Byerly Elem Trts Ellipsoidals

om.oeoe‘Ae .oelenee/oe 1koesee pee:8:a:eees:2 aa,a2 eaaaS : aeeaeee: -0.oS aaefsoS2es aokoea foe aeaao aSo=Re ay a.“we,Caeio*,—_Fa>\ ra%9 usr: : Jspat “x 0,Sg *re ee,va-- Biceey%, 4 : re*,. ‘ %.aof Berk©- veda »,SIYIFS <* on. 7; é om“ae; ;rE ~ §-. Berle7,)My z*sral S<aoest Foe icsrs - %5on 7—* s *—-- %, “ ; oe & *Ng s*: mg!Berkus. 4: |o'%gs % “e,“an - My & wyEEa < ‘”aa / : ¥Gs% Sf o4e onmy,F s *,‘ ol “4,- s7B. s omere - %..: 7 $e* -.%, -%, & »; x¢ Berk % 7i Emy “EESg"p34ae ™& S&S o™ é, Sur. . i- eyyp©i: . ayBerke My s @ ra /, Beckie, “,4 ¢S oe. ie<*"Sind **,*,iKa7Frag b>ca *iue Pecks ee 4nS S "ae4.%,Prya7G** . ,wDsas %- ¢Psckeg i #CS* - .2 .ey= K 7 A s% “aDsSS ha & ib? , Z Syie,% os eSeys picks i 'i »~ o%‘4 enta $i 5 “~ Pry 2va; “ey - 7 rw ¢ os p ANELEMENTARY TEEATISE FOURIER SSERIES SPHERICAL, CYLINDRICAL, ANDELLIPSOIDAL HARMONICS, WITH APPLICATIONS TOPROBLEMS INMATHEMATICAL PHYSICS. BY WILLIAM ELWOOD BYERLY, PH.D., DOVER PUBLICATIONS, INC. New York,NewYork Copyright, 1893, byWilliam Elwood Byerly. ThisnewDover edition firstpublished in1959 isanunabridged andunaltered republication ofthelast edition. Itisreproduced byper mission ofGinn andCompany, theoriginal publishers ofthis text. 61 .3I: ( MATH-STAT. Library ofCongress CatalogCardNumber: 59-3787 Manufactured intheUnited States ofAmerica Dover Publications, Inc. 180Varick Street NewYork 14,NewYork PREFACE. LIBRARY ABOUT tenyears agoIgaveacourse oflectures onTrigonometric Series, following closelythetreatment ofthatsubjectinRiemann s&quot;Partielle Differentialgleichungen,&quot;toaccompanyashort course onThe Potential Function, given byProfessor B.0.Peirce. Mycourse hasbeen graduallymodified andextended until ithasbecome an introduction toSpherical Harmonics andBessel sandLame sFunctions. Two years agomylecture notes werelithographed bymyclass fortheir own- useandwere found soconvenient that Ihave prepared them for publication, hopingthattheymayproveuseful toothers aswell astomy own students. Meanwhile, Professor Peirce haspublished hislectures on &quot;The Newtonian Potential Function&quot;(Boston, Ginn&Co.), andthetwo sets oflectures form acourse(Math. 10)given regularlyatHarvard, and intended asapartialintroduction tomodern MathematicalPhysics. Students takingthiscourse aresupposedtobefamiliar with somuch ofthe infinitesimal calculus asiscontained inmy&quot;Differential Calculus&quot; (Boston, Ginn&Co.)andmy&quot;Integral Calculus&quot; (second edition, samepublishers), towhich Irefer inthepresent book as&quot;Dif. Cal.&quot; and&quot;Int. Gal.&quot; Here, asinthe&quot; Calculus,&quot; Ispeakofa&quot;derivative&quot;rather than a&quot;differential ^ coefficient,&quot; andusethenotation Dxinstead ofr-for&quot;partial derivative withox respectto x.&quot; Thecourse wasatfirst, asIhave said,anexpositionofRiemann s&quot;Partielle Differentialgleichungen.&quot; Inextending it,Idrewlargely from Ferrer s &quot;SphericalHarmonics&quot;andHeine s&quot;Kugelfunctionen,&quot; andwassomewhat indebted toTodhunter(&quot;FunctionsofLaplace, Bessel, andLame-&quot;), Lord Rayleigh (&quot;TheoryofSound &quot;),andForsyth (&quot;Differential Equations &quot;). Inpreparingthenotes forpublication,Ihave beengreatly aided bythe criticisms andsuggestionsofmycolleagues, Professor B.0.Peirce andDr. Maxime Bocher, andthelatter haskindly contributed thebrief historical sketch contained inChapter IX. W.E.BYERLY. CAMBRIDGE, MASS., Sept. 1893. ANALYTICAL TABLE OFCONTENTS. CHAPTER I. PAGES INTRODUCTION 129 ART. 1.Listofsome important homogeneous linear partial differential equations ofPhysics. ARTS. 2-4. Distinction between thegeneral solution andaparticular solution ofadifferential equation. Need ofadditional data tomake thesolution ofadifferential equation determinate. Definition oflinear and oflinear and homogeneous. ARTS-. 5-6 .Particular solutions ofhomogeneous linear differential equations maybecombined intoamore general solution. Need of.development interms ofnormal forms. ART. 7.Problem: Permanent state oftemperatures jnathinrectangular plate. Need ofadevelopment insin& series. Example. ART. 8.Problem :Transverse vibrations ofastretched elasticstring. Adevelop ment insine series suggested. ART. 9.Problem: Potential function duetothe attraction ofacircular ring ofsmall cross-section. Surface Zonal Harmonics (LegendresCoefficients). Example. ART. 10.Problem: Permanent state of temperaturesinasolid sphere. Development interms ofSurface Zonal Har monics suggested. ARTS. 11-12. Problem: Vibrations ofacircular drumhead. Cylindrical Harmonics (BesselsFunctions). Recapitulation. ART. 13.Method ofmaking thesolution ofalinear partial differential equation depend upon solving asetofordinary differential equations byassuming thedependent variable equal toaproductoffactors each ofwhich involves butoneoftheindependent variables. ARTS. 14-15. Method ofsolving ordinary homogeneous linear differential equa tionsbydevelopment inpowerseries. Applications. ART. 16.Application to LegendresEquation. Several forms ofgeneral solution obtained. Zonal Harmonics ofthesecond kind. ART. 17.Application toBessel sEquation. General solution obtained forthecasewheremisnotaninteger, andforthecase wheremiszero. Bessel sFunction ofthesecond kindandzeroth order. ART. 18.Method ofobtaining thegeneralsolution ofanordinary linear differential equation ofthesecond order from agiven particular solution. Application to theequations considered inArts. 14-17. CHAPTER II. DEVELOPMENT INTRIGONOMETRIC SERIES 30-64 ARTS. 19-22. Determination ofthecoefficients ofnterms ofasine series sothat thesum oftheterms shall beequal toagiven function ofxforngiven values ofx.Numerical example. ART. 23.Problem ofdevelopment insine series treated asalimiting case oftheproblem just solved. ARTS. 24-25. Shorter method ofsolving theproblemofdevelopmentinseriesinvolving sines ofwhole multiples ofthevariable. Workingrulededuced. Recapitulation. ART. 26.A&amp;gt; VI TABLE OFCONTEKTS. PAGES fewimportant sinedevelopments obtained. Examples. ARTS. 27-28. Develop ment incosine series. Examples. ART. 29.Sine series anoddfunction ofthe variable, cosine series aneven function, andboth series periodic functions. ART. 30.Development inseriesinvolving both sines and cosines ofwhole multiples ofthevariable. Fourier sseries. Examples. ART. 31.Extension of therange within which thefunction andtheseries areequal. Examples. ART. 32.Fourier sIntegral obtained. CHAPTER III. CONVERGENCE OFFOURIER SSERIES 65-08 ARTS. 33-36. Thequestion oftheconvergence ofthesine series forunity con sidered atlength. ARTS. 37-38. Statement oftheconditions which aresufficient towarrant thedevelopment ofafunction intoaFourier sseries. Historical note. ART. 39.Graphical representation ofsuccessive approximations toasine series. Properties ofaFourier sseries inferred from theconstructions. ARTS. 40-42. Investigation oftheconditions under which aFourier sseries canbedifferentiated termbyterm. ART. 43.Conditions under which afunction canbeexpressed as aFourier sIntegral. CHAPTER IV. SOLUTION OPPROBLEMS INPHYSICS BYTHEAIDOFFOURIER SINTEGRALS AND FOURIER SSERIES 69-134 ARTS. 44-48. Logarithmic Potential. Flow ofelectricityinaninfinite plane, where thevalue ofthePotential Function isgiven along aninfinite straight line; along twomutually perpendicular straight lines; along twoparallel straightlines. Examples. Use ofConjugate Functions. Sources and Sinks. Equipotential linesandlines ofFlow. Examples. ARTS. 49-52. One-dimensional flowofheat. Flow ofheat inaninfinite solid; inasolid withoneplane faceatthetemperature zero; inasolid withoneplane facewhose temperatureisafunction ofthetime (Riemannssolution);inabarofsmall cross section from whose surface heat escapesinto airattemperaturezero. Limitingstate approached when thetem peratureoftheoriginisaperiodicfunction ofthetime. Examples. ARTS. 53- 54.Temperatures duetoinstantaneous andtopermanent heatsources andsinks, and toheat doublets. Examples. Application tothecasewhere there is leakage. ARTS. 55-56. Transmission ofadisturbance along aninfinite stretched elastic string. Examples. ARTS. 57-58.KStationary temperaturesinalong rectangular plate. Temperatureofthebase unity. Summation ofaTrigono metric series. Isothermal linesandlines offlow. Examples. ART. 59.Potential Function given alongtheperimeter ofarectangle. Examples. ARTS. 60-63. One-dimensional flow ofheat inaslabwithparallel planefaces. Both faces at temperaturezero. Both faces adiathermanous. Temperature ofone facea function ofthetime. Examples. ART. 64.Motion ofastretched elastic string fastened attheends. Steady vibration. Nodes. Examples. ART. 65.Motion ofastring inaresisting medium. ART. 66.Flow ofheat inasphere whose surface iskept ataconstant temperature. ARTS. 67-68. Cooling ofasphere in air. Surface condition given byadifferential equation. DevelopmentinaTrigo nometric series ofwhich Fourier sSine Series isaspecialcase. Examples. TABLE OFCONTENTS. Vll PAGES ARTS. 69-70/ Flow ofheat inaninfinite solid with oneplane facewhich is exposedtoairwhose temperatureisafunction ofthetime. Solution foran instantaneous heatsource when thetemperature oftheairiszero. Examples. ARTS. 71-73. Vibration ofarectangular drumhead. Development ofafunction oftwo variables inadouble Fourier sSeries. Examples. Nodal lines ina rectangular drumhead. Nodal lines inasquare drumhead. MISCELLANEOUS PROBLEMS 136-143 I.LogarithmicPotential. Polar Coordinates. II.Potential Function inSpace. III.Conduction ofheat inaplane. IV.Conduction ofheat inSpace. CHAPTER V. ZONAL HARMONICS 144-194 ART. 74.Recapitulation. Surface Zonal Harmonics(Legendrians). Zonal Har monics ofthesecond kind. ARTS. 75-76. Legendrians ascoefficients inaPower Series. Specialvalues. ART. 77.Summary oftheproperties ofaLegendrian. List ofthe first eight Legendrians. Relation connecting anythree successive Legendrians.ARTS. 78-81. Problems inPotential. Potential Function dueto theattraction ofamaterial circular ringofsmall cross section. Potential Function duetoacharge ofelectricity placed onathin circular disc. Examples: Spheroidal conductors. Potential Function duetotheattraction ofamaterial homogeneous circular disc. Examples:Homogeneous hemisphere ;Heterogeneous sphere ; Homogeneous spheroids. Generalisation. ART. 82.Legendrian asasum of cosines. ARTS. 83-84. Legendrian asthemth derivative oftherathpower of x21.ART. 85.Equations derivable from LegendresEquation. ART. 86. LegendrianasaPartial Derivative. ART. 87.Legendrian asaDefiniteIntegral. ARTS. 88-90. Development inZonal Harmonic Series.Integral oftheproduct of twoLegendriansofdifferent degrees. Integral ofthesquare ofaLegendrian. Formulas forthecoefficients oftheseries. ARTS. 91-92.Integral oftheproduct oftwoLegendrians obtained bytheaidofLegendresEquation; bytheaidof Green sTheorem. Additional formulas forintegration. Examples. ARTS. 93- 94.Problems inPotential where thevalue ofthePotential Function isgiven ona sphericalsurface andhascircular symmetry about adiameter. Examples. ART. 95.Development ofapower ofxinZonal Harmonic Series. ART. 96. Useful formulas. ART. 97.Development ofsin?i0 andcosn0 inZonal Harmonic Series. Examples. Graphical representation ofthe firstseven Surface Zonal Harmonics. Construction ofsuccessive approximations toZonal Harmonic Series. ARTS. 98-99. Method ofdealing withproblems inPotential when thedensityis given. Examples. ART. 100. Surface Zonal Harmonics ofthesecond kind. Examples: Conal Harmonics. CHAPTER VI. SPHERICAL HARMONICS 195-218 ARTS. 101-102. Particular Solutions ofLaplacesEquation obtained. Associated Functions. Tesseral Harmonics. Surface Spherical Harmonics. Solid Spherical Harmonics. Table ofAssociated Functions. Examples. ARTS. 103-108. De velopment inSpherical Harmonic Series. Theintegral oftheproduct oftwo Till TABLE OFCONTENTS. PAGES Surface Spherical Harmonics ofdifferent degrees taken over thesurface ofthe unitsphereiszero. Examples. The integral oftheproduct oftwoAssociated Functions ofthesame order. Formulas forthecoefficients oftheseries. Illustra tiveexample. Examples. ARTS. 109-110. Anyhomogeneous rationalintegral Algebraic function ofz,y,and 2which satisfies Laplace sEquationisaSolid Spherical Harmonic. Examples. ART. 111.Atransformation ofaxes toanew sethavingthesame originwillchange aSurface Spherical Harmonic intoanother ofthesame degree. ARTS. 112-114. Laplacians. Integral oftheproduct ofa Surface Spherical Harmonic byaLaplacian ofthesame degree. Development in Spherical Harmonic Series bytheaidofLaplacians. Table ofLaplacians. Ex ample. ART. 115. Solution ofproblemsinPotential bydirectintegration. Examples. ARTS. 116-118. Differentiation along anaxis. Axes ofaSpherical Harmonic. ART. 119.Roots ofaZonal Harmonic. Roots ofaTesseral Har monic. Nomenclature justified. CHAPTER VII. CYLINDRICAL HARMONICS (BESSELSFUNCTIONS) 219-237 ART. 120. Recapitulation. Cylindrical Harmonics (BesselsFunctions)ofthe zeroth order; ofthenthorder; ofthesecond kind. General solution ofBessel s Equation.ART. 121. Bessel sFunctions asdefiniteintegrals. Examples. ART. 122. PropertiesofBessel sFunctions. Semi-convergentseries foraBessel s Function. Examples. ART. 123. Problem: Stationary temperatures ina cylinder (a)when thetemperature oftheconvex surface iszero; (b)when the convex surface isadiathermanous; (c)when theconvex surface isexposed toair atthetemperaturezero. ART. 124. Roots ofBessel sfunctions. ART. 125. The integralofrtimes theproductoftwoCylindrical Harmonics ofthezeroth order. Example. ART. 126. DevelopmentinCylindrical Harmonic Series. Formulas forthe coefficients. Examples. ART. 127. Problem: Stationary temperaturesinacylindricalshell. Bessel sFunctions ofthesecond kind employed. Example: Vibration ofaringmembrane. ART. 128. Problem: Stationary temperaturesinacylinder when thetemperature oftheconvex surface varies with thedistance from thebase. Bessel sFunctions ofacomplex variable. Examples.ART. 129. Problem: Stationary temperaturesin,acylinder when thetemperaturesofthebase areunsymmetrical.Bessel sFunctions ofthenth order employed.Miscellaneous examples.Bessel sFunctions offractional order. CHAPTER VIII. LAPLACE SEQUATION INCURVILINEAR COORDINATES. ELLIPSOIDAL HARMONICS 238-266 ARTS. 130-131. Orthogonal Curvilinear Coordinates ingeneral. LaplacesEqua tionexpressedinterms oforthogonalcurvilinear coordinates bytheaidofGreen s theorem. ARTS, 132-135. Spheroidal Coordinates. Laplace sEquation in spheroidal coordinates, innormal spheroidalcoordinates. Examples. Condition thatasetofcurvilinear coordinates should benormal. Thermometric Parameters. Particular solutions ofLaplacesEquationinspheroidal coordinates. Spheroidal Harmonics. Examples. The Potential Function due totheattraction ofan oblate spheroid.Solution foranexternal point. Examples. ARTS. 136-141. TABLE OFCONTENTS. IX PAGES Ellipsoidal Coordinates. LaplacesEquationinellipsoidal coordinates. Normal ellipsoidal coordinates expressed asElliptic Integrals. Particular solutions of LaplacesEquation. Lamp sEquation. Ellipsoidal Harmonics (LampsFunc tions).Tables ofEllipsoidal Harmonics ofthedegrees 1,2,and 3.Lamp s Functions ofthesecond kind. Examples. Development inEllipsoidal Harmonic series. Value ofthePotential Function atanypoint inspacewhen itsvalue is given atallpoints onthesurface ofanellipsoid. ART. 142. Conical Coordinates. Theproductoftwo Ellipsoidal Harmonics aSpherical Harmonic. ART. 143. Toroidal Coordinates. Laplace sEquationhitoroidal coordinates. Particular solutions. Toroidal Harmonics. Potential Function forananchor ring. CHAPTER IX. HISTORICAL SUMMARY 267-276 APPENDIX. TABLES ; 277-287 TABLE I.Surface Zonal Harmonics. Argument*? 278 TABLE II.Surface Zonal Harmonics. Argument x 280 TABLE III.Hyperbolic Functions 282 TABLE IV.Roots ofBessel sFunctions 280 TABLE V.Roots ofBessel sFunctions 286 TABLE VI. Bessel sFunctions 287 CHAPTER I. INTRODUCTION. 1.Inmany important problemsinmathematical physics weareobliged todealwith partial differential equationsofacomparatively simpleform. Forexample,intheAnalytical TheoryofHeatwehave forthechangeof temperatureofanysolidduetotheflow ofheatwithin thesolid, theequation Dtu=a\D*u+D*u+D,* [i] where urepresentsthetemperatureatanypointofthesolidand tthetime. Inthesimplest case, that ofaslab ofinfinite extent withparallel plane faces, where thetemperaturecanberegardedasafunction ofonecoordinate, [i]reduces toDtu=a*Du, [n] aform ofconsiderable importanceintheconsideration oftheproblemofthe coolingoftheearth scrust. Intheproblemofthepermanentstate oftemperaturesinathinrectangular plate,theequation [i]becomes D*u+D*u=0.[in] Inpolarorsphericalcoordinates[i]islesssimple,itis D,U= Inthecasewhere thesolid inquestionisasphere andthetemperature atanypoint depends merely onthedistance ofthepoint from thecentre [iv]reduces to^(ru}=^^^.[v] Incylindricalcoordinates[i]becomes Dtu=a*[D?u+-Dru+-tD*u +Dfu].[vi] Inconsidering theflow ofheat inacylinder when thetemperatureat anypoint depends merelyonthedistance rofthepoint from theaxis fvi&quot;!becomes,1.LJDtu=a\D}u -f-Dru).[vn] *Forthesake ofbrevity weshall often usethesymbolV2fortheoperation Dx2+DV*+D 22 ; andwith thisnotation equation [i]would bewrittenDtu=a2V2u. 2 INTRODUCTION.[ART.1. InAcoustics inseveral problems wehave theequation D*y= a*D2y&amp;gt;, [vm] forinstance, inconsidering thetransverse orthelongitudinal vibrations ofa stretched elastic string,orthetransmission ofplane sound waves through the air. Ifinconsidering thetransverse vibrations ofastretchedstring wetake account oftheresistance oftheair[vm]isreplaced by D*y+2kDty=a*Dy. [ix] Indealing with thevibrations ofastretched elastic membrane, wehave the equation Dfz=c\D2*+Dz), [x] orincylindricalcoordinates +*Drz+D}z). [xi] InthetheoryofPotential weconstantly meet LaplacesEquation D*V+D*V+D?V= [xn] or Vr=0 which insphericalcoordinates becomes + ^(sinflAF) + D^r=0, [xn,] andincylindricalcoordinates D*V+l-DrV+^2&amp;gt;}r+2&amp;gt;ir=0. [xiv] Incurvilinear coordinates itis St^+^tAy)]-[xvl where /i(*,y,*)=PI ,/2fry,*) =pz, fsfry,*)=p* representasetofsurfaces which cutoneanother atright angles, nomatter what values aregiventopi ,p2,a-nclp3;andwhere V=(ArP.)+(^P.)*+(Apa), and, ofcourse, must beexpressedinterms ofpi,pt,andp8- IfithappensthatV2 pi=0,V2p2=0,andV2p3=0,then Laplaces Equation [xv]assumes thevery simpleform =0. [xvi] CHAP.I.] PARTICULAR SOLUTIONS. 3 2.Adifferential equationisanequation containing derivatives ordifferen tialswith orwithout theprimitivevariables fromwhich theyarederived. Thegeneralsolution ofadifferential equationistheequation expressing the most generalrelation between theprimitivevariables which isconsistent with thegivendifferential equation andwhich does notinvolve differentials or derivatives. Ageneralsolution willalwayscontainarbitrary (i.e.,undeter mined)constants orarbitrary functions. Aparticularsolution ofadifferentialequationisarelation between the primitivevariables which isconsistent with thegivendifferentialequation, butwhich islessgeneral than thegeneral solution, although included init. Theoretically, every particularsolution canbeobtained from thegeneral solution bysubstitutinginthegeneralsolutionparticular values forthearbi traryconstants orparticularfunctions forthearbitrary functions; butin practiceitisoften easytoobtainparticularsolutionsdirectly from thediffer ential equation when itwould bedifficult orimpossibletoobtain thegeneral solution. 3.Ifaproblem requiringforitssolution thesolvingofadifferential equa tion isdeterminate, theremust always begiveninaddition tothedifferential equation enough outside conditions forthedetermination ofallthearbitrary constants orarbitraryfunctions that enter intothegeneral solution ofthe equation;andindealing withsuchaproblem,ifthedifferentialequation can bereadilysolved thenatural method ofprocedureistoobtain itsgeneral solution, andthen todetermine theconstants orfunctions bytheaidofthe givenconditions. Itoften happens, however, thatthegeneral solution ofthedifferential equa tioninquestioncannot beobtained, andthen, since theproblem ifdeterminate willbesolved ifbyanymeans asolution oftheequation canbefound which will alsosatisfythegivenoutside conditions,itisworth while totrytoget particularsolutions and sotocombine them astoform aresult which shall satisfythegiven conditions without ceasingtosatisfythedifferentialequation. 4.Adifferential equationislinearwhen itwould beofthe firstdegreeif thedependentvariable and all itsderivatives were regarded asalgebraic unknown quantities.Ifitislinear andcontains notermwhich does not involve thedependentvariable oroneofitsderivatives, itissaid tobelinear andhomogeneous. Allthedifferential equationscollected inArt.1arelinear andhomogeneous. 5.If&value ofthedependentvariable hasbeenfound whichsatisfies a given homogeneous, linear, differential equation, theproduct formed bymultiply ingthisvalue byanyconstant will also beavalue ofthedependent variable which willsatisfytheequation. i INTRODUCTION.[ART.6. For ifalltheterms ofthegiven equationaretransposedtothe firstmem ber,thesubstitution ofthefirst-named value must reduce thatmember to zero; substituting thesecond value isequivalenttomultiplying eachterm of theresult ofthe first substitution bythesame constant factor, which there foremaybetaken outasafactor ofthewhole firstmember. Theremaining factor being zero, theproductiszeroandtheequationissatisfied. Ifseveral values ofthedependentvariable have beenfoundeachofwhich satisfiesthegiven differential equation,theirsum willsatisfytheequation ;for ifthesum ofthevalues inquestionissubstituted intheequation eachterm ofthesumwillgiverisetoasetofterms which must beequaltozero,and therefore thesum ofthese setsmust bezero. 6.Itisgenerally possibletogetbysomesimpledevice particularsolutions ofsuch differential equationsasthosewehave collected inArt. 1.The objectofthebranch ofmathematics withwhich weareabout todeal isto findmethods ofsocombining theseparticularsolutions astosatisfy anygiven conditions which areconsistent with thenature oftheprobleminquestion. This oftenrequiresustobeable todevelop anygivenfunction ofthevaria bleswhich enter intotheexpressionofthese conditions interms ofnoi-mal formssuited totheproblemwithwhich wehappentobedealing, andsug gested bytheform ofparticularsolution thatweareable toobtain forthe differential equation. These normal forms arefrequentlysines and cosines, buttheyareoften much more complicatedfunctions known asLegendrds Coefficients,orZonal Harmonics;LaplacesCoefficients,orSphericalHarmonics;BesseVs Functions, orCylindrical Harmonics ;Lame sfunctions, orEllipsoidal Harmonics, &c. 7.Asanillustration,letustakeFourier sproblemofthepermanentstate oftemperaturesinathinrectangular plateofbreadth TTandofinfinite length whose faces areimpervioustoheat.Weshall supposethat thetwolong edgesoftheplatearekeptattheconstant temperature zefo, thatoneofthe short edges, which weshall callthebase oftheplate,iskeptatthetempera tureunity, andthat thetemperaturesofpointsintheplatedecrease indefi nitelyaswerecede from thebase;weshall attempttofindthetemperature atanypointoftheplate. Letustakethebase astheaxis ofXandoneendofthebase astheorigin. Then tosolve theproblem wearetofindthetemperatureuofanypoint from theequation D,u+J}JU=Q[m]Art. 1 subjecttotheconditions u=when x=(1) u=&quot;x TT (2) u=&quot;y=oo (3) u=l&quot;y=0. (4) CHAP.I.] RECTANGULAR PLATE. 5 Weshall begin bygetting aparticular solution of[in],andweshall use adevice which always succeeds when theequationislinear andhomogeneous andhasconstantcoefficients. Assume*u=eay+f*x ,where aand(3areconstants, substitute in[in]and divide byeav+ P*,andwehave a2-f/32=0.If,then, thiscondition issatis fiedu=eay+&xisasolution. Hence u=eay* *&quot;*fisasolution of[in],nomatter what valuemaybe giventoa. Thisform isobjectionable, since itinvolves animaginary. Wecan,how ever, readily improveit. Takeu=eaveaxi ,asolution of[in],andu=eP-ver**, another solution of[in];addthese values ofuand divide thesumby2andwehave e*vcosax.(v.Int. Cal.Art.35,[1].)Therefore byArt.5 u=eaycosax(5) isasolution of[in]. Take w=ea*ea!riandu=eaye-axi ,subtract the second value ofufrom the firstanddivide by2iandwehave eaysinax. (v.Int. Cal.Art.35,[2]).Therefore byArt.5 u=eavsinax(6) isasolution of[HI]. Letusnow seeifoutoftheseparticular solutions wecanbuild upasolu tionwhich willsatisfy theconditions(1), (2), (3),and(4). Consider u=e^sinax .(6) Itiszerowhen x= forallvalues ofa.Itiszerowhen x=ITifaisa whole number. Itiszerowhen y=ooifaisnegative. If,then,wewrite uequaltoasumofterms oftheformAe~mvsinmx,wheremisapositive integer, weshall have asolution of[in]which satisfies conditions(1),(2) and(3).Letthissolution be u=A^e~vsinx+Aze~2vsin2x+Ase~3ysinSx+A^e-^ sin4x-\---- (7) AI,Az,AS)A4,&c.,being undetermined constants. When y=(7)reduces to u=AIsinx+A2sin2x-fA8sin3x-j-.44sin4#-f.(8) Ifnow itispossibletodevelop unityintoaseries oftheform(8),our problemissolved; wehaveonlytosubstitute thecoefficients ofthat series for AUA2,A8,&c.in(7). *.This assumption must beregarded aspurely tentative. Itmust betested bysubsti tuting intheequation, and isjustifiedifitleads toasolution. tWeshallregularly usethesymboliforV l. 6 INTRODUCTION.[ART.8. Itwillbeprovedlater that4/1 1 1 x 1=Isinx-f-Q-sin3#+Fsm&r4-=rsin7a;4- I ir\o o (f forallvalues ofxbetween andTT;hence ourrequired solution is 4i- 1 1 1 -iu= e~ysmx+o&~vsin3x-|-~-e~oysin5cc+^e~7ysinIx-\-(9) forthis satisfies thedifferentialequation and allthegiven conditions. Ifthegiven temperature ofthebase oftheplate instead ofbeing unity isafunction ofx,wecansolve theproblemasbefore ifwecanexpress the givenfunction of#asasum ofterms oftheformAsinmx,wheremisa whole number. Theproblemoffinding thevalue ofthepotential function atanypointof along, thin, rectangular conducting sheet, ofbreadthTT,through which an electric current isflowing, when thetwolongedges arekeptatpotential zero, andoneshort edgeatpotential unity,ismathematically identical with the problem wehavejustsolved. EXAMPLE. Taking thetemperatureofthebase oftheplate described above as100 centigrade, andthat ofthesides oftheplateas0,computethetemperatures ofthepoints correct tothenearest degree.Ans.(a)26; (b)15; (c)6. 8.Asanother illustration, weshall take theproblemofthetransverse vibrations ofastretchedstring fastened attheends, initiallydistorted into some given curve andthenallowed toswing. Letthelengthofthestring be I.Take thepositionofequilibriumofthe stringastheaxis ofX,andoneoftheends astheorigin, andsupposethe string initiallydistorted intoacurve whose equation y=f(x)isgiven. Wehave then tofindanexpressionforywhich willbeasolution ofthe equation D?y=a?D%y [vin]Art. 1, whilesatisfyingtheconditions y=when x=(1) y==x=l(2) y=f(x)&quot;t=(3) 2&amp;gt;ty=&quot;*=0, (4) thelastcondition meaning merelythatthestringstarts from rest. CHAP.I.]VIBRATING STRING. 7 Asinthelastproblemlet*y=eax+^tandsubstitute in[vin].Divide byeax+fttandwehave/32=a2a2asthecondition that ourassumed value of ;/shall satisfytheequation. _ 6&amp;lt;ucaat,~ is,then, asolution of(viu)whatever thevalue ofa. Itismore convenient tohave atrigonometric thananexponential form to dealwith,andwecanreadilyobtain onebyusing animaginary value forain(5). Replaceabyaiand(5)becomes y=e(*a0cu ,asolution of[vin]. Replace abyaiand(5)becomes?/=e~(a:a )al ,another solution of[vin]. Add these values ofyanddivide by2andwehave cosa(x at). Subtract the second value ofyfrom the firstanddivide by2iandwehave sina(x at). ycosa(x+at) y=cosa(x at) y=2sina(x+ ?/=sina(x at) are,then, solutions of[vin]. Writing ysuccessively equaltohalf thesum ofthe firstpairofvalues, half their difference, half thesum ofthelast pairofvalues, andhalf their difference, wegettheveryconvenientparticular solutions of[vin]. y=.cosaxcosaat y=.sinaxsinaat y=sinaxcosaat y=cosaxsinaat . Ifwetakethethird form y=.sinaxcosaat itwillsatisfyconditions(1)and(4),nomatter what valuemaybegiven to a,and itwillsatisfy (2)ifa= where raisaninteger. Ifthenwetake .TTX irat . 2jrx 2irat STTX . y=Alsin cos--h^2sinjcos--- 1-A3sinjcos---h&quot; * (6) L Li&amp;gt; & L L whereA15A2,A3 areundetermined constants, weshallhaveasolution of [vin]which satisfies(1), (2),and(4).When t= itreduces to TTX . . 2irx . . .STTX . y=A!sm+Azsm--h^sin-y-+ (7) Ifnow itispossibletodevelop f(x)intoaseries oftheform(7),wecan solve ourproblem completely. Wehave onlytotake thecoefficients ofthis series asvalues ofAj,A2,A8...in(6),andweshall have asolution of FVIII]which satisfies allourgivenconditions. *Seenoteonpage5. 8 INTRODUCTION.[ART.9. Ineach ofthepreceding problems thenormalfunction, interms ofwhich a givenfunction hastobeexpressed,isthesine ofasimple multipleofthe variable. Itwould beeasytomodifytheproblem sothat thenormal form should beacosine. Weshallnowtakeacoupleofproblems which aremuch morecomplicated andwhere thenormal function isanunfamiliar one. 9.Let itberequiredtofindthepotential function duetoacircular wire ringofsmall cross section andofgiven radiusc,supposing thematter ofthe ring-toattract according tothelawofnature. Wecanreadily find,bydirectintegration, thevalue ofthepotential function atanypointoftheaxisofthering.Wegetforit r-JU d) whereMisthemass ofthering,andxthedistance ofthepoint from the centre ofthering. Letususespherical coordinates, taking thecentre oftheringasorigin and theaxisofthering asthepolaraxis. Toobtain thevalue ofthepotentialfunction atanypointinspace, wemust satisfy theequation rD?(rV)+ApnUt?) -h iXVF= 0,[xm]Art.1, subjecttothecondition 7= (C2^i^en=0.(1) From thesymmetryofthering,itisclear that thevalue ofthepotential function must beindependentof &amp;lt;f&amp;gt;,sothat[xm]will redi^ee to rZ&amp;gt;?(rV)+ D.(sinZ&amp;gt;.F)=0.(2) Wemustnowtrytogetparticularsolutions of(2),andasthecoefficients arenotconstant, wearedriven toanew device. Let*F=rmP,wherePisafunction of6only,andmisapositive integer, andsubstitute in(2),which becomes m(m+l)r*P+4^D,(sinD,-P)=0. *Seenoteonpage 5. CHAP.I.] POTENTIAL DUETOWIRE KING. 9 Divide byrmandusethenotation ofordinaryderivatives sincePdepends upon only,andwehave theequation 0, (3) fromwhich toobtain P. Equation (3)canbesimplified bychanging theindependentvariable. Let x=cos6and(3)becomes Assume*nowthatPcanbeexpressedasasum orasaseries ofterms involving whole powersofxmultiplied byconstant coefficients. LetP=2anx11andsubstitute thisvalue ofPin(4).Weget 2O(n I)anx~2n(n+1)anx&quot;+m(HI+1)an]= , (5) where thesymbol 5indicates thatwearetoform allthetermswecanby takingsuccessive whole numbers forn. As(5)must betruenomatter what thevalue ofx,thecoefficient ofany given powerofx,asforinstance#*,must vanish. Hence (k+2)(k+lK+2-k(k+l)a,+m(m+!)*= (6) (m+1)a*+*=+2)&quot;a* Ifnowanysetofcoefficients satisfyingtherelation(7)betaken,P=2a*x* willbeasolution of(4). If k=m,ak+2=Q,afc+4=0,&c. Since itwillanswer ourpurposeifwepickoutthesimplestsetofcoefficients that willobeythecondition(7),wecantakeasetincluding am. Letusrewrite(7)intheform a*~~ (m-k)(m+k+1)*+** Wegetfrom(8),beginning with k=m2, m(m 1)a&quot;-2==~ 2.(2w-1)a &quot; _m(m l)(m 2)(m 3)a&quot;-4=2A.(2m 1)(2m 3)a&amp;gt;n m(ml) (m2)(m 3)(m 4)(m~5)a-6~&quot; 2.4.6.(2m-1)(2w-3)(2m 5) *Seenoteonpage6. 10 INTRODUCTION.[ART.9. Ifmisevenweseethatthesetwillendwitha,ifmisodd,with%. where amisentirely arbitrary, is,then, asolution of(4).Itisfound con venient totakeamequalto (2m 1)(2m 3)1 ml and itcanbeshown thatwith thisvalue ofamP=1when JB=1.Pisafunction ofxandcontains nohigher powers ofxthanxm .Itis usual towrite itasPm(x). Weproceedtocompute afewvalues ofPm(x)from theformula (2^-1)(2m-3).