i+1)8i-l(hew-I'"h,.)}. The anomaly of the lastterm being of theform (1+uo)8i-1(for of course Uo=i),insteadof being u o8i-l,is not alittleremarkable. Of the four sets of Sturmian quantities, namely the residues, thequotients, andthedenominators andnumerators of theconvergents toj:'itwill have been seen thatthefirst and thirdare expressible in termsof the roots and factors by single summations of equal simplicity, thesecond and fourth by double summations, whereof thatwhich corresponds to the numerators is much the more complicated of thetwo. 67. ON AFUNDAMENTAL RULEINTHE ALGORITHM OF CONTINUED FRACTIONS. [Philosophical Magazine, VI.(1853), pp. 297-299.] LET~_1_~&c. be any continued fraction, andletthesuccessive ~+~+a.+ convergenta .!.-,_1_!,&c. be called ND\lftDT lI,&c.,andletD.bedenoted ~~+~ I2 by(~,~...a.)-,thenthefollowing identity obtainswhich I regard 8Bthe fundamental theorem in thetheoryofcontinued fractions, butwhich I have neverseenstatedin any work where this subjectistreated[cf.pp. 530, 618 above]. Theorem. (al...am)X(a...-rl...Um+n)+(~...Um-l)X(amH'" C1.om+n) =(~...am,am+l'"am+n)' Corollary 1. (~,~...am)X(a.,a....am+l) -(~,a,...am)X(~,~...Um+l)=(_)m1. Thisisthewell-known theorem D,Ni+1-Di+IN.=±1, which, however, is only acaseofamuch more generaltheoremeasilydeduced from the fundamental theorem givenabove.Infact,wemay derive im­ mediately fromthelatter,theequation (~,~••.am).X(~,a....Um+.)-(a.,a,...am)X(~,a....a...-ro) =(_)m(am+., Um-ti-l...toi-Iterms). 8. 41 642 HenceOnaFundamental Rulein Dm-1N.,.-DmNm-l =(-)",1, Dm-'JNm-D.,.NfIfr--'J=(-)'"a"" Dm_aN m-DmNm-a=(-)'"(amllm-l+1), Dm-4Nm-DmNm-4=(-)"'(ama".-1~+ am+am-l), &c. &c.[67 Corollary 2. (~...ap,ap+l...ap+f)(~...ap,up+!...up+!:) -(~...up,Up+l'" up+g)(~...up,Up+l•••up+,,) =(-)p{(up+!...uP+f)(Up+l•• ,Up+I;)-(Up+l'"UP+g)(up+!...UP+A)}' Sub-corollary. Ifall the several quantities ~,~,Us...are equal toone another,thequantity DfDI;-DgD"isconstantinmagnitude, butalternating in sign, so long as the differences of the indices f,g,h, kareconstant; and asaneasydeduction fromthissub-corollary, if T'HI=uT"-bTra-l bethecharacteristic equation of arecurrent series, and if f+k=g+h, TfTI;~'[gT"will beconstant iandasaparticular case ofthisdeduction b2 from the sub-corollary to the second corollary of the fundamental theorem, we have thatis which is Euler'stheorem. See Terquem's Nouvelles Annales, Vol.x.p. 357, andNovember 1852. I was led up to a knowledge of thefundamental theorem (be it new or old) by some recentresearches connected with my newRule of Limits, considered with reference to theconditions which must be satisfied when one ofthelimitsfound by therule comes into actualcontactwith aroot; acontactwhich I can demonstrate isalwayspossible, aswell for the superior asfortheinferior limits, and with so much thefewerequations (asdis­ tinguished frominequations) ofcondition between thecoefficients of the assumed auxiliary function which the application oftherule oflimits requires, asthereare fewer pairs of imaginary rootsin the function whose roots are to be limited, 67] theAlgorithm ofOontinued Fractions. 