^1 I&quot;m(m-1) ! L2.2m-lx 2.4.(2m 1)(2m 3)Wehave:-i -(10)PQ(x)=1 orP(cos 0)=1 P^x)=x&quot;P!(COS 0)=cos0 P2(x)= (3.x2-1) P2(cos 0)=i(3cos2-1) A()=iG^3- 3a;)&quot;P3(cos tf)=*(5cos3-3cos0) p^x)=$(35*4-30*2+3)or P4(cos 0)=(35cos430cos2+3)PB(x)=i(63x570x3+lox)or P5(cos0)=^(63cos570cos8+15cos0). Wehave obtained P=Pm(x)asaparticular solution of(4)and P=Pm(cos 0)asaparticularsolution of(3).Pm(x)orPm(cos 0)isa newfunction, known asaLegendresCoefficient, orasaSurface ZonalHar monic, andoccurs asanormal form inmany important problems. V=rwPm(cos 0)isaparticularsolution of(2)and rmPm(cos 0)issome times called aSolid Zonal Harmonic. Wecannowproceedtothesolution ofouroriginal problem. V=ArP (cos 0)+^4 1rP1(cos 0)+^2r2P2(cos 0)-M 3r8P3(cos 0)H---- (11) whereAQ,Al,A2,&c., areentirely arbitrary,isasolution of(2)(v.Art.5). When=(11)reduces to since, aswehave said,Pm(x)=1when x=1,orPm(cos 0)=1when=0. Byourcondition(1) when *=0.F= CHAP.I.] ZONAL HAKMONICS. 11 BytheBinomial Theorem M _^&quot;r-,_1^,1^ r!__i^^...&quot;i (c*+r*)*~TL 2c2&quot; 1&quot; 2.4c* 2.4.6 c&quot;&quot;* &quot;J providedr &amp;lt;c.Hence isourrequiredsolution ifr&amp;lt;c;for itisasolution ofequation (2)and satis fiescondition(1). EXAMPLE. Taking themass oftheringasonepound andtheradius oftheringasone foot,computetotwodecimalplacesthevalue ofthepotentialfunction dueto theringatthepoints (a) (r=.2,0=0); (d) (r=.6,0=0); ( =2=V^(a).98; (ft).99; (C)1.01; (d).86; 2; (e)&amp;gt;9Q. (/)1&amp;lt;00;(^)1.10. Theunitused isthepotential duetoapoundofmass concentrated atapoint andattracting asecond poundofmass concentrated atapoint,thetwopoints being afootapart. 10.Aslightlydifferent problem callingfordevelopmentinterms ofZonal Harmonics isthefollowing: Requiredthepermanent temperatureswithin asolid sphereofradius1, onehalf ofthesurface being keptattheconstant temperature zero,andthe other half attheconstant temperature unity. Letustakethediameter perpendiculartotheplane separatingtheunecjually heated surfaces asouraxisand letususesphericalcoordinates. Asinthe lastproblem, wemust solve theequation rD*(ru)+A(sinD9u)+ D}u=[xm]Art. 1 which asbefore reduces to (sinD9u)=(1) from theconsideration thatthetemperatures must beindependentof Ourequationofcondition is u=1from= to=andu=from= ^to= when r=1. 12 INTRODUCTION.[ART.1.7. Aswehave seenu=rmPm(cos 6)isaparticular solution of(1),mbeing anypositivewhole number, and u=ArPQ(cos 0)+A1rPl(cos 0)-fA2r&amp;gt;Pz(cos 6)+AsrP3(cos 0)-\---- (3) whereAQ,Ai9A*,As-.areundetermined constants,isasolution of(1). When r=1(3)reduces to w=AP(cos 0)+^!P!(cos 0)-f^2P2(cos0)+^43P8(cos 0)H---- (4) Ifthenwecandevelopourfunction ofwhich enters intoequation (2)in aseries oftheform(4),wehaveonlytotake thecoefficients ofthat series asthevalues ofAQ,A^,A Z,&c.,in(3)andweshallhave ourrequiredsolution. 11.Asalastexample weshall taketheproblemofthevibration ofastretched circular membrane fastened atthecircumference, thatis,ofanordinary drum head.Weshall supposethemembraneinitiallydistorted intoanygiven form which hascircular symmetry*about anaxisthrough thecentreperpendicular totheplaneoftheboundary, andthenallowed tovibrate. Herewehave tosolve D?z=c*(D?z+;Drz+iD{^[xi]Art, 1 subjecttotheconditions %=/(?)when t=(1)D(s=&quot;=(2) z= r=a(3) From thesymmetryofthesupposedinitial distortion zmust beindepend entof &amp;lt;#&amp;gt;,therefore[xi]reduces to (4) andthis istheequationforwhich wewish tofindaparticularsolution. Weshallemployadevice notunlike thatused inArt. 9. Assume t=R-TwhereRisafunction ofralone andTisafunction of talone. Substitute thisvalue ofin(4)andweget 01dr Thesecond member of(5)does notinvolvet,therefore itsequalthe first member must beindependentof t.The firstmember of(5)doesnotinvolve *Afunction ofthecoordinates ofapoint hascircular symmetry about anaxiswhen its value isnotaffected byrotating thepoint through anyangle about theaxis.Asurface ha* circular symmetry about anaxiswhen itisasurface ofrevolution about theaxis. tSeenoteonpage5. CHAP.I.] VIBRATING DRUMHEAD. 13 r,andconsequentlysince itcontains neither tnorr,itmust beconstant. Let itequalfJ&amp;gt;z ,whereftofcourse isanundetermined constant. Then(5)breaks upintothetwodifferential equations O(6) 5==0.(7)dr2rdr^ (6)canbesolved byfamiliar methods, andwegetT=cospetandT=sinput assimple particularsolutions(v.Int. Cal.p.319, 21). Tosolve(7)isnotsoeasy.Weshall firstsimplifyitbyachangeofinde pendentvariable. Let r=--(7)becomes 0.(8) xdx^ Assume, asinArt. 9,thatRcanbeexpressedinterms ofwhole powersof x.LetR=2anxnandsubstitute in(8).Weget 2[n(n l)anxn~2+nanxn~2+ nz] , anequationwhich must betruenomatter what thevalue ofx.The coeffi cient ofanygiven powerofx,asxk~2 ,must, then, vanish, and k(k l)ak+kak+*_2= or k*ak+ak_,= whence weobtain ak_2= k*at (9) astheonlyrelation thatneed besatisfied bythe coefficients inorder that R=2akxkshall beasolution of(8). If k=0,%._ 2=0, t_4=0, &c. Wecanthen begin with k asourlowest subscript. ak2From(9)ak=--^~ a Then a2=2 2242 *2&amp;gt;2.42.62 r^ _i^ ^6 &quot;IHence -K= 12*&quot;&quot;^MT~ 22.42.62 where amaybetaken atpleasure,isasolution of(8),providedtheseries is convergent. 14 INTRODUCTION.[ART.11.* Take a=$1,andthenR=JQ(x)where --_LX*X* JQ(x)122224222426222426282**(^) isasolution of(8). JQ(x)iseasily shown tobeconvergent forallvalues realorimaginaryofx, since theseries made upofthemoduli oftheterms ofJQ(x) (v.Int. Cal. Art.30) where risthemodulus ofx,isconvergentforallvalues ofr.Fortheratio &amp;gt; ofthe rc+1stterm ofthis series tothenthterm is _andapproaches4ft zero asitslimit asnisindefinitely increased, nomatter what thevalue ofr. ThereforeJQ(X)isabsolutely convergent. JQ(x)isanewandimportantform. Itiscalled aBessel sFunction ofthe zerothorder, oraCylindricalHarmonic. Equation (8)wasobtained from(7)bythesubstitution ofx=ftr,therefore ,r22222 &quot;22r22,42 22.42.62 isasolution of(7),nomatter what thevalue offt,and z=J(pr)cospet or*=Jo(fir)sinftcisasolution of(4). z=JQ(fir)cosftcsatisfies condition(2)whatever thevalue offt.In order that itshould alsosatisfy condition(3) ftmust besotaken that Jo(fta)=0; (11) thatis,ftmust bearootof(11)regardedasanequationinft. Itcanbeshown thatJ(x)=0 hasaninfinite number ofrealpositive roots, anyoneofwhich canbeobtained toanyrequired degreeofapproxima tionwithout seriousdifficulty. Letx1}z2, , bethese roots. Then if .(12) where^,A2,As,&c.,areanyconstants,isasolution of(4)which satisfies conditions(2)and(3). When t=(12)reduces to =A,J Qfar)+A2Jfar) -fA%JQfar)+.(13) Ifthen/(r)canbeexpressedasaseries oftheform justgiven,thesolution ofourproblem canbeobtained bysubstitutingthe coefficients ofthat series torAl9AI,A9,&Q.,in(12). CHAP. L] DISCUSSION OFMETHODS. 15 EXAMPLE. Thetemperatureofalong cylinderisatfirstunity throughout. Theconvex surface isthen keptattheconstant temperaturezero. Show that thetem peratureofanypointinthecylinderattheexpirationofthetime tis +Aie where/ii;/42,&c.,aretheroots ofJ(^c)=0,andwhere 1=A1J(plr)+A2J(p2r)+AsJ(f^sr)-\---- , cbeing theradius ofthecylinder. 12.Each ofthe fiveproblems which wehavetaken upforces npon usthe consideration ofthedevelopmentofagiven function interms ofsome normal form, andintwoofthem thenormal form suggestedisanunfamiliar function. Itisclear, then, thatacompletetreatment ofoursubjectwillrequire theinves tigationofthepropertiesandrelations ofcertain newandimportant functions, aswell astheconsideration ofmethods ofdevelopinginterms ofthem. 13.Ineach oftheproblems justtaken upwehave todealwith ahomo geneouslinearpartialdifferential equation involving twoindependentvari ables, andwearecontent ifwecanobtainparticularsolutions. Ineach case theassumption made inthelastproblem,that there exists asolution ofthe equationinwhich thedependentvariable istheproductoftwofactors each of which involves butoneoftheindependent variables, willreduce thequestion tosolving twoordinarydifferential equations which canbetreatedseparately. Ifthese equationsarefamiliar ones their solutions canbewritten down at once;ifunfamiliar, thedevice used inproblems 3and5isoften serviceable, namely,that ofassuming that thedependentvariable canbeexpressedasa sum orseries ofterms involving whole powersoftheindependent variable, andthendeterminingthecoefficients. Letusconsider againtheequationsused inthe first,second and third problems. (a) D?u+D&amp;gt;=(1) Assume u=X.YwhereXinvolves xbutnoty,andYinvolves ybutnotx. Substitute in(1),YD*X+XDJY=0, or,sincewearenowdealing with functions ofasingle variable, 1d*Y 1d*X or ~*=~* 16 INTRODUCTION.[ART.13. Since the firstmember of(2)does notcontainx,andthesecond member does notcontainy,andthetwomembers must beidentically equal, neither of them cancontain either xory,andeachmust beequaltoaconstant, saya2 . Then _a2r==(3) d*X and-^r+a2X=0; (4) and if(3)and(4)canbesolved, wecansolve(1).They have fortheircom pletesolutionsY_A&ay+Be-*y and XCsinax-fDcosax .(v.Int.Cal.p.319, 21.) HenceY=eayandY=e~ayareparticularsolutions of(3),X=8inax andX=cosaxareparticularsolutions of(1),andconsequently u=eaysinax,u=e&vcosax,u=e~aysinax,andu=e~aycosax areparticularsolutions of(1).These agree with theresults ofArt. 7. (ft) Dfy=a*D*y (1) Assume y=T.XwhereTisafunction oftonlyand -2&quot;afunction ofx only;substitute in(1)anddivide bya*TX.Weget hence asinthelastcase--risaconstant;call ita2 ,and(2)breaks (3) (4) Thecompletesolutions of(3)and(4)are X=Asinax+Bcosax and T=Csinaa*+Dcosaat, (v.Int.Cal.p.319, 21). y=sinaxcosaatf, y=sinasesinaat,y=cosaxcosaat,y=cosaxsinaa areparticularsolutions of(1),andagreewith theresults ofArt. 8. (c)rDr\rF)+-^D9(sinODV)=Q- (1) AssumeF= -R.where^involves ralone, and involves Balone; sub stitute in(1),divide by.R.,andtranspose;weget R CHAP. L] SOLUTION ASAPRODUCT. 17 Since bythereasoning used in(a)and()eachmember of(2)must beacon stant, saya2 ,wehave a*R(3) and (3)canbeexpandedinto (5)canbesolved(v.Int. Cal.p.321, 23),andhasforitscompletesolution R=Arm-fBrn , where m=\+Va2-fi and n=$Va2+i Hence ?i=m1,anda2maybewritten m(m -f-1),mbeing wholly arbitrary; andR=Arm+J?/---1 . 1R=rm ,and^=^TT are,then, particularsolutions of With thenewvalue ofa2 (4)becomes &amp;lt;+l)-0. (7) which hasbeen treated inArt.9forthecasewheremisapositive integer, andtheparticularsolution =Pm(cos^)hasbeen obtained. Hence V=rmPm(cosO) and F=^TPwl(cos0), mbeing apositive integer,areparticularsolutions of(1).The first ofthese wasobtained inArt. 9,butthesecond isnewandexceedingly important. 14.Themethod ofobtaining aparticularsolution ofanordinarylinear differentialequation, which wehave used inArticles 9and 11,isofvery extensiveapplication, andoften leads tothegeneralsolution oftheequation inquestion. 18 INTRODUCTION.[ART.14 Asaverysimple example,letustake theequation Art.13(a)(4),which weshall write dx*+a*z=.(1) Assume thatthere isasolution which canbeexpressedinterms ofpowers ofx-jthatis,letz=2tanxn ,where thecoefficients aretobedetermined Substitute thisvalue forzin(1)and iveget 2\n(n l)anxn-2+a?anxn ~\=. Since thisequation must betruefrom itsform, without reference tothevalue ofx,thatis,since itmust beanidenticalequation, the coefficient ofeach powerofxmust equal zero,andwehave (n+1)(K+2)whence an=~tf~~a+2 istheonlyrelation thatneed hold between the coefficients inorder that z=2anxnshould beasolution of(1). Ifn-\-2= orn-f-1= ,anwillbezeroandan_2,are_4,&c.,willbe zero. Inthe firstcasetheseries willbegin witharinthesecond with a^. (n+l)(n+2) Ifwebegin with awehave.. a a* a&quot; a2=K-.a, a4= j-.a &amp;gt; 6=,-.a,&c.,... azx* a*x4ax6mand z=aQl----+----rH----(2) or z=acosao;(3) isaparticularsolution of(1). Ifwebegin with a^wehave a&quot;_a* 3]%,&amp;lt;*5= 5! and =aj -~&quot;~ &quot;*&quot;&quot; CHAP. L] SOLUTION INPOWER SEKIES. 19 isasolution of(1);alcanbetaken atpleasure. Leta=a,(4)becomes a3*8 *=ax&quot;^T^T~7T or z=sina# which, then,isaparticularsolution of(1). z=Asinax-\-Bcosax(5) is,then, asolution of(1),and since itcontains twoarbitrary constants itis thegeneralsolution. 15.Asanother example wewilltaketheequation d*z dz x*d^+2xtic~m(m+1&amp;gt;=0, (1) which isineffect equation (6),Art.13(c),and letmbeapositive integer. Assume z=^anxnandsubstitute in(1).Weget 2 |&amp;gt;&amp;lt;&amp;gt;-hi) m(m+1)]ana?=. This isanidenticalequation,therefore \n(n+1)m(m+l)]a n=. Hence an=forallvalues ofnexceptthose which make n(n+1)m(m -f1)= , thatis,forallvalues ofnexcept n=mandn=m 1.Then z=Axm+Bx~m~l (2) isthegeneralsolution of(1)and z=xmand *=5Hn areparticularsolutions. Ifmisnotapositive integer thismethod willnot lead toaresult, andwearedriven back tothatemployedinArt.13(c). 16.Letusnowtaketheequation d which isineffectequation (4),Art.9,and isknown asLegendresEquation. (1)maybewritten \ /7i2 /7i \ / \/ 20 INTRODUCTION.[ART.16 Assume z=S,anxnandsubstitute in(2).Weget ^{n(n !)xn~*+[m(m+1) n(n-f1)]V}= , Hence(n+!)(+2)aw+2+[m(m+1)n(n+1)]an= , or =m/?rc-i-1)M/7^J_i)an+s- (3) Ifan=0,then aw_2=0, M_4=0,&c.;butan=Qif=2or= 1. Forthe firstcasewehave thesequenceofcoefficients ^- m(in m(m 2)(m+1)(m+3) 4J~&quot;^0 m(m 2)(m 4)(m+1)(m-f3)(m-f5) Letustakea,which isarbitrary,as1.Then z=pm(x)where /*//wi_L1\ j/MI 9^/-I-1 &quot;Nf isasolution ofLegendresEquationifpm(x)isafinitesum oraconvergent series. Forthesecond casewehave thesequenceofcoefficients (m-1)(m+2) 3=--3f--ai i (m l)(m 3)(m+2)(m -f4) 5=--5]-- ! (m-1)(m- .3)(m-5)(m+2)(m+4)(m+6) #7=-- j-j Letustake !,which isarbitrary,as1.Then z=qm(x)where -l)(m-3) (m+2)(m-f 4)~~ 3 isasolution ofLegendresEquationifqm(x)isafinitesum oraconvergent series. CHAP.I.] LEGENDKE SEQUATION. 21 Ifmisapositive evenwhole number, pm(x)willterminate with theterm containingx&quot;1 ,and iseasilyseen tobeidentical with ,.^-*Pm(x). [v.Art.9(9)] r(m-\-1)mv/ L v/j For allother values ofm,pm(x)isaseries. The ratio ofthe(n+l)stterm ofpm(x)totherath,whenmisnotaposi tiveeven integer,is (2n2m)(2rc1+m) Itslimiting value, aswisincreased,iscc2 ,andthe series istherefore con vergentif 1&amp;lt;x&amp;lt;1.Itisdivergentforallother values ofx. If 7H.isapositiveoddwhole number qm(x)willterminate with theterm containing x,and iseasily seen tobeidentical with For allother values ofm,gfm(ic)isaseries, andcanbeshown tobecon vergentif 1&amp;lt;x&amp;lt;1,anddivergentforallother values ofx. z=APm(x)+Bqm(x) (6) isthegeneralsolution ofLegendresEquationif 1&amp;lt;a&amp;lt;1,nomatter what thevalue ofw.From Art.13(c)itfollows that areparticularsolutions of nomatter what thevalue ofm,providedcos isneither onenorminus one. Intheworkweshallhave todowithLaplacesandLegendresEquations, itisgenerally possibletorestrictmtobeing apositive integer, andhereafter weshallusuallyconfine ourattention tothat case. 22 INTRODUCTION.[ART.16. With thisunderstandingletusreturn to(3),whichmayberewritten (m n)(m-\-n+1) If an+2=Q,thenM+4=0,an+6=0,&c.; but #+2= if 7i=m,orn=m 1. Ifin(3)webeginwithn=m 2,wegetthesequenceofcoefficients already obtained inArt. 9,andwehave *=Pm(x),where (2m-l)(2m-3)1f_ m(m-l) ~ &quot; m(m 1)(m 2)(m 3)_4 &quot;12.4.(2m 1)(2m 3)^ m(m 1)(m 2)(m 3)(m 4)(m 5) 2.4.6.(2m1)(2m3)(2m5) asaparticularsolution ofLegendresEquation. If,however, webeginwithn=m3,wehave3 2(2m+3) 2.4.(2m+3)(2m+5) (m-f1)(m+2)(m-f3)(m-f-4)(m-f5)(m-f-6) 2.4.6.(2m-f3)(2m+5)(2m-f7)a-m-i m! a_m_lmaybetaken atpleasure,and isusuallytaken as^^^,2m-j.l) and z=Qm(x)where m! 2.4.(2m+3)(2m+5)H2.(2m+3)*+ H J isasecond particularsolution ofLegendresEquation, providedtheseries is convergent. Qm(x)iscalled aSurfaceZonal Harmonic ofthesecond kind. CHAP.I.] ZONAL HARMONICS. 23 Itiseasilyseen tobeconvergentifx&amp;lt; 1orx &amp;gt;1,and divergentif -!&amp;lt;*&amp;lt;!. Hence ifmisapositive integer, x) (10) isthegeneralsolution ofLegendresEquationifx&amp;lt; 1orx &amp;gt;1. Wehave seen that for 1&amp;lt;x&amp;lt;1 *-() (-ifrr(&quot;+1\T*.( ) 2-[r(f+i)] ifmisaneven integer, and ifmisanoddinteger. IfnowwedefineQm(x)asfollows when 1&amp;lt;x&amp;lt;1 T(m-hl) ifmisanoddinteger, and ifwisaneven integer, then(10)willbethegeneral solution ofLegendres Equationifmisapositive integer when 1&amp;lt;x&amp;lt;1,aswell aswhen x&amp;lt;1 orx &amp;gt;1. 17.Letuslastconsider theequation d?z 1dz which isknown asBessel sEquation, andwhich reduces to(8)Art. 11, thatis,to d*z 1dz whenw=0;* (1)canbesimplified byachangeofthedependentvariable. *Thisequation was firststudied byFourier inconsidering thecooling ofacylinder. We shalldesignateitas&quot;Fourier sEquation.&quot; 24 INTRODUCTION. [ART.17. Let 2=xvandweget d*v 2ra+ldv^2+^-^+v==0 & todetermine v. Assume v=S&amp;lt;*#&quot;,andsubstitute in(2).Weget 2 [&amp;gt;(2m-fri)anxn-2+ &quot;]=0; whence an_2=n(2m+ra)an. Ifwebegin with n=0,then afl_2=0,an_4=0,&c.,andwehave the setofvalues 22 (ra__ *= 2.4(2m+2)(2m+4)~~ 24 .2!(w+l)(m+2)_OP_ ___p_ __. a=&quot;&quot; 2.4.6(2m+2)(2m -f4)(2m+6)~26.3!(m+l)(m+2)(m+3) rcc2__ ^_whence *=am |^1- 2(m+1}+ 2*.2!(m+l)(m+2) -i J isasolution ofBessePs Equation.aisusually taken as ^^&quot;jifmisaP08 itive integer,oras 2OTr/m+^ifmisunrestricted invalue, andthesecond member of(3)isrepresented byJm(x)and iscalled aQesseVsFunction ofthe wth order,oraCylindricalHarmonic oftherath order. Ifm= ,Jm(x)becomes 7(x)and isthevalue of*obtained inArt.11 asthesolution ofequation (8)of-that article. Ifinequation (1)wesubstitute x~mv.m placeofxmvforz,wegetinplace of(2)theequation d*v lZmdv, dx*+F-^+&quot;= andinplaceof(3) - 22 (1m)24 .2!(1 m)(2-w &quot;&quot; 26 .3!(1-m)(2-m)(3 m)&quot;^J CHAP. I.] BESSEL SEQUATION. 25 If istaken equalto-- -thesecond member of(4)isthesame 2i 1(1~ wi) function ofmandxthatJm(x)isof+man(ixan(imay^ewritten *.,() Therefore z=AJm(x)+BJ_m(x) (5) isthegeneralsolution of(1)unlessJm(x)andJ_m(x)should prove nottobe independent. Itiseasilyseen thatwhenm=0,J_m(x)andJm(x)become identical and(5)reduces to andcontains butasingle arbitraryconstant and isnotthegeneral solution of Fourier sEquation (8)Art.(11). Itcanbeshown thatJ-m(x)= (V)mJm(x)whenever misaninteger, andconsequentlythat thesolution(5)isgeneral onlywhenmifreal isfrac tional orincommensurable. Thegeneralsolution fortheimportantcasewherem=is,however, easily obtained. LetF(m, x)bethevalue which thesecond member of(3)assumes when a=1;then thevalue which thesecond member of(4)assumes when a=1willb-^( w,aj), and ithasbeenshown that z=F(m,x) and z=F(m,x)aresolutions ofBessel sEquation;z=F(m,x) F(m,x) is,then, asolution, asisalso F(m,x) F(m,x). 2m^} F(m.x) F(m.x)butthelimiting value which--approachesasmapproaches &amp;gt;TH zero is[DmF(w,^)]^,)andconsequently Q (7) isasolution oftheequation d?z Idz a*+x;fe+= W andthegeneral solution of(8)is z=AJ(x)+B[D mF(m,z)]^. r a;2F(m,x)= 26 .3!(w+l)(m+2)(m+3) INTRODUCTION.[ABT.17. i *mF(m, x)=xmlogx t,-|.2,j.4_. 1~ 2\m+1)+24 .2!(m+l)(m+2)+* J Thegeneral term ofthelastparenthesis canbewritten 2&quot;.k\(m+l)(w -f2) (m+k) and itspartialderivative withrespecttomis ^ ^2**7T!^(m+l)(m -f2) (m-fk) Take the Z&amp;gt;mofbothmembers andwehave Dm(m-j-1)(m+2) (m+As) +2) (m a;2 *_i i &quot;1 -f&quot;1&quot; t &quot;^+2&quot;tw+A;J 22(mH-1)24 .2!(m+l)(m+2)26 .3!(m+l)(w+2)(m -f3) _^_^_ ~&quot;22(m+l)2 &quot;~24.2!(m+l)(m+2) ___r26.3!(m+l)(m +2 andwehavei rJ i )(m+2)Lm-f 1~rm+ L+1_J_n4+lm+Z^m+Sj^ 16I11 -- 28 (4!) and z=AJ(x)+BK(x), (9) 2a;4/I 1\ /I 11\=7(a5)log2+^-^ VI+2/+232U+2+3/where isthegeneralsolution ofFourier sEquation (8). XQ(X)isknown asaBesseVs Function oftheSecond Kind. CHAP.I.] GENERAL SOLUTION. 27 18. Itisworth while toconfirm theresults ofthe lastfew articles by getting thegeneralsolutions oftheequationsinquestion byadifferent and familiar method. Thegeneralsolution ofanyordinarylinear differential equationofthe second order canbeobtained when aparticularsolution oftheequation has beenfound[v.Int. Cal.p.321, 24(a)]. Themost general form ofahomogeneous ordinarylinear differentialequa tionofthesecond order is o(i) wherePandQarefunctions ofx.Supposethat y=v(2) isaparticularsolution of(1).Substitute y=vzin(1)andweget -O.(3) Call-=.Then(3)becomes adifferential equationofthe first order inwhich thevariables canbesepa rated. Multiply bydxanddivide byvz1and(4)reduces to Integrate andwehave log*+logv*+Cpdx=C or zv*=ecfpd*=Be-fpd* , rf* e~fpdx dx; (/VfPdx \A+^J!^- dx)(5) isthegeneral solution of(1),theonly arbitrary constants inthesecondmem berof(5)being thoseexplicitly written, namely, AandB. (a)Applythisformula to(1)Art. 14, ~+a2*= ; (1) 28 INTRODUCTION.[ART.18. given:=cosoa;,asaparticularsolution.Substitutingin(5)wehave sinceP= z=cosaxIA+B J^j / ,B\ =cosax(A+tanaxj =Acosax+Blsinax, (2) asthegeneralsolution of(1),andthisagrees perfectly with(5)Art. 14. (b)Take equation (1)Art. 15. ad*z dz dx2dx \ /&amp;gt; \/ given:z=cm ,asaparticularsolution. 2 / fPdx 1 HereP=~, IP&amp;lt;&=2loga=logx2 ,and e=-2.Henceby(5) T&amp;gt; that is isthegeneralsolution of(1),andagrees with(2)Art. 15. (c)Take LegendresEquation, (2)Art. 16. (1-x*)^-2x^ +m(m+l)z= ; (1) given:s=Pm(x),asaparticularsolution. HereP=^~__^a,CPdx=log(1-x*) ,ande-/p&amp;lt;te= 1_^ Hence by(5)*=Pm(x)(A+3J(1_^p^j)(2) isthegeneralsolution of(1)andmust agree with(10)Art. 16,ifmisan integer, andtherefore where (7isasyetundetermined, andnoconstant term istobeunderstood with theintegralinthesecond member. (d)Take BesseFs Equation, (1)Art. 17. d*z 1dz given:z=Jm(x) ,asaparticularsolution. CHAP. L] GENERAL SOLUTION. 29 HereP=- ,fPdx=logx,ande~fpdx=- .Hence by(5) (2) isthegeneralsolution ofBessel sEquation. Ifm=(2)becomes andmust agree with(9)Art. 17. Therefore ,(4) where Cisatpresent undetermined, andnoconstant term istobetaken with theintegral. The first considerablesubject suggested bytheproblems which wehave taken upinthisintroductory chapteristhat ofdevelopmentinTrigonometric Series(v.Arts. 7and8). CHAPTER II. DEVELOPMENT INTRIGONOMETRIC SERIES. 19.Wehave seen inChapterI.that itissometimes important tobeable toexpressagiven function ofavariable x,interms ofthesines orofthe cosines ofmultiples ofx.Theprobleminitsgeneral formwas first solved byFourier inhis&quot;Analytic TheoryofHeat&quot;(1822),and its.solutionplays a veryimportant partinmost branches ofmodernPhysics. Series involving onlysines and cosines ofwholemultiplesofx,that isseries oftheform &oH~#1cosx+&zcos2x++ isinx-{-aasin2x-f- aregenerally known asFourier sseries. Letusendeavor todevelop agiven function ofxinterms ofsinx,sin2, sin3x,&c.,insuchawaythat thefunction andtheseries shall beequalfor allvalues ofxbetween x=andx=TT. Tofixourideas letussupposethatwehaveacurve, ?=/(*), given, andthatwewish toform theequation, y= _!sinx+#2sin2x+ 3sin3x-J- , ofacurve which shall coincide with somuch ofthegiven curve asliesbetween thepoints correspondingtox=andx=TT. Itisclear that intheequation y=#!sinx(1) !maybedetermined sothat thecurve representedshallpassthrough any given point.For ifwesubstitute in(1)thecoordinates ofthepointinques tionweshall have anequationofthe first degreeinwhich axistheonly unknown quantity andwhich will therefore give usoneandonlyonevalue forOj. Inlikemanner thecurveyt y=axsinx-}azsin2x maybemade topassthrough anytwoarbitrarily chosenpoints whose abscissas liebetween and TTprovidedthattheabscissas arenotequal; and y=a-Lsinx+ 2sin2x-f 3sin3x+-+ansinnx maybemade topassthrough anynarbitrarily chosen points whose abscissas liebetween and TTprovidedasbefore that their abscissas arealldifferent. If,then, thegiven function f(x)isofsuchacharacter that foreachvalue ofx between x=andx=TTithasoneandonlyonevalue, and ifbetween x=andx=TTitisfinite andcontinuous, orifdiscontinuous hasonly finitediscontinuities(v.Int. Gal.Art. 83,p.78),thecoefficients in y=jsinx-f-azsin2x-\-assin3x{+a,Hsinnx(2) PRELIMINARY STUDY OFAFINITE SUM. 31 sonbedetermined sothatthecurverepresented by(2)willpassthrough any narbitrarily chosenpointsofthecurve y=f(x) _(3) whose abscissas liebetween and TTandarealldifferent, andthese coefficients willhavebutonesetofvalues. Forthesake ofsimplicity supposethatthenpointsaresochosen that their projectionsontheaxisofJTareequidistant. Q&\1- T--J=Ax;then thecoordinates ofthenpointswillbe[Ax,/(Ax)], [2Ax,/(2Ax)J, [3Ax,/(3Ax)], |&amp;gt;Ax,/(raAx)].Substitute them in(2)and wehave =0,1sinAx-f-a2sin2Ax-fassin3Ax-\ f-ansin =0,1sin2Ax -f-a2sin4Ax-f-assin6Ax++ sin2 r(^) /(3Ax)=isin3Ax+azsin6Ax -f- ssin9Ax-j f-ansin /(nAx)=axsin?iAx+azsin2wAx+ass^113?iAx-{-+&amp;lt; coequationsofthe firstdegreetodetermine thencoefficientsi,az,3, an. Notonlycanequations (4)besolved intheory,buttheycanbeactually solved inanygivencasebyavery simple andingenious method due to Lagrange. Letustake asanexamplethesimple problemtodetermine thecoefficients alta2,a3,a*,and 5,sothat y=atsinx-\- 2sin2x-f-a3sin3x+a4sin4x-|-a6sin5x(5) shall passthroughthefivepointsoftheline 32 DEVELOPMENT INTRIGONOMETRIC SERIES. [ART.20. 7T 2-7T Multiplythefirstequation by2sing,thesecond by2sin -g-,thethird O__ Arrr O7T by2sin-g-,thefourth by2sin -g-,the fifthby2sin -g-andadd the equations. The coefficient of2is 7T 27T 2-7T 4?T ,SlT ,6?T,9.4?T ,87T 2sin -gsm -g-+2sin~6~sin &quot;6&quot;&quot;*&quot;SmITSm &quot;6&quot; ~6~smT STT.107T -f-2sin-g-sing-; TT 2?r TT STT but 2singsin-g-=cosgcos-g-,&c. Hence thecoefficient ofazbecomes TT 2-7T ,3-7T 4?r 5?r cos 5-+cos-~-+cos-^-+cos--+cos-x- D D D D v 3?r 6?r OTT 12?r 15?r cos -7, cos -p,-cos -7. cos-7;cosj-6 D D D O and thismaybereduced bytheaidofanimportant Trigonometric formula which weproceedtoestablish. 20.LEMMA. Ilsin(2n+l)2 cos+cos20-fcos30 -i hcosnO=+o z C1) sin- ForletA^=cos+cos20+cos30H hcosnOandmultiply by2cos0. 2/Scos=2cos2+2cos cos20+2cos cos30H h2cos cosrc0 =1-fcos+cos20++cos(n 1) +cos20+cos30+cos40H hcos(n+1)0 -_2S+1+cos(n+1)cos cosnO .Hence 1 cosn cos(n+1) &quot;==~~2~2(1cos0) ^..sin(2n+!)K sin 77 CHAP.II.] NUMERICAL EXAMPLE. 33 21.Applying (1)Art.20to(7)Art.19thecoefficient ofaareduces to UTT.337T ll7T_7T337T_37T but-jo^T2a~12~ T2 therefore 2sm122sm12 and 2vanishes. Inlikemanner itmaybeshown that the coefficients ofa8)a4,and vanish. The coefficient of%is 2sinJ+2sin^J+2gin^+2sin^+2sin^O O D O v 2?r 4?r GTT STT 10-7T cos---cos--cos--cos--cos -r- 6 D O O O 2sin^2sinF6 D The firstmember ofthefinalequationis 2-7T TT 2?r 2?r,^3?r .3?r,rt4?r .4?r ._STT .5?r TT iix2-8in28in 28in, Hence &quot;i= 1XTsiuT= I&quot;(2+^=2approximately- t=i Ifwemultiply the firstequationof(6)Art.19by2sin,thesecond by A d ft 2sin~ ,thethird by2sin~ ,thefourth by2sin,the fifth 1OTT by2sin,addandreduce asbefore weshall find 2 &amp;lt;r+ JCTT .2&7T 7T 34 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.22. andinlikemanner weget 2^-sktr .Skir TT _2 &amp;lt;^kir .4&7T__7ry/3_a*- 62,IfSm~6~-&quot; 18-&quot; 2 Therefore y=2sinx 0.9sin2x-f0.5sin3cc 0.3sin4a+0.1sin5x(1) 7T 2-7T 37T cuts thecurvey=xatthe fivepoints whose abscissas are /r&amp;gt;~~ ;-rr &amp;gt; 5?r 22.Theequations (4)Art.19canbesolved byexactly thesame device. Tofindanycoefficient ammultiplythe firstequation by2sinm&x,the secqnd by2sin2m&amp;lt;\x, thethirdby2sin3m&x, &c.andadd. The coefficient ofanyother aasakintheresulting equation willbe 2sink&xsinm&x+2sin2k&x sin2mAa?+2sin3&A# sin3mAa;-}- -f-2sinTI^AX sinnm&x =cos(?/i A;)Aaj+cos2(i &)Aa;+cos3(w ^)AxH-----hcosw(m A;)Ao; cos(/H-A;)Aaj cos2(??t+^)^iccos3(m.-|-A;)AiP-----cosn(m-}-k)AOJ -sn~-(m A;)Ace sin-(m+ &quot; ,.2sini-r-^ 2sin 1j / i &amp;lt;\A -and(n+1)As=TT . Hence thecoefficient ofakmaybewrittentfmk)&x~l r. .,N (w-hA;)Aa:n (m-QTT-^-2~ \Bm I^^&quot;~ 2J (m A;)Ax.(m2sin*- - 2sin* butthis isequalto--or- -|-^according asmkisoddoreven aa andsoiszero ineither case. CHAP.II.] DETERMINATION OFCOEFFICIENTS. 35 The coefficient ofamwillbe 2sin2raAx-(-2sin22raAx+2sin23wAx-| -j-2sin2rcraAx 1 -f 1+ 1++ 1 cos2wAx cos4wAx cos6?w.Ax cos2nm&x __ 2 2smraAxJv But(2n-}-l)wAx=2m(n+l)Ax raAx=2m7rraAx, thereforesin(2n+l)mAx=sin(2ra?r-raAx)__1 2sinwAce 2sinmAx 2 andthecoefficient ofamis ?i-f-1 . The firstmember ofourfinalequationwillbe t=n 2 kkx sin Hence t=n m=^q- andthecurve y=!sinx-}-azsin2a;-|- -f-ansin ?ia;, (2) where thecoefficients aregiven by(1)willpassthrough thenpoints ofthe curve y=f(x) whose abscissas areAx,2Ax,3Ax, rcAx.Axbeing ^jr- Itshould benoted that since thenequations (4)Art.19areallofthe first degree there will exist onlyonesetofvalues forthenquantities al,az,a9, anthatcansatisfytheseequations. Consequentlythesolution which we have obtained istheonlysolutionpossible. 23.Theresultjustobtained obviouslyholds good nomatter howgreat a value ofnmaybetaken. Ifnowwesuppose nindefinitelyincreased thetwocurves(2)Art.22and y=/(x)willcome nearer andnearer tocoinciding throughout thewhole of theirportions between x= andx=TT,andconsequently thelimiting form thatequation (2)Art.22approachesasnisindefinitely increased will represent acurve absolutely coinciding between thevalues ofxinquestion withy=/(x). 3 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.24. Letusseewhat limiting value amapproachesasnisindefinitely increased. 2*=&quot; am= ^-j j-^/(&Ax)sinkm&x(1)Art. 22. i=i k=n2AT ^/(&Ax)sin;bmAx 2r- -i=/(Ax)sinmAx.Ace+/(2Ax)sin2mAx.AxH----- f-/(rcAx)sinraraAx.Ax _2 |~/{Ax)sin?raAx. Ax+/(2Ax)sin2mAx.Ax+-i TTL +/(TT Ax)sinm(7T Ax).AxJ TT sinceAx= r As 7iisincreased indefinitely Axapproaches zero asalimit. Hence the limiting value ofamasnincreasesindefinitelyis &quot;I TTAx=(&amp;gt;&quot;f/C71&quot;~~Ax)sinm(ir Ax).AxJ2limitl~/(Ax)sinmAx.Ax-f/(2Ax)sin2mAx.AxH 2=-I/(x)sinmx.dx.[v.Int.CaLArts.80,81.] &quot;J Hence/(x)=axsinx+a2sin2x+ ssin3x+ , (2) where anycoefficient amisgiven bytheformula 2/am=I/(x)sinmx.dx, (3)^r isatruedevelopmentof/(x)forallvalues ofxbetween x=andx=TT providedthat theseries(2)isconvergent,for itisinthatcaseonlythatwecan assume thatthelimitingvalue ofthesecond member ctf(2)Art.22canbeob tained byadding thelimiting values oftheseveral terms. When x= andwhen x=TTevery term inthesecond member of(2) iszero,andthesecond member iszeroandwillnotbeequalto/(x)unless/(x) isitself zerowhen x= andx=TT;butevenwhen/(x)isnotzero for x= andx=TTthedevelopment given above holds goodforanyvalue ofxbetween zeroand TTnomatter hownear itmaybetaken toeither ofthese values. 24. Instead ofactually performingtheelimination inequations (4)Art. 19and gettingaformula foraminterms ofn,andthenletting nincrease indefinitely, wemight have saved labor bythefollowing method. *Weshall usethesign=forapproaches. Ax= isreadAxapproaches zero. CHAP. II.J ABRIDGED METHOD. 37 Return toequations (4)Art.19andmultiplythe firstbyAa;sinm&x, thesecond byAa;sin2mkx, andsoon,that ismultiply eachequation byAa; times thecoefficient ofaminthatequation, andthenaddtheequations. Wegetasthecoefficient ofak sinkAxsinm&x. Ao;+sin2/vAa; sin2m&x. Aa;-| (-sinnk&x sinnm&x. Aa;. Letusfind itslimiting value asnisindefinitelyincreased. Itmaybe written, since (n-\-1)Ax=TT, limit rsin&AxsinraAa;. Aa;-|-sin2&Aa;sm2w&Aa;. AaH Aa;==L 4~sin&(TT Aa;)sinm(TT Aa;).Aa;J jr = Jsinkxsinmx.dx; o IT JT butIsinkxsinmx.dx=iI[cos (in k}xcos(in+k)x]dxJ J = ifmandkarenotequal. The coefficient ofamis Ax(sin2raAa;-{-sin22mbx+sin23mAa; -f+sin2nm&x). Itslimiting value im .1 Qsin2mAa;.Aa;+sin22mAa;.Aa; -{-+sin2 m(7T Ax)Aa; /7Tsin2mx.dx= 4 The firstmember is /(Ax)sinwAa;.Aa;+/(2Ax)sin2mAx.Ax+ +/(nAx)sinmwAx.Aa; and itslimiting value is J/(a;)sinmx.dx . o Hence thelimiting formapproached bythefinalequationasnisincreased is IT Csn am=-T/(^)si Whence a=-/^sinmx.cte asbefore. Thismethod ispracticallythesame asmultiplyingtheequation f(x)=!sinx+ asin2a;-f ssin3a;-f- (1) bysinmx.dxandintegratingbothmembers fromzero toTT . 38 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.25. Itisexceedingly importanttorealize thattheshort method ofdetermining anycoefficient amoftheseries(1)which hasjustbeen described inthe itali cized paragraph,isessentiallythesame asthat ofobtaining ambyactual elimination from theequations (4)Art. 19,andthensupposing ntoincrease indefinitely,thusmaking thecurves(3)Art.19and(2)Art.19absolutely coincide between thevalues ofxwhich aretaken asthe limits ofthe definiteintegration. 25.Wesee,then, thatanyfunction ofxwhich issingle-valued, finite, and continuous between x= and x=TT,orifdiscontinuous hasonlyfinite discontinuities each ofwhich ispreceded andsucceeded bycontinuouspor tions, canprobably bedevelopedintoaseries oftheform f(x)=jsinx-f~ 2sin2x+a8sin3x+*** (1) 2/* 2/where am= If(x)sinmx.dx=I/(a)sinma.da; (2) andtheseries andthefunction willbeidentical forallvalues ofxbetween x= andx=TT,notincluding thevalues x=andx=TTunless thegivenfunction isequaltozero forthose values. Anelaborate investigation ofthequestionoftheconvergence oftheseries (1),forwhich wehave notspace, entirelyconfirms theresult formulated above*andshows inaddition that atapointoffinitediscontinuity theseries hasavalue equaltohalf thesum ofthetwovalues which thefunction approachesasweapproachthepointinquestion fromoppositesides. Theinvestigation which wehavemade inthepreceding sections establishes thefact that thecurve represented byy=f(x) need notfollow thesame mathematical lawthroughoutitslength, butmaybemadeupofportions of entirelydifferent curves. Forexample, abroken lineor1alocusconsistingof finitepartsofseveral different and disconnectedstraight lines canbe represented perfectlywellbyy=asine series. 26.Letusobtain afewsinedevelopments. (a)Let f(x)=x. (1) Wehave x=atsinx+ asin2x+ 8sin3x-}- (2) 2~ where am=-Ia;sinmx.dx(3) *Provided thefunction hasnotaninfinite number ofmaxima andminima intheneigh borhood ofapoint,v.Arts. 37-38. CHAP.H.J EXAMPLES OFSINE SERIES. 39/^xsinmx.dx=(sinmxmxcosmx).m?v w Ixsinma;, cfa;=f-l)CT7r 771 andina; sin2# sin~~~ ~ ) Let am= tsinmx.dx ;(1) (2) cosma; sinmx.dx=, /sin Hence=(l cosWTT)=-[1 ( I)&quot;1] v L v yJ = ifmiseven = ifwisodd.m sinSx,sin5* (3) Itistobenoticed that(3)givesatonceasinedevelopment foranyconstant c.Itis, _4c/sinx .sin3x sin5a; \ TTV1* 3 5 / Ifwesubstitute x= in(4)(a)or(3)(b)wegetafamiliar result, namely f=i-|+M+&quot;- &amp;lt;5) aformula usuallyderived bysubstituting x=1inthepowerseries for tan-1*,(v.Dif. Cal.Art.135.) (4)(a)doesnotholdgoodwhen x=TT,and(3)(b)failswhen x=and when x=TT,forinallthese cases theseries reduces tozero. (c)Letf(x)=xfrom a= tox=^ and/(*)=TT a;from cc= -^tox=TT. Thatis,lety=/(*)representthebroken lineinthefigure. Asthemathematical expressionfor f(x)isdifferent inthetwohalves ofthe curvewemust break UD 10 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.26. * w&quot;i T Cf(x)sinmx.dx intoCf(x)sinmx.dx -fCf(x)sinmx.dx. -.Wehave, then, JT I W aw=\xsinmx.dx+J(TT a)sinmx.dx(1) 7T o 4 . 7T But sinm^=1 ifm=1or4&+1 =&quot;m=2&quot; k+2 = i TO=3 4A;+3 =&quot;m=4 4A . Hence ify=/(^) representsourbroken line, 4sin a; sin3x .sin5x sin7x, When x=^/(x)= ^andwehave 8= 12+32+52+72+ (d)Asacasewhere thefunction hasafinite discontinuity,let /(x)=1from x= tox=^and 7T y=s /&quot;(x)willinthiscaserepresentthelocus inthefigure. YAsbefore/{&amp;gt;, it 9 Cf(x)sinmx.dx=Cf(x)sinmx.dx IT -{-r/(x)sinmx.dx . 2 IT 22C Csinmx.dx+-^J0*siam=- |sin??Kc.dx 4--I0*sin CHAP.II.] EXAMPLES 41 IT am=-Isinmx.dx= (1cosmJ)7r%TTm\ 27 But cosm= ifm=lor4&-}-l &quot;m=3&quot; 1 m=4&quot; Hence 2/sina; ,2sin2x sinSx sin5a; .2sin6z .sinffx &quot;&quot;~~ ~~~ &quot;~~~~ If cc= thesecond member of(2)reduces to- ,for 2/111 1 . andweseethattheseriesrepresents thefunction completelyforallvalues of 17Txbetween x= and x=TTexceptforx-and there ithasa value which isthemean ofthevalues approached bythefunction asx approaches from oppositesides. EXAMPLES. Obtain thefollowing developments: 2r/7T867T\ /7T867T\-[(T-^)smx~ (J--)s7T8 .sm2x&quot; sm3x .,.2Fsinx .TT sinSa; 2?r . sin5o? (3)/( ^sin6z---- J, 42 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.27. iff(x)=xfrom x= tox=-andf(x)=from x= to a;=TT /JN2 .rsin a; 2sin2a; .3sin3x 4sin4a; (4)sm^=-s 222 if/Aisafraction. (5)e*=|[i(1+*&quot;)sinx+|(1-e)sm2x+jg(1+e&quot;)sin3* ., 2sinh TTrl 2 ..3 4 1 (6)sinhx=- -smxrsm2x+ sin3a;7^sm4aH---- . 7T l_J O 10 17 21~~1 2 (7)coshx=-- (1-j-coshTT)sina+^(1~coshTT)sin2a; 3 ~I+r^r(1+coshTT)sin3ic+. 27.Letusnowtrytodevelopagiven function of a;inaseries ofcosines. Asbefore suppose thatf(x)hasasingle value foreachvalue ofxbetween x= and x=TT,that itdoesnotbecome infinite between x= and x=TT,andthat ifdiscontinuous ithasonlyfinite discontinuities. Assume f(x)=+&icos aj+^2cos2x-f 3cos3x+ (1) Todetermine anycoefficient bmmultiply (1)bycosmx.dx andintegrate eachtermfrom toTT. /bcosmx.dx=0. bkcoskxcosmx.dx=^ |[cos(m k)x-fcos(w+tyx^dx o= ifmand A;arenotequal. /bmcos2mx.dx=r-2*C?x+cosmxsinwx), m2mv n bmcos2mx.o^= 6m,ifmisnotzero. Hence bm=-T/()cos Twx.efce=-T/(a)cosma.da, (2) ifmisnot ero. CHAP.II.] EXAMPLES OFCOSINE SEKIES. 43 Togetbmultiply (1)bydxandintegrate from zero toIT. f&quot;*** ir Ibkcoskx.dx=0. =^Cf(x)dx=^Hence 5=~(f(x)dx=-f(a)da, (3) which isjusthalfthevalue thatwould begiven byformula(2)ifzerowere substituted for ra. Tosaveaseparate formula(1)isusuallywritten f(x)=%b+#1cosx-f#2cos2x+bacos3x+ (4) andthen theformula 2/ 2 /= I o/=-I/(a)cosma.da(2) o willgivebQaswell astheother coefficients. Itisimportanttoseeclearlythatwhatwehavejustdone indeter mining the coefficients of(1)isequivalenttotaking n-f-1terms of(4), substitutingin y=^b -f-bicosx+ 2cos2a-J-+bncosnx(5) inturnthecoordinates ofthen+1pointsofthecurve whoseprojections ontheaxisofJTareequidistant, determiningb,bi}bt)bn byelimination from then+1resulting equations, andthentaking thelimit ingvalues theyapproachasnisindefinitelyincreased,(v.Art.24.) IT IfAx=n-=theabscissas ofthen+1points used are0,Ax,2Ao;, wAcc, sothatweshould expectourcosine developmenttohold for x= aswell asforvalues ofxbetween zeroand TT. 28.Letustakeoneortwoexamples: (a)Let /(*)=* (1) 2 2/ 2 2 /=- J.xcos 7w*.&amp;lt;*e=^(cosmir1)=^[(-1)-- 44 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.28. Hence x=(cosx-\1- 27r\ 325* (2)holds good notonlyforvalues ofxbetween zeroand ITbutforx= andx=TTaswell, since forthese values wehave and ,.+ which aretruebyArt.26(c)(3). ()Letf(x)=xsin a?.(1) b=IXSilLX.dx= TTJ 7T 2r 1r 1 0i=Isinxcos ie.cfo=-Ixsin 2o%&amp;lt;&;= - ,V &quot;V bm= ja;sinxcos w#.&amp;lt;&c= |[ajsin(m+1)a;sin(m l)ic]c?a; ^IT V ifmisodd 2 r- rIT ifmiseven. Hence cosx2cos2x,2cos3x 2cos4# -.____ 42 1.3 3.5 5.7. -I- If=wehave EXAMPLES. Obtain thefollowing developments: ics6a? ,cosIQx ,cos14a; ,&quot;] &quot;F&quot; ~5^~ ~J1~~&quot;J if/()=a;from a;= to a;=and/(cc)=TTxfroma= tox=TT. CHAP.II.] EXAMPLES. 45 ,n\ jy\1 t2[&quot;COSXCOS3x .COS5x COSIx (2)f&=+ ~--- 5 iff(x)=1from x= tox=and/(#)=from x=-tox=TT. 7T2 .1/57TA 2 &quot;I+s2(T~ /cos&quot;&quot; 62cos 6a;&quot; J if/(a;)=from x= tox=~and/(a;)=from a=-tox=TT. (6)*=(^-1)- *(+!)cosx+ i(*- 1)cos2x (7)- + cos4*--- J. 2Fl 1 (8)sinhx=--(coshTT1)^(cosh-TT+1)cos a; -f--(coshTT1)cos2x(coshTT+1)cos3xH----1 . 2ftsinftTrr1 cosx ,cos2x cos3x TT L2ft2 /*21 cos4x(9)cosf^-I2^2- 2_i22_22u2-3a ifyu.isafraction. 29.Although anyfunction canbeexpressedboth asasine series andasa cosineseries, andthefunction andeither series willbeequalforallvalues of xbetween zeroandTT,there isadecided difference inthetwoseries forother values ofx. Both series areperiodicfunctions ofxhaving theperiod27r. Ifthenwe letyequal theseries inquestion andconstruct theportionofthecorrespond- 46 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.29. ingcurve which liesbetween thevalues x= TTand x=ITthewhole curve willconsist ofrepetitionsofthisportion. Since sinmx= sin(mx)theordinatecorrespondingtoanyvalue of xbetween TTandzero inthesinecurve willbethenegative oftheordinate correspondingtothesame value ofxwith thepositive sign. Inother words thecurve y=&!sinx-\-azsin2x-{- asin3x-J- (1) issymmetricalwithrespecttotheorigin. Since cosmx=cos(mx)theordinatecorrespondingtoanyvalue ofx between TTandzero inthecosine curve willbethesame astheordinate belongingtothecorresponding positivevalue ofx.Inother words thecurve y=%1Q-\-^cosx-f-bzcos2x-\-bacos3x-\-- (2) issymmetricalwithrespecttotheaxisofY. Ifthenf(x)=/( x) ,that isiff(x)isanoddfunction thesine series correspondingtoitwillbeequaltoitforallvalues ofxbetween TTandTT, except perhapsforthevalue x forwhich theseries willnecessarily be zero. Iff(x)=f(x),that isiff(x)isaneven function thecosine series cor respondingtoitwillbeequaltoitforallvalues ofxbetween x= TTand x=TT,notexcepting thevalue x=0. Asanexampleofthedifference between thesineandcosine developments ofthesame function letustake theseries forx . y=2sinxsin2.r,sinSa; sin4a; (3) cos3x~cos5x cos7x [v.Art.26(a)andArt.28(a)]. (3)representsthecurve and(4)thecurve ^ Y CHAP.II.] FOURIER SSERIES. 47 Both coincide with y=xfrom x= tox=TT,(3)coincides with y=xfrom x= TTtox=TT,and neither coincides withy=.xfor values of#lessthan TTorgreater than TT.Moreover(3),inaddition to thecontinuousportionsofthelocus representedinthefigure, gives theiso lated points (7T,0) (7T,0) (37T,0)&C. 30.Wehave seen that iff(x)isanoddfunction itsdevelopmentinsine series holds forallvalues ofxfrom TTtoTT,asdoes thedevelopmentof f(x)incosine series iff(x)isaneven function. Thus thedevelopmentsofArt.26(a),Art.26Exs.(2), (4),(6);Art.28(fl) Art.28Exs.(3), (7),(9)arevalid forallvalues ofxbetween TTand TT. Anyfunction ofxcanbedevelopedintoaTrigonometricseries towhich it isequalforallvalues ofxbetween TTand TT . Letf(x)bethegiven function ofx.Itcanbeexpressed asthesumofan even function ofxandanoddfunction ofxbythefollowing device. identically; but-* ~-^--isnotchanged byreversing thesignofxand istherefore aneven function ofx\andwhenwereverse thesignofx, s/x\_s/_x\ &amp;lt;&quot;-*^- -isaffected onlytotheextent ofhavingitssignreversed anda isconsequently anoddfunction ofx. Therefore forallvalues ofxbetween TTand TT \cosx-{-bzcos2x-\-#3cos3x-f- i A 2f*f(x) -4-f(x)where bm=I:L^^g*fcosmx.dx; andV -f- 2sin2a;-f-a9sin where am= sin bmandamcanbesimplifiedalittle. J=^ IT =- ICf(x)cos7x.c?a;+/(&quot;&quot;x)cos ^&quot;^-^ &amp;gt; 48 DEVELOPMENT INTRIGONOMETRIC SERIES.[ART.30. but ifwereplacexby x,weget W -IT I/( %)cosmx.dx=Cf(x)cosmx.dx= \f(x)coswx.dce, IT 1/^andwehave bm= I/(a;)cosmx.dx . 7T/ Inthesamewaywecanreduce thevalue ofamto 1r* Ijn[x)sinmx.dx . IT Hence (/()= 2&o+&icosx-f&2cos2-fb8cos3#-f ) ( +aisin cc+a2sin2x+ 8sin3aj+ ) where bm=Cf(x)cosmx.dx=-Cf(a)cosma.da.(3) IT IT and am=- J/(#)sinwx.rfcc=- J/(a)sinma.c?a .(4) IT IT v andthisdevelopmentholds forallvalues ofxbetween TTand TT. Thesecond member of(2)isknown asaFourier sSeries. EXAMPLES. 1.Obtain thefollowing developments,allofwhich arevalidfrom x= TT tox=TT: ... 2sinh7rrl 1 .1 _ 1 1~| (1)e*=-- --cosx.+gcos2cos3a;4- cos4a;+&quot; .2sinh-TT rl 2 . .3 4~|H--- -sm a?-sm2x-hsm3x sin4x+ (2)2sinh-TT rl 2 . .3 4- -sm a?-sm2x-hsm3 TT2r .cos3x .cos5a; .cos7a; 4--|_s*++^+ .^sin a; sin2x ,sin3a; sin4a; ,~r~^~ ~3~ ~T~ wheref(x)=from= TTto a?=andf(x)=a;fromx= to a;=TT. CHAP.II.] EXTENSION OFFOURIER SSERIES. 49 (3)f(x)=-+- acosx+- 2cos2*-h- 2cos3z+-cos5xa 2 2 2 62I,cos6*+] 37T ../37T &quot;I- J, where/(x)=xfromx= TTto=0,f(x)=from sc= tox=- , and /()=x -rfrom x=-tox=TT .^ *j 2.Show thatformula(2)Art.30canbewritten f(x)-ccosj8+cicos(x ft)+c2cos(2aj ft) -f-c8cos(3x ft)H---- where Cm=(a*+*andft^tan-1 ^. 3.Show thatformula(2)Art.30canbewritten /(*)= |cosinft+Clsin(a;-fft)+c2sin(2x+ft)+c8sin(3x+ft)H---- where cm=(a*+6J)* and/3m=tan-15a. am 31.Indevelopingafunction of ccintoaTrigonometricseries itisoften inconvenient tobeheldwithin thenarrow boundaries x= TTandx=TT . Letusseeifwecannot widen them. Let itberequiredtodevelopafunction ofxintoaTrigonometricseries which shall beequaltof(x)forallvalues ofxbetween x= cand a;=c. Introduce anewvariable 7T =*&amp;gt; which isequalto TTwhen x= candtoTTwhen x=c. /(x)=/(- )canbedevelopedinterms of*byArt.30(2), (3),and(4). Wehave y/^z J=-b+&!cos2+ 2cos2z+ cos3z+ f,^ + isin2+#2sin2z+a&sin3*+ ) where 6m=Cf( z)cosmz.dz .(2) 50 DEVELOPMENT INTRIGONOMETRIC SERIES. [ART.31. and am=- If\~%\sinmz.dz .(3) IT and(1)holds goodfrom z= TTtoz=IT . Replacezbyitsvalue interms ofxand(1)becomes 1 7TX, 27TCC . 3lTX, f(x)=-ft+^cos-*+ acos--f&scos -+C C C TTX . .27TX . .37TX .+ !sin h2sm|-a8sin rc c c(4) The coefficients in(4)arethesame asin(1),and(4)holds good from 0:= ctox=c. Formulas(2)and(3)canbeputintomore convenientshape. 1Cdc\ 7 l/&quot;^/\ mirx*jbm=If(z}cosmz.dz=-If(x)cos--ctemTTjJ\7T I TTJ^CC bm=()cos dx= c c Inlikemanner wecantransform(3)into c c 1/^ x mTra; _ 1/-/%x.WTrA .. am=cJ/(*)fr&amp;lt;&= ;J/(*)8in & Bytreatinginlikefashion formulas(1)and(2)Art.25andformulas(4) and(2)Art.27weget .v .TTX . . . . .^ f(x)=!sm--hsin-f-asm--+ (7) where =?Cf(x)sin==? &amp;lt;fo=? f/(A)sin^-XA . c*/ c c*/t/(8) 1 irx . ZTTX ..w^/ ( ^^vand j(xj==o^o T&quot;*icos r^2cos r^cos r(&quot;/ ^2 c c&quot; where J.=? J/^)cos2=? &amp;lt;te= J/(X)cos rfX .(10) and(7)and(9)holdgoodfrom x= tox=c. CHAP.II.] EXTENSION OFFOURIER SSERIES. 51 EXAMPLES. 1.Obtain thefollowing developments: 4r .TT.T 1 STTX,1fax . &quot;1 (1)i=-LsmT+g8m+gsm+J from x= tox=c. 2Cr .7TX 1 2-TTX ,1 37TX 1 47TZ . &quot;| (2)x= sm-- -sin--hsm--7sm--h* I v/7rL_e2 c 3c4 c J from x= *ctoaj==c. TTX .1 STTX .1 STTO; 1 TTTCC ST+^COS+PCOS+7COS from ic= tox=c. 4\ 7TX 7T2 .27T ./7T24\ 7T2 .47T3:./7T24\ 57TX -Tsin+(&quot;5 )sm from a;= tox=c. c24c2Tra; 1 27rz .1 STTJC 1 4?ra;+32cos-j2cos from #= cto=c. 4-ecTTO;.2(1ec ).2 in 4(1 47T2 cos 1- from x= tox=c. /Kv-,v4c |~&quot;.frx 1 37rx .1 .OTTX ,()/(*)~ 2|_sln~ 32sm~7~&quot;r52S1~ J from x= tox=c, where/(x)=xfrom x= tox= |and/(x)=cxfrom x= ^to 52 DEVELOPMENT INTRIGONOMETRIC SERIES. [ART.32. 2.Show thatformula(4)Art.31canbewritten f(x)=-CQcosft+c\cos (ft)+cacos ( ftJ /STTX+C3COS I ft where b*W+W)and^=tan~1^ 3.Show tliatformula(4)Art.31canbewritten f(x)=|c&amp;lt;&amp;gt;sinft+G!sin(~-fft)+c2sin(-^p+ft) -fc8sin(+ft where cm=(a*+&JDandfim=tan-x^ 32.Intheformulas ofArt.31cmayhave asgreatavalue asweplease, sothatwecanobtain aTrigonometricSeries forf(x)that will representthe givenfunction throughasgreat aninterval aswemaychoose totake.If, then,wecanobtain thelimiting formapproached bytheseries(4)Art.31as cisindefinitelyincreased theexpressioninquestion oughttobeequaltothe givenfunction ofxfor allvalues ofx.Equation (4)Art.31canbewritten asfollows ifwereplace6,bft,, i,o, bytheir values givenin Art.31(5)and(6). fj/(X)cos=*cos=dX +J&amp;gt;)cos cos=A+... c c + j&amp;gt;(X)sin2JsinSdX+//Wsin^sin^rfX+] TT\ TTX . .7T\ .7TX 4-cos cos hsm sin 7T 2-Tra; ,2?rX .2-Tra; 4-cos-cos--hsm- sin-+ c c c c CHAP. II.] FOURIER SINTEGRAL. 53 +cos- c(\-x)+cos^(X-x)+- cos x~a cos x&quot;* +COS(-^(X-3)+COS(- )(A-X)+ since cos(&amp;lt;)=cos &amp;lt;f&amp;gt;. f)(*-*&amp;gt;cos-*~* cos .TT OTT, x ,TT TTxH--COS-(A X)H--COS (X X)c cvc cv +Zcos^(X-x)+. ](1) As cisindefinitelyincreased the limiting value approached bythe parenthesisin(1)is (X Icosa(A. x).da. co Hence thelimiting formapproached by(1)is Jcosa(X- a:).da , (2) andthesecond member of(2)must beequaltof(x)forallvalues ofx. Thedouble integralin(2)isknown asFourier sIntegral, and since itisa limiting form ofFourier sSeries itissubjecttothesame limitations asthe series. Thatis,inorder that(2)should betruef(x)must befinite, continuous, and single valued forallvalues ofx,orifdiscontinuous, must haveonlyfinite discontinuities.* (2)issometimes giveninaslightlydifferent form. co oo Since(cosa(X x).da= jcosa(X x).da+Jcosa(A. x).da 00 SB00 Tcosa(X x).da=Tcos(a)(A. x).d( a)= Jcosa(X x).da -00 00 * jcosa(\ x).da=2 |cosa(\ x).daand o *Seenoteonpage38. 54 DEVELOPMENT INTRIGONOMETRIC SERIES, and(2)maybewritten f(x)=i J/(X)&amp;lt;*xJcosa(\-x).da.(3) oo Iff(x)isanevenfunction oranoddfunction(3)canbestillfurthersimpli fied. Let/(*)=-/(-) Since thelimits ofintegrationin(3)donotcontain aorXtheintegrations maybeperformedinwhichever orderwechoose. That is oo oo Nowf/(X)dX fcosa(X x).da=Cdaf/(X)cosa(X x).d\. -oo-so uu f/(X)cosa(X x).d\ f/(X)cosa(X x).d\+f/(X)cosa(X cc).dX. oo oo f/(X)cosa(X a).dX=f/( X)cosa(Xx).d( X) oo oo oo =~&quot;f/(X)CS &quot;(^+C)^X and(3)becomes oo oo f(x)=-Cdaf/(X) [cosa(X x)cosa(X+x)~\.d\ 00 00 =CdaCf(\)sinaXsinax.d\ &quot;JJ oo OD f(x)=-f/(XXXTsinaXsinax.da .(4) or Iff(x)=/( x) (3)canbereduced inlikemanner to f(x)=-Cf(\)d\Ccosa\cosax.da .(5) Although (4)holds forallvalues ofxonlyincasef(x)isanoddfunction, and(5)onlyincasef(x)isaneven function, both(4)and(5)hold for all positivevalues ofxinthecaseofanyfunction. EXAMPLE. (1)Obtain formulas(4)and(5)directly from(7)and(9)Art. 31. CHAPTER III. CONVERGENCE OFFOURIER SSERIES. 33.ThequestionoftheconvergenceofaFourier sSeries isaltogether too largetobecompletelyhandled inanelementarytreatise. Wewill,however, consider atsome length oneofthemost importantoftheserieswehave obtained, namely 4f~. .sm3x .sin5aj .sinIx~\ O^//\T - ^smx+ h51^1J,[v.(3)Art.26(5).] andprovethat for allvalues ofxbetween zeroand TTitssum isabsolutely equaltounity ;thatis,thatthelimit approached bythesumofnterms ofthe series sinxAina.da+sin2xAin2a.da -fsin3xAinSa.da+ , asnisindefinitely increased,is1,providedthatxliesbetween zeroand TT. Let Sn= sinxAina.da+sin2xAin2a.da -fsin3xAin3a.da-\ JT +sinnxAinna.da .(1) Then Sn= (&quot;[sin asinx-f-sin2asin2#+sm^asm3a;+ hsinnasinnx~\daV 1*&quot;=-T[cos (a 05)cos(a-f )+cos2(a cc)cos2(a+x)-\ +cosn(a x)cosn(a+)]^ =-T[cos (a x)+cos2(a x)+cos3(a x)-\ hcosn(a x)~\da o T[cos (a+x)+cos2(a+*)+cos3(a+SB)+ hcosn(i4 56 CONVERGENCE OFFOURIER SSERIES. {ART. Therefore byArt.20(1) 1-sin(2,1+1)1,sin(2n+l) *&quot;5?J-~T=^- ^a~2^J ~^T^-da o sin-o sin- Inthe firstintegral substitute(3for- ,andinthesecond integralsub- a-4-x stituteftfor~- . Weget ?_ E+ iVsin^g ikn(2. +l)^ n Itremains tofindthelimit approached bySnasnisindefinitelyincreased. 34. r r_sin_(27J sisin8 2 For sin o Letusconstruct thecurve= .+cos2/3+cos4^8H----- \-cos2n/8 , byArt. 20. smx Wehaveonlytodraw thecurve y=sin(2n-fl)icandthen todivide thelength ofeach ordinate bythevalue ofthesine ofthecorresponding abscissa. Iny=sin(2n+l)zthesuccessive arches intowhich thecurve is divided bytheaxisofXareequal, andconsequentlytheir areas areequal. CHAP. III.] SUMMATION OFASINE SERIES. 57 CONVERGENCE OFFOURIER SSERIES.[ART.35. Ineither caseeachparenthesisisanegative quantitysince and itfollows thataisgreater than. 2 Again 7T_ ,,x ,,, , N.ai-f(&aas;&quot;r(4%;-r ~t-(&n_2an-i)~ran if7tisevenand 7T_ ^ ifnisodd. Ineither caseeachparenthesisispositive and itfollows that aais lessthan . Since aandaa-^differ from bylessthan theydiffer from each other, that is,bylessthan av. Inlikemanner wecanshow that aQa:anda%+azdiffer from bylessthan a2;and ingeneral that a%-f-aza8+ akdiffers from bylessthanak\oreven that Zi 7TP differs from bylessthan aknomatter what thevalue ofp,provided pis greater thanunity. 35.From what hasbeenprovedinthelast article itfollows that 6 *-: dx . sin a; where bissome value between7^:-and-^ ,differs from bylessthan 2n-\-\2iA thearea ofthearch inwhich theordinate ofy=i-^-: correspond-smx ingtox=bfalls ifthisordinate divides anarch, orbylessthan thearea ofthearchnextbeyondthepoint (b,0)ifthecurve crosses theaxisofJTat thatpoint. CHAP.III.] SUMMATION OFASINE SERIES. Thearea ofthearch inquestionislessthan- - ,itsbase,Ztij1 ,avaluegreater than thelength ofitslongest ordinate. sinfb- -j-r) Therefore fsm(2 .ro+*)*dxJsmx differs from bylessthan59 sin /b sinft w sin/3 __1rsin(27i+l)/3 ?rJsin/8^ 2 This lastvalue forSncanbesomewhatsimplified. Substituting y=(3weget XIfnownisindefinitely increased- :r approaches2ri4-1 / TT\ sin Ift^-T: I zero asitslimit, andwegettheveryimportantresult 6 limitFCsin(2n+l)a?,n_TT^\n=oo|_J sinx~ 2 o 36 .4=33 _* p L sin(2.1+1)0, 1 sin(2n+I)/? &quot;J&quot; sin/?&quot; rsm(2n+l)fi rsm(2n+l)yd__pn(n ^ J sin^8&quot;J sinyyJ sin/8x x 60 CONVEKGENCE OFFOUJRIEfi SSEiilES.[ART.37. Substituting y=TTftin IT a; pin(2+l)flwehayeJ smM IT i ?+* ?_? ?22 22 2 /sin(2?i+!)/? , /*sin(2n-fl)y , /*sin(2+!)/? ,^ dp= I *-: ay=I *-: dpsm/3 J siny/ sin/? 7T 22 TTir_x 2 22 sin(2n+l)/3/sin(2n+l)/8 J sin/2pJ sin0u Hence 2psin(2n+1)/?2Vsin(2+1)^2Asin(2n+I)/? = siu^^+^J^J~*^J&quot;~* limitFrsin(2n+1)/? 7_~|TT .A^^, ...,,OJ? _._ I^^dft\=-zif &amp;lt;x &amp;lt;TTby(1)Art.35 .^l_/ sinu ^-J and limitrf.n(2+l)g&quot;I IT . f 0&amp;lt;;c&amp;lt;w by(1)Art.3g. =oo|J sm/8 ^J 2 -o Therefore limitr^I=1+i-i=1ifO &amp;lt;TT and 7i=00LJ 4F,sin3^c .sin5x sin7a? ~1 -Lsm*++++]=! forallvalues ofxbetween zeroand TT. 37.Byasomewhat long butnotespeciallydifficult extension oftherea soning just givenitcanbeshown that iff(x)issingle-valued andfinite between x= TTandx=?r,andhasonlya^/rnite number ofdiscon tinuities andofmaxima andminima between cc= TTandx=TTthe Fourier sSeries o&o+*icos*+^2cos2.x+^3cos3xH + isin a?+.asin2ic+assin3o?-j CHAP.III.] DIKICHLET SCONDITIONS. 61 7T 1Cwhere am=-I/(a)sinma.da TT\J n it 1rand 6m= If(a)cosma.da, Fourier sSeriesonlyisequaltof(x)for allvalues ofxbetween x= TTandx=TT,excepting thevalues ofxcorrespondingtothediscon tinuities of/(),andthevalues TTand TTif/(TT)isnotequalto/( TT); andthat ifcisavalue ofxcorrespondingtoadiscontinuityoff(x),thevalue oftheserieswhen x=cis andthat if/(TT)isnotequalto/( TT)thevalue oftheserieswhen x= TT andwhen a;=TTis Iff(x)whilesatisfying theconditions named inthepreceding paragraph exceptforafinitenumber ofvalues ofx,becomes infinite forthose values, the series isequaltothefunctionexceptforthevalues ofxinquestion provided JT thatCf(x)dxisfinite anddeterminate,(v.Int. Cal.Arts. 83and84.) IT 38.Thequestion oftheconvergencyofaFourier sSeries andthecondi tions under which afunction maybedevelopedinsuch aseries was first attackedsuccessfully byDirichlet in1829, and hisconclusions have been criticised andextended bylater mathematicians, notably byRiemann, Heine, Lipschitz, andduBoisReymond.Itmaybenoted that thecriticisms relate nottothesufficiency buttothenecessityofDirichlet sconditions. Anexcellent resume oftheliterature ofthesubjectisgiven byArnold Sachse inashort dissertation published byGauthier-Villars, Paris, 1880, entitled &quot;EssaiHistoriquesurlaRepresentationduneFonction Arbitraire duneseule variableparuneSerieTrigonome trique.&quot; 39.Agood dealoflightisthrown onthepeculiaritiesoftrigonometric seriesloytheattempttoconstruct approximatelythecurves correspondingto them. Ifweconstruct y=alsinxand y=azsin2xandadd theordinates ofthepoints having thesame abscissas weshall obtain pointsonthecurve 62 CONVERGENCE OFFOURIER SSERIES.[ART.39. y=&!sinx-{-#2sin2x . Ifnowweconstruct y=a&sin3xandaddtheordinates tothose of y=.0,1sinx+ asin2#weshallgetthecurve y=CL-Lsin ic+&amp;lt;z2sin2#-f-a8sin3x . Bycontinuingthisprocess wegetsuccessive approximationsto y &!sin#+#2sin2a;+assin3cc+#4sin4a?+ Letusapplythismethod toafewoftheseries which wehave obtained in ChapterII. Take y=sinx-}-- sin3x-\-- sin5x-f- (1) =when o;=0,-from x= to CC=TT,and when X=TT, v.Art.26[&](3). y=2/sin#-sin2z-fasin3x-sin4z-fj(2) =xfrom x= tox=7r,and when cc=7r, Art.26[a](4). y=- Ijjsinx- 2sin3x+- 2sin5x- 2sin7x-f---- I(3) =xfrom ic= tox=,and TTxfrom #=-oto#=7r, Art.26[c](2). 2/=Tsin a;+-sin2+-sin3z+Psin5x-sin6x-\--sm7x-\----(4)1 A O O D / =when x= ,-from x=tox= ,andfromx=-tox=TT. _ Z ^ v.Art.26 Itmust beborne inmind thateach ofthese curves isperiodic havingthe period 2?r,and issymmetricalwithrespecttotheorigin. Thefollowing figures I,II,III,andIVrepresentthe first fourapproxima tions toeach ofthese curves. Ineach figure thecurve y=theseries, andtheapproximationinquestion aredrawn incontinuous lines, andthepreceding approximation andthecurve correspondingtotheterm tobeadded aredrawn indotted lines. CHAP.III.]SUCCESSIVE APPROXIMATIONS TOASINE SEKIES. 68 \ ,-~, y , ---^^^^X^X T^ \ --** X-N ,-&amp;gt;A 64 CONVERGENCE OFFOURIER SSERIES. [ART.39*. x\IV \ * ;\\ /x\ CHAP.III.] PROPERTIES OFFOURIER SSERIES. 65 Tigs. I,II,III,andIVimmediately suggest thefollowingfacts : (a)Thecurve representingeach approximationiscontinuous evenwhen thecurve representingtheseries isdiscontinuous. (b)When thecurve representingtheseries isdiscontinuous theportionof each successive approximatecurve intheneighborhoodofthepoint whose abscissa isavalue ofxforwhich theseries curve isdiscontinuous approaches moreandmore nearlyastraightlineperpendiculartotheaxisofXandcon necting theseparate portionsoftheseries curve. (c)Thecurves representingsuccessive approximationsdonotnecessarily tend tolose theirwavy character, since each isobtained from thepreceding onebysuperposing uponitawave linewhose waves areshorter eachtimebut donotnecessarilylose their sharpnessofpitch.This isthecase inFigures I,II,andIV. InFig.Illthewaves ofthesuperposedcurves grow rapidly flatter. Itfollows from thisthat insuch cases asthose representedinFigures I,II, andIVthedirection oftheapproximatecurve atapoint having agiven abscissa doesnotingeneral approachthedirection oftheseries curve atthe corresponding point,orindeed, approach anylimiting value, astheapproxima tion ismade closer and closer; and that thelengthofanyportionofthe approximatecurve willnotingeneral approachthelengthofthecorrespond ingportion oftheseries curve. Analyticallythisamounts tosayingthat thederivative ofafunction ofx cannot ingeneral beobtained bydifferentiating termbyterm theFourier s Series which representsthefunction. (d)Theareabounded byagiven ordinate, theapproximate curve, theaxis of X,andanysecond ordinate willapproachasitslimit thecorrespondingareaof the series curve iftheseries curve iscontinuous between theordinates in question; andwillapproachtheareabounded bythegiven ordinate, theseries curve, theaxisofX,anysecond ordinate, andalineperpendiculartotheaxis ofX,andjoiningtheseparate portionsoftheseries curve ifthelatter hasa discontinuity between theordinates inquestion. Analyticallythisamounts tosayingthat theFourier sSeries corresponding toanygiven function canbeintegratedtermbytermandtheresultingseries willrepresent the integralofthefunction evenwhen the function is discontinuous(v.Int. Cal.Art.83). Wemaynote inpassingthat ifthefunction curve iscontinuous acurve representing theintegralofthefunction will becontinuous andwill not changeitsdirection abruptlyatanypoint;while ifthefunction curve isdis continuous thecurve representingtheintegralwill stillbecontinuous butwill changeitsdirection abruptlyatpoints correspondingtothediscontinuities of thegiven function. 66 CONVERGENCE OFFOURIER SSERIES.[ART.40. 40.The facts thatthederivative ofaFourier sSeries cannot ingeneral be obtained bydifferentiating theseries termbytermandthat itsintegral canbe obtained byintegrating theseries term byterm aresoimportant that itis worth while tolook atthematter alittle moreclosely. Letusconsider the differentiation oftheseriesrepresentedinArt.39FigureI. Let Sn=sinx+Qsin&e+-=sin5x++.1sin(2n+1)#.o o 2iii~p1 Then -r-2=cosx-f-cos3x+cos5x+ -fcos(2n-fl)x.dx dx andthecurve isparalleltotheaxisofXforx= nomatter what the value offt. Ifx= orx=TT andthecurvey=S nbecomes more nearly perpendiculartotheaxisofX attheorigin andforx=TTasweincrease ?i. That is =&quot;ft=2 ft= Consequently when a?=~doesnotapproach anylimiting value asftis indefinitelyincreased. Indeed, inthe successive approximationsthepoint whose abscissa is issuccessively ontherear,onthefront, andonthecrest o orinthetroughofawave, andalthough thewaves aregettingsmaller theydo notlosetheir sharpnessofpitch.7Cf Ifxhasanyother value between and TTnwillchange abruptlyasnis changed andwillnotapproach anylimiting value asnisincreased. CHAP.III.] DIFFERENTIATION OFFOURIER SSERIES. 67 41.Ingeneralifwedifferentiate aFourier sSeries 1S=~#o~r^icosx-\-b2cos2x-f-bscos3x+ + isinx-|-a2sin2#-f-a3sin3#+** weget b:sinx2b2sin2x 3bssin3x +aicosx-f2a2cos2x-f-3a8cos3#-f-*** Differentiate again andweget &!cosx2262cos2x32bscos3x !sinx22a2sin2x3\sin3x Weseethateachtimewedifferentiate wemultiply thecoefficient ofsinkx andofcoskxbykwhile theterm stillinvolves coskxorsinkx . Since theseries cosx-f-cos2x-{-cos3x-f-** -f-sinx-f-sin2x-\-sin3x-f- isnotconvergent, andaFourier sSeries converges onlybecause itscoefficients decrease asweadvance inthe series, thedifferentiation ofaFourier sSeries mustmake itsconvergencelessrapidifitdoesnotactually destroy it,and repetitionsoftheprocesswillusually eventually make thederived series diverge. Itistobeobserved thatthederived series areFourier sSeries, butofsome whatspecial form, that istheylacktheconstant term.(v.Art.30.) Ifnowweintegrate aFourier sSeries o#o~h^icosx-\-b2cos2x-f-^3cos3x-f-** 2 +%sinx-f- 2sin2x+assin3x-f- weget C+o^o35~i~^is^n^H~7T^2sin2x-\--b ssin3x+ i Z O 1 I !cosx- 2cos2,x7?a3cos6x,A O aTrigonometricSeries which converges more rapidly than thegivenseries. Itistobeobserved that theseries obtained byintegrating aFourier s Series isnotingeneral aFourier sSeries owingtothepresenceoftheterm $box. (v.Art.30.) 42.Wearenowreadytoconsider theconditions under which afunction of xcanbedevelopedintoaFourier sSeries whose termbyterm derivative shall beequal tothederivative ofthefunction. 68 CONVERGENCE OFFOURIER SSERIES. Letthefunctionf(x) satisfytheconditions stated inArt. 37.Then there isoneFourier sSeries andbutonewhich isequaltoit.Call this series S. Letthederivativef(x)*ofthegivenfunction alsosatisfy theconditions stated inArt. 37.Thenf(x) canbeexpressedasaFourier sSeries. ByArt. 39(d)theintegralofthis latter series willbeequaltotheintegralof/(#), that istof(x)plusaconstant, andoneintegralwillbeequaltof(x). Ifthis integral which isnecessarilyaTrigonometricSeries isaFourier s Series itmust beidentical with S.ItwillbeaFourier sSeries onlyincase theFourier sSeries forf(x)lacks theconstant term%bQ. But =1Cf(x)dx by(3)Art. 30. Therefore b=[/(TT)-/(- TT)] ; andwillbezero if/(TT)=/( TT). Inorder thatf(x)shallsatisfy theconditions stated inArt.37f(x)while satisfyingthesame conditions must inaddition befinite andcontinuous between x= TTandx==TT. If,then, f(x)issingle-valued, finite, and continuous, andhasonlyafinite number ofmaxima andminima, between x= TTandx=TT,(thevalues x= TTandx TTbeing included), andif/(TT)=/( TT)f(x)canbe developedintoaFourier sSeries whose termbyterm derivative willbeequal tothederivative ofthefunction. Itwillbeobserved that inthiscasetheperiodiccurve y=Siscontinuous throughoutitswhole extent. 43. Since aFourier sIntegralisalimitingcase ofaFourier sSeries the conclusions stated inthischapter hold, mutatis mutandis foraFourier s Integral. Forexampleifafunction ofxisfinite andsingle-valuedforallvalues ofx andhasnotaninfinite number ofdiscontinuities orofmaxima andminima in theneighborhoodofanyvalue ofxitwillbeequaltotheFourier sIntegral /(A)cosa(A. x).d\ andtothatFourier sIntegral only,andtheintegralwith respecttoxofthis Fourier sIntegralwillbeequaltoCf(x)dx. Ifinadditionf(x)isfiniteandcontinuous forallvalues ofxthederivative oftheFourier sIntegral with respecttoxwillbeequalto *Weshall regularly usethenotation f(x)for-^.v.Dif. Cal.Art. 124. CHAPTER IV. SOLUTION OFPROBLEMS INPHYSICS BYTHEAIDOFFOURIER S INTEGRALS ANDFOURIER SSERIES. 44.InArt.7wehavealready considered atsome lengthaproblemin Heat Conduction whichrequired theuseofaFourier sSeries.Weshall begin thepresent chapter with aproblem closely analogousinitstreatment tothat ofArt.7,butcallingfortheuseofaFourier sIntegral. Supposethatelectricityisflowinginathinplanesheet ofinfinite extent andthatthevalue ofthepotential function isgivenforevery pointinsome straightline inthesheet, requiredthevalue ofthepotential function atany pointofthesheet. Letustake thelineastheaxisofXandconsider atfirstonlythosepoints forwhich yispositive: Wehave, then, tosatisfy theequation (1) subjecttotheconditions F=0 wheny=oo(2) r=f(x) y=(3) wheref(x)isagiven function, andwearenotconcerned with negative values ofy. AsinArt. 