643 I may add thatthefundamental theorem is animmediate resultofthe representation ofthetermsoftheconvergents toacontinued fractionunder theform ofdeterminants. Thus.forexample, thedeterminant a,1 - 1,b,1 - 1,c.1 - 1.d,1 - I.e,1 -1,1 is obviously decomposable into a,1 xd,1 - 1,b,1 -1,c-1.e,l -1,f+Ia,1 ,xIe.1I -1,bI-1.1 orinto orintoIa.1Ixc.1 I-1,b-1.d,1 - 1,e,1 -I,}+ax d,1 I - 1,e.1i -1,f I thatisaxb,1+c,1 -1,c,1 -I,d,1 -1,d,1 - 1, e,1 -1,e,1-l,f -1,f (abodef) =(abc)(def)+(ab)(ef) =(ab)(cdef)+a(def) =a(bcdef)+(cdef). Thusthewhole of theproperties ofcontinued fractions arededuced without algebraical calculation from atheorem whichitselfspringsim­ mediately byinspection fromthewell-known simple rule for the decom­ position of determinants. Ifinsteadof asimplesetatriplesetofquantities betaken.as {llJIII...li-I} ~,mll···mi ' nl,nll•••fii_1 644OnaFundamental RuleinOontinued Fractions. [67 which, when i=1,i=2,i=3,i=4, &c.isto beinterpreted to mean ?nt;I?nt,~I;?nt,i; ?nt, ~ -'It, -nl> ~,t;-nl,'mj,It -n., 'mt -~,m"l, '-n.,m4, &c.respectively, thevalue of thedeterminant represented by any such set being called T"we have in general T,='I'niTi-l+4."",T'-I' which, when 'I'niand4."",areconstant, becomes the characteristic equation toanordinary recurring aeries, The theorem corresponding to the funda­ mentaltheorem for suchtriplesetswillbe 1~'~···4.H1fllt~li-I1flHI>lHt...l,+>,1 ?nt,m,'l'niH'+1=l?ntJ~m,xl'11li+ltffl&+t·..'I'ni+,'+! ~.~ ~+" nl,'11t ""'-I ~+I' ~ll...~+" {llt~li-t}flH'l,+"1 +lin,?nt,~ffl&-IX1'11li+ll 'I'ni+i'+1 • ~,nt""'-I""'H ~+i' 68. ON AGENERALIZATION OFTHELAGRANGIAN THEOREM OFINTERPOLATION. [Philosophical Magazine, VI.(1853),pp.374--376.] THEREis a well-known theorem ofLagrange fordetermining theform of a rational integral function of one variable of thedegreem,when its values corresponding to m+1values of the variable areassigned. M.Cauchy, in hisOoursd'Analyse del'EcolePolytechniq:ue, hasextended thistheorem to thecaseof arationalfraction, of which values corresponding to asufficient numberof values of the variable are given;butthesolution of thequestion theregiven,although of course correct, 'isunsatisfactory, as it presents the numerator anddenominator underforms not strictlyanalogous. The theorem of Lagrange, in respect of its subjectmatter,may be best generalized as follows. . Suppose any numberofrationalintegral functions of xoftheseveral degrees m l-1,1ns-1...'"'"- 1,sayUllU8...Ui,andthattheequation IIUI+t«U,+ +liU,=0 is known to be satisfied for m l+~+ +'"'"-1 (say)p.-1 assigned values ofthesystem of quantities ~,~...li,x;therewillthenbep.-1linear equations connecting thep.coefficients comprised in UllU8...Ui,and therefore theratios of these coefficients, and consequently of the functions to oneanother, may bedetermined. Thereis no difficulty in representing, by aid of themethod of determinants, theresultof solving these equations whatever be thenumberoffunctions; butforthesake ofgreatersimplicity, I shall suppose threeonly of the several degrees, e-1,i-I, CI)-1ine, which I shall call U, V,W.Nowsuppose thatlU+mV+nW=0 is known to be satisfied for l=["m=ffl.t,n=1lf,x=XI,ttakingall possible values from1toe+i+CI)-1,sayT-1;lettheindices1,2,3...T-1bepartitioned in every possible wayinto threegroups,containing [when it is thefunctionU which is tobedetermined] respectively e-1,iand CI)indices, as BI,B~...B.