7wehave e~aysinaxand e~aycosaxasparticular values ofV whichsatisfy (1)and(2).Wemustmultiply them byconstant coefficients andsocombine them astosatisfycondition(3). By(3)Art.32 - 00 00 f(x)=-Cdaf/(A)coso(X x).d\. (4) 00 Wewish tobuild upavalue ofVwhich willreduce to(4)when y=0. Thisrequires alittle carebutnotmuchingenuity. 70 SOLUTION OFPROBLEMS INPHYSICS.[ART.44. Take e~avcosaxand ^&quot;^sinaa; andmultiplythe firstbycosaX,and thesecond bysinaX;theyare stillvalues ofVwhichsatisfy (1).Add these andweget e~aycosa(X x), stillavalue ofVwhich satisfies(1),nomatter what thevalues ofaand X. Multiply byf(\)d\andwehave e-VMcosa(X-x).d\ (5) asavalue ofVwhich satisfies(1). V=Ce-ayf(\)cosa(\ x).d\ (6) 00 isstillasolution of(1)since itisthelimit ofthesumofterms covered by theform(5);andfinally .00 oo V=-CdaCe~*yf(\)cosa(Ax).d\ (7) isasolution of(1)asitismultiplied bythelimit ofthesum ofterms formed bymultiplyingthesecond member of(6)bydaandgivingdifferent values toa. But(7)must beourrequiredsolution since while itsatisfies(1)and(2),it reduces to(4)when y=andtherefore satisfies condition(3). Iff(x)isanevenfunction wecanreduce(7)totheform 00 00 V=-CdaCe-vf(\)cosaxcosa\.d\(8) and iff(x)isanoddfunction totheform 9V=-CdajVay/(X)sinaxsina\.d\.(9) (7), (8),and(9)arevalid onlyforpositivevalues ofy,butastheproblemis obviously symmetricalwithrespecttotheaxis ofX,(7), (8),and(9)enable ustogetthevalue ofthepotentialfunction atanypointoftheplane. EXAMPLES. 1.Obtain forms(8)and(9)directly bytheaidof(5)and(4)Art. 32. 2.State aprobleminstatical electricityofwhich thesolution givenin Art.44isthesolution. CHAP. IV.]FLOW OFELECTRICITY INANINFINITE PLANE. 71 45.Asaspecialcaseunder Art.44letusconsider theproblem: Totind thevalue ofthepotential function atanypointofathinplanesheet ofinfinite extent where allpointsofagivenlinewhich lietothe leftoftheorigin are keptatpotential zero,and allpoints which lietotherightoftheorigin are keptatpotential unity. Heref(x)= if x&amp;lt;0 andf(x)=lif x&amp;gt;0. (7)Art.44givesustherequiredsolution. Itis .00=-Cda ^cosa(\ x).d\ ; (1) butthiscanbemuchsimplified. Wehave V- Cd\Ce~avcosa(\ x).da. /* aNow e~axcosmx.dx= -J-T- \f a&amp;gt;0.(Int.Cal.Art. 82,Ex.8.) Hence|e-aycosa(\ x).da=J andconsequently K.(2)7T\ ?/ 7T C Sincelog=log(x-fyi)=-log(x2+y2 )+itan~x |, [Int.Cal.Art.33(2)], 7T 7T 47T \ 7T X and 1--tan~Jand log(x*-f-y2 )areconjugate functions, (v.Int. f7T 3& *Jr Cal.Arts. 209and210.) Hence ^-^log^+y2 ) (3) isasolution oftheequation O; (4) 72 andthecurves andSOLUTION OFPROBLEMS INPHYSICS.[ART.45. -log(3*+^=ft(6) cuteach other atright angles. Ifweconstruct thecurves obtained bygiving different values toain(5)we getasetofequipotentiallines fortheconducting sheet described atthebegin ningofthis article, andthecurves obtained bygivingdifferent values to(b)in (6)willbethelinesofflow. Moreover since _12. 2?r isasolution ofLaplacesEquation (4),thelines offlowjustmentioned willbe equipotentiallines foracertain distribution ofpotential,forwhich theequi potentiallines above mentioned willbelines offlow. V=a, that is reduces to =xtanair . Ifnowwegivetoavaluesdiffering byaconstant amount wegetasetof straightlines radiating from theoriginanalatequal angularintervals. Vi=b,that is ft, (6) reduces toe~.(8) Ifwegivetobasetofvalues differing byaconstant amount wegetaset ofcircles whose centres areattheorigin andwhose radii form ageometrical progression. Theyaretheequipotentiallines forathinplanesheet ofinfinite extent where thepotential function iskept equaltogivendifferent constant values onthecircumferences oftwogivenconcentric circles orwhere wehave asource atthe origin; and forthis systemthelines(7)arelines offlow, and(3)isthecompletesolution. The figure givestheequipotential lines and lines offlow foreither sys tem,butonlyforpositivevalues ofy. Thecomplete figurehastheaxis ofX =i asanaxisofsymmetry. CHAP.IV.]FLOW OFELECTKICITY INANINFINITE PLANE. 73 EXAMPLES. 1.Solve theproblem ofArt.44forthecasewhere /()=!ifx&amp;lt;0 and/()=1if x&amp;gt;0. Ans., F=-tan-1-. TTy 2.Solve theproblem ofArt.44forthecasewhere f(x)=aifx&amp;lt;0 andf(x)=l&amp;gt;if x&amp;gt;0. Ans.,r=i a 3.Reduce(7), (8),and(9)Art.44totheforms =-(fjr+J ^ respectively. 46.Anespecially interesting case ofArt.44isthefollowing where /(aj)= ifx&amp;lt;-l, /()=!if- 1&amp;lt; a; &amp;lt;1,and/(x)= if x&amp;gt;l. Here F=-rtan-1^^+tan-1in^l. mTrL y yJ Now ilog[(1- )i]=ilog[(1-a;- yi)i]=ilog[y+(1- *&amp;gt;] =log[(1- a;)+y]+ and 7T 7T i, 12i, (i z)2+?/2 ,ir,1+z, ,1n -log =log^j:;7 4--tan-1 ! \-tan-1 . TT 1a2?r(1-f&quot;*)TyTTL y 2/-J 74 SOLUTION OFPROBLEMS INPHYSICS.[ART.46. Hence 1L IJ+S . i1~^\ IT (1 X)2+?/2 -(tan-1--htan-1- )andlog^--. (. \\7r\ y y/ 27r& (1+xY+y2 areconjugate functions;* and tan-1=a(2)7T y y/ isanyequipotential line,and anylineofflow forthesystem described atthebeginningofthis article;and &amp;gt; isthesolution ofanewproblemforwhich(3)represents anyequipotential lineand(2)anylineofflow. *Thefunction conjugate to might havebeenfound asfollows. If &amp;lt;/&amp;gt;istherequired function and\f/thegiven function we havebyInt.Cal.Arts. 211,212,and213therelations Dx&amp;lt;f&amp;gt;=Dy^andDy&amp;lt;f&amp;gt;=Dxf. 1~l+x 1x Ifnowweintegrate Dy^with respect toxtreating yasaconstant andaddanarbitrary function ofyweshallhave &amp;lt;f&amp;gt;.Sothat =~ j{log[(1+x)2+y2 ]-log[(1- i Comparingthiswith itsequal Dx\f/abovewefind-&quot;and/(y)= aconstant therefore2^(1 +)++C whereCmaybetaken atpleasure,isourrequired conjugatefunction. CHAP.IV.] SOUKCE ANDSINK INANINFINITE PLANE (2)reduces to 2.2 -i=*ana7r and(3)to z2+y2+2^75 (5) +1=0 or or(a;+ctnh&7r)2-fif=(6) (5)and(6)arecircles. The circles(5)have their centres intheaxis ofYf andpassthrough thepoints(1,0)and(1,0);andthecircles(6)have their centres intheaxisofX. (4)isthecomplete solution, (6)isanyequipotentiallineand(5)anylineof flow foraplanesheet inwhich thepointsinthecircumferences oftwogiven circles whose centres arefurther apart than thesumoftheir radii arekeptat different constantpotentials,orwhere asource andasink ofequal intensity areplacedatthepoints (1,0)and(1,0).Animportant practical ex ampleiswhere twowires connected with thepolesofabatteryareplaced with their freeends incontact with athinplane sheet ofconducting material. Thefigure shows theequipotentiallinesandlines offlowofeither system. Thecomplete figure would have theaxis ofXforanaxisofsymmetry. 1.Show that if/()=axwhen JB &amp;lt; *,/()=a2when * &amp;lt;x &amp;lt;b, f(x)=a8when a; &amp;gt;b, A 1 /-v^ 2aj)tan-1 1-(2 s)tan&quot;1 76 SOLUTION OFPROBLEMS INPHYSICS.[ART.47. 2.Show that if/(aj)=0ifx&amp;lt;0,f(x) =a1if&amp;lt;x&amp;lt;bltf(x)=azif b1&amp;lt;x&amp;lt;bz,f(x)=a3if62&amp;lt;x &amp;lt;b3.&c., F= attan&quot;1- -|-(i 2)tan&quot;11-(a,a,8)tan&quot;1 TTL y y y ,y-I 3.Show that iff(x)= 1ifx &amp;lt; 1,/(re)=reifl&amp;lt;re&amp;lt;l, 4.Show that iff(x)= 1ifa &amp;lt; 1,/()=0 if 1&amp;lt;a &amp;lt;1, f(z)=lifaj&amp;gt;l, T^1T,1+ ,1-aHF=- tan&quot;1!--tan&quot;1- .wL y yJ Show thattheequipotentiallines areequilateral hyperbolas passing through thepoints (1,0)and(1,0),andthat thelines offlow areCassinian ovals having (1,0)and(1,0)asfoci. The lines offlowareequipotentiallines andtheequipotentiallines arelines offlow forthecasewhere thepoints (1,0)and(1,0)arekeptatthesame infinitepotential,orwhere verysmall ovals surrounding thesepointsarekeptatthesame finitepotential. Thecase isapproximatelythat ofapairofwires connected with thesamepoleofa battery whose otherpoleisgrounded, andthen placedwith their ends incon tactwith athinplane conductingsheet. 5.Show that iff(x)=Qifx&amp;lt;0,f(x)= 1if0^x&amp;lt;a, f(x)=0 ifa&amp;lt;x&amp;lt;b, and/(re)=1ifx&amp;gt;b, 1r-TT .ax ,bx .x~\V=-\-tan-1--tan-1-- tan-1- . TTL2 y y yj Theconjugatefunction V=2^log[(a-aO isthesolution forthecasewhere asinkandtwosources ofequal intensitylie ontheaxisofX,thesink attheorigin andthesources atthedistances aand btotherightoftheorigin. Oneofthelines offlow iseasilyseen tobethe circle x2 -\-y2=ab . 47. Iftheplane conducting sheet hastwostraight edgesatright angles witheach other andone iskeptatpotentialzerowhile thevalue ofthepoten- CHAP.IV.] EXAMPLES. 77 tialfunction isgivenateachpointofthesecond, that isifF= when z= andV=f(x) when2/=0,thesolution isreadilyobtained. Itis V=Ida(e~avf(\)sinaxsin v.(9)Art. 44. This reduces to v.Ex.3Art. 45. EXAMPLES. 1.IfF=0 wheny=andV=F(y)when ce=show that 2rrV~~\da\e~axF(X)sinaysinaA.c^A. o o 2.IfV=f(x) when y=andV=F(y) when o;=show that 3.IfF(y)=6theresult ofEx.2reduces to 4.IfF(y)=lfor &amp;lt;y&amp;lt;1andF(y)=Qfory&amp;gt;l while/() for &amp;lt;a &amp;lt;1andf(x)= for a; &amp;gt;1 n-i^=^-tan-*i+2tan-i2 y y x +tan-^-tan- 78 SOLUTION OFPKOBLEMS INPHYSICS.[ART.48. 5.Ifoneedgeoftheconducting sheet treated inArt.47isinsulated, sothatDXV=0 ifx=andV=f(x) when y= cosaxcosaX.dX 48. Iftheconducting sheet isalong stripwithparallel edges oneofwhich isatpotentialzerowhile thevalue ofthepotential function isgivenatall pointsoftheother, that isifV=0 when y=andV=F(x) when y=btheproblemisnotaverydifficult one. Since wearenolonger concerned with thevalue ofVwhen y=ooF= eaysinaxandV= ea*cosaxareavailable asparticularsolutions ofthe equation D*r+D&amp;gt;v=Q(i) aswell asV=e~aysinaxandV=e--y cosax . (&y_l_p-y Consequentlysinax=coshaysinax[Int.Cal.Art.43(2)] e*ye-*v and sinax=sinhaysinax[Int.Cal.Art.43(1)] and coshaycosazand sinhaycosa# arenowavailable values ofFand canbeusedpreciselyase~aycosaxand e~aysinaxareused inArt. 44. Following thesame course asinArt.44weget oo asasolution of(1)which willreduce toV=F(x)when y=b and to V=0 when y= ,since sinh=-= , and(2)istherefore ourrequiredsolution. IfVistobeequaltozerowhen y=bandtof(x)when y=wehave onlytoreplace ybybyandF(x) byf(x)in(2).Weget CHAP. IV.]FLOW INALONG STRIP WITH PARALLEL EDGES. 79 IfV=f(x)when y=andV=F(x)when y=bthen 00 Thiscanbeconsiderably simplified bytheaidoftheformula .WTT ., sin- Csinhpx TT q I ..cosrx.dx=Jsinhqx 2q PTT , ,iir o cos(-cosh ifp*&amp;lt;q2 .[BierensdeHaan, Tables ofDel Int.(7)265]andbecomes d\ ^ 7r(b ?/). .TT-* cosx ,J;4-cosh-(\ 17T?//_,/xN -^sm-fIm)2b bJ^ -* cos-f- -}-cosh(A. x]or 2*S1 &quot; *i.koshf(X-x)-cos7+ coshf(A-*)+cos EXAMPLES. 1.Given theformula f..**,==tan- tanh ifb &amp;gt;aJa+bcoshxv^^^ V^^+&quot;&quot; &quot; 2/ show that ifV=I when?/=andV= when y=^F=- (i y). 2.Show that ifF=0 wheny=i,F= 1when y= and a; &amp;lt;,andV=1when?/=andx &amp;gt; .TTXtanh-- &quot;n Thesolution fortheconjugate system, thatis,forastrip having asource at (0,0)andaninfinitely distant sink is 80 SOLUTION OFPROBLEMS INPHYSICS.[ART.48. 3.Show that ifV \when y=andx&amp;lt;0,F=l when y= and x &amp;gt;,F= 1when y=band x &amp;lt;,andF= 1when y=and .r &amp;gt;, F=-tan-1 (tan^(b y)tanh^)+-tan-1(tan^ytanh^7T \Zt&amp;gt; AU 7T \Zc&amp;gt; Zb =tan&quot;1 7TTTiC Thesolution fortheconjugate system, thatis,forastriphaving asource anda sink atthepoints (0,0)and(0,b)is .TTX .TrycoshT+cos V=-log-. 7T L_ TTX TTVJcosh cos-f1&quot; b b 4.IfF=0 when x=Q,V=f(x)when y=andx&amp;gt;0, andF=0 when y=band a? &amp;gt;, :[cosa(X x)cosa sinh 1 Tryfr =2rmtJ[oosh(X-x)-cosScosh(X+a)-cos forpositivevalues ofxandforvalues ofybetweeen and b. 5.IfFi=when cc^O, ^1=^(0;)wheny=6andx&amp;gt;0, and F!=when y=and cc &amp;gt; ^ =1sin cosh X+x+cos forpositive values ofccandvalues ofybetween and b. 6.IfF2=0when 3=0,F2=/()when y= and a; &amp;gt;0,and Vt=F(x)wheny=band aj &amp;gt; .F2=F+F! for x&amp;gt;0 and 0&amp;lt;7/&amp;lt;6. (v.Exs.4and5) 7.Ifoneedgeofthestripdescribed inArt.48isinsulated sothatwehave F=/()when y=andDyV=b when ?/=show that =1CdaftrjJ cosh/(X)cosa(X- CHAP.IV.] FLOW OFHEAT INONEDIRECTION. Bytheaidoftheformula fCOShpX 7T : cosrx.ax= coshqx q,T7T cosh cos 2/ P7Trii . .?7T cos hcosh q q [BierensdeHaan, Def. Int.Tables(6)265] , reduce thisto81 ifP&amp;lt; 1 .7T?7/*&quot;x ftSm 2ftJ&quot;~^_aocoshj(\ x) cosy- 8.IfF=0 when7/=056and x&amp;lt; a,V=l when?/= orb anda&amp;lt;x&amp;lt;a, andF=0 when y= or Z&amp;gt;andx &amp;gt;a V=- 7Tsinh il sm7T?/4-tan-., It Itt-^ JL, I sinh^- 11?ry _|sin^i- ft 9.IfK=0 when?/= orband x&amp;lt;,V=1when y=and a&amp;lt;z&amp;lt;a, T=0 when y= orband&amp;gt;, andV= 1when=ftand a &amp;lt;ic&amp;lt;Ca r=- 7Ttanh TTt/tan-tan 10.Asystem conjugate tothat ofEx.9isF=+ccwheny= or ft andx=a,F= oowhen y= or ftandx=a.Inthiscase sin2-+sinh2^- - Tr 1 -ift ftF=--log--.^rinia+rinhil&ltf) ft ft 49.Letustakenowaproblemintheflow ofheat.Suppose wehavean infinite solid inwhich heat flowsonlyinonedirection, andthat atthestart the temperature ofeachpoint ofthesolid isgiven. Let itberequiredtofindthe temperature ofanypoint ofthesolid attheendofthetime t. Herewehave tosolve theequation (i) =0.(2)[v.Art. 1(n)] subject tothecondition u=f(x) when 82 SOLUTION OFPROBLEMS INPHYSICS. [ART.49. Astheequation (1)islinear with constant coefficients wecangetaparticu larsolution bythedevice used inArts. 7and 8. Letu=ePt+axand substitute in(1).Weget astheonlyrelation which needholdbetween(3anda. Hence u=e*+&quot;2a2&amp;lt;=e2 &quot;2e**(3) isasolution of(1)nomatter what value isgiventoa. Togetatrigonometricform replaceabyai. Then M=er-^e*1 . Ifin(3)wereplace abyaiweget AsinArts. 7and8wegetfrom these values u&~a2a2&amp;lt;sinaxand w=e~a2*2cosaz asparticularsolutions of(1),abeing whollyunrestricted. From these values wewish tobuildupavalue ofuwhich shall reduce to f(x)when=andshall stillbeasolution of(1). Wehave f(x)=1CdaCf(\)cosa(A-x).d\ (4) 30 v.Art.32(3),andbyproceedingasinArt.44weget u=-Cdafe-a&amp;gt;aW /(A)cosa(A x).d\ (5) TJV/*/ oo asourrequiredvalue ofu. Thiscanbeconsiderably simplified. Changingtheorder ofintegration oo 2a2cosa(A x).da. (6) la=\le~ 4a2 (7) +J ZtU *u bytheformula Ce-acosbx.dx=^e-5 [Int.Cal.Art.94(2)] I 2a o 00 Hence&quot;=2ofe//(X)r^* k &quot; ^ CHAP.IV.] EXAMPLES. 88 \ .y.Letnow ft= -^, then X=x+2a&amp;lt;fi.ft EXAMPLES. 1.Letthesolid beofinfinite extent and letthetemperature beequaltca constant catthetime t=. Then u= v.Int. Cal.Art.92(2). 2.Letu=xwhen=0. Then u=4=(x+2afi.ft)e-Pdft=x. 3.Letu=x*when t=0. Then u 4.Letu= if x&amp;lt;b,u=l ifb&amp;lt;x&amp;lt;b, andu= ifx&amp;gt;^,when t=. Then 6 a; M=J_re-e^__j_TJ^8+3&ry+io^^&amp;gt;+5fa*_ .1 ^rJ V^Ua 32o&amp;gt; 5.2!2a^6 J 5.Letw= if a&amp;lt;0 andw=lif ic&amp;gt;0 when=0. 6.Aniron slab10c.m.thick isplaced between and incontact withtwo very thick iron slabs. The initialtemperatureofthemiddle slab is100,and ofeach oftheouter slabs0.Required thetemperature ofapointinthe middle oftheinner slab fifteen minutes after theslabs havebeenputtogether.Given a3=0.185 inC.G.S. units.Ans., 21.6. 84 SOLUTION OFPROBLEMS INPHYSICS.[ART.50. 7.Twoverythick iron slabs oneofwhich isatthetemperature andthe other atthetemperature 100throughout areplaced together face toface. Find thetemperatureofeach slab10c.m.from theircommon face fifteen minutes after theyhavebeenplaced together. Ans., 70.8, 29.2. 8.Find aparticularsolution of Z&amp;gt;tu=a2Du ontheassumptionthat it isoftheform u=T.X whereTisafunction oftalone andXisafunction ofxalone. 50.Ifoursolid hasoneplanefacewhich iskeptattheconstant tem perature zero,andwestartwithanygivendistribution ofheat, theproblemis somewhat modified. Take theoriginofcoordinates intheplaneface. Thenwehave asbefore theequationDtu=a*D2u, (1) butourconditions are u=when x=Q(2) u=f(x)t=(3) andweareconcernedonlywithpositivevalues ofx. Wemaythenusetheform(4)Art.32 oc oo f(x)=-Cdaf/(A.)sinaxsina\.d\, (4) o o andproceedingasinthelastsection weget u=-CdaCe-^fQ^)sinaxsina\.d\(5) o o asourrequiredsolution. Thismaybereduced considerably-. 1/? ru=-i /(*)&amp;lt;&.J&-&amp;lt;**[cosa(\ x)cosa(A+x)&quot;]da, 00 or u=l f/(X) (e~^Sr er(J ^lr)d\ (6) 2aV/7rto by(7)Art. 49,andthismaybereduced totheform u=a.2a.-e -x+2a.^)^-(7) EXAMPLES. 1.Lettheinitial temperaturebeconstant andequaltoc. CHAP. IV.] EXAMPLES. 85 Then 2aVt =^\^_ i_+_^^+ VTTi-2a^t 3.(2a^)3 5.2!(2ay)6 7.3!(2ay^)7 2.Assuming thattheearthwasoriginallyatthetemperature 7000 Fahren heitthroughout, andthatthesurface waskeptattheconstanttemperature 0, find(1)thetemperature 10miles below thesurface 10,000,000 years after the cooling began; (2)thetemperature 1milebelow thesurface atthesame epoch; (3)thetemperature 10miles below thesurface 100,000,000 years after thecooling began; (4)thetemperature1milebelow thesurface atthesame epoch; (5)therate atwhich thetemperature wasincreasing with thedistance from thesurface ateachpointateachepoch. Neglect theconvexityoftheearth ssurface andtake SirWm.Thomson s value ofa2 (400)thefoot, theFahrenheitdegree, andtheyearbeing taken as units.(Thomson andTait sNat. Phil. Vol. II.Appendix.) Ans., (1)3114; (2)329.5; (3)1036; (4)103; (5)1forevery 20feet,3 forevery 50feet, 1forevery 50feet, 1forevery 50feet. 3.Lettheinitialtemperature beconstant andequalto b,thenbyEx.1 X 2aVi 2bre. 4.Letthetemperature oftheplane facebebinstead ofzero,and letthe initialtemperature bezero. Thenwehaveonlytoaddbtothesecond member ofthesolution inEx. 3, aswemay since u=bisasolution of(1)Art.49,andweget SaVt 5.Letu=bwhen x=andu=f(x)when=0. Then ZaVt (A- by(6)Art. 50. 86 SOLUTION OFPROBLEMS INPHYSICS. ART. 51, 6.Letu bwhen x=andu=cwhen t^=Q. Then w== &+(&amp;lt;._&) 7.Iftheearth hasbeencooling for200,000,000 years from auniform tern perature, prove that therate ofcoolingisgreatest atadepthofabout 76 miles, andthat atadepthofabout 130miles therateofcooling hasreached itsmaximum value foralltime. Letaz400. 8.Show that iftheplane faceofthesolid considered inArt.50instead of being keptattemperature zero isimpervioustoheat tt=i=J/(A)(fl**+e~** )d\.v.(6)Art. 50. 51. Ifthetemperature oftheplane faceofthesolid described inArt.50 isagiven function ofthetimeandtheinitialtemperatureiszero, thesolution oftheproblem canbeobtained byavery ingenious method duetoKiemann. Herewehave tosolve theequation Dtu=a*Dxu(1) subjecttotheconditions u=F(t)when x=^ i(2)u=*=0.) Weknow that isasolution of(1),v.Ex.1Art. 50. Itiseasily shown that (3) where cisanvconstant,isasolution of(1). For 2x 1* x _|J* g=_ \7r2aVt c and CHAP.IV.]TEMPERATURE OFFACEAFUNCTION OFTHE TIME. 87 Let &amp;lt;f&amp;gt;(x, t)beafunction ofxand Iwhich shall beequaltozero iftis negative andshall beequalto iftisequaltoorgreater thanzero;sothat ifa= &amp;lt;f&amp;gt;(x, i)=land if t= &amp;lt;f&amp;gt;(x,t)=Q. Weshallnowattack thefollowing problem,tosolveequation (1)subjectto theconditions u= if t= u=F(0)&quot;x= and &amp;lt;t &amp;lt;T u= F(kr)&quot;x= kr&amp;lt;t&amp;lt;(k +l)r, where kisanywhole number andTisanyarbitrarily chosen interval oftime. Ifweform thevalue u=F(kr) [&amp;gt;(*,t-kr) -&amp;lt;j&amp;gt;(x tt-(k+l)r)] (4) uwillsatisfy equation (1)since zero, unity and arevalues ofuwhichsatisfy (1).?willbezero if t&amp;lt;kr bythedefinition ofthefunction &amp;lt;j&amp;gt;(x, *);if#=w= if&amp;gt;(&-J-l)randu=F(kr)if Therefore *=00 (5) isthesolution oftheproblem stated above. (5)canbesimplified somewhat from theconsideration that foragiven value oft &amp;lt;(&amp;gt;,tkr)=0if kr&amp;gt;t.If,then, nristhegreatest wholemultiple ofTnotexceeding t, k=n u=^ F(kr )[&amp;gt;(*,*-kr)-j(x,t-(k +l)r)].(6) Ifnowwedecrease Tindefinitely thelimiting form of(6)willbethesolu tionoftheproblem stated atthebeginningofthis article. (6)maybewritten 88 SOLUTION OFPROBLEMS INPHYSICS.[ART.51. and ifrisindefinitelydecreased thelimiting form of(7)is t Since tXispositivebetween thelimits ofintegration and(8)maybewritten t & &quot; **4^zTt TV ^~w*^t A)/i^^\\ _7\/O\ orifwelet ft= , 8.(10) EXAMPLES. 1.If t/=ntwhen cc=andw=when t= 2.Athick iron slab isatthetemperaturezerothroughout, oneofitsplane faces isthen keptatthetemperature100 Centigrade for5minutes, then at thetemperaturezero forthenext5minutes, then atthetemperature100 for thenext 5minutes, andthen atthetemperaturezero. Requiredthetem peratureofapointintheslab5c.m.from theface attheexpirationof18 minutes. Given; a2=.185. Arts., 20.l. 3.Ifu=F(t)when x=andu=f(x)when t= ,then /t v.(6)Art. 50. CHAP. IV.]TEMPERATURE APERIODIC FUNCTION OFTHETIME. 89 4.IfinArt.(51)F(t)isaperiodic function ofthetime ofperiodTitcan beexpressed byaFourier sseries oftheform m=oo 1 OTT F(t)=-b+5)\_amsinma*+bmcosma*]&amp;gt;where a=, m=l or where/3mcos\m=amandpmsinXm=6m. v.Art.31Ex.3. Show thatwith thisvalue ofF(t) (10)Art51becomes oo m= oo oo^pm[sin(mat+^/e&quot;!cos m=1 x -cos(mat+XJe-^8sin H andthat astincreases uapproaches thevalue n=oo _ 17Ix^ xi/ma. x Given that ^sin= r^&quot;8injV2;e-3*cos dx=e^^ cosbV2. v.Riemanri) Lin.par. dif. gl.54. 5.Ifwearedealing with abarofsmall cross-section where theheat not onlyflows along thebarbutatthesame timeescapesatthesurface ofthe barinto airatthetemperaturezerowehave tosolve the differential equation Dtu=a?Dxub*u . v.Fourier, Heat 105. Show that forthiscase u=e~(W+alal)tsinaxand u=er&amp;lt;**+alat)cosax areparticular solutions, andthat ifu=f(x) when t= =C \7TJ cf.(8)and(9)Art. 49. 90 SOLUTION OFPROBLEMS INPHYSICS.[ART.51. Ifu=when x=andu=/(#)when t= u= cf.(7)Art. 50. Ifu=e~^when t=andu=when x= 00 00 u=4=[*(V(6vr+ &amp;lt;*- &amp;lt;r?(V and ifu=1when x=andw=when t=wehave onlytoadd e~&quot;? tothesecond member ofthelastequation,since u=e~~z satisfies the equation Ifu=F(i)when x=andu=when=wecanemploy the method ofArt. 51. +(*t-A)=e-T+~ and u= cf.(9)Art. 51, u==e-^-^F(t-jj)i cf.(10)Art. 51. IfF(t)isperiodic andhasthevalue taken inEx. 4,show that thevalue approached byuastincreases is snma*- where p=(62+V/&4+m2a2 )andgr= (6+Vfl4+iV CHAP.IV.] ANGSTKOM SMETHOD. 91 Given and where=--e~ /a 7T ^sin- &amp;lt;2dx= -e~^sin2d dx= e~2ccos Angstromsmethod ofdetermining theconductivityofametal isbased on theresultjustgiven (v.Phil.Mag. Feb.1863), and isdescribed bySirWm. Thomson(Encyc.Brit. Article&quot;Heat &quot;)asbyfarthebest that hasyetbeen devised. 52. Ifuisaperiodicfunction ofthetimewhen x= asinArt.51Ex.4 andweareconcerned with thelimiting valueapproached byuastincreases wecanavoidevaluating acomplicateddefiniteintegralifwetakethefollowing course. Since aswehave seen inArt.49u=e^t+axisasolution of provided onlythatI3=a2azwehave M=^ asasolution. Replacing /?by f&thisbecomes =e*^ or tt=e*P since =1(1) and Hence areparticular solutions of(1). 92 SOLUTION OFPROBLEMS INPHYSICS.[ART.53. From thesewegetreadily x\f^i./ xIma .\ =pmeaVTsin(mat- -\-g-+Xmj(4) asasolution.(4)reduces to u=pmsin(mat+A.m)when #= andto M=pOTe-fTsinXfB-7when #=0. Ifweaddatermwhich satisfies(1)andwhich isequaltozerowhen x= andtopme-^Ysin fA.TO-- \~o~)wnen *= (v.Art.50)weshall have asolution of(1)which iszerowhen t= andwhich is pmsin(mat -\-\m)when a;=. Theterm inquestion approacheszero astincreases[v.(7)Art.50]andwe have atoncethesolution giveninArt.51Ex.4,asourrequiredresult. EXAMPLE. Show thatu=ef&quot;+a*isasolution ofDtu=a^ubzuif(3=a*az b*, andhence that -^T=-)&amp;gt;andu=e*aV/2cos((ft -p),7 \ / VS&amp;gt; where p=[V/82+b*-f62]iand q= aresolutions. Hence JE2L So* V3u=pme~^-sin(pt*-. \ &v isasolution. If/?=ma this lastresult reduces tou=pOTsin(mat -f-Am)when x= andbythereasoningofArt.52itmust bethevalueuapproachesastincreases ifwehave thesame conditions asinthelastpartofArt.51Ex. 5. 53.Thewhole problemoftheflowofheat istreated bySirWilliam Thom son(v.Math, andPhys. Papers, Vol.II),andother recent writers from adif ferent anddecidedly interesting pointofview, which weshallbrieflysketch inconnection with theproblemofLinear Flow. Suppose wearedealing with abarhaving asmall cross-section andanadia- thermanous surface, andtake asourunitofheattheamount requiredtoraiseby aunitthetemperatureofaunit oflengthofthebar. Ifatapointofthebara CHAP. IV.] INSTANTANEOUS HEAT SOURCES. 93 quantity Qofheat issuddenly generatedthepointiscalled aninstantaneous heat source ofstrength Q. Iftheheat instead obeing suddenly generatedisgenerated graduallyand ataratethatwould giveQunits ofheatperunit oftime thepointiscalled a permanentheat source ofstrength Q. Thetemperatureatanypointofthebaratanytimeduetoaninstantaneous source ofstrength Qatthepointx=Xiseasilyfound bytheaidofformula (8)Art.49asfollows: IfaquantityofheatQissuddenly generated along theportionofthebar from x=Xto a-=X-fAX,where AX isanyarbitrary length, thetem peratureofthatportionwillbesuddenlyraised to,andweshall haveby (8)Art.49 A+AA Qu= asthetemperatureofanypointofthebaratanytime tthereafter. Ifnowwewrite uequaltothelimitingvalue approached bythesecond member of(1)asAA.ismade toapproachzeroweget (2) asthesolution forthecasewhere wehave aninstantaneous source atthe pointx=X . Itistobeobserved that in(2)u=when t= and u== 2a\l-jrt when .x=Xand t&amp;gt; . Ifwehave several sources wehave onlytoaddthetemperatures duetothe separatesources. Formula(8)Art.49maynowberegardedasthesolution forthecasewhere westart with aninstantaneous heat source ofstrength /(X)c?Xinevery element oflength ofthebar. Asource ofstrength Qiscalled asink ofstrength Q-,and(6)Art.50 mayberegarded asthesolution forthecasewhere wehave atthestart an instantaneous source ofstrength /(X)e?Xinevery element ofthebarwhose dis tance totherightoftheoriginisX,andaninstantaneous sink ofstrength /(X)(/Xinevery element ofthebarwhose distance totheleftoftheoriginisX. Ifwehaveaninstantaneous source attheorigin (2)reduces to u=j=.erM (3) 2a\/7rt 94 SOLUTION OFPROBLEMS INPHYSICS.[ART.54. Forapermanent source ofconstantstrength Qattheorigin (3)gives t Q /*a#iu=1= Ie~4a}(t-T) (t r)adr (4) 2a\7rJ^ o andforapermanent source ofvariable strength f(t) M=_j_A ^__^ 2a^rJ In(4)and(5)^obviously reduces tozerowhen t=andx &amp;gt;,but its valuewhen x= isnoteasilydetermined. Wecanavoid thedifficulty by introducing theconceptionofadoublet. 54.Ifasource andasink ofequal strength Qaremade toapproach each other whileQmultiplied bytheir distance apartiskept equaltoaconstant P thelimitingstate ofthingsissaid tobeduetoadoublet ofstrength Pwhose axis istangenttotheline ofapproach andpoints from sink tosource. A doublet ofstrength Pdiffers from adoublet ofstrength Ponlyinthat its axishastheoppositedirection. Letusfindthetemperature due toaninstantaneous doublet ofstrength P placedattheorigin. Forasource ofstrength Qatx=^andanequalsink atx=??wehave ...^/., .,&quot; \ 4a2&amp;lt;t?4e2&amp;lt; )j orif2vQ=P, ~P /! -I--rZ * ...Tf,x e~~2a*t) If17ismade toapproachzero Px &,^and u=-==e~55 (1) isthesolution forthetemperatureatanytimeandplaceduetoaninstantane ousdoublet ofstrengthPplacedattheorigin.Foradoublet atanyother point x=\wehave P(x- CHAP. IV.] PERMANENT DOUBLET. 95 Forapermanentdoublet ofconstantstrength Pplacedattheoriginwe have t 6^^ (t~T)~*dr5 (3) andforapermanentdoublet ofvariablestrength f(t)^ (t~r &amp;gt;~*-&amp;gt;w* , (4) ifsc&amp;gt;0, and ifx &amp;lt;0,ifwelet(3=== From(5)and(6)weseereadilythat u=when t= andthat u= 7j2w^enx ifweapproachtheorigin from theright andthat f(t}u=^-jwhen x= ifweapproach theorigin from the left. Ifthepointx= iskeptattheconstant temperaturebandwearecon cerned onlywithpositivevalues ofxwecangetfrom(5)thesolution given in Art.50Ex.4bysupposingapermanentdoublet ofstrength2a?bplacedat theorigin. Tosolve theproblemtreated inArt.51wehaveonlytosupposeapermanent doublet ofstrength 2a?F(t) placedatx= andfrom(5)wegetatonce (10)Art. 51. EXAMPLE. Show that ifDtu=a?Du tfuandaninstantaneous source ofstrength Qisplacedatx=\ -* t-**ZZT v.Art. 51,Ex. 5. Show that ifaninstantaneous doublet ofstrengthPisplacedatthepoint Px * 96 SOLUTION OFPROBLEMS INPHYSICS.[ART.55. Ifapermanentdoublet ofstrength f(t)isplacedatx= whence w=when t= and #&amp;gt;0 orx &amp;lt;andw=*y-fwhen2a2 a=0. Hence ifweplaceatx= apermanent doublet ofstrength 2a*F(t) we getthesolution giveninArt.51Ex.5forthecasewhere u=F(t)when x=andu=when=provided weareconcernedonlywithpositive values ofx . IfF(t)=cthisreduces to c/*w=-j=Ie~p2 7rJ 55.Asanother exampleoftheuseofFourier sIntegral weshall consider thetransmission ofadisturbance along astretched elasticstring. Suppose wehave astretched elastic stringsolong thatweneed notconsider what happensatitsends, that issolongthatwemaytreat itslengthas infinite. Letthestring beinitiallydistorted intosome given formandthen released;toinvestigateitssubsequentmotion. LetustakethepositionofequilibriumofthestringastheaxisofXand anygiven pointasorigin. Wehave, then,tosolve thedifferential equation V*y=a*Dy (1) [v.(vin)Art.1]subjecttotheconditions y=f(x)when *=(2) Dttj=&quot;t=Q.(3) AsinArt.8wefind y=cosa(x at)andy=sina(x at) asparticularsolutions of(1). From thesewemust buildupavalue that willreduce to * * f(x)=-CdaCf(\)cosa(\-x).d\ (4) CHAP. IV.] INFINITE STRETCHED ELASTIC STRING. 97 when t=andwill atthesame timesatisfy (3). y=cosaA.cosa(x -f-at)+sinaA.sina(x -\-at) or y=cosa(A.xat) isasolution of(1). Hence y=-Cdaf/(A)cosa(Axat).d\ (5) -00 isalsoasolution of(1). (5)reduces toy=f(x)when t=but itgives oo QO Dty=-Cada Cf(\)sina(A-x).d\ oo when t=andconsequentlydoesnotsatisfy equation (3). Ifinforming (5)weuse cosa(x at)and sina(x at) instead of cosa(x -f-at)and sina(x -f-at)weget y=-CdaCf(\)cosa(A.x+a).e& (6) _oo which isasolution of(1),andreduces toy=f(x)when= ,but itgives 00 00 Dty^CadaCf(\)sina(X x).d\ -oo when= anddoesnotsatisfy (3). If,however, wetake one-half thesumofthevalues ofyin(5)and(6)we get y=-~daf(\)cosa(Axat).d\ =-r~CdaC 00 00 +-CdaCf(\)cosa(\-x+at).d\], (7) oj; asolution of(1)which satisfies both(2)and(3),and is,therefore, ourrequired solution. This result canbeverymuchsimplified. Ifwesubstitute z=x+at V. &amp;lt;K -CdaCf(\)cosa(Axat).d\ -x =-CdaCf(\)cosa(X- *).r/A=/() =/(aj+ ; 7T.7 / 98 SOLUTION OFPROBLEMS INPHYSICS.[ART.56 andinlikemanner wecanshow that -CdaCf(\)cosa(Xx+at).d\=f(x at). _oo Hence oursolution becomes y= |[/(*+at)+f(x -at)-]. (8) This result isofgreat importanceinthetheoryofelasticstrings and it shows thattheinitial disturbancesplitsintotwoequal waves which runalong thestring, onetotherightandtheother totheleft,withauniformvelocity a, andthatthere isnothing likeaperiodic motion orvibration ofanysortunless theends ofthestring produce some effect. 56. Ifthestringisnotinitiallydistorted butstarts from itspositionof equilibrium withagiveninitialvelocity impressed upon eachpointwehave to solve theequation Vfy=a*l)iy (1) subjecttotheconditions yQwhen=(2) Dty=F(x)*=0.(3) Wegetbytheprocess used inArt.55 y=1Cdaf y27raJ J butrsina(X-x+at)^Tsina(\-x-at)da=. rao if rea^ &amp;lt;A &amp;lt;x+a^5and isequaltozero forallother values ofX;since =-ifm&amp;lt;0 1= if v.Int. Cal.Art.92(3). Hence y=FXd\ (4) isourrequiredsolution. CHAP.IV.] LONGRECTANGULAR PLATE. 99 EXAMPLES. 1.Ifthestringisinitiallydistorted andstarts with initialvelocitysothat y=f(x)andDty=F(x)when= y= \[/I*+ &amp;lt;)+A*~ *)]+Ta x 2.Iftheinitial disturbance iscaused byablow, asfrom thehammer ina piano, which impresses uponallthepointsinaportionofthestring oflength canequaltransversevelocitybshow thatthefront ofthewavewhich willbe seen toruntothe leftalong thestring willbeastraight linehaving aslope equalto an(^alength equalto V4a2+b* -Ofcourse awavehaving afront ofthesame length with aslope equalto willbeseen torunto 2iCL theright along thestring, andtheeffect ofthetwowaves willbetoliftthe be string bodily andpermanentlytoadistance above itsoriginal position.Z(L 57.We shallnowtakeupafewexamples oftheuseofFourier sSeries. IntheproblemofArt. 7letthetemperature ofthebase oftheplate bea given function ofx,theother conditions remaining unchanged. Sincef(x)=^(amsinmx) TO=1 7T where am=-f/(a)sin &amp;lt;*.da m=oo v wehave u=^e~mvsinmx (/(a)sinma.da .(1)m=l Ifthebreadth oftheplateisainstead ofTT a &quot;?/ .rmrx /.mirXTx~ism-^-J/(X)sin-rfX .(2) -&quot;2u=- =l 58. Ifthetemperatureofthebase isunity andthebreadth oftheplateis TTthesolutionis,aswehave seen inArt. 7, I u=- \e~vsinx+-e~^sin3x+- e&quot;51sin5#H---- | H)7TL O 5 _| This series canbesummed withoutdifficulty. Wehave thedevelopment ,Nz zz ,z** ifthemodulus of*islessthan 1.Int. Cal.Art.221(4). 100 SOLUTION OFPROBLEMS INPHYSICS.[ART.58. Hencelog(1-)=- |-|-|- |---- ifmod. 2 &amp;lt;1 . andf[log(l+*)-log(l-z)]=;[+f8+|V--(2) ifmod. 2 &amp;lt;1 . But log(!+)=log[1+r(cos&amp;lt;f&amp;gt;-fisin &amp;lt;)] and [Int.Cal.Art.33(2)] and(2)becomes 1[~11-f-2rcos 2|_2g12rcos1[&quot;l^.i.-t-^cuag-rr-,..,._-! ;u r(cos&amp;lt;+*si11 &amp;lt;)fs (cos3&amp;lt;j&amp;gt;-f*sin 3&amp;lt;ft) , Q. _i:. _|_- --j-...^jj From(3)wegettwoequations 1 1-}-2rcos &amp;lt;^&amp;gt;+r2rcos&amp;lt; ,r8cos 3&amp;lt; ,r5cos 5&amp;lt;ft, ^^ Ig l-2rcos4&amp;gt; +r2=1 3 5 1 .2rsin d&amp;gt;rsin &amp;lt;i .r8sin 3&amp;lt;f&amp;gt; ,r5sin 5&amp;lt;^&amp;gt; , ,K\ -tan&quot;1 1_rr= ][ 3~~ 5 both valid forallvalues of &amp;lt;#&amp;gt;providedr &amp;lt;1. e~yislessthan 1ifyispositive. Hence from(5) r^sinx+r*sin3x+e^sin5x+...=1tm_1 1_12sin a;_1 _,sin a; and(1)maybewritten 2, .since^=tan~* ., TT smhy CHAP.IV.] STATIONARY TEMPERATURE. 101 Ifwereplacerbye~yand &amp;lt;j&amp;gt;byxin log[1+r(cos&amp;lt;f&amp;gt;+isin &amp;lt;f&amp;gt;)] itbecomeslog[1+e~&quot;cosx-\-i e~vsina;] orlog[1+cos*-fisin2] v.Int. Cal.Art.35(3)and(4) afunction ofzasawhole; and log[1 r(cos&amp;lt;-fisin &amp;lt;)] becomeslog(1cosz isinz)j hence byInt. Cal.Arts. 212and213, 1,1+2e~ycos a;-fe~2y1 2e~ysin a?lQ 1 coshy+cosxandI tan_ 4 cosh ?/cosx 2 sinhy areconjugate functions, and 1.cosh?/4-cosx Ui=-log~^(7)TT coshycosx isthesolution fortheproblem where theisothermal lines arethelines offlow ofthepresent problem andthelines offlow aretheisothermal lines ofthe present problem. Forourproblem, then, theisothermal lines aregiven bytheequation 2 sinxtan-1 .=a TT sinhy andthelines offlowbysinx airor.,=tan -:--sinhy 2 1coshjH-cos* TT coshycosx coshy+cosxor :L_ !=eb./9)coshycosx^ EXAMPLES. = ,and u=1wheny= ,andu=when a5aax=andwhen x=a, 102 SOLUTION OFPROBLEMS INPHYSICS.[ART.59. 2.Ifu= &amp;lt;j&amp;gt;(x)when y=0,u=f(y)when x=Q,and u=F(y) when x=a .ra-TrX ,d\2^ 22K m7r xxNm&amp;lt;7r*u=- y^e&quot;asin I ^&amp;gt;(X)sma** aJ am=l .1 ,TTXrr 1 1 n+sm I2a aJI .7T, NTTOJ .7T .N TTX ocosh (X y) cos cosh (X+V) cos-1 a^a a^a QO +i-f/[^it,. . TTX .TT/x. . T cosh- (X y)+cos cosn(A-H~2/)~fcos 6Z- CL CL v.Art. 48,Exs. 4,5,and 6. 59. Ifthree sides ofaplane rectangular sheet ofconducting material be keptatpotentialzeroandthevalue ofthepotentialfunction atevery pointof thefourth sidebegiven;tofind thevalue ofthispotentialfunction atany pointofthesheet. Toformulate: 0.(1) F=0 when z=0.(2) Y=0 x=a.(3) F=0 y=b.(4) V=f(x) y=0.(5) WorkingasinArt.48weget .WTTxsmh- (J y)av .nnrx-r--sm- .,rmrb asmh- a asavalue ofFwhich satisfies equations (1), (2), (3),and(4)ifraisaninteger. Therefore sinh isourrequiredsolution., x-/^-y\a a-v ^x .mTrxC . .m?rX | 7-sm-I/(X)sm-dX ..irnrb a/vJinh- CHAP. IV.] RECTANGULAR PLATE. 103 EXAMPLES. 1.Iff(x)=1Eq. (6)Art.59reduces to sinh- (b y) sinh(b y)4I,TTX ,1 a^ .3jrxV=-\- --sm--\----sin- 77L.,7Tt a3 ..37rb asmh smn- a a .sinh (b?/),1 av JJ .5-rrx .~| -f-- --sm--h5 .,5?r6 asmh- a 2.IfF=0 when ce=0,F=0 when a=a,F= when y=0, andV=F(x) when y=b,then .=sinh- - ^X^F a m7rx Cr,,^\ WMTAF=-V-rsm- I^(X)sin-d\. a^-/1nra? aJ^am=ismh-o a 3.If^JB=1theanswer ofEx.2reduces to .iry,TTV _smh - smh smh4I a TTX.1 a 3?rx.1 a OTTCC .F=- -sm--ho-51sm--r-^-7sm-- ----. TTL .,irb a 3 .,3?r6 a 5 .,Strb asmh smh- x smh- a a a 4.IfV=Q when cc=0,F=0 when x=a,V=f(x)when y=0, andV=F(x) when y=b,then . ..WITTX7 xm=oo sjnh-(by)__2vF .mTTic / a^ y/rN.ra-TrA.,_F=- &amp;gt;,sm- (- 1I/(A.)sm-d\a^L a\ ..mirbJJva..smha ..rmrysmhaC-n, NmirX ,\~\ rIF(\}sm dX}. ..mirbjva/Jsmh o a 5.If/(cc)=F(x)theanswer ofEx.4reduces to a in^^ (/(A.)sin^^ &amp;lt;fX. aJ d _|raTT 2^1 ~^~ .mirb COSh-jr2a SOLUTION OFPROBLEMS INPHYSICS. [ART.59. 6.Iff(x)=F(x)=1theanswer ofEx.5reduces to- ;-sin------ -- _sinL nn&amp;gt;,wb a 3 i&amp;lt;&quot;&amp;gt;^cosh cosh^a 2a -p2a 1.IfV=f(x) wheny=0,V=F(x) when7/=^,^=^(y) when=0,andV=x(y)when= ,then .sinh^ sinh o a ^T^-^r^^ Jw- osm-j*^|*(A) sin^&amp;lt;*A -i&quot;~ sinh 6 sinh ,__o /* ,.N.mir\ ,A~1H-- Iv(A)sm;c?AjJ^y5 /Jsinh70 8.If/(a;)= &amp;lt;(y)=andF(x)=x(y)=1theanswer ofEx.7maybe reduced to 8mT+2-~ .,3?r/a \ 4?r/a \ sinh-T-(- xj cosh-r(-eel _V2 / .S7TV.1 ftV2 /.47TV o--o- sm~T^+7- *-sm^---- o .Sjra64,4?ra J J CHAP.IV.] FLOW OFHEAT INASLAB. 105 9.Find thetemperatureofthemiddlepointofathinsquare platewhose faces areimpervioustoheat; 1st,when threeedgesarekeptatthetem perature and thefourth edgeatthetemperature 100; 2d,when two opposite edgesarekeptatthetemperature andtheothertwoatthetem perature 100; 3d,when twoadjacent edges arekeptatthetemperature andtheother edgesatthetemperature 100. Seeexamples 3,6,and 8. An*., (1)25; (2)50; (3)50. 60.Letuspassontotheconsideration oftheflowofheat inonedimension. Suppose thatwehaveaninfinite solidwithtwoparallel plane faceswhose distance apartisc. Take theorigininonefaceandtheaxisofXperpendiculartothe faces. Lettheinitial temperature beanygiven function ofxand letthetwofaces be keptattheconstant temperature zero; tofindthetemperatureatanypointof theslab atanytime. Wehave tosolve theequation (1) subjecttotheconditions u=Qwhen x= (2) u=Q&quot;x=c(3) u=f(x)*=0.(4) InArt.49wehavefound u=eraa2&amp;lt;sinax and u asparticular solutions of(1). e-a2 &amp;lt;*2*sinax satisfies(2)whatever value isgiventoa.Itsatisfies(3) ifa=-provided misaninteger. Letustrytobuild avalue ofuoutof c terms oftheform Aer^T sin^^which shallsatisfy (4).C Wehave 2-r-\r .rmrxC,*\ m7rX, =- cZfIsmTj/(A)sm~T~dxJJ sm m=lmTTX/*_.%x,m7T\,~| J/(A)sm dlJ, (6) reduces to(5)when t=and isourrequiredsolution. 106 SOLUTION OFPKOBLEMS INPHYSICS, [ART.61. EXAMPLES. 1.Iff(\)=b,aconstant, (6)Art.60reduces to 4ftr a2*2* .7TX^1 9a27T2t .37TX.1 23ff2a( .5^0?,&quot;Iu= e^~sin 4--e^~~sin-- h-e^T~sin--p TTL c3 c 5 c J 2.Aniron slab10cm.thick isplaced between and incontact withtwo other iron slabs each 10cm.thick. Thetemperatureofthemiddle slab isat first100throughout, andoftheoutside slabs throughout. Theouter faces oftheoutside slabs arekeptatthetemperature 0.Requiredthetemperature ofapointinthemiddle ofthemiddle slab fifteen minutes after theslabshave been placed incontact. Given a2=0.185 inC.G-.S. units. Ans., 10.3. 3.Two iron slabs each20cm.thick oneofwhich isatthetemperature andtheother atthetemperature 100throughout,areplaced togetherface to face,and their outer faces arekeptatthetemperature 0.Find thetem peratureofapointintheircommon faceand ofpoints 10cm.from thecom mon face fifteen minutes after theslabs havebeenputtogether. Ans., 22.8; 15.l; 17.2. 4.Onefaceofaniron slab40cm.thick iskeptatthetemperatureand theother face atthetemperature 100 until thepermanentstate oftem peraturesissetup.Each face isthenkeptatthetemperature 0.Required thetemperatureofapointinthemiddle oftheslab,andofpoints10cm.from thefaces fifteen minutes after thecooling hasbegun. Ans., 22.8; 15.6; 16.7. 61.Ifthefaces oftheslabtreated inArt.60instead ofbeing keptatthe temperaturezeroarerendered impervioustoheat, thesolution oftheproblem iseasy. Inthiscasewehave tosolve theequation subjecttotheconditions Dxu=when x= Dxu=&quot;x=c u=f(x)&quot;*= Wehave onlytousetheparticularsolution u-=e~(fa?tcosax asweused u= e-&quot;**2*sinax inArt. 60.Weget CHAP. tV.]FLOW OFHEAT INASLABWITH ADIABATIC FACES. 107 EXAMPLES. 1.Solve example2Art.60supposingthat theouter surfaces areblanketed after theslabs areplaced togethersothat heat canneither enter norescape. Find inaddition thetemperatureoftheouter surfaces fifteen minutes after theslabs areplacedincontact. Ans., 33.3; 33.3. 2.Solve example3Art.60onthehypothesis juststated, gettinginaddition thetemperaturesofpoints ontheouter surfaces. Ans.,50;33.9; 66.l; 27.2; 72.8. 3.Solve example 4Art.60supposingthat heat neither enters norescapes attheouter surfaces after thepermanentstate oftemperatures hasbeen set up.Find alsothetemperaturesofpointsintheouter surfaces. Ans.,50;39.7; 60.3; 35.5; 64.5. 4.Show that ifu=when x=Q,Dxu=when x=c,andu=f(x) when t= , sn Asn Suggestion:Assume u=when x=2cand f(2c x)=f(x), and see (6)Art. 60. 62.Ifthetemperatureoftheright-hand faceoftheslabconsidered inArt. 60isaconstantyinstead ofzerowehaveonlytoadd tothesecond member of(6)Art.60aterm u^which shallsatisfytheconditions (1) w1==0when x=(2) Ml=t=(3) 1=yX=C.(4) ut=2-obviouslysatisfies(1), (2),and(4);tomake itsatisfy (3;aswell wemust addatermu2which shall beequal tozerowhen x=andwhen ViCa=candto *-when=0,while always satisfying (1).Itisgivenc immediately by(6)Art.60and is sinmTTX/\.miT\ ^\ ,. -j-JXsm d\J.(5) c /.mir\ c2 ,c2 \sin-d\=--cosmir=(l)m+l &amp;gt; c mir^mir 108 SOLUTION OFPROBLEMS INPHYSICS.[ART.63. , 2ym ^T/( l)mm*aWt .m7TX\/Axand u2=-V(i-e--sin-) (6) Taj .2-+- \_C 7T. &amp;lt;r- w .Hence %=---e^~sm C Iftheleft-hand face oftheslabconsidered inArt.60istobekeptata constant temperature ftandtheright-handface atthetemperaturezerowe cangetthetermuswhich must beadded tothesecond member of(6)Art.60 byreplacing ybyftandxbycxin(7).Wethenhave _re X2x-\/l mairt . ft\--- &amp;gt;J~e^~sLo7r^\msm c EXAMPLES. 1.Show that if.^=/3when x=Q,u=ywhen cc=c,andu=f(x) when ^= sm irf(A) fllsmmTTX - -7-Jc/w 2.Show that if ^^=ftwhen a:= ,u=when ,t=0,andDxu when c=c 4/a*ir*t .7TX .1ga^f .STTOJ 1 25a7Tt .&7TX~^Sm+ 6~^&quot;sm+ e&quot;-Sin~ 63. Ifthetemperatureoftheright-handfaceoftheslabjustconsidered is afunction ofthetime instead ofaconstant andthetemperatureofthe left- hand face iszerotheproblem canbesolved byamethod nearlyidentical with thatofArt. 51. CHAP. IV.] TEMPERATURE OFONEFACE VARIABLE. 109 Let &amp;lt;J&amp;gt;(x,i)beafunction ofxand twhich shall bezero iftislessthan zero andshall beequalto OW=* /-\\m \ HsA*e~ma czsin I c TT*-l\m c/ m=l [v.(7)Art.62]iftisequaltoorgreater than zero. Sothat &amp;lt;f&amp;gt;(x,t)= if t &amp;lt; $(x.t)=1&quot;t= and x=c 4&amp;gt;(x,t)=l&quot;x=c 4&amp;gt;(x,t)=0&quot;x=Q. PreciselyasinArt.51weget *= _limit^A|~. [&amp;lt;f&amp;gt;(x,t kr) &amp;lt;f&amp;gt;(x,t (k-f-I)T)]T|/-i\ astherequiredsolution ofourproblem, nbeing asinArt.51thelargest integerin-where tisanygiven value ofthetime. Onourhypothesisthelastterm of(1),thatis,F(nr)&amp;lt;l&amp;gt;[x,t (n-f-l)r]=0; thenext tothelasttermF(nT)&amp;lt;^(x^t ^r)hasforitslimiting value JH=1 while asinArt.51thelimiting value oftherestofthesum is t CF(\)DI&amp;lt;J&amp;gt;(X,t o m= oo~ &quot; vvsin Hence m=l 110 SOLUTION OFPROBLEMS INPHYSICS.[ART.63. (2) Ifwesubstitute/?= ^(t X)weget EXAMPLES. 1.Ifthetemperatureoftheleft-hand face isafunction ofandthetem peratureoftheright-handface iszeroandtheinitial temperatureiszero u=- - sn/ 2.Ifthetemperatureoftheleft-hand face isafunction of*,the initial temperatureiszero,andtheright-handface isimpervioustoheat 3.IfinArts. 60-63wearedealing with abarofsmall cross-section andof lengthcandheat isradiating from thesurface ofthebarinto airatthetem peraturezero sothatDtu=a*D*u b*u,show that:(a)thesecondmem bers of(6)Art.60and(1)Art.61must bemultiplied bye~m ;(b)equation (7)Art.62becomes sinh CHAP. IV.]VIBRATION OFASTRING FASTENED ATTHEENDS. Ill (c)equation (2)Art.63becomes . Bin 64.Theproblemofthemotion ofafinite stretched elasticstringoflength Ifastened attheendsanddistorted atfirst intosome given curve y=f(x), andthenallowed toswing, hasbeen treated andpartiallysolved inArt. 8. Thecompletesolution iseasily seen tobe cosratC*,*\ ^TrA. ^ Jf(\)sm-j-d\ .(1) Thesecond member of(1)isaperiodic function of Ihaving theperiod 21 .Themotion, then, unlike that inthecase ofaninfinitestring (Art. 55)is^21atrue vibration, aperiodic motion. Theperiodisthetime ittakes adis- Ob turbance totravel twice thelengthofthestring (v.Art.55).Acareful examination of(1)willshow thattheactual motion isagood deal likethat inthecaseconsidered inArt. 55.Theoriginal disturbance breaks upintotwowaves oneofwhich runs totheright until itreaches theend of thestring and isthen reflected, andrunsback totheleftortheunder sideof thestring, while theotherwave runs totheleftand isreflected atthe left- handendofthestring andrunsback totheright under thestring and is again reflected, runsback totheleftoverthestring andsoonindefinitely. Ifthecurve intowhich thestringisdistorted atthestart isoftheform T.W17TX ...,. y=osin thesolution is ,nnrx rmrat ybsmj-cos-.(2) Nomatter what value tmayhave thecurve isalwaysoftheform . .TtlTTX y=A&w.j-\ thatis,fordifferent values oftwehave asetofsinecurvesdiffering onlyin theamplitude andnotatallintheperiod ofthecurve. Inthiscase either thewholestringifm=1,oreachmth ofthestringifraisnotequalto one, risesand falls,andthere isnoapparent onward motion. When this is thecasewearesaidtohave asteadyvibration. 112 SOLUTION OFPROBLEMS INPHYSICS.[ART.64 Ifm=1wegetsteady motion ofthestringasawhole and ifthevibration israpid enoughtogiveamusical note thenote issaid tobethepure funda mental note ofthestring.Ifm=2thevibration istwice asrapidaswhen ra=1,themiddlepointofthestring doesnotmove and iscalled anode, the twohalves ofthestringareinopposite phasesofvibration atanyinstant, and thenotegivenisanoctave higher than thefundamental noteand iscalled its pure firstharmonic. Ifm=3thevibration isthree times asrapidasinthefirstcase, there are I 21twonodes x=-andx= ,andthenote isthepuresecond harmonic of o o thefundamental note. Foranyvalue ofmthevibration ismtimes asrapidaswhenm=1,there arem 1nodes atthepoints x=,x,x- I,andwegetthe mm m m 1stharmonic ofthefundamental note. Itisclearfrom(1)thatnomatter what theoriginal form ofthestring the resulting vibration canberegardedasacombination ofsteadyvibrations each ofwhich alone wouldgivethefundamental note ofthestring oroneofits harmonics, andthat thecomplexnote resultingisreallyaconcord ofthefun damental noteandsome ofitsharmonics. Afinelytrained earcanoften recognizeinacomplexnotethefundamental note ofthestring andsome ofitsharmonics and iscapableofanalyzing a complexnote into itscomponent purenotespreciselyasFourier sTheorem enables ustoanalyzethecomplexfunction representingtheinitial form ofthe stringintothesimplersine-functions which must becombined toform it. EXAMPLES. 1.Show that ifapointwhose distance from theendofaharp stringis -ththelengthofthestringisdrawn asidebytheplayersfingertoadistance bfrom itspositionofequilibrium andthen released, theform ofthevibrating stringatanyinstant isgiven bytheequation 2bn2 (n1 ^TT .mjrx m7rat\ -sm-sin;COS- ;I &amp;gt;* n l l Show from this that alltheharmonics ofthefundamental note ofthe string which correspondtoforms ofvibration havingnodes atthepoint drawn aside bythefingerwill bewantinginthecomplexnote actually sounded. CHAP. IV.]VIBRATION OFASTRING INARESISTING MEDIUM. 113 2.Ifastretched stringstarts from itsposition ofequilibrium, each ofits points having agiveninitialvelocity,sothatwehave y= when t= thesolution oftheproblem ofitsvibration iseasyandgives m=oo l 2^\/1 .mTTX .miratC-^ m7r\ 2/=Y(-sm-sin Im)sin ( air*-4\mI IJ^Im=l 3.Write down thesolution forthecasewhere thestringisinitiallydis torted andeachpointhasagiveninitialvelocity. 65.Ifwedonotneglect theresistance ofthe airintheproblemofthe vibration ofastretchedstring thedifferentialequationisrather morecompli cated andthesolution isnotsoeasilyobtained. Theequationisgiven as(ix) Art. 1. Letussolve theproblemforthecasewhere there isnoinitialvelocity. Herewehave D?y-f2kDty=a2Dy.(1) ?/=when x=(2) y= x=l(3) y=f(x)t=Q(4) Dty=&quot;t=0.(5) Wegetparticular solutions of(1)intheusual way. Assume y=e*+P&amp;lt; andsubstitute in(1).Wehave astheonlynecessaryrelation between/?and a.Thisgives /3=-k^a*a2+k*._ Hencey=eax-**vla+*2 (6) isasolution of(1)nomatter what thevalue ofa. Tothrow itintoTrigonometric formreplace abyai,andsince inactual problems k,which isproportional totheresistance,isvery small, take 1 outasafactor oftheradical. Wehave 114 SOLUTION OFPROBLEMS INPHYSICS.[ART.65. Sinceamaybepositiveornegative wecanget y=e~ktsin(axtYaV A;2 ) and y=erktcos(axt0*2a2 A;2 ) assolutions of(1),orbycombining these y=e~ktsinaxcos tV^a2a2k*(7) y=e~ktsinaxsin tVaV k*(8) y=e~ktcosaxcosV2a2k*(9) (7)and(8)satisfy (1)and(2)for allvalues ofa.They satisfy (3)if a=.Letusseeifoutofthemwecannot buildupavalue thatwillsatisfy (4)and(5)aswell. m=l 2 &quot;^T/ y=- e-*&amp;lt;y(sin^-cos *A/^ ^I/(*)sin&quot;^^) (I2) fc \ft t ^7 fm=l reduces to(11)when t=andtherefore satisfies(4). i ^ 2mirx .72-/xN.#=- B-H-sm -KA)sm :^/.mTrx -s2)(sm-7-o i cos t\l , k*. I/(A) sin *x When ^= the first lineofthesecond member of(13)vanishes butthe second linereduces to Wemust, then, introduce into(12)anadditional termwhich shall equalzero when t=andwhose derivative withrespecttotshall cancel thetermabove when t=0. CHAP. IV.]VIBRATION OFASTRING INARESISTING MEDIUM. 115 This iseasily seen tobe Hence ourcompletesolution is -&2 * i&quot; k+,sin tV^TT^~ *&quot;)sin^Tf/Wsin^7^^1 (14)/m27T2a2 ^6 JtJ i Here thefactthate~kt ,which decreases rapidlyastincreases,isafactor of thewhole second member shows that theamplitudeofthevibrationrapidly decreases. Comparingthissolution with that giveninArt.64forthecasewhere there isnoresistance weseethattheperiodofanygiven term ,Asin cos tv ^-- , isgreater than that ofthecorresponding termA1sin cos-inArt. 64. i I Inother words the effect oftheresistance ofthe air istoflatten some what eachcomponent partofthenote given bythestring. More than this since theperiodsofthedifferent terms of(14)arenolonger exactsubmultiples oftheperiodofthe firstterm, thecomponent notes arenolongerinperfect harmony with thefundamental note ofthestring, andtheidealperfecthar mony between thefundamental noteand itsharmonics isnotquiterealized in anyactual case. When kisvery small, asinthecase ofafinestring, thedeparture from perfect harmonyisvery slight; butinthecaseofacoarsestringorworse still ofanelastic ribbon, where theresistance ofthe air isconsiderable, the unmusical character ofthesound isverynoticeable. EXAMPLES. 1.Solve Ex.1Art.64allowing fortheresistance oftheair. 2.Solve Ex.2Art.64allowingfortheresistance oftheair; 2.,v-*/1 mrrx .mWa? ./ mTrX - .p tf 116 SOLUTION OFPROBLEMS INPHYSICS.[ART.66. 3.Find aparticularsolution of(1)Art.65ontheassumptionthat itisof theformyT.X, whereTisafunction of talone andXafunction ofx alone. 66.Wepassonnow toacoupleofproblemsthatrequire themodification andextension ofFourier sTheorem, thecooliny ofasphereinair,andthe vibration ofastretched rectangular membrane, butasanintroduction tothe former weshall firstconsider thefollowing verysimple problem;tofind the temperatureofanypointofaspherewhose initial temperatureisanygiven function ofrthedistance ofthepointfrom thecentre, andwhose surface is keptattheconstant temperatureb. Herewearetosolve Dt(ru)=a*V*(ru), (1) see[v]Art. 1,subjecttotheconditions u=f(r)when t=Q(2) u=b&quot;r=o(3) ifcistheradius. Letv=ru,thenourequationsbecome DtV=a*D*v(4) v=rf(r)when=(5) v=bc&quot;r=o(6) V=Q r=0.(7) Ourproblemisnowpreciselythat ofArt.62andwehave asoursolution ru/m*a***. .Wrr / .=- 2,/e^~&amp;lt;sin-IX/(X) m=l.^sm d\ r-9//-j\OT228 m7rr\~\ -fb\r+y\(^-e-^ sin)(8) 1 TT^\m c/Jm= 1 EXAMPLES. 1.Iff(r)=b(8)Art.66reduces tou bandthere isnochangeof temperature. 2.Iftheinitial temperatureisconstant andequalto/3 TTT^L c2 c 1 9a*r .STTT--em-J CHAP. IV.] COOLING OFASPHEKE INAIR. 117 3.Aniron sphere40cm.indiameter isheated tothetemperature10CP centigrade throughout;itssurface isthenkeptattheconstant temperature 0. Find thetemperatureofapoint10cm.from thecentre, and findthetem peratureofthecentre, 15minutes after cooling hasbegun. Given a2=0.185 inC.G.S. units. Ans., 2.l; 3.3. 67. Ifinstead ofhaving thetemperatureofthesurface ofthesphere constant, thesphereisplacedinairwhich iskeptattheconstant tem perature zero, theproblemismuch more complicated. Forinthiscasethe surface temperaturecannolonger besimply expressedbut isgiven byanew differential equationDru-\-hu=Qwhen r=c, (1) where hisanexperimentalconstant depending uponwhat iscalled thesur faceconductivityofthesphere. Ourequations, then, are ) (2) u=f(r)when=(3) Dru-irku=when rc.(4) AsinArt.66letv=ru ;thenwehave Dtv=a*D*v(5) v=rf(r)when t=(6) v=&quot;r=(7) -v=Qwhen r=c.(8) v=e~2a*cosarand v=e~a2&amp;lt;x2sinarhavealready been found aspar ticular solutions of(5)(seeArt.60). v=e-Msinar(9) satisfies(7)forallvalues ofa. Substitute thisvalue ofvin(8)andwehave accosac+(Jic 1)sinac=.(10) Ifakisavalue ofawhich isaroot ofthetranscendental equation (10) v=e~aiakfsinakr(11) willsatisfy (5), (7),and(8). Itremains toseewhether outofterms oftheform givenin(11)wecan buildupavalue ofvwhich willsatisfy (6). 118 SOLUTION OFPKOBLEMS INPHYSICS.[ART.67. When= thesecond member of(11)reduces tosmakr.Ifthenwe canexpress rf(r)asasumofterms oftheform bksinakrwhere akisaroot of(10) v= .bke-2&amp;lt;x fcsinakr(12) willsatisfyalloftheequations (5), (6), (7),and(8),andwillbetherequired solution. Here, then,wehave anewproblem analogoustothat ofdeveloping ina Fourier sSeries, butrather morecomplicated, namely,todevelop anyfunction ofxinaseries oftheform^?amsinamxwhere amisarootoftheequation (10);orifwecallac= &amp;lt;and hcl=p, where am= ,&amp;lt;OTbeing aroot oftheequation &amp;lt;cos &amp;lt;f&amp;gt;+psin &amp;lt;f&amp;gt;=(13) ormoresimplyof &amp;lt;fr+^tan$ =0; (14) remembering thattheseries andthefunction must beequalforallvalues ofx between zeroand c. If &amp;lt;f&amp;gt;misarootof(14)&amp;lt;f&amp;gt;misalsoaroot. Since sin-x= sin f x)theterms oftherequired development which correspondtonegative rootsmaybecombined with thosecorresponding topositive roots, andtherefore weneed consideronly positiveroots. &amp;lt;/&amp;gt;= isaroot of(14)butassin0=0 there willbenocorresponding term inthedevelopment. Ifweconstruct thecurve y=-|*(15) andthecurve y=tanx(16) theabscissas oftheirpointsofintersection arevalues ofxwhichsatisfy |-tanaj=0,thatis,areroots ofequation (14).Itiseasytoseethat there willalways beaninfinite number ofrealpositive roots, oneforeach of thebranches oftheperiodic curve y=tanxwhich lietotheright ofthe origin. Thenumerical values ofthese roots canbeobtained byaneasycom putation. The construction suggested above shows that asmincreases &amp;lt;f&amp;gt;m willrapidly approach thevalue(2m 1)-ifpispositiveorifpisnegative andnumericallylessthanunity, and(2m -f-1)ifpisnegative andnumer ically greater thanunity. CHAP.IV.] COOLING OFASPHERE INAIR. 119 There exist, then, aninfinite number ofpositive realroots of&amp;lt;-{-ptan &amp;lt;f&amp;gt;= andconsequentlyof accosac-J-(lie 1)sinac=. 68.Thedevelopment called forinthe last article canbeobtained very easily from asimpler onewhichweshallnowconsider, namely,todevelop f(x) intoaseries oftheform f(x)=axsinfax+azsinfax+a3sin &amp;lt;f&amp;gt;3x-\---- (1) where ^u&amp;lt;2, &amp;lt;f&amp;gt;3&quot;areroots oftheequation (ftcos&amp;lt;-\-psin&amp;lt;= ,(2) thedevelopmenttoholdgood forallvalues ofxbetween x=andx=1. Letusproceed asinArts. 24and27. Call =Axandform nequa- ft-f-1 tionsbysubstitutingforxinturn intheequation f(x)=ttlsin &amp;lt;foaj+ 2sin&amp;lt;2x-f 8sin &amp;lt;3xH----- 1-ansin &amp;lt;#)raic(3) thevalues Aa,2Ax,3Ax, 7iAx;thisbeing equivalent tomaking thevalues ofthesumandthefunction coincide forthenvalues ofxsubstituted. Todetermine anycoefficient ammultiply thefirstequation byAx.sin (&amp;lt;mAx), thesecond byAx.sin (2&amp;lt;^&amp;gt; ?raAx), thethird byAx.sin (3&amp;lt;mAx),andsoon,the nthequation byAx.sin 0&amp;lt;mAx) ;addtheequations andcompute thelimit ingvalues oftheterms oftheresulting equationasnisindefinitely increased. This asinArt.24isseen tobeequivalent tomultiplying (3)bysin^m andintegrating between thelimits x=andx=1. The firstmember oftheresulting equationis Cf(x)si1 /sin o The coefficient ofakis i sin&amp;lt;j.xsin &amp;lt;f&amp;gt;mx.dx, andofamis / x.e?x . J*sin2 &amp;lt;f&amp;gt;T 120 SOLUTION OFPROBLEMS INPHYSICS.[ART.68. l i Tsin^fXsin &amp;lt;f&amp;gt;mx.dx=-i[cos(fa&amp;lt;j&amp;gt;m)xcos(fa-\-Q^x^dx if ^ ^F8*11(fa ^m)sin(fa -f-&amp;lt;f&amp;gt; facosfasin&amp;lt;m &amp;lt;f&amp;gt;msin &amp;lt;fo.cos&amp;lt;m 4&amp;gt;k fan But ^&amp;gt;fccosfa-\-psin^= and &amp;lt;f&amp;gt;mcos &amp;lt;^&amp;gt;m+psin&amp;lt;m=by(2). Hence thenumerator ofthesecond member of(4)iszero,andthecoefficient ofakvanishes ifkisnotequaltom. Jsisn x.x=-+m- sn*cos &amp;lt;^m=- Therefore am=.^ (f(x)sin &amp;lt;f&amp;gt;mx.dx .(6) sm2&amp;lt;^&amp;gt; mj^v/ 1 ^-^c The coefficient oftheintegralin(6)canbetransformed asfollows soasnot toinvolve trigonometricfunctions. &amp;lt;f&amp;gt;mcos &amp;lt;l&amp;gt;m+psm&amp;lt;l&amp;gt; m=0, by(2) &amp;lt;f&amp;gt;mCOS2 (j)m-f&quot;o^^^^m== &amp;gt; sin 2&amp;lt;^&amp;gt;m__cos2 &amp;lt;frm^-^ 2&amp;lt;^&amp;gt;m p Hence by(7)and(8) _ Therefore ourrequired developmentis &quot;&quot; snxa sin CHAP.IV.] COOLING OFASPHERE INAIR. 121 From(10)iteasily follows that forvalues ofxbetween and c f(x)=a-Lsina\x+azsinazx-f-agsinazx-f-(11j andamisarootoftheequation accosac-\-p sinac=.(13) Itistobeobserved that ifpisinfinite(13)reduces tosinac=0,am becomes-and(11)and(12)giveourregulation Fourier sine series (V.Art.c ^ 31),andtherefore theordinary Fourierdevelopmentinsine series ismerely a special case oftheproblem just solved. Moreover since theFourier method ofdetermining thecoefficients ofsucha seriesrequires that c (sinamxsinanx.dx= , o that isthatrin(a,-a&amp;gt; _sin^+ajc=Qam~an aman aoorreduping, thatsmamc sinanc orthatamandanshould beroots oftheequation accosac_ sinac wherepissomeconstant,itfollows thatwehave obtained in(11)themost general sinedevelopment thatcanbeobtained byFourier smethod. EXAMPLES. 1.Show thatthesolution oftheproblemofArt.67is andaisarootof accosac+(Ac 1)sinac= 122 SOLUTION OFPROBLEMS INPHYSICS.[ART.68. 2.Iftheinitial temperatureofthesphereisconstant andequaltoft a2c24-(he I)2sinamc=2h- .----- L. 3.Ifthetemperatureoftheairisaconstantyinstead ofzero thesurface equationofcondition is Dru-{-h(u y)=when r=c. Thesubstitution oful=uy,however, brings theproblem under Ex.1 andweget r(u y)= bme-aZa^sinamr 7M=1 where bm=- .&quot; fA[/(A)-y]sinamcan2c2+/^^1)J1) 4.Anironsphere 40cm.indiameter isheated tothetemperature 100 centigrade throughout;itisthenallowed tocool inairwhich iskeptatthe constant temperature 0.Find thetemperatureatthecentre; atapoint10 cm.from thecentre; and atthesurface; 15minutes after cooling hasbegun. Given a2=0.185 andh=^-inC.G.S. units,(v.Ex.3,Art.66.)oOO Ans.,97.67;97.36;96.46. 5.Show that ifintheslabconsidered inArt.60oneface isexposedtoair atthetemperature zero, sothatwehaveDtu=a2J)u, u=when x=0, u=f(x)when t=Q,andDxu+hu Qwhen x=c,then snamx =2__/Xsin ambeing arootofaccosac-\-hesinac=. CHAP. IV.] TEMPERATURE OFAIRVARIABLE. 123 6.IfintheproblemofArt.57heatescapes from onesideoftheplateinto airatthetemperaturezero sothatwehaveD%u+D*u= ,u=when x=0,uf(x) when y=0,andDxu-\-hu=Qwhen x=a,then sn ambeing arootofaacosaa-f-^asinaa=. 7.IfintheproblemofArt.59there isleakageatoneside ofthesheet so thatwehaveD*V+D*V=Q, F=0 when a?=0,F=0 when y=b, V=f(x)when y=0,andDxV+hV=0 when xa,then MTlSlnhamb where amhasthevalue giveninEx. 6. 69.Ifwehaveaninfinite solidwithoneplane facewhich isexposedtoair atthetemperatures U=F(t)andheatcanflowonlyatright anglestothis face,wecansolve theproblem readilyforthecasewhere the initial tem peraturesarezero.Wehave subjecttotheconditions u=when t= and Dxu-\-h(U u)=when x=. Let v=u-Dxu.(1) Then vwillsatisfy theequation andweshall alsohave v=Uwhen x=. Since U=F(t)v=Ce~^Flt~~i)d/3 (2) X byArt.51(10).Dxuhu hvby(1). Hence ue~hx=h \&amp;lt;r**vdx-fC5 v.Int. Cal. 4,page 314. 124 SOLUTION OFPROBLEMS INPHYSICS.[ART.70. Determining Cbythefactthat ue~hx=when x oowehave 00 u=hehxCe~hxvdx .(3) X Substituting thevalue ofvfrom(2)wehave asourrequiredsolution. Foranextension ofthismethod totheflow ofheat intwoandthreedimen sions and fortheinterpretationoftheresults bytheaidofthetheoryof linages,seeE.W.Hobson, Proc. Lond. Math. Soc., Vol.XIX. EXAMPLES. 1.Ifthetemperatureofthe air isaperiodic function ofthetime, say pmsin(mat+Xm)andwecareonlyforthelimiting value ofuastincreases, show that thisvalue is (1\in,a\A+a\T/ma ma / xma v.Art.52andArt.51Ex. 4. C eax(asinbx bcosbx)Note that Ieaxsmbx.dx=- *- , 72--J a~-\-b2 .&amp;lt; C e&quot;*(acosbx+bsinbx)and Ieaxcosbx.dx=*- 2-- *J a2+62 v.Int. Cal.Table ofInt.(235)and(236). 2.IfD*V+D*V=Q, V=0 when y=and^F-f h{F(y) V]= v^hen x=show that v.Art.47Ex. 1. 70.The solution foraninstantaneous heat source ofstrength Qatthe point x.=\ifheatescapesattheorigininto airatthetemperature zero, so thatDxuhu=when x= ,canbeobtained bytheaidofArt. 53. CHAP. IV.] TEMPERATURE OFAIRZERO. 125 Letu=ul-{-uzwhere uisthetemperature thatwould beduetothegiven source ifwehadnoboundaryattheorigin,sothat M!=^7=e-^f5 .[Art. 53(2)1 J)xuhu=DxUi hui -j-DXU*^2==when x=. Therefore Dxu*hu2=(Dxu^ hu^) (1) when x=. when x=0. This iseasilyseen tobethevalue towhich _BVa reduces when x= ,andthis lastexpressionis (A+ andtherefore satisfies theequation t x&amp;gt;\) Q U+X)2 1=-e4a2&amp;lt; isthetemperature duetoasource atx=A.. If,then,wedetermine ?&amp;lt;2from thecondition that QA+sc \ (A+a D,u,-hu,=--=(-^-- h)e-^r taking carenottointroduce anyarbitrary constant orarbitrary function oft inourintegration, u2willsatisfy equation (2)andcondition(1). Integrating (3)[v.Int. Cal. 4,page314]anddetermining theconstants of integration suitably weget Therefore thesolution ofourproblemis 00 I/&quot;* f\ -4-a?^2 ~1 (5) 126 SOLUTION OFPKOBLEMS INPHYSICS.[ART.71. Ifwereplace Qbyf(X)d\andintegrate from tooowegetasthesolution forthecasewhereuf(x) when=andcc&amp;gt;0, andDxuhu=Q when x= /(\\ I I,/i*vy*vI^&amp;lt;U*l|^ WZ&1 **iv\j v 4OZ w/ai/ I. IO) U\ITrt* Foraninterpretationofthis result bythetheoryofImages and the extension ofthemethod totheconduction ofheat inndimensions see G-.H. Bryan, Proc. Lond. Math. Soc., Vol.XXII. EXAMPLE. Show that ifuf(x)when= andDxu+h[F(t) u]=when x=wemust takeuequaltothesumofthesecond members of(6)Art.70 andof(4)Art. 69. 71.Asanother problem requiringaslight extension ofFourier sTheorem letusconsider thevibration ofarectangular stretched elastic membrane fastened attheedges, that isofarectangular drumhead. Iftwoofthesides aretaken asaxesandtheplaneofequilibriumofthe membrane astheplaneofXYtheequationforthemotion ofthemembrane is see[x]Art. 1. Letthemembrane bedistorted atthestart intosome given form z=/(, y) andthenallowed toswing. Ourequationsofconditions arethen z= when x=Q(2) *= x=a(3) *=/(*, 0&quot;*=(6 &amp;gt; Wecangetaparticularsolution of(1)byourusual device. Assume and substitute in(1).Weget y2=c2 (a2+/32 )astheonlyrelation that need hold betweena, ft,andy,inorder that=eaa!+ft+*may bea solution. This gives Therefore *=e**+* isasolution of(1)nomatter what values aregiventoaandft. CHAP.IV.] VIBRATION OFARECTANGULAR DRUMHEAD, 127 Keplaceaandftbyaiandftiandwehave (8)asasolution, andfrom thisweget z=sin(ace+ftyct&amp;lt;Ja2+ft2 ) and z=cos(ax-}-ftyct Va2+ft2 ) asparticularsolutions of(1),aandftbeing unrestricted. (8)and(9)wecangetsolutions ofthefollowing forms z=sinacesinftysinctVa2+ft2 z=sinacesinftycosct\ja2 -\-ft2 z=sinacecosftysinctVa2-fft2 z=sinacecosftycosctVa2+ft2 cosacesinflysinctVaz+ft2 \ z=cosacesinftycos z=cosacecosftysin 3=cosacecosftycos each ofwhich willsatisfy equation (1).Thesecond ofthese willsatisfyalso (2), (4)and(7)whatever values betaken foraandft.Itwillsatisfy (3)and (5)ifaandftareequal andrespectively. If,then,wecansocombine terms oftheform(10) .sm sina bcoscirt astosatisfy (6)ourproblemwillbecompletelysolved. Thiscanbedone ifwecanexpress f(x,y)asasum ofterms oftheform mirxniry Asin sin,thesumand thefunction being equalwhen xlies between andaandybetween and b. f(x,y)canbeexpressedinterms ofsinbyFourier sTheorem ifwe regard yasconstant. Wehave (11) m=l 128 SOLUTION OFPKOBLEMS INPHYSICS.