-I;B.,BHI•••BO+>-IjBo+i...B"-I 41-3 646 Generalization ofaLagrangian Theorem. [68 (thetermsinanygroupmay bearranged indifferently inanyorder,butare not to be permuted). Let~i(p, q,r...s)denoteingeneral (p-q)x(p-r) x(p-s) x(q-r)x(q-s) x(r-s), andwrite s,=I(1){l'l...l'.-l;m••'"mh+i-l; nle+i•••n.....1 }. ~i(e,x'l...x'..-1) ~i(x••,X'~l...x'''tH)~i(x'l+4...X"'_I) Themark(1) is used to denote(-)raisedtoapowerwhoseindexisthe numberofexchanges of placewhereby thearrangement I, 2...(T-1)canbe shiftedintothearrangement (JIJ(J,...(JF-l' Inlikemanner,let K2-=I(1){l'l...l•.;m'~l...m'I+4_I;n'I+4'"n"'_1 }, 'i(x61'x8t...x••)~i(x,X9,+l...X'I+4-1) ~i(x'l+4...X"_I) andKI=I(1){lStl,,;m'~l•..m'l+4;nle+i+1'"n.....1 } ~i(xx.....xsJ~i(X9,+l' x,~...x.~ ~i(x,X'l'H+l...X"_I). Then,usinge todenoteanyarbitrary constant, weshallhave U=cK I,V""'(-)'cK" W=(_)HiCKI; andso, ingeneral,theratiosto oneanother ofanynumber offunctions of onevariable, of which thelinearconjunctives forasufficient numberofgiven valuesofthevariable andofthecoefficients of conjunction are known to vanish, may be expressed intermsofthosevalues. EDITOR'S NOTE ON SYLVESTER'S THEOREMS FOR DETERMINANTS INTHISVOLUME. INSylvester's paperNo. 87, p.241above,besidetheerrorsnoticedbySylvester himself, pp.261and401 ofthisvolume, thesubstitution ofb.I... b...fora.I...a.,inline22of p.2« andthesubstitution ofa8)...ae,.forb91...be,.in line8of p. 245, thereisthemorefundamental errorthatinformula (2). p.244,andtheformulaatthe foot of p. 247 the suffixesof the b's shouldbelei'''!:'and11...1"andthesuffixe8of thea'ashouldbe(JI...(J,andtf>:J...t/>,.Itmay be aconvenience tothereaderto haveathandanotherview ofSylvester's threemaintheorema on determinants inthisvolume (pp. 247,258,249). 1.Amatrixof type(Ill.n)isanobjectofcalculation depending onmnnumbers which we supposearranged asarectangle ofIIIrowsandncolumns. By theproduct(a)(b)oftwomatricea (a).(b)ofrespective types(n!,m), (m, ~)ismeantthematrixof type (~.fit)whiohhsefor its (p,q)thelement, thatis theq.thelement ofitsp-throw, the number ap1b1q+...+ap".b".q, whereap.,brqarerespectively the(p.r)thand(r,q)thelements of(a)and(b). Ifidenoteaparticular one of the(:1)possibleaelections ofrnumbers fromI,2, ...,~,aay il...i"andjdenoteaparticular one of the(;)possible selections of ,numbers from1,2,....fIt. aayjl...j"we may piok outfrom the product matrix(a) (b)aminormatrixofrrowsand, columns consisting oftheelements ofthiscommon to the rows i1...i,andtheoolumnsjl'..j.; thisisolearlygiven by ((a)(b»u=(a~~:.:::.~.,~~) (~:~~ ~:~.)=(a), (b)J, alrla""" .. b",j,b",J. where(a),is thematrixof type(r,III)constituted bythei-rowsof(a),and(b)Jthematrixof type(m,,)constituted bythej-columns of(b).When.