[ART.71. where =-A sin /(A,y)in(12)isafunction ofyandmaybedeveloped byFourier sTheorem. n= oo Wehave/(A,y)=4.sin^(13) n=l 6 = fJ/(A,/*)sin^rf/*.(14) Substituting for/(A,y)in(12)thevaluejustobtained wehave .?w,7rA. .WTrtt _\n A*)sm-^-sin-fdp)sin-where n= ooa 6 n=1 and snX,sn sn .m,7rx .rnry ra2 ,n\ ,*r^ ,nsm-^-sm-^cosCTT^^/~+-\ , (16) Hence where Amn=Id\(f(\,a)sin sin dii . (17) a!)J J*\*&quot;-/ a brv isourrequiredsolution. EXAMPLES. 1.Show that ifthemembrane starts from itspc-sitionofequilibrium but with agiveninitialvelocity impressed upon eachpointsothat z=when t=andDtz=F(x, y)when t= thesolution is mirx .mry.sm-sm-* a 6 4/^_Cn,^ Ic?A IF(\,i ^_n,^ \.where ^4=Ic?A IF\ sin-sm CHAP.IV.] RECTANGULAR DRUMHEAD. 129 2.Ifthere isboth initial distortion and initialvelocity 4-^-\ x-v fttTTX .mrv r~. \ml .n2 . m2 .n z=&amp;gt; &amp;gt;,sin Binr*\AmncosCTrt\l =+75-+.Bmnsmc7rt\ +77ao^**^ a b*a o \a bm=ln=l a 6 /** /^* 7^t7T\.where ^t TO&amp;gt;n= fd\ I/(A.,p)sin sin a b 1 vssn sn 3.Obtain aparticularsolution of(1)Art.71byassumingz=T.X. Y. whereTisafunction oftalone,Xof#alone, andYof?/alone. 72.Anumber ofinteresting conclusions canbedrawn from theresults of Art.71andExs. 1and 2. (a)Nooneofthethree values ofzisingeneral aperiodic function oft, andconsequentlyavibrating rectangular membrane willnotingeneral givea musical note. (b)Astretched rectangular membrane canbemade togiveamusical note bystarting thevibrationproperly. For ifthe initial circumstances aresucli thatthesolution reduces toasingle term, aswillbethecase iftheinitial dis tortion intheproblemofArt.71besuch thatf(x,y)=AmMsin sin -, Ob O orthe initialvelocityinEx.1besuch thatF(x, y)=Bm^nsin sin^^ , ortheinitial distortion and initialvelocityinEx.2bethevaluesjust given, then thevibration willbeperiodic andwillhave theperiod In** Va*^b2 SinceTisafunction ofmandnandmandnareanywhole numbers, the samemembrane iscapableofgiving agreat varietyofmusical notes ofdiffer entpitches.Ifmandnarebothunitywegetthelowest notethemembrane cangive,which iscalled itsfundamental note. Itsperiod (2) If7ftandnarebothequaltokweget 2^ (3) 130 SOLUTION OFPROBLEMS INPHYSICS.[ART.73. therefore themembrane canbemade togiveanyharmonic ofitsfundamental note. More thanthis, since aswehave seen 2T= ^ \m ,n1 \*+P istheperiodofanynote themembrane cangive,and since ifmandnare replaced bymkandnkweget 2T-*mk,nk ck/ n^a2^ ft* themembrane cansound alltheharmonics ofanynotewhich itcangive. (c)Inthecaseconsidered above, where thesolution reduces tothesingle term .imrx .mry |~ m2 .K? . m2-n?~]Z=Sm~7~8m 6L&quot;CS^+ ft2+mnSm^U2+ ft3J (mor-. ,, . . z= for allvalues or #,anda 2a 3a if cc=,or ,or inmm a 2a (m 1).,, .,. . , thelines x=,x=,x=*- remain atrestduring thewhole in m m vibration andarenodes. Thesame thingistrue ofthelines b 2?&amp;gt; 3b 73. Ifthemembrane issquareitmayhavemuch morecomplicated nodes than ifthelength andbreadth areunequal,asinthis,case theperiodofany term ofthegeneralsolution reduces to T= andthere willingeneralbetwoterms having thesame period, andamusical note ofthepitch correspondingtothat period maybeproduced byinitial cir cumstances thatbringinboth terms. Thus ,mirx .mry \~CTrt=sin-sinMAmncos- d (L\_Cb,r,+BC7r^ / 2&quot;! 2 I ,nsin-Vm2+n* -J sin2sin^EI[a acos sn CHAP.IV.] NODES OFASQUARE DRUMHEAD. 131 isaform ofvibration that willgiveamusical note. Letuswrite this cirt Ir.r ,mirx .mry.,nirx .miry~\z=cos \m? -[-nAsin-sin-4-Bsin-sin-- a [_ a a a a_j C7r^r~r~i of^rwwro; .mry. .nirx .miri/~\+sinVmfl+n* I(7sin-sin 2+#sm-sm(2)a [_ a a a aJv and instudyingtheforms ofmusical vibration ofwhich themembrane is capable wemaytakeA,B,C,andDatpleasure. Consider thesimplecase whereA=CandB=D\then(2)reduces to (,mirx .niry. mrx .miry\/ cirt , _Asm-sin-+Bsin-sin--IIcos \mz-4-nz a a a a/V a Values ofxandythat willreduce the firstparenthesisin(3)tozero willcor respondtopointsofthemembrane remaining motionless during thevibration. Letusconsider afewcases atlength. (a)Ifm=1andn=1,the firstparenthesisin(3)becomes in sia which isequaltozeroonlywhen x= ory= ,orx=aorya, thatis,forthefouredgesofthemembrane.If,then, themembrane issound ingitsfundamental note ithasnonodes. (b)Ifm=1andn=2,wehave ,irx .27ry. 2irx .iryAsm sin--4-Bsm-sin-= a a a a togive thenodes. LetB= ,then sin sin-= ,which issatisfied byy=- ;anda aa inaddition totheedges theliney= isatrestand isanode. IfA= IfA=BIfAQx=-isanode. .irx .2/iry. 2irx .irysmsm --Usm sm-= a a a a n.irx .iry iry..... irx irx .iry.2smsm-cos-+2sin cossm-= a a a a a a .irx .iry/ iry. irx\smsm- (cos-+cos1=0.a a\a a/ SOLUTION OFPKOBLEMS INPHYSICS.[ART.73. The first factorgives thefouredges ofthemembrane. Thesecond written equal tozerogives COSvy trx / TTX\ -^= COS-=COS(7T I# a \ a/ Try TTX 7Ta a which isadiagonal ofthesquare. IfB=A .7TX.2-7T?/.27TX.7TI/sinsm sin sin-=a a a a iry TTX cos=cos which istheotherdiagonal ofthesquare. Other relations between AandBwillgiveTrigonometric curves oftheform ITUB 7TX COS=-- -COS- aA a which areeasily constructed andwhichobviouslyallagree inpassing through themiddlepointofthesquare.Wegivethefigures forafewofthecases CHAP.IV.] NODES OFASQUARE DRUMHEAD. (c)Ifm=n=2wehave (A+B)sin sin^= a a togivethenodes, which aremerely thelines a ax=- ,andy=-. Thisformgives theoctave ofthefundamental note. (d)Ifm=1andn=3wehave . .TTX.STH/,STHC .TT?/Asin sin -+Bsm sin= togivethenodes. IfA=wegeta a ,x== and*=133 =-andy= - O OIfB=weget IfA=Bweget .7TX .37Tt/.37TX.7T?/smsm:sm sin-=a a a a .TTX .iryrtsmsm-4a aL(2) or/ Try 7TX\/Try. 7TX\ Icos cosjlcos-4-cos I=V a. aJ\ a a/ xy=and IfA=Bweget cos2^-fcos ora2 27TI/. 2-TTXCOS+COS= 1,a a(3) (4) aTrigonometric curveeasily constructed. Forother relations between AandBwegetmorecomplicated Trigonometric curves coming under thegeneral form A2&amp;lt;7r.y, r&amp;gt; 27TCC A+BAcos - -{-Bcos= 134 SOLUTION OFPROBLEMS IKPHYSICS. which allagreeincontaining thepoints aaa2a\/2aa\,/2a2a\ o&amp;gt;T )&amp;gt; (-TT&amp;gt; o)&amp;gt;an(i l&quot;S-&amp;gt;-5&quot;l- A-B MISCELLANEOUS PROBLEMS. I.LogarithmicPotential Polar Coordinates. 1.Show thatD*V-\-DjV=Qbecomes ifwetransform toPolar Coordinates. 2.Ifin weletF= -K.3&amp;gt;weget &amp;lt;$=Acos ad&amp;gt;-j-Bsin ad&amp;gt;) whence Y=racos(i) &amp;lt;+J5sinad) -^Aea&amp;lt; t&amp;gt;-{-Bea* | Blr~a)R=Alcos(alogr)+^!sin(alogr) ;) F=rasin ad&amp;gt; V= cos ad&amp;gt; r* F= sin ad&amp;gt;/MAV=ea &amp;lt;t&amp;gt;cos(alogr)F=cosh ad&amp;gt;cos(alogr) V=e-4&amp;gt;sin(alogr) V=er* cos(alogr) F&quot;=e-* sin(alogr)V=cosh ad&amp;gt;sin(alog ?*) Y=sinh ad)cos(alogr) F==sinh ad&amp;gt;sin(alogr) areparticularsolutions of(1). 3.Show that ifFsatisfies(1)Ex.2andF=/(&amp;lt;)when r=a and wherefor w TO=-Cfty)cos andn aTO=T/C^) 136 MISCELLANEOUS PROBLEMS. 4.Show that ifVsatisfies(1)Ex.2andV=f(r)when 4&amp;gt;=and J coshcosaX- -sinCOSh9(X&quot;~ &quot; TT 2 cosh(X-logr)-cos4 5.IfF=l when &amp;lt;#&amp;gt;=and OO&amp;lt;1, andT=0 when &amp;lt;#&amp;gt;=and sin* 6.IfF=/(r)when &amp;lt;^&amp;gt;=andF=0 when &amp;lt;/&amp;gt;=/? cosa(X-log mh/Sa MZ*cosh(X-logr)-cos if &amp;lt;&amp;lt; &amp;lt;/3. 7.IfF=0 when&amp;lt;=andV=F(r)when&amp;lt;= =^sin cosh(X-logr)+cos 8.IfF=xWwhen &amp;lt;/&amp;gt;= and r&amp;lt;,F=0 when ^=j8,and F=0 when r=a o coshI(X+logI)-cos^ LOGARITHMIC POTENTIAL. 137 9.IfF=0 when&amp;gt;=!,F=l when&amp;lt;=0,F=0 when&amp;lt;= 10.IfF=0 when ?=!,F=l when&amp;lt;= ,F=1when 4&amp;gt;= 11.IfF=/(&amp;lt;)whenr=,F=0 when &amp;lt;J&amp;gt;=0,andF=0 when . m7rd&amp;gt; . -xsm-i r&amp;lt;a where am= /(&amp;lt;/&amp;gt;)sin^and0&amp;lt;&amp;lt;^&amp;gt;&amp;lt;^. 12. IfF=/(&amp;lt;)when r=a,F= when r=6,F^O when ^=0, andF= when &amp;lt;^&amp;gt;=^8,then ifa &amp;lt;r &amp;lt;band &amp;lt;&amp;lt; &amp;lt;(3 where am= &amp;lt;#sin/ am= gj 13.If V=F(&amp;lt;t&amp;gt;)when r=zZ*,F=0 when ra,F=0 when andF=0 when &amp;lt;#&amp;gt;=)8,then if a&amp;lt;r&amp;lt;b and &amp;lt; &amp;lt;^&amp;gt;&amp;lt;/3 _ sn 2/nrr 2mir&quot;* where aw=- F&amp;lt;fisin 14.IfV=\(r)when ^=0,F=0 when ^=^3,F=0 whenr=a. andF=0 when r=b,then if a&amp;lt;r&amp;lt;b and &amp;lt; &amp;lt;#&amp;gt;&amp;lt;/8 sinh__ log6log.mTT(logilog a)_ _!2^_log*-log* logbloga 138 MISCELLANEOUS PKOBLEMS. log-* 2/&quot; mirx where am=--- --Ix(aeX)sm logaJAV ?- logblogaJAVlog&loga 15.IfV=\f/(r) when&amp;lt;=0,F=0 when &amp;lt;j&amp;gt;=0,F=0 when r=at andF=0 when r=b,then if a&amp;lt;.r&amp;lt;b and &amp;lt; &amp;lt;f&amp;gt;&amp;lt;/? ra7r&amp;lt;f&amp;gt; &amp;gt;n=oo sinh^ 5 logftloga .m7r(logrloga) r~/ , &quot;Stf/n _ bill sinh log6loga , 2 /.where am== : I$(aexsm dx. mlogb logajr^logbloga \j II. Potential Function inSpace. 1.Show that 00 00 00 00 cosax-*cos- forallvalues ofxand?/. 2.Findparticularsolutions ofDX2F+7&amp;gt; I?F+D?V=intheforms V=sinh zVa2 -!-/?2 .sin(a^c ^y) F=cosh ^V^aM-&quot;/?2 .sin(aa; /9^) &c. 3.GivenI&amp;gt;X2F+DV2V+D?V=Q, andV=f(x,y)when s=0,solve for positivevalues ofz. Result: V- c 4.Confirm theresult ofthelastexample byshowingthat iff(x ty)isinde pendentofy F=1Czf(xtidx (v.Ex.3Art.45). POTENTIAL FUNCTION INSPACE. 139 5.IfDlV+D*V+DtV=, andV=l when *= forallpoints within therectangle bounded bythe lines x=a,x=a,y=b, and y=&;andV=Q when z= for allpointsoutside ofthisrectangle, then .qn (a-x)\ba-x)*+(b-y)2+.Z2 ] +*)2+(ft~y)2+*2 ]) a+x?+(b-yf+^2 ]&amp;gt; 22 r2 if a &amp;lt;# &amp;lt;a,and &amp;lt;(b-y) _sin-i(&quot;+x)\b-y)2-z\(a+x?+(l- yY+*a ,^+yjsin_ifa a;)2 (Z&amp;gt;+y)2zz [(a x)2 T ^(6+?/)21( ^)2 (^+2/)2+*2 [( ^)2+ +.t (a+x)\b+ 6.Ifthevalue ofthepotential function Visgivenatevery pointofthebase ofaninfinite rectangular prism and ifthesides oftheprismareatpotential zerothevalue ofVatanypointwithin theprismis I/&quot;2 -4-n2 mTTX .WITTI/ /7x/*5sm ^sin -^-1^1 m=ln=l 00 IfF= 1onthebase oftheprismthisreduces to 16-^r-\^A =^2*-&quot;smsin sin sm 7.Ifthevalue ofthepotential function onfive faces ofarectangular parallelepiped, whoselength, breadth, andheight area,6,andc,iszero,and 140 MISCELLANEOUS PROBLEMS. ifthevalue ofVisgivenforevery pointofthesixthface, then forany point within theparallelepiped .^^\- * a&quot; IP .mTrx .mry ~4~4mn/2 nn* en b sinh TTC\/-4- i 4C7v/%/x \wrA .TWTU .where A m&amp;gt;n=IaA. I/(A, yu)sin sin~-dp. 8.Ifthevalue ofthepotentialfunction isgiven ontwoopposite faces ofa rectangular parallelepiped and iszeroonthefourremaining faces, thenwithin theparallelepiped sm sm sinhTTC\ r+ m2 ,w2 \/ 7+77 2^J .mirx .nsm--sm 4/* ,r , Nm?rX .mra where ^OT,n=JcOf/(A, /*)sm-^-sin^ o o a b and J5 m&amp;gt;n=-CdXF(\,p)sin^-sin^^. o o 9.Ifthevalue ofthepotentialfunction isgivenatevery point onthesurface ofarectangular parallelepiped, what isitsvalue atanypointwithin the parallelepiped? III. Conduction ofHeat inaPlane. 1.Findparticularsolutions ofDtu=a2(D^u+D*u)oftheforms u=r-W+Pysin(ax #/) u=e~**a*+^cos(ax py). 2.Given theinitial temperatureofevery pointinathinplane plate,find thetemperatureofanypointatanytime, CONDUCTION OFHEAT INAPLANE. 141 7*e~ 3.Foraninstantaneous source ofstrength Qat(A,ft) Q (A a)+0*-y)f M=i3:~4^ v.Art. 53.47ra2 Foraninstantaneous doublet ofstrength Pat(0,ft)with itsaxisperpen dicular totheaxisofT v.Art. 54. Forapermanent doublet ofstrength Pat(0,ft)with itsaxisperpendicular totheaxisofY ~P&amp;gt; 2 Ifthestrengthofthedoublet werePd/jt,andtheheatwereuniformly generated andabsorbed along theelementd/j,oftheaxisofYbeginningat (0,ft)weshould have _ P_sn=xj. -*p yn- be 4 2&amp;lt; ~T~\ /- \Q==o- 9e 4a=t atanA- &amp;gt;27ra* ic2+(/-t i/)22?ra2x andsince dtan&quot;1--istheangleARA,whereAandAarethepoints (0, and(0,ft+&amp;lt;fyt)and^isthepoint (x,y),w=when x= unless JD ft&amp;lt;y&amp;lt;Cft+d/JL ,inwhich case u= ifxapproacheszerofrom the &Q, positive side;andu=Qwhen, ^=exceptintheelement dp. Ifthen u=when t=andu=f(y)when x=wehaveonlytosupposea doublet ofstrength 2a2 f(x)dx placedineach element oftheaxisofFand then tointegrate ;weget 1C-**+&amp;lt;&amp;gt;-&amp;gt;/)* xfM ju=-Ie- 4a2&amp;lt;2 i/ Ts^TTj iC24-(U, V)2a-+(^-y)i Forapermanent doublet ofstrength F(t)at(0,ft)wehave u= xF(r) 142 MISCELLANEOUS PROBLEMS. From the^reasoning above thismust bezerowhen t=exceptatthepoint (0, IJL),must be2a?F(t)atthepoint (0,fi),and ateveryotherpointofthe axisofYwhen tisnotzero. Hence ifu=when t=andu=F(y, t)when x= mry/*7x/V,.^ N.mirX. .mru, -jj*I cZAJ/(X, /i)sin sm^dp.1rxF(lL rfyai+Qt-y)* , ,I/*, /*xDTF(fJL,r)sS+Q-y)*w=~ Io ./ To 4az ayLtH Ia/*I2.xNr2e^(t-r)&r TTjX24-(fJh ?/)2TTj^JiC24-(/i ?/)2 00 00 Foranextension ofthissolution bythemethod ofimagestothecasewhere there areother rectilinear boundaries andforitsapplicationtothecorrespond ingproblemsintheflow ofheat inthree dimensions seeE.W.Hobson inVol. XIX Proc. Lond. Math. Soc. 4.Iftheperimeterofathinplane rectangular plateiskeptatthetem peraturezeroandtheinitial temperaturesofallpointsoftheplatearegiven, then, foranypointoftheplate m= ;n=oo be 4:^^ 2-2(* j.2Nwmirx .n Ag ^^ yjQriT\+pGin Qm U&amp;gt;?jeVft* &amp;lt;?S1117S111 be^4^ b m=ln=l ifbisthelength and cthebreadth oftheplate. 5.Alargemass ofiron atthetemperaturecontains aniron core inthe shapeofalongprism40cm.square. Thecore isremoved andheated tothe temperatureof100throughout andthenreplaced. Find thetemperatureofa pointintheaxis ofthecore fifteen minutes afterward. Given a2=.185 in C.G.S. units. Ana., 52.9. 6.Iftheprismdescribed inEx.5after being heated to100has itslateral faces keptfor15minutes atthetemperaturefindthetemperatureofapoint initsaxis. Ans., 20.8. IV. Conduction ofHeat inSpq.ce. 1.Show that oo oo oo oo oo GO CdaCd(3CdyCdXCdp T/(A, p,v)cosa(A x)cosP(p y)cosy(v z).dv000 oo oo oo =/(*&amp;gt;y&amp;gt;*) forallvalues ofx,y,and z. 2.Show that ^-\ -\^\.,mTrx .mry.ptrz /(*, y,*)=2)2,2,4-.sm smVsmV m=ln=lp=l 8/*/*./.mTTA .ftTTU ./?7Tlwhere Am^=jd\jdpi /(A,^v)sm sm-^sm^ CONDUCTION OFHEAT INSPACE. 143 3.Obtainparticular solutions ofDtu= a\I&amp;gt;u-\- D*u+D?u) ofthe forms u=er^+P+i*sin(ax fly yz) cogax z&amp;gt; 4Given theinitial temperature ofevery pointinaninfinite homogeneous solid findthetemperatureofanypointatanytime. C7C )*Je~ 5.Ifthesurface ofarectangular parallelopipediskeptatthetemperature zeroandtheinitial temperaturesofallpointsoftheparallelopiped aregiven, then foranypointoftheparallelopiped . ,,72 sin-j-sin sin ln=lp= 1 where Amn=r~7I^Idf*I/(^^&amp;gt;^)gin;sin sin ocajJJbed 000 6.Anironcube40cm.onanedgeisheated totheuniformtemperature of 100Centigrade andthentightly enclosed inalarge ironmasswhich isatthe uniformtemperatureof0.Find thetemperature ofthecentre ofthecube fifteen minutes afterwards.Am., 38.4. 7.Anironcube40cm.onanedgeisheated totheuniformtemperature of 100andthen itssurface iskeptforfifteen minutes atthetemperature 0. Required thetemperature ofitscentre.Ans.,9.5. CHAPTER V.* ZONAL HARMONICS. 74.InArt.16weobtained [v.(6)Art.16]asthegeneral solution ofLegendresEquation ^ ^dx2(1) z mbeing whollyunrestricted invalue andxlying between 1and1;where Pm(*;=i- and ym\x)==x andwefound2. 4! m(m-2)(m-4)(m+1)(m+3)(m+5) 6!-a-h- (d) -l)(m+2) 3!2)(m 5! (m-l)(m-3)(m-5)(m+2)(m+4)(m+6) 7,*-1 -IT (COS#), mbeing unrestricted invalue, asparticularsolutions ofthespecial form assumed byLaplacesEquationinsphericalcoordinates whenVisindepend entof &amp;lt;;thatis,oftheequation rD?(r V)+-^Do(sin6D9T)=0. (6) *Before readingthischapter thestudent isadvised tore-readcarefully articles 9,10,13(c), 15,10,and18(c). SURFACE ZONAL HARMONICS. Fortheimportant casewheremisapositive integer wefound145 (7) [v.(10)Art.16]asthegeneral solution ofLegendresEquation (2),whence V=rmPm(cos B) V (8) areparticular solutions of(6)ifmisapositive integer. |^(m-l)(m-2)(^-3)..&quot;1 2.4.(2m l)(2w 3) J [v.(8)Art.16]and isafinitesumterminating with thetermwhich involves xifmisoddandwith theterminvolving xifmiseven. Itiscalled aSurface Zonal Harmonic, oraLegendresCoefficient,ormore briefly aLegendrian. i m-fi) W,-f2)i (2m-f1)(2m 1)1_x&quot;&amp;lt;+ *&quot;2.(2wi+3)xm+8 (m+l)(^+2)(/^+3)(m+4)1 H 2.4.(2w/.+3)(2m+5)cc+5^J^} if a:&amp;lt; 1or x&amp;gt;l.[v.(9)Art.16.] Itiscalled aSurface Zonal Harmonic ofthewmid hind. __!!+!2.4.6....(m ~~^3.5.7. ...m [v.(13)Art.16]ifmisoddand 1&amp;lt;x&amp;lt;1. 2.4.6.... 1.3.5.... (m- [v.(14)Art.16]ifmisevenand 1&amp;lt; a;&amp;lt;1.(12) 146 ZONAL HARMONICS.[ART.75. Inmost ofthework thatimmediatelyfollows weshall regard xinPm(x)as equaltocosandtherefore aslying between 1and1.* 75. InArticle 9theundetermined coefficient amofxminPm(x)was arbitrarily written intheform*-^ .- -- forreasons which shallm nowbegiven. InArticles 9and16z=Pm(x)wasobtained asaparticularsolution of LegendresEquation bythedevice ofassumingthat zcould beexpressedasasumoraseries of terms oftheform anxnandthendeterminingthe coefficients. Wecan,how ever, obtain aparticularsolution ofLegendresEquation byanentirelydiffer entmethod. Thepotentialfunction duetoaunit ofmass concentrated atagiven point (*uy\&amp;gt;*i)is V- ,1 (2)^x-xtf+^-ytf +(*-*,? andthismust beaparticularsolution ofLaplacesEquation Q, (3) asiseasilyverified bydirect substitution. Ifwetransform(2)tosphericalcoordinates using theformulas of transformation x=rcos6 y=rsin cos&amp;lt; z=rsin sin&amp;lt; weget -==(4) Ol+sin sin0!cos(&amp;lt;&amp;lt;fo)]+n asasolution ofLaplacesEquationinSphericalCoordinates rDr\rV}+ A(sinAF)+^D|F= [xm]Art. 1. Ifthegiven point (xl}yl}zly)istaken ontheaxis ofX,asitmust bethat (4)maybeindependentof &amp;lt;f&amp;gt;,Oi= ,and (5) *Englishwriters onSpherical Harmonics generallyuseninplaceofxforcos 0.We shall follow them, however, onlywhenweshould thereby avoid confusion. CHAP. V.]PARTICULAR SOLUTION OFLAPLACE SEQUATION. 147 isasolution of rDr\rV)+^A(sinBD9V)=V.(6) Equation (5)maybewritten VI 2scos+z2isfinite andcontinuous forallvalues realorcomplex of K.Itisdouble-valued butthetwobranches ofthefunction aredistinctexcept forthevalues ofzwhich make 12zcos9-\-z?=namelyz=cos+isin and z=cos isin0,both ofwhich have themodulusunity andwhich are critical values. , =isfinite and continuous except forthevalues of VI 2zcos8+2* z=cos isin and *=cos -fisin forwhich itbecomes infinite ;itis double-valued buthasascritical valuesonlythese values ofz.Itisthen holomorphic within acircle described with theorigin ascentre andtheradius unity, andcanbedeveloped intoapower series which willbeconvergent for allvalues ofzhaving moduli lessthan one.(Int.Cal.Arts. 207, 212,214, 220.) Ifthen r &amp;gt;^ =canbedevelopedintoaconvergent series involving whole powersof . Let^pm~bethis series,pm,ofcourse, being afunction ofcos 0.Then [v.(7)]isasolution of(6).Substitute thisvalue ofFin(6)andweget Asthismust holdwhatever thevalue ofrprovidedr &amp;gt;rvthe coefficient of eachpower ofrmust bezero,andhence theequation sinOdB must betrue.(sin^)+m(m+l)Pm=(9) 148 ZONAL HARMONICS. [Airr.75. Butaswehave seen inArt.9thesubstitution ofx=cos6in(9)reduces itto andtherefore z=Pm isasolution ofLegendresEquation (1). Ifr &amp;lt; /-i-canbedevelopedintoaconvergentseries V2ii rcose+- involving whole powersof- rm r\ Let bethis series. Then (v.8)isasolution of(6);substitutingin(6)weget whence itfollows asbefore that isasolution ofLegendresEquation \ more brieflyitisthecoefficient ofthemthpoweroizinthedevelopmentof /I_2xz+z2)-?accordingtopowersof,standingforcos 6. (1-2xz+s2)-i=[1-2(2*-s)]-i andcanbedeveloped bytheBinomial Theorem;thecoefficient ofzmiseasily pickedoutand is (2m l)(2w 3)---l|~_w(m-1) 2 ml L 2(2m-l)- 2.4.(2m l)(2m 3)_H _T Butthis isprecisely Pm(x). [v.Art.74(9)] HencePm(x)isequaltothe coefficient ofthemth powerofzin thedevelopmentof[l2+ **]&quot;*intoaPower series, themodulus of beinglessthan unity. CHAP.V.] PROPERTIES OFSURFACE ZOXAL HARMONICS. 149 76. Ifx=lPM(x)=l.For ifx=1(1-2.vr:+z*)- reduces to (12z+z2)-?that isto(1 z)-1 ,whichdevelops into andthecoefficient ofeachpowerofzisunity. Therefore Pm(l)=l.(1) Wehave seenthat ifmisevenPm(x)containsonlyevenpowers ofxand terminates with theterminvolving x,that iswith theconstant term. Thevalue ofthisconstant term canbepicked outfrom theformula for Pm(x)[v.Art.74(9)].Itis(-1)5l^l^L-J);01.itcanbefound as follows: Itisclearly thevaluePm(x)assumes when #=();itis,then, the coefficient ofzminthedevelopmentof(1-f3*)-i;but andthecoefficient ofzm ,mbeing anevennumber,is(1)2&quot;^-^ . 2.4.6 m IfmisoddPm(x)containsonlyoddpowersofxandterminates with the terminvolving xtothe firstpower. The coefficient ofthisterm canbe picked outfrom(9)Art.74and is(-l)^*3 1&quot;m-; oritcanbe ^.4.O. (Wi Lj found asfollows :Itisclearly thevalue assumed by-^^when x= dx Itis,then, the coefficient ofzminthedevelopmentofz Z _z=^--U 2.4 m-l andthe coefficient ofzminthisdevelopment is (1)2ri 2.4.6&quot;(m 1)mbeing anoddnumber. 77.Torecapitulate: ,mtm-i^m-^m-^ 2.4.(2m-1)(2m-3) _m(m-l)(m-2)(m-3)(m-4)(m 5)m_ H 2.4.6.(2m l)(2m 3)(2m 5) JW 150 ZONAL HARMONICS.[ART.77. mbeing apositive integer,isaSurface Zonal Harmonic orLeyendrian ofthe mth order. Itisafinite sumterminating with the firstpowerofxifmis odd,andwith thezeroth power ofxifmiseven. Pm(a)isthe coefficient ofthemthpowerofzinthedevelopmentof (12xz-+-22)~intoapowerseries. Hence if*&amp;lt;1 x).z&amp;gt;+-.(2) Whence +- .-]ifr ifpo(cos tf)+-P^cos 6)+-P2(cos0)+ PIL ^i ^i .if isasolution ofLegendresEquation whenmisapositive integer.F= r&quot;IPIB(cos d) and r=_J_Pwi(cos^ aresolutions oftheform ofLaplacesEquationinSphericalCoordinates which isindependentof &amp;lt;,namely rJ9r2 (rV)4--T^JA(sinDeV)=0. (4)sinv 2.4.6.- --2m(8) (9) CHAP.V.] TABLE OFSURFACE ZONAL HARMONICS. 151 rdP^x)-}3.5.7. (2w+1) LdxJ e^ ;2.4.6.- &quot;2m(10) Forconvenience ofreference wewrite outafewZonal Harmonics. They areobtained bysubstituting successiveintegersforminformula(1). P1(x)=x P4 (&amp;lt;c)=(35z4-30z2+3) P.(a;)=(231*6- P7(a?)=i (429,-c7- Ps(x)=i105;r2-5) -35x)(11) AnySurface Zonal Harmonic maybeobtained from thetwoofnextlower orders bytheaidoftheformula (n+l)P n+l(x)-(2n-}-l)xP H(x)+nPn_1(x)=:0 (12) which iseasily obtained and isconvenient when thenumerical value ofxis given. Differentiate(2)withrespecttozandweget whence Hence by(2) -x)(P (oj)+PI(X).Z --)= (13) 152 ZONAL HARMONICS. [AiiT.78. (13)isidentically true,hence thecoefficient ofeachpowerofmust vanish. Picking outthecoefficient ofznandwritingitequaltozerowehaveformula (12)above.* 78.Wearenowable tosolvecompletelytheproblemconsidered inArt. 9. Wewere tofindasolution ofthedifferential equation rD*(r F)+De(smOD eV)=0 (1) subjecttothecondition Weknow(v.Art.77)that and aresolutions of(1). Forvalues ofr &amp;lt;c (3) G 2c22.4c42.4.6 c8 Therefore forvalues ofr &amp;lt;G (cos d)- \ isourrequired solution; because eachterm satisfies equation (1),andthere forethewhole value satisfies(1),andwhen= [v.(5)Art.77],andhence(4)reduces to(3)and(2)issatisfied. Forvalues ofr &amp;gt;c M_Mr. _l* l^^_1^6c!,&quot;1~~L 2r2&quot;t &quot;2.4r42.4.6r&quot;rJ 2r82.4r62.4.6 r7 *Fortables ofSurface Zonal Harmonics v.Appendix Tables Iand II. CHAP. V.] PROBLEMS INPOTENTIAL 153 Therefore forvalues ofr &amp;gt;c isourrequired solution. For itsatisfies(1)andreduces to(2)when=. 79.Asanotherexampleletussuppose aconductor intheform ofathin circular disccharged withelectricity, and let itberequiredtofindthevalue ofthepotential function atanypointinspace. Ifthemagnitude ofthechargeisMandtheradius oftheplateisathe surfacedensityatapointoftheplateatadistance rfrom thecentre is M and allpointsoftheconductor iir.jatthepotential (v.Peirce sNew-&k tonian Potential Function, 61.) Thevalue ofthepotential function atapointintheaxisoftheplateatthe distance xfrom theplateiseasily seen tobe d/M.x2a*\M m I _,/&quot;*r)^~~-_I-- dx\2a x*+a2/ a*+x _M ifx&amp;lt;ia, .i if x&amp;gt;a. Integrating andthendetermining thearbitrary constant wehave M.x*a* M\~7r x .xsx5 ,a-7 &quot;I__ (&amp;gt;Qg1_- ._ I________ _J____I 2a x*+a2a[_2 a^3a* 5a6^la1J ifx &amp;lt;a, =^1&quot;--- 4- 4-&quot; a[_x 3x*^5x6Ix*+ if x&amp;gt;a. 154 ZONAL HARMONICS. [ART.79. Wehave, then, tosolve theequation rDr\rV)+~D9(sinBD6F)= subjecttotheconditions V-EVl-L^.^. ^4.-^-...1 a|_2a^3355^77J when=and r&amp;lt;a , M\~aa8a6a7 and r-7U&quot;l?+ S=i&quot;7i5+ when=and r &amp;gt;a. Therequiredsolution iseasilyseen tobe ifr &amp;lt;aand &amp;lt;- , andr-- if r&amp;gt;a. EXAMPLES. 1.Given that ifachargeJK&quot;ofelectricityisplacedonanellipsoidalcon- ductor thesurface densityatanypointPoftheconductor isequaltoj^; c&amp;gt; wherepisthedistance from thecentre oftheconductor tothetangent planeat P(V.Peirce, New. Pot.Func. 61) ;findthevalue ofthepotentialfunction at anyexternal pointwhen theconductor istheoblate spheroid generated bythe rotation oftheellipse- 2+^2=1about itsminor axis. Ans.(1)Ifthepointisontheaxisofrevolution Mr ._,/bx+^-& a;beingthedistance from thecentre. (2)Ifthepointisonthesurface ofthespheroid CHAP. V.] EXAMPLES. 155 (3)Ifthedistance rofthepoint from thecentre islessthan^a2b2and (4)Ifthedistance rofthe-point from thecentre isgreater than^a2b2 P2(cos0) 2.Iftheconductor istheprolate spheroid generated bytherotation ofthe x2y2 ellipse 2+7}=1about itsmajor axis,show that ifthepointisanexternal point and isontheaxis atadistance xfrom thecentre, r= Ifthepointisnotontheaxisand r &amp;gt;V^2b2 80.Asathird example wewillfindthevalue ofthepotential function due toathinhomogeneous circulardisc, ofdensity p,thickness k,andradius a. Thevalue ofFatapointintheaxis ofthedisc atadistance xfrom its centre isreadily found andprovestobe and F=?^fi- - ~~ a[_2x 2. 2.4.6 x-62.4.6.87&quot;I 156 ZONAL HARMONICS. [ART.80. If x&amp;lt;a *21.1a* 1.1.3 a;8 _2J[fr iala21.1g* ,1.1.3 a;6 _1.1.3.5 x ~| F~TL a+2^2~~2.4*+2.4.6a62.4.6.8a8~t&quot; J* Hence thesolution foranyexternal pointis ifr &amp;gt;a,and if ^&amp;lt;a and EXAMPLES. 1.The potentialfunction duetoahomogeneous hemispherewhose axis is taken asthepolar axis,is ifr &amp;gt;a,and is if r&amp;lt;a and 2.The potentialfunction due toasolid spherewhose densityispropor tional tothedistance from adiametral plane is,atanexternal point, CHAP. V.J ZONAL HARMONICS. 157 3.Thepotentialfunction duetothehomogeneousoblatespheroid generated x2y2 -bytherotation of2+j-2=1about itsminor axisis,atanexternalpoint, .^_.(sin~l\a a &quot;&quot;&quot;I +sin~l jxF3* 2(a2-b2 )_2( ifthepointisontheaxisofthespheroidatadistance xfrom itscentre. 3 r~^5 r8-2p*(GOs6) i(a-y)ir5.7 r5 ifr &amp;gt;(a*b2)^ ,and 3M r~7r r .., TT r2 F=^ir^iL4-(^^iPl(cos^+4?^^p2( 1 r8cos a2-^2iand if x generated bytherotation of~ 2-\~~2==4.Thepotentialfunction due tothehomogeneous prolate spheroid 2 2 tsmajoraxisis,atan externalpoint, r*fii2lnQ^+i (&quot;-^SPrcosw (oJ)tLl.3r^3.5 r8 if 81.Themethod employedinthe lastthree articles may bestated in generalasfollows: Whenever inaproblem involving thesolving ofthe special form ofLaplacesEquation thevalue ofVisgivenorcanbefound for allpoints ontheaxis ofXand thisvalue canbeexpressedasasumoraseries involving onlywhole powers positive ornegative oftheradius vector ofthepoint,thesolution forapoint 158 ZONAL HARMONICS. [ART.82. notontheaxiscanbeobtained bymultiplyingeachtermbytheappropriate Zonal Harmonic, subject onlytothecondition that theresult ifaseries must beconvergent. Itwillbeshown inthenext article thatPm(cos 0)isnever greater than onenorlessthanminus one. Hence theseries inquestionwillbeconvergent forallvalues ofrforwhich theoriginalseries wasabsolutely convergent. 82.Inaddition totheform givenin(1)Art.77forPm(x)other forms areoften useful. Itoughttobepossibletodevelop PTO(cos 0),which mayberegardedasa function of6,intoaFourier sSeries, andsuch adevelopment maybeobtained, though withmuch labor, bythemethods ofChapterII. Thedevelopmentinterms ofcosines ofmultiplesofmaybeobtained much moreeasily bythefollowingdevice. Wehave seen inArt.75thatPm(cos 0)isthecoefficient ofthemthpower ofzinthedevelopmentof(12cos6+2)~iinapower series, andthat ifmod z &amp;lt;1(12zcos-f-#2)~icanbedevelopedintosuchaseries. We know bytheTheoryofFunctions thatonlyonesuch series exists,sothat the method bywhich wemaychoose toobtain thedevelopmentwillnotaffect the result. (1-2zcos+*2)-i=(1- z(eGi+-&quot;)+z2)-i nze*)-^maybedevelopedintoanabsolutely convergentseries if mod z &amp;lt;1,bytheBinomial Theorem. Wehave Theproductofthese series willgiveadevelopmentfor(12zcos -fz2 )-% inpowerseries. The coefficient ofzmiseasily picked out,andmust beequal toPm(cos 0).Wethus get CHAP. V.]ZONAL SURFACE HARMONIC ASASUMOFCOSINES. 159 1.3.5 m(m l)(m 2)&quot;1,.,. 1273(2-l)(2-8)(2-6)cos(m-6)0+J.(1) Ifmisoddthedevelopmentrunsdown tocos0;ifmiseven tocos(0),but inthat casethecoefficient ofcos(0),thatis,theconstant term, willnotcontain thefactor 2which iscommon toalltheother terms, butwillbesimply ri.3.5&quot;-(m1)~|2 L 2.4.6.wJ Wewrite outthevalues, ofPm(cos 6)forafewvalues ofm P(cos0)=l pl(cos 0)=cos P2(cos 0)- (3cos20-f1) P3^cos 0)=(5cos30+3cos0) P4(cos 0)=(35cos40+20cos20+9)t)4 P5(cos 0)= Tog[63cos50+35cos30+30cos0] P6(cos 0)= gigC231cos6#+126cos4+1Q5cos20+50] P7(COS 0)=j^j C429COS7^+231COS5^+189COS30+175COS^] P8(cos 0)=[6435cos80+3432 cos60+2772 cos40 +2520 cos20+1225].(2) Since allthecoefficients inthesecond member of(1)arepositive, andsince each cosine hasunityfor itsmaximum value itisclear thatPm(cos 0)has itsmaximum valuewhen=0;butwehaveshown inArt.76thatPm(1)=1. Therefore Pm(cos 0)isnevergreater thanunityif isreal. Itisalsoeasily seenfrom(1)thatPm(cos 0)cannever belessthan 1. 160 ZONAL HARMONICS.[ART.83. 83.Pm(x)canbeverysimply expressedasaderivative. Wehave m(m-l)(m-2)(m-3) H 2.4.(2m l)(2m 3) J aJ (m+1)! 2.(2m-l) (m+l)m(m-l)(m-2) m_8_ ~] T2.4.(2m-l)(2m-3)&quot; J m(x)dx (2m-l)(2m-3)&quot;-i .(2m-l)-3)&quot;-ir (m+2)(m, 2)I L 2.(2m- 2.4.(2m l(2m-3)t_2_H &quot; J CmP(x^dxm-(2m-l)(2m-3)--l r_2m(2m-l) J*m(?(2m)! L 2(2m-l) 2m(2m l)(2m 2)(2m--3) 2m_4_&quot;1 &quot;*2.4.(2m-1)(2m-3)&quot;J __(/m )(m~~ )&quot;|^2m_mx2 &amp;gt;n-2m(m ) ^_4 (2m)! L 2! m(m l)(m 2) 2m_6,~] 31~X &quot;*J* Thequantityinbrackets obviouslydiffers from(or2 l)mbyterms involving lowerpowersofxthan themth. This importantformula isentirely general and holds notmerely when xcos0,butforallvalues ofx. CHAP. V.]EQUATIONS DERIVED FROM LEGENDRE sEQUATION. 161 84.The lastresult issoimportantthat itisworth while toconfirm itby obtainingitdirectly fromLegendresEquation v.(1)Art. 75. Let usdifferentiate(1)withrespecttoaafewtimes representin &quot;by*&quot;&c -Weget J2-1flj~ **&amp;gt; i~22x^+[m(m d^z&quot; da&quot; ^ (1~^2 ) 5T~2-4*+[^(^+1)-2(1 andingeneral or(1x2 )^~-2(/iH-l)z-y-+[m(m+1)w(n+l)](n)=.(2) Following theanalogyofthesestepsitiseasytowrite equationsthat will differentiate into(1). will differentiate into(1), ifdifferentiated twice willgive (1), (1- x&quot;)^+2.2* |j!+[m(m+1)-2(1+2)&amp;gt;.= ifdifferentiated three times willgive (1),andingeneral (1~x2 )+2(n-l)x+[m(m+1)-n(n-1)]zn=(3) ifdifferentiated ntimes withrespecttoxwillgive (1). Ifn=m+1(3)reduces to (4) 162 ZONAL HARMONICS. [ART.85. andthe(m+l)stderivative withrespecttoxofanyfunction ofxwhich satisfies(4)willbeasolution of(1). (4)canbewritten andcanbereadily solved byseparating thevariables andintegrating,v.Int. Cal.(1)page314. Itgives zm=C(x*-1). ._ Hence ==C---(5)dxmdxm isasolution ofLegendresEquation (1)andagrees with thevalue ofPm(x) obtained in^.rt. 83. 85.Theequationsobtained inArt.84aresocurious andsosimplyrelated that itisworth while toconsider them alittlemorefully. Wehave seenthat differentiates into Cl-T&amp;gt;+&amp;gt;(-!&amp;gt;I+ *&quot;-&amp;lt;&amp;gt;! W that ifwedifferentiate(2)mtimeswegetLegendresEquation (1-x*)^j2-2x^+m(m+1&amp;gt;=0; (3) that ifwedifferentiate(2)2mtimesweget , (l-^g-2(+l)x|=05 (4) that ifwedifferentiate(2)mntimeswehave (1-x2 )g+2(7i-1)3|-+[m(m+1)-n(n-1)]*=0; (5) andthat ifwedifferentiate (2)m+ntimeswehave =0.(6) Bytheaidof(1)wefound inthe last article aparticularsolution of(2), namely CHAP. V.]GENERAL SOLUTIONS OFTHEDERIVED EQUATIONS. 163 Ifwesubstitute in(2)z=u(x2 l)mfollowing themethod illustrated fullyinArt. 18,wegetasthegeneral solution of(2) - 1( (7) AandBbeing arbitraryconstants. /dx- 1\M+1iseasilywritten out[v.formula(42)page6.Table ofInte- (X L) grals.Int. Gal.Appendix].Ifx &amp;lt;1itvanishes when x=0.Ifx &amp;gt;1it vanishes when x=oo .Ifthenx &amp;lt;1(7)canbewritten z=A(x* l)&quot;&amp;gt;+B(x* !)C-~ J&amp;lt;*- and ifx &amp;gt;1dx 00 =A(x*-1)+B(x* l)mCd* m+l (9)*/(x l) .(5 andinthese forms unnecessary arbitrary constants areavoided. From(7)wecangetthegeneral solutions of(3), (4), (5),and(6). isthegeneralsolution of(3). ^^_1 rfx2 &quot; isthegeneral solution of(4). .lmn/^2--J\m &amp;lt;=Ad-^^- + isthegeneral solution of(5). isthegeneral solution of(6). Ineach ofthese formsAandBarearbitrary constants andtheintegralis tobetaken from toxifx &amp;lt;1andfromxtoooifx &amp;gt;1. Ofcourse(10)must beidentical with theformsalready obtained inArts. 16 and18asgeneral solutions ofLegendresEquation. Equation (4)issosimplethat itcanbesolveddirectly, andwegetits solution intheform which must beequivalentto(11). 164 ZONAL HARMONICS. [ART.85. Comparing (14)with(7),thesolution of(2),weseethateverysolution of(4) canbeobtained from asolution of(2)bydividing thelatter by(x* l)m ,or inother words that ifwewrite(2) and(4)as(1x2 )-^ 2(m+l)xy-1=(4) z=z^(x2 l)m ;andthesubstitution ofthisvalue in(2)willgive (4),and thesubstitution ofzt^T--min(4)willgive (2). (x&amp;gt;~~~ _l_ ) Wehave, then,twowaysofobtaining (4)from(2);wemaydifferentiate(2) 2mtimes withrespecttox,orwemay replacezin(2)byz^(x~ l)w . Ifweusethe firstmethod wehave seen thatLegendresEquation (3)is midway between(2)and(4).That isifwedifferentiate(2)mtimesweget (3)and ifwethen differentiate(3)mtimes weget (4).Letussee ifthe half-way equationinoursecond processisLegendresEquation. T-P /2 *1\ and y=z,(x2l)f Sothat ifin(2)wereplace byy(xil)fandthenrepeattheoperation ontheresulting equation weshall get (4).Making tjiefirstsubstitution we find, = )(15) notLegendresEquationbutasomewhat more general form. Ofcourse its solution is (2)and(4)arespecial forms of(5)and(6).Letustrytheexperimentof bstitutingin(5)=y(l#2 )fand both substitutions givethesame equationsubstitutingin(5)=y(l#2 )fandin(6)*=y Wefindthat(LX) (17) CHAP.V.] ZONAL HARMONIC ASAPARTIAL DERIVATIVE. 165 Thesolution of(17)canbeobtained from either(12)or(13)and is or which ofcourse must beequivalent. 86. tInaddition tothevalue ofPm(x)givenin(1)Art.83there isanother importantderivative formwhich weshall proceedtoobtain. Itis Wehave seen inArt.75that canbedeveloped into rITI r* aconvergentseries if1\&amp;lt;randthat the(m+l)stterm ofthat series is Pm(cos fl)/-!Letusobtain thistermbyTaylorsTheorem. 1 _2H1cose4-~~ t&amp;gt;rcos Regardingthisasafunction of(x v^)anddeveloping accordingtopowers ofi\byTaylorsTheorem wegetasthe(m+l)stterm 1 orm Hence =D.-. r&quot;l+1w! \r/ 87.Wehavenowobtained four different forms forourzonal harmonic, apolynomialinx,anexpression involving cosines ofmultiplesof0,aform involving anordinaryrath derivative withrespecttox,andaforminvolving- apartialrath derivative withrespecttox.We shallnowgetaformdue toLaplace, involving adefiniteintegral. C Ja bcos&amp;lt;(a2 if a*&amp;gt;b2 [v.Int. Cal.page 68]. 166 ZONAL HAIIMONICS.[AiiT.87. 1 1 .*o^,^icanbeexpressedintheformibytaking a=1zx and b=z\Jx2landnomatter what value xmayhave zcanbetaken sosmall thata2willbegreater than P.Thenby(1) 1__!_fd$ _1rd^ (1-2xz-fz2 )^~7rJ izx-zs/^Tl. cos^~ 7^Ji_3(3_|_v/^Tl. cos$) 1^ , =- J[1+(x+Vz21.cos &amp;lt;$)z+(a;+Va;21.cosT/ .cos if istaken sosmall thatthemodulus ofz(x+Va;21.cos^)islessthan 1.But byArt.77(2)Pm(x)isthecoefficient ofminthedevelopmentof hence Pw(a)=^J*|&amp;gt;+^2-l. cos &amp;lt;^&amp;gt;]-^.(2) Byreplacing&amp;lt;byTT &amp;lt;/&amp;gt;in(2)weget -!.cos &amp;lt;H&amp;lt;ty.(3) and ifmod- &amp;lt;1orinother words if(l-2^+a )*~&quot;*/1__2a.l+lU mofl 2 &quot;&amp;gt;1 - -*canbedevelopetlintoaconvergentseries involv- V1 2a! *+ i /IYB ingpowersof-,andthecoefficient of(-) will beP^(ic);but thiswillbe thecoefficient ofz~m~linthedevelopmentof-p JJTIaccordingto (_L ,/,V~p*^ )2 descending powersofz,mod 2being greater than 1. Ifnowweleta=zx 1and b=z\lx21,a2&2=1 2-rs4-2and gmaybetaken sogreat that a2i2 &amp;gt;0.Thenby(1) 2xz-f-2 )&quot;2&quot; TTJ~x_i_zyx*i.cos I o(x \ix* 1.cos ^&amp;gt;)17==== .if L_ ~&quot; ?rJ /r__i/^ITloI*V&quot;6 *cos &amp;lt;/&amp;gt;)L(a;Va;21.cos 1 (xsix* 1.cos&amp;lt;) CHAP. V.] DEVELOPMENT INZONAL HARMONIC SERIES. 167 n andthecoefficient ofz~m~lis-J j==^&amp;gt;&amp;gt;[+i Hence Pm(x)=^f=J*- WWJ[ZV*2I-COS &amp;lt;/&amp;gt;]&quot;+ Replace&amp;lt;byTT &amp;lt;f&amp;gt;andweget *.(*)=4/rr&quot;/\X~~cos 88.Intheproblemsinwhich wehave already used Zonal Harmonics (v.Arts. 78-81) wehave been able tostartwith thevalue ofthePotential Function atanypoint ontheaxis ofX,and ithasbeennecessarytodevelop theexpressionforVonthat axis interms ofascendingordescending powers ofx.If,however, westartwith thevalue ofVinterms offorsome given value ofr,that isonthesurface ofsomesphere, wemust develop thefunction ofinterms ofzonal harmonics ofcos(v.Art.10),andourproblem becomes thefollowing: Todevelopagivenfunction ofcos interms ofzonal har monics ofcos0,ortodevelopagivenfunction ofxinterms ofthefunctions Pm(x),xlyingbetween 1and 1. Theproblemresemblescloselythat ofdevelopinginaFourier sseries, which wehave alreadyconsidered atsuch length. Let f(x)=AP(x} -4-APfx) -4-Ai&amp;gt;P&amp;lt;,(x*} -4-AP(x) -f- (1) forallvalues ofxfrom 1to1and let itberequiredtodetermine the coefficients. If/(x)issingle-valued andhasonlyfinite discontinuities between x= 1 andx=1wemayproceedasinArt. 19. Letustake n-\-1 terms of(1)andattempttodetermine thecoefficients. Take n-\-l values ofxatequalintervals Axbetween x= 1andx1 sothat(n+2)Ax=2;/(1-fAx), /(1+2Ax), /(1-f3Ax), /[1+(,+l)Ax]willbethecorrespondingvalues of/(x).Substitute these values in(1)andwehave /(-1-fAx)=APo(-1+Ax)+-4iPi(1+Ax) +A,P,(-1+Ax)+-+AnPn(-1+Ax) /(1+2Ax)=APo(1+2Ax) -fA^(1+2Ax) -fA,P 2(-1+2Ax)++APn(-1-f2Ax) /(I Ax)=APo(l Ax)-fAiPi(i Ax)+^2P2(1Ax)+ thatis,n-f-1equations from which intheory then-f-1coefficients AQ,Al}--Ancanbedetermined. 168 ZONAL HARMONICS.[ART.89. Following theanalogyofArt.24letusmultiply the first equation byPm(1+Ax).Ax, the second byPm(1+2Ax).Ax, the third byPw(1-|-3Ax).Ax, &c.,andaddtheequations. The firstmember ofthe resulting equationis 2/(-1-fMx)P m(-I+Mx).Ax , (3) fc= i andthecoefficient ofanyAas^4Zinthesecond member is fc=n+l 2)Pm(-1+&Ax)P,(-1+*Ax).Ax. (4) it=] Ifnownisindefinitelyincreased(3)approachesasitslimiting value i Cf(x)P m(x)dx (5) JL i and(4)approaches Cpm(x)Pl(x)dx.(6) ~i Wehavenow tofindthevalue oftheintegral (6)orasweshall write itforthesake ofgreaterconvenience KQ~mv~/-wv-/-2m+nmlnlJ dxmdxn -i -i by(1)Art. 83. li J dxmdxnLdx ax I -i |^ :^ dx(1) -I byintegration byparts. Now ifz=X(x2 l)w //&amp;gt;^ 6^*&amp;gt;C1 tt-U./_J Hence thej9thderivative with respecttoxofanyfunction ofxcontaining (x3l)nasafactor willcontain(x2l)n~pasafactor ifp&amp;lt;n. CHAP. V.] DEVELOPMENT INZONAL HARMONIC SERIES. 169 /n-l/T2-]\n- - &amp;gt;then, contains (xz 1)asafactor and iszerowhen x=1dxn~ andwhen x= 1,sothat(1)reduces to z &quot;(.r2 1)&quot;dn (x* 1)&quot; _^+ fo-3 !) ^&quot;-ifa2 1) &amp;lt;fo--i 1 1/\Z &quot;(.r2 !)dn (x* 1)*__ftp*J dxmdxnJ i 2lItfollows that a.-l ^^a;a~) d*-g(s-l) ^a;- 1)&quot;tf*^a*-1)^ i If m&amp;lt;.n wegetfrom(3) Af(s -i)&quot; ^(.r2-i) 1&amp;gt;^2m ,7 J&quot; &?- cto-^J&quot; If Af J If,then,misnotequaltoni n i Ifm=wwehave tofindTCpm(x)P n(x)dx=0.(4) (3), 170 ZONAL HARMONICS. [ART.90. i i C(x* l)mdx=C(x l)m (x-fl)mdx=^rC(x l)m~\x+1J J in~T-U/ ml _ * Hence22+1m! or22m(m\y(m+V)(m+2) (2m+1) l-mfa/J dx= : 2?^+l 90.The solution oftheprobleminArt.88isnowreadily obtained, and wehave ,(x)+A,P,(x)+ (1) where Am=f(x)P m(x)dx. (2) Thefunction andtheseries areequalforallvalues ofxfrom x= 1to x_^^an(jy/x^jssubjecttonoconditions savethosewhich would enable us todevelopitinaFourier sSeries,[v.Chapter III.] Ofcourse(1)canbewritten /(cos 0)=AP (cos 0)+AiP^cos 0)+^2P2(cos 0)H where Am=y/(cos0)Pm(cos0)^(cos 0) i orif/(cos 0)=^(0) F(ff)=APQ(cos 0)+-4iPi(cos 0)+^2P2(cos 0)-\ (3) where Am=2-^F(0)P m(cos 0)sin0.^0(4) o andthedevelopmentholds goodfrom *= to TT. If/(x)isaneven function, that is,if/()=/() (1)and(2)canbe somewhat simplified.Forinthatcase itcanbeeasily shown(v.Art.77)that CHAP.V.] DEVELOPMENT INZONAL HARMONIC SERIES. 171 i andthat Cf(*)*** +1(x)dx=0-, ? sothat if/(- aj)=/(*) f(x)=AP(x)+A,P,(x)+A,P4(a?)+4,Pe(a?)+ (5) where An=(4*+l)J/(aj)Ptt(aj)daj.(6) If/(#)isanoddfunction, thatis,if/( x)=f(x)itcanbeshown in likemanner that f(x)=AyPi(x)+AaP,(x)+ABPt(x)+A,P,(x]+ (7) i where J 2fc;h,=(4/c+3)^f(x)P K+1(x)dx. (8) Ifitisonlynecessarythatthedevelopment should hold for &amp;lt;x &amp;lt;1any function maybeexpressedinform(5)or(7)atpleasure. i 91.Wecanestablish thefactthat |Pm(x)P n(x)dx=byamore gen- i eralmethod than thatused inArt. 89. LetXmbeanysolution ofLegendresEquation [p-*2)I]+? &quot;(m+v*= i&amp;gt;wArt -16J- which with itsfirst derivative withrespecttoxisfinite, continuous, and single-valuedforvalues ofxbetween 1and1, 1and1being included. Then and(1_x2)+M(B +lXn=.(2) Multiply (1)byXnand(2)byXmandsubtract andintegrate andweget [m(m+i)_n(n+i)]^^^=Jxm[(1-^)^ 172 ZONAL HARMONICS. [ART.91. Integrate byparts, Whence ^^A= unlessm=n. (3)givesatoncetheimportantformula CYYJJV^= m(_..a\TdXmYdX dx dx fromwhich come asspecialcases IPm(x)P n(x)dx=JX andsincePc(aj)=1 ,. m(m+1) unlessm=. EXAMPLES. i 1.Show thatCpm(x)dx= ifmisevenand isnotzero. a=i 1 3.5.7.K+)2.4.6.... o odd. v.Art.91(7)andArt.77(10). 2.Show that i Cpm(x)P n(x)dx= ifmandnarebotheven orboth odd. m+n+1 , tn!=(-1) ifmisevenandnodd. v.Art.91(6)andArt.77(8), (9),and(10).cf.J.W. Strutt (Lord Kayleigh)Lond. Phil. Trans. 1870, page579. i 3.Show that f[Pw()]2^=V-rv.Art.89(5)J 2m-4-1 CHAP. V7 .] DEVELOPMENT INZONAL-HARMONIC SERIES. 173 92.Formula(4)Art.91canbeobtaineddirectly from LaplacesEquation bytheaidofGreen sTheorem(v.Peirce sNewt. Pot.Func.48). Take thespecial form ofGreen sTheorem[(148)48Peirce sNewt.. Pot. Func.] fff(VV2V-VV2U}dxdydz =J&quot;(UDnV-VDnU)ds (1) whereV2stands for(D+D*+Z&amp;gt;22 ),Dnisthepartialderivative along the external normal, andtheleft-hand member isthespace-integral through the spacebounded byanyclosed surface, andtheright-hand member isthesurface integral taken overthesame surface,(v.Int. Cal.Chapter XIV.) If7andFare solutions ofLaplacesEquation V2F=V27=0 and(1) reduces to (UD nV-VDnU)ds=.(2) f Now rmjmand rnXnaresolutions ofLaplacesEquationifxcos (v.Art.16). Iftheunitsphereistaken asthebounding surface andU=rmXmand V=rnXn(1)and(2)willhold good. DnU= 27T and(2)becomes Cd$C(nXmXnmXmXn)smO.dO 2ir(n m)CxmXnsinO.dB=.(3)or o Since x=cos6,sinO.dO=dxand(3)reduces to W 1 unlessm=n. 93.Wecannow solve completelytheproblem ofArt.10which was in that article carried tothepoint where itwasonly necessarytodevelop a certain function ofintheform *Itshould benoted that thisproofisnomore general than that ofthelastarticle, for, in order thatGreen sTheorem should apply tormXm,thisfunction and itsfirstderivatives must befinite continuous andsingle-valued within andonthesurface oftheunitsphere, (v.Peirce, Newt. Pot.Func.48.) 174 ZONAL HARMONICS.[ART.94- 7T given that/(0)=1from= to=- and/(0)=0from=^toO=TT. Thisamounts tothesamethingasdeveloping F(x)intotheseries F(x)=APQ(x)+A.P^x)+A,P 2(x)+A3P,(x) +. where^X35)=from x= 1tox= andF(x)=lfrom z= to aj=l. ByArt.90(1)and(2) A=fPO(S)&amp;lt;&=2jdx= 2 2m+1/*andanycoefficient Am= r IPm(x)dx. ByArt. 91,Ex.1 Pm(x)dx= ifmiseven &quot;i^l 1 3.5.7. &quot;m . .,, =(1)2_____ _ifmisodd. m(m+1)2.4.6.(ra 1) Hence Am= ifmiseven ^-^^Mi-;-;-- (-;)ifmisodd .2w+22.4.6.(m 1) and u= I+|rP^cos 0)-^4^^(cos 0)+ii.^|^P^cos 0)+--(2) foranypointwithin thesphere. 94. IfinaproblemonthePotential -Function thevalue ofVisgivenat every pointofasphericalsurface andhascircular symmetry*about adiameter ofthatsurface thevalue ofVatanypointinspacecanbeobtained. Wehave tosolve LaplacesEquationintheform ,2/ry\_j__L_ &amp;gt;0(sinei) 9V)=(1)sinQ *Seenoteonpage12. CHAP.V.] POTENTIAL GIVEN ONASPHERICAL SURFACE. 175 subjecttotheconditions F=/(0)when r=a V=Q r=oo. Wehave/(0)==APo(cos 0)+A^P^cos 0)+J2P2(cos 0)-\ where Am=^^f/(0)P m(cos 0)sinO.dB. v.Art.90(4). Hence V=4-fAl(jjP^cos 0)-fJ2(-)P2(cos 0)+J3/- jP3(cos0)4 (2) istherequired solution forapoint within thesphere, and 0)-fJ2(^)8p2(cos 0)4-J30)4 A(cos 0)+-.. (3) istherequired solution foranexternalpoint. EXAMPLES. 1.Ifonthesurface ofasphereofradius cVisconstant andequal toa show thatV=^a foranypoint within thesphere andV= forany externalpoint. 2.Twoequal thinhemispherical shells ofradius cplaced together toform aspherical surface areseparated byathinnon-conducting layer. Chargesof staticalelectricityareplaced onthetwohemispheres oneofwhich isthen found tobeatpotential aandtheother .atpotentialb.Find thevalue ofthe potential function atanypoint. I-Qr ^is -P^COS 0)-.--P3(COS 0) foraninternalpoint v=nr ; foranexternalpoint. 176 ZONAL HARMONICS. [ART.94. 3.IfF!=/(cos 0)when r=aandVl=when r bshow that for a &amp;lt;r&amp;lt;& where An 4.IfF2=.F(cos 0)when r=bandF2=when r=athen for where ^m 5.Ifthevalue ofthepotentialfunction isgiven arbitrarily onthesurfaces ofasphericalshell buthascircular symmetry*about adiameter V= Fi+F2 (v.Exs.3and4). 6.Two concentric hollowspherical conductors areinsulated andcharged. Theinner oneofradius aisatpotential p,andtheouter oneofradius bisat potential q.FindFforanypointinspace. V=pifr&amp;lt;a, ba\r / ba = if r&amp;gt;b. r_if a&amp;lt;r&amp;lt;b , 7.IfF=0 onthebase ofahemisphere andF=/(cos0)ontheconvex surface, show that forapointwithin thehemisphere where A2k+l=(4A+3)/()P tt+1(a;)^ [v.Art.90(8)]. 8.Iftheconvex surface ofasolid hemisphereofradius aiskeptatthe constant temperature unity andthebase attheconstant temperaturezero show that after thepermanentstate oftemperaturesissetupthetemperature ofanyinternal pointis *Seenoteonpac:e12. CHAP. V.] DEVELOPMENT OFXn . 177 9.Asphereofradius aandwithblackened surface isexposedtothedirect raysofthesuninairatthetemperaturezero. Find thestationary temperature ofanyinternalpoint. Suggestion: Dru-J-huMf(6)=when r=.a. Letu=^Am^mPm(&amp;lt;x&amp;gt;*$)&amp;gt;andf(0)=^mPm(cos 6). Thenwehave 4*Pm(cos 0)+h%AmPm(Gos0}-M^SmPm(Gos6}= , MBmwhence A=-- HereffO)=cos if 0&amp;lt; &amp;lt;and/(0)= if A Zi f(0)= 1+1pi(cos 0)+^P2(cos6)-j^P4(cos 6)+ v.Art.91Exs.(2)and.(3).cf.J.W.Strutt(Lord Kayleigh),Loncl. Phil. Trans, vol.160,page587. 95.Theformulas ofArt.90enable ustodevelopagiven function ofxin terms ofZonalSurface Harmonics, thedevelopment holdingtrue forvalues of xbetween 1and-\-1.If,however, wecanshowbyoutside considerations thatagiven function ofxcanbeexpressedinZonal Surface Harmonics, the development holdingtrue forallvalues ofx,theformulas ofArt.90willgive usthedevelopmentinquestion. Forexampleifnisapositive integer xncanbeexpressedinterms 6fZonal Surface Harmonics nomatter what thevalue ofx,andnoHarmonic ofhigher order thannwill enter. Fortheformulasgiving thevalues ofPI(#),P2(V),Pn(x)(v.Art.77)mayberegardedasnalgebraic equationsofthe firstdegree interms ofx,x2 ,x8 ,xnandPI(X),Pz(x)j&quot;-Pn(x)- From these equations then1quantities x,x2 ,x*,--xn~l ,canbeelimi nated, andthere will result anequationofthe first degreeinxnandPI(X),P2(V),Pn(x),which willenable ustoexpressxnintheform A,+A^x)+A2P2(x)++AnPn(x), nomatter what thevalue ofx,andweshall have thesame formula when 1 &amp;lt;x &amp;lt;1aswhen x &amp;gt;1orx &amp;lt; 1. 178 ZONAL HARMONICS. [ART.95. Letusobtain thisdevelopment. ByArt.90(1)and(2) xn=AP(x)+AlP1(x)+AzP,(x)+ (1) 2m+1C m&amp;gt; ,\j ,&amp;lt;&amp;gt;\where Am= -xnPm(x)dx.(2) Byintegration bypartsweget fajrf&amp;lt; &quot;^7^dx=n(n-l)(n-2)--(n-m +l)xn~m (l-x2 )mdx,(3) -i -i ifm&amp;lt;Cn+1, = ifm &amp;gt;n. Byintegration bypartswereadilyobtain thereduction formula /.ox, Q/&quot;oP+2/1 rr%\q l^/y -wTlPTlPA y,p/IXi^aXI* I-*- *^ ^^ WIlcIlLo J9-+ 1 1 /* 1^ / 2 I+mdx= :-r-. ifn+^ iseven,J7i-fm+1 = ifn+m isodd. ,. (2m+ l&amp;gt;(tt-l)(n-2)-(n-m+1) m~ (nm+1)(nm+3)(wr/i+5)-(w-fm+1) ifm &amp;lt;?i+1an(lWi4-wiseven, = ifm &amp;gt;norifm+nisodd. Therefore -1!) thesecond member ending with theterm :P^,(a;)ifnisevenandwith o n-\-L theterm ,Px(a;)ifnisodd- 7i~pa CHAP. V.] USEFUL FORMULAS. Forconvenience ofreference wewrite outafewpowersofx.179 128 643564iLp^+^p/^+i Ifagiven function ofxcanbeexpressedasaterminating powerseries itcan bedeveloped intoaZonal Harmonic Series bytheaidof(4).Given that f(x)=(t-fa^x+^2ai2+^s^3H j let /()=BQ+^P^z)+tf,P 8(a:)+^3P8(x-)+ ; thenpickingoutcarefullythecoefficient ofPm(x)wehave Bm= 1.3.5.-(2m 1)2.(2m+3) +2.4.(2m+3)(2m+5)I] dPn(x) dx96.Thedevelopmentof-g-*isuseful and iseasily obtained. Let ^_^^ ^P(:r)+^^(a;) 2m+1/i_ %dPjx]Then byArt.90(2); idPm( dx(2) 180 ZONAL HAKMONICS.[ART.97. [Pm()P.(*)] -0 ifm+niseven x= 1 =2ifm+w&amp;gt;isodd. dPJx) ,isan SincePn(x)isanalgebraic polynomialofthenthdegreeinx, CL*Ct algebraic polynomialofthen1stdegreeinx.Therefore in(1)misless dPm(x).than M;consequently thannandisanalgebraic polynomialina;oflower degree dPm(x] dx Wegetthen Am=2m+1ifm-fnisoddandm&amp;lt;.n, ifin-f-niseven orin &amp;gt;n1 ; dxnl\ \ n-3VbyArt.95(3). and &amp;gt;-.()+ &amp;lt;) thesecond member ending with theterm3Pi(x)ifnisevenandwith the termP(x)ifnisodd. From(3)anumber ofsimpleformulas arereadilyobtained. Forexample [T.(4)andArticle 77(12)]. (x2-1) [v.(5)andArticle 91(7).=xPn(x)-(5) (6) (7) 97.Bytheaidoftheformulas ofArt.96anumber ofvaluabledevelop ments canbeobtained. LetusgetcosnOandsinnOnbeing anypositivereal. zcosnOand z=sinn&aresolutions oftheequation ^HAP.V.] ADDITIONAL DEVELOPMENTS, ifweletx=cos6,oftheequation dx181 (1) Let stherequired developmentofcosnOorofsinnO. ThenVam[(1-x2 )*L*&)-x^M+n*Pm(x)1=0 by(1). m=0L J 2=Pm(x)isasolution ofLegendresEquation (v.Art.77). Hence and(1)becomes &quot;Tr-dPm(x) -| Formulas(4)and(6)ofArt.96enable ustothrow(2)intotheform I ft dx_ri2(m+I)2 &amp;lt;lPni_1(x}-\_ (3) (3)must beidenticallytrue. Therefore thecoefficient ofm+1^must equal zero,andwehave 2m-f-5 n2m2... m+ln*(m-f S)2^* IfwearedevelopingcosnO 1CaQ=-IcosnBsinB.dB o ir =- j[sin (?i-f1)^sin(n o 11+COS 717TbyArt.90(4), and ;osw^cos sin0.c?0 31cosmrbyArt.90(4), (6) 182 ZONAL HARMONICS.[ART.1)7. (4), (5),and(6)giveus COSn6=- Ifftisawhole number 1-j-cos TITTor1cosWTTwillvanish andtheseries willendwith theterminvolving Pw(cos 0).Forthiscase(7)maybe rewritten .1 2.4.6. cosnv=-- 23.5.7. IfwearedevelopingsinnO 1r. 1sinUTTa=-IsmnvsmB.dB=- &amp;gt;2j 2tn~~~1 o 1=? Jsinn0cos sin0.rf0= |-^and sin=-1.J^[P(cos )+5^2P,(co6 0) Ifnisawhole number sinWTT= ,and alltheterms of(9)vanish except those involving Pn_1(cos 0),Pn+1(cos 0),Pn+3(cos ff)&c.,which become inde terminate. Forthiscase itisnecessarytocomputean_1independently. CHAP.V.] EXAMPLES. 183 Wehave 2nlran-i=n Is^n^0-fn_i(cos 0)sinO.dO o 7T =- I[cos(n 1)0cos(n Hence and EXAMPLES. 1.Show that =^I1+5(-JP2(cos 0)4-9(^-7JP4(cos 0)-f13 whence [v.Art.90(4)andArt.82]. 2.Show that whence (D [v.Art.90(4)andArt.82]. 3.Byintegrating theresult ofEx.1andsimplifying bytheaidofArt.96 (5),obtain thedevelopment sin-x=I[3(!)pl 184 ZONAL HARMONICS.[ART.97. whence =||~Po(cos0)-3fflp^cos0)-7(^P 4.Byintegratingtheresult ofEx.2andsimplifying bytheaidofArt.96 (5)obtain whence sin8=|PiP(cos 0)-5Q(|)2p2(cos 0)-9 (|)(j$P*(cos*)J Tomake clearer theanalogyofdevelopmentinZonal Harmonic Series with developmentinFourier sSeries wegiveonpage 185acutrepresentingthe firstseven Surface Zonal HarmonicsPj(cos 6),P2(cos 0),-P7(cos 0),which areofcourse somewhat complicated Trigonometriccurves resembling roughly cos0, cos20, cos70; andonpage 186, the first four successiveapproxi mations totheZonal Harmonic Series |+|p1(cos(9)-|.|pa(co8(?)+i|.||p5(co8(9)-...[i] [v.(1)Art.93],and |[PO(COS ff)-3(!)Pi(cos 0)~7(^^(cos0) (v.Ex.3Art.97). [i]isequalto1from=to=^andtofrom=^to=7r;and [n]isequalto6from=to6&quot;--TT . The figuresonpage186areconstructed onpreciselythesameprincipleas those onpages63and64,withwhich theyshould becarefully compared. 98.ByapplyingGauss sTheorem(B.0.Peirce, Newt. Pot.Func.31)or thespecial Form ofGreen sTheorem, CCTV2Vdxdydz=&nVds=-TTpdxdydz, CHAP.V.] ZONAL HARMONIC CURVES. o w185 186 ZONAL HARMONICS.[ART.97. \ v.page184. CHAP. V.] THIN SPHERICAL SHELL. 187 [Peirce,N.P.F.49(149)]toaboxcutfrom aninfinitelythin shell of attracting matter byatube offorcewhose end isanelement ofthesurface of theshellwereadilyobtain theimportantresult irPK=DnV,-D nVz.(1) wherepisthedensity andKthethickness oftheshell, V\thevalue ofthe potentialfunction duetotheshell ataninternal point andV2itsvalue atan externalpoint, andwhereDnisthepartialderivative along theexternal normal totheouter surface oftheshell. Ifwehave todealwith asurface distribution ofmatter wehaveonlyto replace pKin(1)bycrwhere oristhesurfacedensity, whence 4,7ro-=DnVl-DnV2(2) (v.Peirce, K.P.F. 45,46,and47). Formulas(1)and(2)enable ustosolve problemsinattraction whenwe know thedensityoftheattracting mass, andproblemsinStaticalElectricity whenweknow thedistribution ofthecharge, bymethodsanalogoustothat of Art. 94. Forexampleletusfindthevalue ofthepotential function due toathin material sphericalshell ofdensity pandradius a. SinceVmust beasolution ofLaplacesEquation andmust befinite both when r=and r=oowehave tandVzmust approachthesamelimiting values asrapproachesa.Hence or DnV,=DrV,= \v, Therefore by(1)Aa2m+lDnVz=DrVz=-(m+1)-- +Pm(cos 0). ifKisthethickness oftheshell. 188 ZONAL HARMONICS.[ART.99. Letp=/(cos 0)=2}&amp;lt;7mPm(cos 0) i where Cm=m ^~Cf(x)P m(x)dx byArt.90(2). Then 4irKCm=(2m+l)Amam-1 , and 47T/CCL,_ 4-7TK _- andFl=4TOK -*.( *), (3) 99.Wecannowgetthevalue ofthepotentialfunction duetoaspherical shell offinite thickness, providedthat itsdensity canbeexpressedasasumof terms oftheformO*Pm(cos 6). Letabetheradius oftheouter surface and bbetheradius oftheinner surface oftheshell. 1st. Letp=O*Pm(cos 6).Then fortheshell ofradius sandthickness ds f&amp;lt;&amp;lt; rm Fl= andF2=4^ 5___Pm(cos^ by(4)Art. 98. Then ifr &amp;lt;b if r&amp;gt;a (gk+m+S l&amp;gt;k+m+ 3JPCT^COg and if b&amp;lt;r&amp;lt;a r a -fFi= TO(cos 0).(3) 2d. Ifp=C^P^cos 6}thesolutions will consist ofsums ofterms of theforms givenin(1), (2),and(3). 3HAP. V.] ZONAL HARMONICS OFTHESECOND KIND. 189 EXAMPLES. 1.Iftheshell ishomogeneous V=2Trp(a?b2 )ifr&amp;lt;b, -= 77 Iif b&amp;lt;r&amp;lt;a. 2.Ifthedensityisanygiven function ofthedistance from thecentreMV= ifr&amp;gt;a, andV=aconstant if r&amp;lt;b. 3.Ifthedensityatanypointofasolidsphereisproportionaltothesquare ofthedistance from adiametral plane if&amp;gt; 4.Ifthedensityatanypointofasolid sphereisproportional toitsdistance from adiametral plane M\~a.1a3_ 1.1a5_ 1.1.3 a7 V=+6PP*(COS *&amp;gt;- 63 ;&amp;gt;~|Se &amp;gt;~ J if r&amp;gt;a.CompareEx.2Art. 80. 100.Wehave seen inArt.18(c)(3)that gro(.)=CPCT(.)/(1_^ (a)]i,(1) noconstant term being understood with j-. ^,/(1a^L-^wC^/J isarational fraction andbecomes infiniteonlyforx=1, (1#/[*(*/ J x=1,and fortheroots ofPOT(ar)=0,allofwhich arerealand lie between 1and1,ascanbeproved bythe aid ofthe relation P,&= 2mm\ dxm Ifx2 &amp;gt;1 I-isfinite anddeterminate andcontains no constant term. Hence if cc2 &amp;gt;1 fortheconstant factor of ahasbeenchosen sothatC= 1. 190 ZONAL HARMONICS.[ART.100. Ifx2 &amp;lt;1thesecond member of(2)isnot finite anddeterminate, andwe arethrown back totheform(1),andCprovestobeunity. (1)givesusreadily Qo(*)=log- (3) ifz2 &amp;lt;l. (2)gives us Q(x)= \log|i if &amp;gt;!. From Art.85(10)itfollows that if a? _ (7canbedetermined and isequalto*- (2m)\-^^^* /_I)m2mm l t0-(4)!-lf^^ (I}m+I2mmldmr,_cdx1Hence QJ*)=L-J^^[^&quot;1}/(^Ip-&quot;1J ifz2 &amp;lt;l, (l and Qm(x)=( ( if x*&amp;gt;l. (7)and(8)giveusforQ(x)and^(sc)thevalues alreadywritten in(3), (4), (5),and(6). Bytherepeated applicationoftheformula (m+1)Qm+l(x)-(2m+X)xQ m(x)+mQm^(x)= , (9) which maybeobtained forthecasewhere z2 &amp;lt;1from Art.16(13)and(14), andforthecasewhere x* &amp;gt;1from Art.16(9),anySurface Zonal Harmonic oftheSecond Kind canbeobtained fromQ(x)andQi(x)asgivenin(3), (4), (5),and(6). CHAP. V.] EXAMPLES. 191 Analogous,formulas forpm(x)andqm(x)canbeobtained withoutdifficulty from Art.16(4)and(5).Theyare (m+l)*qm+l(x)-(2m+l)xpm(x)-m*qm_Jx)=(10) and Pn+,(x)+(2m+1)xqm(x)-p^x) =(11) andtheyholdgoodforanyvalue ofm. EXAMPLES. 1.Confirm thevalues ofQ(x)andQ{(x)giveninArt.100(3), (4), (5),and (6)byexpanding them andcomparing them with Art.16(13), (14),and(9). 2.Ifthevalue ofVonthesurface ofacone ofrevolution canbeexpressed interms ofwhole powers positiveornegative ofr,Vcanbefound forany pointinspace,cf.Art. 81. Ifr-jr--f when=athen 3.IfF=A/m+-riwhen 6=a,andF=0 when=0, 4.FindVforpoints correspondingtovalues of6between aandftwhen Vcanbegiveninterms ofwhole powersofrfor6=aandfor=ft. 5.Findbythemethod ofArt.16solutions ofLegendresEquationofthe form .tr IN|(m-l)m(m +l)(wt+2K1V22(2!)2 28 (3!)2 (m-2)(m-l)m(m+l)(m+2)(m+3) T&quot;28 (3!)2 Ifmisawhole number, lPm(x)=Pm(x)and^P^a)= (l)mPm(x).No matter what thevalue ofm,iPTO(#)isabsolutely convergent for 1 &amp;lt;x &amp;lt;3, and_iP TO(#)isabsolutely convergentfor 3 &amp;lt;x &amp;lt;1. 192 ZONAL HARMONICS.[ART.100. 6.Bytheaidof(7)Art.16show that F= -r-sin(nlogr)kn(cos 0) , Vr F=-=cos(nlogr)kn(cos 0) , aresolutions ofLaplacesEquation rDr\rV)+--1V-^sin(nlogr)ln(Goa 0) , F= -=.cos(nlogr)ln(cos 0) , if *)=*-*+-&amp;lt;*)=i+ + x* and + 3! 5! 7! ,(.r)and ln(x)areconvergentifce2 &amp;lt;1,butaredivergentif sea=1. 7.Show bytheaidofExample 5that F= -r=sin(^logr)7f w(cos ^) F=-icos(TIlogr)Kn(cos 0)1F= -j=.sin(wlogr)K n(cos^), F=-pcos(nlogr)Kn(cosd), aresolutions of rJ)r\rF)+-7^De(sinsin (/ if [_7i2+(2/JL712 &quot;^^ _._ 22 (2!)2 28 (3!)1 CHAP. V.] EXAMPLES. 193 and &quot;+-(x . 7fn(cos 0)isconvergent exceptfor6=TT,and 7if ra(cos0)isconvergent exceptfor=0. kn(x),ln(x),Kn(x),andKn(x)aresometimes called Conal Harmonics. Theyareparticularvalues ofzwhichsatisfy LegendresEquationwritten in theform Foranelaborate treatment ofthem seeE.W.Hobson on&quot;AClass ofSpherical Harmonics ofComplex Degree.&quot; Trans. Camb. Phil. Soc., Vol.XIV. 8.IfV=f(r)when B=P, cos[a(x~10^if 9.IfV=f(r)when=(3andr&amp;lt;a, andF=0 when r=a, 10.IfF=/(r)when=^8anda &amp;lt;r &amp;lt;b,andF= when r==a andwhen r=b, V^A ^M(COS 0) rm7r(logr loga)&quot;I^~^m ^-(co S)8)SinLlog6-logaJMl where m=--^- and logbloga 1. logbloga^rjy logblogadx;if 194 ZONAL HARMONICS. 11.If$ &amp;gt;ftcosemust bereplaced by(-cos0)inexamples 8,9,and10. 12.IfV=f(r) when 6=0,andF=0 when=y, C&amp;lt;J\C-ff*X\ ^afcOS 0)4 ~ irtfjdKJC*f(e } *.(cos /3)4 if fi&amp;lt;0&amp;lt;y. 13.IfF=/(r) when=anda&amp;lt;r&amp;lt;6, F=0 when ^=yand jandF= whenr=.a andwhen r=b, sy)^.(cos y)a.fcos g) .m7r(logrlogq) s7)-7^(cosr)^(cos^)S1 log6-loga where =.=^1 and logologa mirxSmlogb-logar-Sm logi-logaS it /3&amp;lt;0&amp;lt;y and a&amp;lt;r&amp;lt;b. 14.IfF=/(r) when ^=^and&amp;lt;/&amp;lt;*, andF=0 whenr=a and Z&amp;gt;rF+AF=0 when r=5, a(COS ff) r. sina-log where -loga)+AA[W(log&-loga)Sm andamisarootoftheequation acosalog- J+AJsin^alog- j= v.Art.68Ex. 5. CHAPTER VI. SPHERICAL HARMONICS. 101.When wearedealing withproblemsinfinding thepotential function duetoforces which have notcircular symmetry*about anaxisandareusing Spherical Coordinates, wehave tosolve LaplacesEquationintheform )+D e(SmODeV)+ D*V=()(1) [v.(xin)Art.1]. Togetaparticularsolution of(1)weshall assume asusual thatVisa productoffunctions each ofwhich involves butasinglevariable. LetV=R.&,whereRinvolves ronly, involvesonly,and&amp;lt; &amp;lt;only. Substitute in(1)andweget rsin2d\rR)sind R dr2dO Asthe firstmember does notcontain&amp;lt;thesecond member cannot contain &amp;lt;,andasitcontains noother variable itmust beconstant;call itn2 .Equa tion(2)isthen equivalenttothetwoequations +____ =0Rdr2rsin0 dO sin2^ (3)hasbeen solved before andgives us &amp;lt;$=Acos n&amp;lt;f&amp;gt; -\-Bsin n(f&amp;gt; (5) [v.Art.13(a)]. The firstterm of(4)doesnotinvolve andthesecond andthird terms do notinvolve r. *Seenote, page12. 196 SPHERICAL HAEMONICS.[ART.101. -^--^-r-must, then, beaconstant; weshall call itm(m-fl)asinArt. 13(c).Then(4)breaks upinto dr* ^sin^l (6)wassolved inArt.13(c)andgives X=Ali+ Ifin(7)wereplacecosbypweget theequivalentof [v.(17)Art.85],which wassolved inArt.85forthecasewheremandnare positive integers andn &amp;lt;m+1. v.(18)and(19)Art. 85. From(19)Art.85wegetasaparticularsolution of(9) ifwerestrict ourselves towhole positivevalues ofwand n,asweshall do hereafter unless thecontraryisexplicitly stated, andsuppose mnot less than n. Asecond butlessusefulparticularsolution of(9)is Combiningourresults wehave asimportant particularsolutions of(1) &amp;gt; V=rm(Acos n&amp;lt;j&amp;gt;+Bsinnj)sin&quot; and V=-jL(Acos n&amp;lt;j&amp;gt;+3sin 7i&amp;lt;)sin&quot;d&quot;^&amp;gt; (13) wheremand ?iarepositive integers and &amp;lt;m+1. CHAP. VI.] TESSERAL HARMONICS. 197 -isanewfunction of//-,that is 102. sinra6 or(1 p?)l ofcos6}andweshall representitbyP,(ft)*and shall call itanassociated functionofthenthorder and ?&amp;gt;ith degree.Itisavalue ofsatisfying equation (9)Art101. Bydifferentiating thevalue ofPm(x)givenin(9)Art.74wegettheformula Py^-(2&quot;0!Bing F (m-n}(m-n-1) _2-r &quot;W2-m!(ra n)![_/2.(2ra-1) (ra tt)(m nl)(m?t2)(ran3)m_n_4_ 2.4.(2m-1)(2m-3) theexpressionintheparenthesis ending with theterminvolving /Aifmnis even andwith theterm involving pifmnisodd. Forconvenience ofreference wegiveonthenext pageatable fromwhich P?(At)canbereadily obtained forvalues ofmandnfrom 1to8. cosn&amp;lt;^P^(fi)and sin n&amp;lt;f&amp;gt;Pn?(/Lt),thatis, cos sinn &quot;^and sin TI&amp;lt;/&amp;gt; sin&quot;T&quot; arecalled Tesseral Harmonics oftherath degree andnthorder, andare values ofVwhichsatisfytheequation oritsequivalent(2) (3) There areobviously2ra-f1Tesseral Harmonics oftherathdegree, namely sin &amp;lt;/&amp;gt;sin sin 2&amp;lt;sin2 sin sn cos Ifeach ofthese ismultiplied byaconstant and theirsum taken,thissum iscalled aSurface Spherical Harmonic oftherathdegree, and isasolution of equations (2)and(3).WeshallrepresentitbyYm(p,&amp;lt;j&amp;gt;)orbyYm(8,&amp;lt;). Most oftheEnglish writers represent thisfunction by j&amp;lt;f&amp;gt;)and arecalled Solid Spherical Harmonics ofthe mthdegree, andaresolutions ofLaplacesEquation (1)Art. 101. Toformulate : ,+)=24,cosn+sin +B.sinn^sin or cos Bnsin(5) isaSurface SphericalHarmonic oftherathdegree. ATesseral Harmonic isaspecialcase ofaSurface Spherical Harmonic, and aZonal Harmonic aspecialcaseofaTesseral Harmonic; Pm(fi)being the Tesseral Harmonic ofthezeroth order andtherath degree;itmight be writtenP. EXAMPLES. 1.Show that reduces to (1-x*)-2(n+l)x+[ra(ra+1)-n(n+l)]y= ifwesubstitute(1 x*)\yforz,evenwhen raandnareunrestricted. CHAP. VI.] TABLE FORASSOCIATED FUNCTIONS. 199 2.Show that ifinthesecondequationofEx.1welety=^ akxkweget a(--*)(. ++!