=rthismatrixissquareand,ifIedenotes aselection ofrnumbers fromI,2,...,III,itadeterminant is given by Ie(a)(b»U1=2:I(a)tII(b)~I, ~ whereI(a)tI,which we maydenotebyI(a)~I,denotesthedeterminant of theminorof(a) 648 Note. formed with igi-rowsanditsk-oolumllll, andI(b)kiIorI(b)kildenoies ~hedeterminan~ of the minor formed with ~hek·rows and ~hej-columns of(b),and~hesummauon eXWndlltothe pouible (~)significations of k.Similarlyif(a),(b).(e)bemamOllll ofreapecUve types(ftl,fl), ('I,m), (m,/Is).theminorwith the rows iand~hecolumns jof theproductmatrix(a) (b)(e)of type("I,fit)is givenby «a)(b)(e))'i=«a) (b»cle)i=(a), (b)(e)i, and when,=r,itlldeterminant is I«a)(b)(e»jJ I=2:I«a)(b))lkll(e)ki1 k =2:2:I(a)llil(b)MII(e)kil,k" wherekisas before and hdenotesaselection ofrnumbers from 1,2,...,ft,~summation extending tothe(~)(;)possibleBignitlcatiOIlll ofk,h. 2.Renoe the theorem of Sylvellter on theminordeterminants oflinearlyequivalent quadratic functionll, pp.244,247above.Forifbythe8ubBUtntion ZI=I'u!l1 + ...+1'o1.Y,.....,Z.=~l!11 ++~!I., thequadratic form0.11% 1' +... +2a,,'%IZ, + ...becomebIlYI'+ +2bltYIY,+....weat onoe find •b"v=2:P-oP(al1l'o1q+... +alll'1q+ ...+tIa/ol"q),.-1 sotha~themamxof~henew form isgivenby (b)=(il)(a)(,u), wherea,.,,lolpq,IA-lJ"bJlllare the (P.q)thelements respectively of the matrioea (a).(,u),(,ii),(b), whichareallof type ('I,fl).Supposing the numbers "I,11,m,n,of§1allequaltofl,the determinant ofthe(i,j)thminorof orderrin(b)is 2:2:I(,ii)llil(a)MII(,u)kiI, k" or 2:2:I(a)MII(,uJ.,.,II(,u)kiI, k" thisbeing the reaultwhich in the notation of Sylvester would be written :::2:(a",a",...a",,)(,u",""'"JJ."")(,ukl,uk•....ut,.,, "kak,O.k•...altr!J.l.!J.l,IJ.'r\.ui.JA;•...JJ.ij thefirstrow giving the rows used to form any minor determinant. Itwillbe noticed thatthe columns of the matrix(,u)which come intoconsideration are those of the sameenumeration as the rows and columns of the minor of the matrix(b)which is tobeexpressed; thisiscontrary to Sylvester's formula of p. 247above. 8.When the product of two, squarematrices (a).(b).eachof type(II,fl),is~heso-called unitmatrix,in which every element is zero savethosein thediagonal whichareeachunity.the matrices are called invene; andwehave(a)(b)=I=(b)(a). Denoting by tJ.;.ithedeterminant of a minormatrixof type(r, r)formed with rows il...irandcolumnsil...jrfrom(a),andbya."i thedeterminant of thecomplementary matrixof type(fl-r,fl-r),wehave,if.d=I(a)I,,u=(;), byLaplace's rule for the expansion of adeterminant a.,)a.'Jl+...+a;,.a.'il'=.d,or, 0, according asi=jori*j.Thusthetwomatricea of type (,u,JJ.),in which the (i,j)thelement of thefirstistJ.;.jIand the(i,j)thelement of the secondisj',are inverse tooneanother,sothat we have (a.)(~)=(~)(a.)=(~)(ii)=(ii)(~)=1, Note. 649 where the barabove the symbol for amatrixindioates thetransposed matrixdiffering from the originalinhavingits first, second, ... rows those respectively which were the first,second,... columns of theoriginal. Fromthisequation it iseasyto prove thatthedeterminant of the matrix(a)isthe'!:!.th power of A..If(b)beinverseto(a),andfl;jbe thedeterminant of typen (r, r)formed from (b)aswasalifrom(a),it follows by considering (eee§1)thedeterminant of theproductofthematrix(a)1of type(r,n)formed by the i-rows of (a)andthematrix(b)iof type (n, r)formed by thej-columns of(b),that a,1flli+a12{J,.i+...+aljJofljJoi=I,or, 0, according asi=jori*j;hencethematrices (a)and(fl)areinverse; andthus,by the above a'Jj flli=""""I ' or in words, anyminordeterminant of the inverse of 110givenmatrixisequaltothe comple­ mentary determinant formed from the transposed of theoriginal matrixdivided by the deter­ mine.ntof theoriginal matrix.Inparticular thisgives the elements of theinversematrix expressed by minorsof theoriginal. If(a),(b)beany twomatrices of type(n,n) wecanform 110matrixof type(n,n) byreplacing thei-thselection ofrrows in(a),bythej-thselection of rrows of(b);thismatrixbeingcalled (a,b)jJand itsdeterminant la, bi;j,wehave,byLaplace's rule Ia,bI;j=a/(Jflil+a/f.lfli2+...+a/ljJoflijJo; hencethematrix,of type (1-',1-'),of whieb the(i,j)thelementis'a,~lo:,is given by thus,bymeansof(~)(~)=I,(~)(ei)=1, wehave andthematricea(Ia~b')(I\a')=(~)(p)(~)(ei)=I, ea~bI),(Ib~~I), areinverse; thisisSylvester's theorem p. 253above. Weremark,using 1 for the unitmatrix,therelations, where(e)is oftype(n,n), Ia,llli=a';j,II,al;j=«;,,(a,b);j(e)=(ae, be)lit of which the laet gives, if (b),=(a)-I,beinverseto(a), (a-I,1);j(a)=(1,a)li,andhencefJ';j=~, asproved above. 4. Letn>r>m, and(a)beoftype(n, n).A fixed minorMmof type (m,m) from (a) determines acomplementary minorof type (n- m, n - m),sayM,,_.. Fromthen-mnumbers, sayp!...p,,_,enumerating the rows of M,,_,makeaselection 81...8.-.,andfrom the numbers, sayql'"q,,_,enumerating ihecolumns ofM,,-mmakeaselection"'1'""'r-..;thenforma minorMrof(a),of type(r,r),whose rows areenumerated bythose of M..togetherwith 81..•8.-., andcolumns by those of M.together with"'1'""'r-..;let the rows andcolumns of(a)not now enumerated begivenrespectively by8{...8',,_rand"'1/•.•",/,,-.-;let(b)beinverseto(a).Then thedeterminant ofMris equal to the determinant formed from (b)withthe rows 81'...8',,-.-and thecolumns 1f>I'•.•",/..-.-,multiplied by.d.Now suppose 81.•.8.-.tobecome in turnallthe 650 Note. ("-m)possible selections from PI...p,,_,andsimilarly tfJ1...tfJr-mallfromql...q,,-m;the r-m ( )determinants Mr80obts.ined form 110matrixHofn-mrowsandcolumns, which is in flloCt 110r-m minorof thepreviously oonsidered lll&trix(II).Wewishtodetermine thedeterminant ofthis matrixH.Now the determinants (II~''"II',,-r'tfJr'...tfJ',,-1')of(b),complementary in indices to thematrices Mr,areminorsof thematrix(PI...P,,-m'ql...q,,-m)of(b),andthe ma.trix of order ("-m)formed from themhastherefore foritsdetermina.nt A/"whereAlis thedetermina.nt of ~- m . _ ("_m_I)A.thlsmatnx(PI...P,,-m'ql'"q..-)of(b),and).= r-tr1;hence, a.s A1= l 'whereAISthe determinant of the fixed ma.trix Mmof(a),thedeterminant H,of order(n-m),offueminorsr-m Mrof(a),is equa.lto h(n-m)(n-1II-l)were~=r-m,17=r-m-I .(~r.4"=A>-.4O", AndthisisSylvester's theorem, p.249above. CAMBRIDGE: PRINTED BYJ.AND C. F. CLAY, ATTHEUNIVlilRSITY PRESS. Siv.t 548s‘ | QA 3Sqg v.I Stanford University Librari Stanford, California _Q9zndb~Cooglt::...,..IWanathfabookonorbeforedatedue... J