+*) i+2 /;i -i\//o\ **jfc I*^&quot; ---^ whence=_p(a?)and=gr^(a)aresolutions ofthe firstequation ofEx.1, nomatter what thevalues ofwand?i,if (m-n)(m-n-2)(m and4!H ,H Ifmnisapositive integer, p(x)ory&quot;(a;)willterminate with theterm involving xm~n ,andinthatcase (m ri)(m n &quot; 2.4.(2m-1)(2m-3) 200 SPHERICAL HARMONICS.[ART.103. theparenthesis ending with aterm involving xifmnisevenandxif mnisodd,isasolution ofthe first equation ofEx. 1.Ifmandnare integersthisvalue ofzis-^.P(x). (jjYfl)\ 103.Wehave seen inthelastchapterthat inmany problemsitisimport anttobeable toexpressagivenfunction ofcos6,that isof/JL,interms of Zonal Harmonics ofJJL.Soitisoften desirable toexpress agivenfunction of p,and &amp;lt;f&amp;gt;interms ofTesseral Harmonics ofpand &amp;lt;. If,forexample, wearetryingtofindthePotential Function duetocertain forces andhave thevalue ofthefunction givenforsome given value ofr, thatis,onthesurface ofsome given sphere whose centre isattheoriginof coordinates, ofcourse thegiven value willbeafunction of6and&amp;lt;and ifwe canexpressitinterms ofSpherical Harmonics ofand&amp;lt;wehave onlyto multiply eachterm bytheproper powerofrtogettherequiredsolution of theproblem.Forweshall then have avalue ofVsatisfying Laplaces Equation andreducingtothegiven function ofand&amp;lt;onthesurface ofthe given sphere. 104. Supposethatwehave afunction offiand&amp;lt;givenfor allpointson theunitsphere,thatis,forallvalues ofJJLfrom 1to1and forallvalues of &amp;lt;j&amp;gt;from to2-7T, /*and &amp;lt;f&amp;gt;being independent variables, andthatwewish to expressitinterms ofSurface SphericalHarmonics. Assume that snw*p (/*&amp;gt;)J Letusconsider firstafinite case,andattempttodetermine thecoefficients sothat m=p *(ti)J shall hold goodatasmany pointsofthesphereaspossible. Theexpression inbrackets inthesecond member of(2)isaSurface SphericalHarmonic of themthdegree andcontains 2m -j-1constant coefficients. Thewhole number ofcoefficients tobedetermined isthen thesumofanArithmetical Progression ofJ9-J-1terms the firstterm ofwhich is1andthelast is2p+l,and is therefore equalto(p+I)2 - Lettheinterval from/*= lto/&amp;lt;i=lbedivided intop+2partseach of which isA/Isothat(p+2)A/u=2,andlettheinterval from &amp;lt;j&amp;gt;=to &amp;lt;f&amp;gt;=2?r bedivided intop+2parts each ofwhich is A&amp;lt;sothat(p+ 2)A&amp;lt;=2?r. CHAP.VI.] DEVELOPMENT INSPHERICAL HARMONIC SERIES. 201 Then ifwesubstitute inequation (2)inturn thevalues(1-}-A/*, - -[-1+(p+l)A/z,A|; (-1+A [_1+(p+1)A/*, 2A*]; [-1+A/z,(p [-1+2A/&, (/&amp;gt;+1)A), [-1+(p+1)A/*, (p+1)A|;since the first member ineach case willbeknown weshall have(p-fI)2equations ofthe firstdegree containing nounknownexcept the(p+I)2 coefficients, andfrom them thecoefficients canbedetermined. Whentheyaresubstituted inequa tion(2)itwillholdgoodatthe(p+I)2points oftheunitsphere where p-\-l circles oflatitude whose planesareequidistant intersect p-\-l meridians which divide theequatorinto equalarcs. Ifnowpisindefinitely increased thelimiting values ofthecoefficients will bethecoefficients inequation (1), and(1)willholdgoodallover thesurface oftheunitsphere. Todetermine any particular constant wemultiply each ofour(p+1)2 equations byAftA&amp;lt;times thecoefficient oftheconstant inquestioninthat equation andadd theequations and theninvestigate thelimiting form approached bytheresulting equationaspisindefinitely increased. Aspisindefinitely increased thesummation inquestionwillapproach an integration; andsinced^d^= sinO.dO dcf&amp;gt;istheelement ofsurface ofthe unitsphere, and asthelimits 1and1of/*correspondtoTTand of6the integrationisasurface integration overthesurface oftheunitsphere. Indetermining anycoefficient asAn^min(1)the firstmember ofthelimiting form ofourresulting equationwillbe !*, &amp;lt;)COS Inthesecond member weshallcome across terms oftheforms 2T1 27T1 (* f* f* f* I d&amp;lt;}&amp;gt;Isin l&amp;lt;f&amp;gt;cos n&amp;lt;^&amp;gt;P^f/jLjP^f/jLjdfjL, jd&amp;lt;f&amp;gt;Icos l&amp;lt;f&amp;gt;cosJJ JJ 01 0-1 2r1 2n-1 oi oi andother terms allofwhich come under theform 2*i whereYm(p,&amp;lt;)andrj(/t, &amp;lt;)areSurface Spherical Harmonics ofdifferent degrees. Ifwearedetermining acoefficient Bnmtheonlydifference isthat sin n&amp;lt;f&amp;gt; andcos n&amp;lt;f&amp;gt;willbeinterchangedintheformsjust specified. 202 SPHERICAL HARMONICS.[ART.105. 105. The integraloverthesurface oftheunit sphere oftheproduct oftwo Surface Spherical Harmonics ofdifferent degreesiszero. 2rr 1 //*d&amp;lt;f&amp;gt;IY^JJL,&amp;lt;j&amp;gt;)Ym(}JL, &amp;lt;f&amp;gt;)d(ji=Q.(1)X Foraswehave seenU=?JY l(fji, &amp;lt;f&amp;gt;)andV=rmYm(n,&amp;lt;f&amp;gt;)aresolutions of LaplacesEquation. Hence byGreen sTheorem C(UD nVVDnU)ds= v.Art. 92. UDnV-VDnU=(m-iy+~* Y&, *)Ym(pt&amp;lt;#&amp;gt;), =(m onthesurface oftheunitsphere ;and 2ir (m- Oj,(/*,&amp;lt;^&amp;gt;)rm(/*,fids=(m- I) Hence unless I=m 2rr1 1 EXAMPLES. 1.Obtain(1)Art.105directly from theequation m(m-j-1)J v.(3)Art. 102,andArt. 91. 2.Show thattheintegraloverthesurface oftheunitsphereoftheproduct oftwoTesseral Harmonics ofthesame degreebutofdifferent orders iszero. Suggestion: /sinkd&amp;gt;cos ld&amp;gt;.d&amp;lt;j&amp;gt;=Tsin k&amp;lt;j&amp;gt;sin l&amp;lt;f&amp;gt;.d&amp;lt;j&amp;gt;=(cosk$cos / s1 i 106 C.Pid^Pmd^dp==^unless l=m A 2(m+ri)\ ifI-2m+I(m n)! CHAP. VI.]DEVELOPMENT INSPHERICAL HARMONIC SERIES. 203 For -1 &amp;lt;*/*&quot; d/t1 byintegration byparts. Replacingwbyrc1inequation (2)Art.84andremembering that &quot; nisapossiblevalue of s:(n~1)weget orifwemultiply by(1 (m =0, or Hence follows thereduction formula Usingthisformula wtimesweget = unless l= 2(HI-fn)! . v.Art.89(4)and(5).2m+1(i n)!if 204 SPHERICAL HARMONICS.[ART.107. 107.Wearenow able tocompletethesolution oftheprobleminArt.104 2ff 2* 2ir andsince Tcos2 n&amp;lt;f&amp;gt;.d&amp;lt;f&amp;gt;=Tsin2 n&amp;lt;f&amp;gt;.d&amp;lt;}&amp;gt;=TTand Cd&amp;lt;f&amp;gt;=2-7Twegetasthe coefficients in(1)Art.104 osn- whence TM=OO n=m cos w&amp;lt;&amp;gt;+s-sninn*)pWJ andthedevelopmentholds goodfor allvalues ofpand&amp;lt;correspondingto pointsontheunitsphere, provided onlythat thegivenfunction satisfies the conditions thatwould have tobesatisfied ifitwere tobedevelopedinto a Fourier sSeries. Ifweuse^jandfainplaceofftand &amp;lt;#&amp;gt;in(1), (2),and(3),wecanwrite(4) intheform COSw(*~ Formulas(1), (2), (3),and(4)areconvenient foractual work; (5)israther more compactlywritten. 108. Asanexampleletusexpresssin2cos2sin &amp;lt;f&amp;gt;cos &amp;lt;j&amp;gt;interms of Surface SphericalHarmonics. Here f(u, &amp;lt;f&amp;gt;)=/*a (l~~ /**)s^n 2&amp;lt; vz 1in ,Isin 2&amp;lt;f&amp;gt;.dd&amp;gt;= , J CHAP. VI.] ILLUSTRATIVE EXAMPLE. 1205 2n,m=2m+1(m ri)l 4?r(m+n)!iIsin 2&amp;lt;sin n&amp;lt;j&amp;gt;.d&amp;lt;f&amp;gt; , / = unless n=2. Ifn=2 Jsin 2&amp;lt;sin n&amp;lt;f&amp;gt;.d&amp;lt;f&amp;gt;= |si sin 2&amp;lt;j&amp;gt;sin n&amp;lt;f&amp;gt;.d&amp;lt;f&amp;gt;= |sin2 2^&amp;gt;.c?^&amp;gt;=TT, and -M=2m+l(m 2)! 4(m 2-m! 4rf/*. byrepeated integration byparts,= ifm&amp;gt;4, =7201(V- 192! 4096_1 6! 7~105 Byalikeprocess wefind .#28= and 2?2,2=TO 2,8 sin2cos26sin&amp;lt;cos &amp;lt;#&amp;gt;= TJ^(A1)sin 2&amp;lt;f&amp;gt;+P42^)siifw=4, Hence (1) =-sin2^sin sn sn(3) The required expression might have been obtained withoutusing the formulas ofArt.107,byavery simple device, asfollows : (4)sin2cos2sin&amp;lt;cos &amp;lt;f&amp;gt;==n*sin2sin 2&amp;lt;f&amp;gt; 206 SPHERICAL HARMONICS. Ifnowwecanexpress /u2intheform ,JL **~4.3[ART.109. wor^w^bedone. 8 354.r7 o *\i/ iwhence /*= -r^rz j-\rQI andsubstitutingthisvalue in(4)weget(2). EXAMPLES. 1.Show that cos8sin86sin &amp;lt;icos2 d&amp;gt;= 2.Show that cos 2d&amp;gt;=2cos2&amp;lt;/&amp;gt; |T-. ataninternal pointandcos cossn 3.IfinaproblemonthePotential Function F= shall obviously havewhen r=awe ^)J sn atanexternal point,whereA &amp;gt;m,Jn,m,and5 n&amp;gt;mhave thevalues givenin(1), (2),and(3)Art. 107. 4.Solve problems (3), (4),and(5)ofArt..94 forthecasewhere Visnot symmetricalwith respecttoanaxis. 109.AnySolid SphericalHarmonic rmYm(n,&amp;lt;f) being avalue ofVthat satisfies LaplacesEquationinSphericalCoordinates willtransform into a function ofx,y,and satisfyingV2F= ifwechangetoasetofrectangular CHAP. VI.]ANOTHER DEFINITION OFASPHERICAL HARMONIC. 207 axeshaving thesameorigin andthesame axisofJTas thepolar system. Moreover thenewfunction willbeahomogeneous rationalintegral Algebraic function ofx,y,z,oftherathdegree. Foreachterm ofrmcosnPjC isoftheform Crmcosn~2k &amp;lt;J&amp;gt;sin2*^sin&quot; Gosm-2l~n where 2k&amp;lt;n+land 2l&amp;lt;m n+l. Thismaybewritten Cr21 .rm-2l-cosm-2l-n0.rM-2*smn-2*0cos*-2fc &amp;lt;.r2*sin2ksin2* &amp;lt; which becomesC(x2-fy*-fzz )1xm-2l~nif-z2k &amp;gt; and isahomogeneous rationalintegral Algebraic function ofx,y,andzofthe rathdegree. Thesame thingmaybeshown ofeachterm ofrmsin n&amp;lt;J&amp;gt;P(/jL). Consequently rFm(/z,&amp;lt;)isahomogeneous rationalintegral Algebraic func tionoftherathdegreeinx,y,and z. 110.Anyhomogeneous rationalintegral Algebraic function Sm(x,y,z)of therathdegreeinx,y,and2,which isavalue ofVsatisfying V2F= con tains2m-f-1arbitraryconstant coefficients. ForSm(x,y,z)will ingeneral consist of^-terms andwill ,, . , .(m+!)(/+2)..therefore contain-1-- -coefficients. \72Sm(x,y,z)willbehomogeneousofthe(ra 2)ddegree andwill contain ^--- -coefficients, which, ofcourse, willbefunctions ofthecoefficients in Sm(x,y,z).SinceV2m(,y,)=independentlyofthenumerical values ofx,y,andzthe-coefficients inV2m(#,y,z)must beseparately zero,andthat factwillgiveus^vequations ofcondition between the (m+l)(ra-f 2). ./ ^-^- !-originalcoefficients andwillleave^ --22 or2?n+1ofthem undetermined. Sm(x,y,z)contains, then, thesamenumber ofarbitrarycoefficients asrmYm(iJ,,&amp;lt;f&amp;gt;). Wecanthen choose thecoefficients inrmYm(fj,}&amp;lt;)sothat itwilltransform intoanygivenSm(x,y,z). ConsequentlyaSolid Spherical Harmonic oftherathdegree might be denned asahomogeneousrational inter/rat Algebraic function ofx,y,andz, $m(x &amp;gt; y&amp;gt; *)&amp;gt;ftfiemthdegree satisfyingtheequation \/*Sm(x,y,z)=Q; and a SurfaceSpherical Harmonic oftherathdegree assuch afunction divided by that isbyr&quot;1 . 208 SPHERICAL HARMONICS.[ART.111. EXAMPLES. 1.Show that ifSm(x,y,z)isaSolid Spherical Harmonic ofthemthdegree x,y,z)~]=n(2m+n+l)r-*S m(x,y,z). Suggestion: 2.Show that iffn(x,y,z)isarational integral homogeneous function ofx, y,andzofthenthdegreeitcanbeexpressedintheform fn(x,y,z)=Sn(x,y,z)+r*Sn_2(x,y,z)+i*Sn_4(x,y,z)+- , (1) terminating with rn~lSl(x,y,z)ifnisodd,andwith rnS(x,y,z)ifniseven. Suggestion:Ifaterm r/Sn_jwere presentinthesecond member of(1),and wewere tooperatewithV2onbothmembers weshould byEx.1haveaterm n_iwhich would beirrational when alltheother terms oftheresulting equationwere rational. Nosuchterm, then, could occur. Inthesameway itmaybeshown byoperatingtwice on(1)withV2that there canbenoterm rsSn_3in(1);andthus stepbystepwecanreach theresult formulated in(1). 3.Expressx2yzintheform /S4+rzSz-\-r*SQ. Suggestion:let x*yz 4+ andtakeV2ofbothmembers weget Operate againwithV2 .=120 . Whence y*,andS^ 4.Expresssin2cos2sin&amp;lt;cos&amp;lt;interms ofSurface SphericalHarmonics. Suggestion: sin2cos2&sin &amp;lt;f&amp;gt;cos&amp;lt;=j- Forresult v.Art.108(3). 111.Atransformation ofcoordinates toanew setofaxeshaving thesame originastheoldsetwillchange agivenSurface Spherical Harmonic into another ofthesame degree.Forsuchatransformation does notchange the form ofLaplacesEquation V2T=0 ifboth sets ofaxes arerectangular, and itiseffected byreplacing x,y,andzintheSolid Harmonic correspond ingtothegivenSurface Harmonic byxcosax+ycosa2+zcosa8, xcosft+ycos/32+zcosySg, andxcosyt-f-ycosy2+zcosy3respectively, where thecosines arethedirection cosines ofthenew axes,and itwill leave CHAP.VI.] LAPLACIANS. 209 thefunction ahomogeneousfunction oftherathdegreeinthenew variables, andondividingthisbytherathpoweroftheunchanged radius vector weshall have aSurface Spherical Harmonic oftherathdegree. 112.Wehave seen inArt.75that if(xlfy1;&amp;gt;%)arethecoordinates ofa given point F=*=(1) V(3-x,Y+(y-yO2+(z-ztf isasolution ofLaplacesEquationV2F=0, andtransforming tospherical coordinates that (2) , V/r22n-1[coscos6l+sin sin1cos (&amp;lt; fa)]+r* isasolution of rD(rF)+ 2*(sin6AF)+ItfF= .(3) Ifyistheangle between theradii vectores rand r^ofthepoints (a*,y,2) and(x1?2/!, z-^)(1)canbewritten ~~ which must beequivalentto(2),andhence cosy=cos cos0^-f-sin^sinXcos (&amp;lt;^^^. (4)which isasolution of(3)isofthesame form as(5)Art.75andby developingitaswedeveloped (5)Art.75wefindthat F= isasolution oftheequation __l_A(si andthat V rmPm(cosy)andF=^Pm(cosy) aresolutions of(3). Ifwetransform ourcoordinates keeping theorigin unchanged andtaking as ournewpolaraxis theradius vector of(xltylt%)ybecomes ournew 6and P^cos y)reduces toP^cos 0) ,aSurface Zonal Harmonic, oraLegendrian,* of therathdegree.Itisthen aLegendrian havingforitsaxisnottheoriginal polar axisbuttheradius vector of(xityi, z-^). Since aLegendrianisaSur faceSpherical Harmonic, Pm(cosy)=Pm[coscos0!+sin sinOlcos (&amp;lt; fa)] isaSurface Spherical Harmonic oftherathdegree. *v.Art. 74. 210 SPHERICAL HARMONICS. [ART.113. Itis,however, ofvery special form, since being adeterminate function of fji, (f&amp;gt;, fjLlfand &amp;lt;iitcontains buttwoarbitraryconstants ifweregarditasa function ofyuand &amp;lt;,instead ofcontaining 2m+1. Itisknown asaLaplacesCoefficient,orbrieflyasaLaplacian,ofthemth degree. We shall soon expressitintheregulationform ofaSurface Spherical Harmonic. Theradius vector of(xlfylt i)iscalled theaxis oftheLaplacianandthe pointwhere theaxis cutsthesurface oftheunit sphereisthepoleofthe Laplacian. Weshall representtheLaplacian Pm(cosy)byLm(p,&amp;lt;/&amp;gt;,/*1? &amp;lt;i).Ofcourse Lm(p,&amp;lt;M&amp;gt; 4&amp;gt;i)=pmW=pm(GOS*)and isreally^dependentof &amp;lt;. 113. Iftheproduct ofaSurface SphericalHarmonic ofthemthdegree bya Laplacian ofthesame degreeisintegratedover thesurface oftheunitsphere,the result isequal to-^multiplied bythevalue oftheSphericalHarmonic at 2m+1 thepoleoftheLaplacian. Thatis, Transform totheaxis oftheLaplacianasanewpolar axis,and letZm(p,&amp;lt;) bethetransformed SphericalHarmonic. Lm(p,&amp;lt;#&amp;gt;,A*I, &amp;lt;fc)willbecomePm(/i), and(1)willbeprovedifwecanshow that sn (v.(5)Art. 102). and Art*)J ButZm(l,0)=J,sincePm(l)=1andP,(l)contains(1- 1)Jasafactor and isequaltozero. Hpnee(2)isproved. .VI.] LAPLACIANS. 211 114.WecannowexpressaLaplacianintheregulation form asaSpherical Harmonic, bytheformulas ofArt. 107. Lm(ji,&amp;lt;/&amp;gt;,/*!, 4&amp;gt;0=Pm(coa y)=POT[cos6cosOl-fsin sinBlcos (&amp;lt; -cos** -*sn n=l -2rr where sin bxArt - andAQJ=^4Kti.=Bnk=byArt.105unless k=m.Hence -*-(i) Eachterm ofaLaplacian involves anumericalcoefficient, afactor which is afunction of/*,asecond factor which isthesame function of/AJ,andathird factor which isoftheform cos /r(&amp;lt;&amp;lt;j).Wegiveonthenextpageatable ofthe firstfewLaplacians,taken fromMinchin sStatics, omittingineachterm forthesake ofbrevitythefunction ofIJLI. Bytheaidof(1)wecanwrite(5)Art.107morecompactly.Itbecomes r= m-O 1 &quot;=2ir rrf^J/0*!,4&amp;gt;i)mO&amp;gt;&amp;lt;kMi,*i)*Ah (2) ir rr or,&amp;lt;#,)=^(2m+1)V&amp;gt;(015^)Pm(oosy)sin^^. (3) m=0 212 SPHERICAL HARMONICS.[ART.114. -a. coico I o I II HS CHAP. VI.] SOLUTION BYDIRECT INTEGRATION. 213 EXAMPLE. Work theproblemsofArt.108andArt.108Exs. 1and2bytheaidof(3) Art. 114. 115. Suchproblemsaswehavehandled inArts. 98and99,andalsoprob lems differing fromthem innothaving circular symmetry about anaxis, can nowbesolved bydirectintegration. Forinstance letitberequiredtofindthevalue atanexternal point ofthe potentialfunction duetotheattraction ofasolidsphere whosedensityatany pointisproportionaltotheproductofanypoweroftheradius vector bya Surface Spherical Harmonic. Let p=Cr{Y m(fji1J^. Then using ourordinarynotation wehave =Cdr,frffcC^t^^^rfd 1? tf _&quot;(W ZrriCoay+r ______=-[~Vo(COS y)-f-P^COS y)2_2rriCosy+r 12r[_rr if r&amp;gt;rlt Consequentlysince 27T 1 =0, Vreduces tothesingle term byArt. 113. 214 SPHERICAL HARMONICS.[ART.115. EXAMPLES. 1.Solve bydirect integration theproblems worked inArts.- 98and99and Examples 1,2,3,and4ofArt. 99. 2.The densityofasolid sphereisproportionaltotheproductofthe squaresofthedistances from twomutually perpendicular diametralplanes j findthevalue ofthepotentialfunction atanexternalpoint. Ans.p=krfcos2 0!sin26lcos2 &amp;lt;f&amp;gt;t iP2(Ml)+1cos -cos 3.SolveExample2byanextension ofthemethod ofArts. 98and99. 4.Aconducting sphereofradius aconnected with theground byawire is placedinthe field offorce duetoanelectrified pointatwhichmunits of electricityareconcentrated. Find thevalue ofthepotentialfunction dueto theinduced charge. Suggestion: LetFIbethepotentialfunction duetothepoint, andVzthat duetotheinduced charge, and letbbethedistance ofthepoint from the centre ofthesphere. Then 1~~ )/l&amp;gt;* 2brcosB4-T2 if r&amp;lt;b. if r&amp;gt;b. -if r&amp;lt;a. --if r&amp;gt;a. When r=aV1+V 9=Q.Hence m ma CHAP. VI.] AXES OFASPHERICAL HARMONIC. 215 and^r- M *& ~i if r&amp;lt;a if r&amp;gt;a. Hence theeffect oftheinduced chargeispreciselythesame atanexternal pointasifthesphere were replaced byunits ofnegative electricity con centrated atthepointr= ,6=.v.Peirce, Newt. Pot.Func., 66. 116. IfthetwopointsPandParetaken onthelineOHwhose direction cosines areA, /JL,andv,and ifuanduarethevalues atPandPofanycon tinuous function ofthespace coordinates, then . iscalled the partialderivative ofualong thelineOHandwillberepresented byDhu. LetX)y,zbethecoordinates ofPandx-\-Ax,y+Ay,z-f-Athecoordinates ofP;then where isaninfinitesimal ofhigher order than thefirst ifAJC,Ay,andAare infinitesimal(v.Dif. Gal.Art.198). u}u_ Ace ,_Ay A*Hence -^^-=Dxw.-j^+Z&amp;gt;yw.^+Dzu. J Therefore Aw=XD,w-f /-tJ&amp;gt;yw+vZ&amp;gt;2u.(1) IfV2^=0,DD$Duisasolution ofLaplacesEquation, For \f\DDD;u)=D*D*DZ(V2 M)=. Hence ifV2^= Z&amp;gt;Auisasolution ofLaplacesEquation, and ifOHlyOH2,OHS,&quot;areasetoflines through theoriginD hlD hzDh^&quot;uisasolution ofLaplacesEquation. 117. IfHkisarational integral homogeneous Algebraic function ofx,y, andzofthekthdegree JTand isoftheform 216 SPHERICAL HARMONICS.[Art.118. /J-T\ /TT\ Thesame thing canbeprovedofDy(j*)andDzIf\andtherefore holds (TT\ \lf \I 77- Ifuisahomogeneous function ofxyy,and zofthedegree m 1and V2w=thenV2 (r}m+1^)=0. W&quot;+1u)=(2m+1)(2m+2)r*&quot;-x1* -f2(2m+ I)?-8^ajZ^M -f2/Z&amp;gt;yw+zD, =0, since xDxu-\-yDyu-\-zD zu(m-}-T)u byEuler sTheorem(v.Dif. Cal.Art.220). 118.= isasolution ofLaplace sEquation AT.Art.75)r TT and isoftheform- r }isthenasolution ofLaplacesEquation byArt.116; TT itisoftheformtbyArt.117and isahomogeneous function ofthe degree m 1. ^ Therefore yam+1A1Aa-^)A8&quot;^*m( /^saso^u^onofLaplacesEquation, and isarational integral homogeneous Algebraicfunction ofx,y,andzofthe mth degree, and isconsequentlyaSolid SphericalHarmonic ofthemth degree (v.Art.110);and^+1A1A2A3&quot; *AOT()isaSurface Spherical Har monic ofthemthdegree.,- Moreover since thedirection ofeach ofthelinesOH^OH2,OHmdepends upon twoangles which maybetaken atpleasure,these angles andMare /M\ 2m+1arbitraryconstants andmaybesochosen that rm+lD h}Dh^-Dhm^J maybeanygivenSurface SphericalHarmonic. Consequently anygivenSurface SphericalHarmonic mayberegardedas formed bydifferentiating successively alongmdeterminate linesOH^Off 2 Offm,and isgiven exceptfortheundetermined factorMwhen these lines are given. ThelinesOH^OH^OHS,---OHmarecalled theaxes oftheHarmonic, and thepointswhere theymeet thesurface oftheunit spherethepolesofthe Harmonic. Themaxes ofaZonal Harmonic coincide with theaxis ofcoordi nates(v.Art.86)andconsequentlythemaxes ofaLaplaciancoincide with whatwehave called theaxisoftheLaplacian (v.Art.112). CHAP. VI.] BOOTS OFZONAL ANDTESSERAL HARMONICS. 217 119.AnySurface Zonal Harmonic Pm(ft)isequaltozero formrealand distinct values offtwhich liebetween 1and1 ;andanyAssociated Func tionP*(p)isequaltozero formnrealand distinct values offtwhich lie between 1and1. df*&amp;gt;contains(ft2 I)&quot;1-*asafactor, v.Art. 89. From Eolle sTheorem,&quot;Iff(x)iscontinuous andsingle-valued and isequal tozero fortherealvalues aand bofx,jisequaltozero foratleastone ctoc realvalue ofxbetween aand b&quot;(v.Dif. Cal.Art.126)itfollows that since (,.*1\m (ft2 l)m=whenft1andwhen/JL=!-^ ^-= foratleast d(u? i\mdP onevalue offtbetween 1and 1.j cannot beequaltozero for more thanonevalue offtbetween 1and1,for itcontains(ft2 I)&quot;1&quot;1asa factor and isarational Algebraic polynomialofthe2m 1stdegree. Inlikemanner wecanshow that^ =hasm2roots equal to aft 1,m2rootsequalto1andtwo real roots between 1and 1which separatethethree distinct roots of^=0;andingeneralif&&amp;lt;w-|-l that~7~it=nas TW-krootsequalto1,mkrootsequalto1, andkrealrootsseparating thek-f-1distinct roots ofjk-i= HencePm(/*)= or f.?m=hasmrealand distinct roots between 1and1,and ithasnomore since itisoftherathdegree. dm+n(u? T)m Thesamereasoning shows thatjm+n=^asm~ndistinct real rootsbetween 1and1,andtherefore thatP^(ft)isequaltozero formn distinct realvalues offtbetween 1and 1.SinceP^(ft)contains sin&quot; asa factor itisalsoequaltozerowhenft= 1andwhenft=1 . cos n&amp;lt;f&amp;gt;isequaltozero for2nequidistant values of &amp;lt;f&amp;gt;,andsin n&amp;lt;j&amp;gt;isequalto zero for2nvalues of &amp;lt;f&amp;gt;.Hence anyTesseral Harmonic sin n&amp;lt;f&amp;gt;P(/jL)or cosn&amp;lt;P,(ft)isequaltozero for2nequidistant values of &amp;lt;f&amp;gt;,forft=1,for ft=1,andformnrealanddifferent values offtbetween 1and 1. Itfollows that thevalue ofanySurface Zonal Harmonic Pm(fi)atapoint onthesurface oftheunitspherewillhave thesamesign solongasthepoint remains ononeofthezones intowhich thesurface ofthesphereisdivided by 218 SPHERICAL HARMONICS. themcircles oflatitude correspondingtothemroots ofPm(p)=0,andwill change signwhenever thepoint passes from oneofthese zones intoanadjoin ingone;andthatthevalue ofanyTesseral Harmonic sin n&amp;lt;f&amp;gt;P^(fi)atapoint onthesurface oftheunitspherewillhave thesame signsolongasthepoint remains onanyoneofthe tesserae intowhich thesurface ofthesphereis divided bythemncircles oflatitude correspondingtotheroots ofP^(fj)= andthe2nmeridians correspondingtotheroots ofsin n&amp;lt;j&amp;gt;=0,andwillchange signwhenever thepoint passesfrom oneofthese tesserae intoanadjoining one. CHAPTER VII.* CYLINDRICAL HARMONICS(BESSELsFUNCTIONS). 120. InArts. 11and17weobtained ) (1) asthegeneralsolution ofFourier sEquation where Jo(x)=1-+--.-+...(3) and iscalled aCylindrical Harmonic orBesseVs Function ofthezeroth order; andwhere and iscalled aCylindrical Harmonic orBesseVs Function oftheSecond Kind, andofthezeroth order. InArt.17wefound that z=Jn(x) isaparticularsolution ofBesseVs Equation where ifnisunrestricted invalue n [~ i&amp;gt;2 v* jXN__x-i x _j 2; &quot;W2r(7i+1)L 22 (7i+1)^24.2!(n+l)(w+2) ....] and iscalled aCylindrical Harmonic orBesseVs Function ofthe rathorder; andthatunless nisaninteger isthegeneralsolution ofBessel sEquation. *The student should re-read carefully Arts. 11,17,and 18(d) before beginningthis chapter. 220 CYLINDRICAL HARMONICS. [ART.121. Ifnisanintegeritcanbeshown that J-B(*)=(-l)V_ n(aO, (v.ForsythsDiff.Eq.Art.102),andthen isthegeneralsolution ofBessel sEquation and k=n 1 (Kn(x)}=Jn(x)logx- \(|)&quot;2)fk=n I kl)\ 2\2/ -/(n+k)\klL 23 v.M.Bocher, Ann.Math. Vol.VI,No.4 121.Auseful expressionforJn(x)asadefinite integral canbeobtained without difficultyfrom Bessel sEquation [(5)Art.120]byaslightmodifica tionofthemethod given byForsyth (Diff. Eq.Art. 136). Itwasshown inArt.17that z=xnvisasolution ofBessel sEquationif vsatisfies theequation ,+2nldv^ ^ (1) dx* xdx b Assume v=*T cos(xt)dt (2) where xand *areindependent, Tisanunknown function oft,andaand areatpresentundetermined. Then and = Substitutingin(1)after multiplying through byx,wehave b* f(1*a)Tzcos(xt)dt-J(2+l)*Tsin (srt)cft=0.(3) ^ CHAP.VII.] BESSEL SFUNCTIONS ASDEFINITE INTEGRALS. 221 Byintegration bypartswefindthat b b f(1 t*)Txcos(xt)dt= |(1t*)Tsin (art) | a a b -J[(l-f)*j-2tT^sin(*0*. a and(3)reduces to & & l~(12 )Tsin(art)~1f [~(12)^+(2?i l)tT\sin(a;)cft=0.(4) a Ifwedetermine Tsothat fJT (l-V^+V&quot;-VtT=0, (5) & andaand &sothat |~(1 P)Tsin(art)1=0 (6) a (4)willbesatisfied andourproblemwillbesolved.(5)gives T=C(1-*2 )&quot;-*, (7) and(6)willobviouslybesatisfied ifa= 1and 6=1. Hence .=C faasoiutio]1 of and ,=Ca.- (8) y.^^ isasolution ofBessel sEquation. Ifwelet t=cos &amp;lt;#&amp;gt;in(8)weget n z=CxnTsin2 &quot; &amp;lt;cos(xcosf Expandcos(xcos &amp;lt;)intoaseries involving powersofxcos &amp;lt;,integrate termbytermbytheaidoftheformulas CS [Int.Cal.(1)Art.99], 222 CYLINDRICAL HARMONICS.[AKT.122. 2 | sin&quot;xcos7&quot;x.dx= (Int.Gal.Art.99Ex.2),andcompare with(6)Art.120,andweget irxn/*Jn(x)=- --TT- Jsin2 &quot; &amp;lt;j&amp;gt;cos(a;cos &amp;lt;j&amp;gt;)d&amp;lt;j&amp;gt;. (9) 2^r(n+pr If7iisapositive integer (9)reduces to **C3 &amp;lt;1.3.5..(2-I Let 7i= in(9)or(10)andweget J(x)=-fcos(xcos &amp;lt;f&amp;gt;)d&amp;lt;}&amp;gt;. (11) EXAMPLES. 1.Obtain Formula(11) directly from Fourier sEquation, (2)Art. 120. 2.Prove byintegration bypartsthat ifn &amp;gt;- Tsin271 &amp;lt;cos&amp;lt;sin(cccos&amp;lt;)cty= %n-\-\C82&quot;** 4&amp;gt;cos(xcos 3.Prove byintegration bypartsthat ifw &amp;gt;5 7T Tsin2 &quot; &amp;lt;^&amp;gt;cos&amp;lt;sin(a;cos &amp;lt;#&amp;gt;)^ =-C[2nsin2&quot; &amp;lt;f&amp;gt;(2n 1)sin2&quot;-2 &amp;lt;]cos(a;cos &amp;lt;f&amp;gt;)d&amp;lt;j&amp;gt;. =-C 122.Wecannowreadilyobtain anumber ofuseful formulas. Differentiate(11)Art.121with respecttoxandweget M=--fcos&amp;lt;f&amp;gt;sin(xcos Cfo 7TJ -fsin2 &amp;lt;#&amp;gt;cos(xcos&amp;lt;)cty byEx.2Art.121. CHAP.VIL] PROPERTIES OFBESSEL SFUNCTIONS. 223 Hence by(10)Art.121^^=Jl(x). (1) Inlikemanner bytheaidofExs.3and2,Art.121,wecanobtain the relations tftt&amp;gt;! Jr~n T/~\~\ dx (2)canbewritten X 1 tf&amp;gt;f- (2)and(3)canbewritten and xdx and whence 2=Jn_,(x)-Jn+l(x) (7) and^e7n(x)=7w_1(aJ)+Jn+1(x). (8) Therepeateduseofformula(8)willenable ustogetfromJ(x)andJt(x) anyofBessel sFunctions whose order isapositive integer. Forexample, we have J2(x)=^J,(x)-J(x) 224 CYLINDRICAL HARMONICS.[ART.122. From atable givingthevalues ofJ(x)andJi(x), then, tables forthe functions ofhigherorder arereadily constructed. Such atable taken from KayleighsSound(Vol. I.,page265)willbefound intheAppendix (Table VI.). Bytheaidof(5)and(6)anyderivative ofJn(x)canbeexpressedinterms ofJn(x)andJn+1(x).Forexample n(n- 1) Ifwewrite e/&quot;(a;)for inFourier sEquation [(2)Art.120],thenmultiply through byxdxandintegrate from zero tox,simplifying theresulting equa tionbyintegration byparts, weget dJQ(x).C -r,\7x+\xJ(x)dx= ; o a; whence by(1)JxJ(x)dx=xJ^x).(9) Ifwewrite t7&quot;(oj)forzinFourier sEquation, then multiply through by^dx wegetN^ tZicandintegrate from zero- tox,simplifying byintegration byparts = ; whence by(1) x(J,(x^dx=[(7 (a;))a+(/i (*))&quot;]. (10) Inlikemanner wecangetfrom Bessel sEquation [(5)Art.120]theformula which(6)enables ustoreduce totheform )Jn+l(x). (12) Formulas(9),(10), (11),and(12)willproveuseful whenweattempt to developinterms ofCylindricalHarmonics. CHAP. VII.] PROPERTIES OFBESSEL sFUNCTIONS. 225 Values ofJn(x)forlargervalues ofxthan those giveninTableVI., Appendix, maybecomputed very easilyfrom theformula v.Lommel, Studien liber dieBessel schen Functionen, page59. The series terminates if2nisanoddinteger, butotherwise itisdivergent. Itcanbeproved, however, thatinanycasethesumofmterms differs fromJn(x) bylessthan thelastterm included, andconsequentlytheformula cansafely beused fornumerical computation. EXAMPLES. 1.Confirm(1), (2),and(3),Art. 122,byobtaining them from(3)and(6), Art. 120. 2.Confirm(1),Art.122,byshowing thatFourier sEquationwill differ entiate intothespecialformassumed byBessel sEquation when n=1. 3.Show that(9),Art. 122,isaspecialcase of(4),Art. 122. 4.Show that thelimit approached byJn(x)asnincreasesindefinitelyis zero,andbytheaidofthis factandof(8),Art. 122,prove that 5.Prove that 6.Show that thesubstitution of (1)forxinLegendresEquation willreduce ittotheform A 0) andthat thelimiting form approached bythisequation asnisindefinitely increased isFourier sEquation, andhence thatJQ(x)canberegarded assome constant factor multiplied bythelimiting value approached byPn(l~~~l) asnisindefinitelyincreased. 226 CYLINDRICAL HARMONICS.[ART.123. 123. Tocomplete thesolution ofthedrumhead problem taken upin Art. 11,wefound that itwould benecessarytodevelop agiven function ofr intheform f(r)=AvTtfar)+A^I^r)+AzJ,(^r)+ where/i^, /A2,//-8,&c.,aretheroots ofthetranscendental equation 7(/xa)= ; and inArt. 11,Ex.thedevelopmentofunityinaseries ofprecisely the sameformwasneeded. (a)Letusconsider another problem. Theconvex surface andonebase ofacylinderofradius aandlengthIare keptattheconstant temperature zero, thetemperatureateach pointofthe other base isagivenfunction ofthedistance ofthepoint from thecentre of thebase;requiredthetemperatureofanypointofthecylinder after the permanent temperatureshavebeen established. Herewehave tosolve LaplacesEquationinCylindrical Coordinates ([xiv]Art.1). D*u+iDru+iDin+Dlu=(1) subjecttotheconditions u=when z= u&quot;r=a u=f(r)&quot;z=b, andfrom thesymmetryoftheproblem weknow thatDfa=0. Assumingasusualu=E.Zwebreak(1)upintotheequations whence u=sinh(fjLz)Jo(pr) (2) and ucosh(/jLz)JQ(fj,r) (3) areparticularsolutions of(1). Iffj,kisarootofJo(pa)=(4) u=sinh(/vOJoCAV) satisfies(1)andtwoofthethree equationsofcondition. Ifthenf(r)=AJ^r)+AzJ(^r)+AzJQ(^r)+ (5) A*uA*, /fa,&c.,beingroots of(4), Jsinh r/ii) ,..sinhCu2)T.sinh satisfies(1)and alloftheequationsofcondition, and istherequiredsolution. CHAP. VII.] FLOW OFHEAT INACYLINDER. 227 (b)Ifinstead ofkeepingtheconvex surface ofthecylinderatthetempera ture zerowesurround itbyajacket impervioustoheat, theequationofi condition u=when r=awillbereplaced byDru=when r=a,orif u=sinh(pz)Jo(pr)}aw=0 when rJdr that isby M(M=*or(v*WArt122) by 7i(/*a)=0. (7) Ifnow in(5)and(6)PI,fa,p9,&c.,areroots of(7), (6)willbethesolu tionofournewproblem. (c)Ifinstead ofkeepingtheconvex surface ofthecylinderatthetempera turezeroweallpw ittocool inairatthetemperature zero, thecondition u=Q when rawillbereplaced byDru-+-hu=when r=a,orif u=sinh(fj,z)J (pr) by pJo(pr)+hJ(iJ,r)= when r=a that isby /W (/^)+ahJ(t*a)= or(v.(1)Art.122) by fJLaJi(fJLa)ahJ(fjia)=0.(8) Ifnow in(5)and(6) fil}p2,/*8,&c., areroots of(8), (6)willbethe solution ofourpresent problem. 124. Itcanbeshown that J(x)=(1) Ji(*)=(2) and xJ(x)+XJo(x)=0 (3) have eachaninfinite number ofrealpositiveroots(v.Riemann, Par. Dif.Gl., 97). The earlier roots ofthese equationscanbecomputed without serious difficulty from thetable forthevalues ofJ(x)(Table VI.,Appendix). The firsttwelve roots ofJ(x)=andJi(x)= aregiveninTable IV., Appendix,atable duetoStokes. Largeroots ofe7(a?)=andofJi(x)= maybevery easily computed from theformulas ^L_ og-050661 .053041 .262051 TT= ~4s- 1(4s-I)3&quot;*&quot; (4s-I)5 x&amp;lt;? .151982 ..015399 .245270 .+ 8-* + givenbyStokes inCamb. Phil. Trans., Vol. IX.,x^representingthe5throot ofJ(x)=0,anda*?thesthrootofJv(x)=0. *Weshall find itconvenient tousethefamiliar notation of/(*)== (v.Dif. Cal., p.119). 228 CYLINDRICAL HARMONICS.[ART.126. 125.Wehave seen inArt.123that U=sink(fa.z)J (iJ,kr)andV=sinh(/jL^J^r)aresolutions ofV*?7=0 andV2F=0 ifweexpress LaplacesEquationinterms ofCylindrical Coordinates(v.(1)Art.123). Hence,iffdS representsthesurfaceintegral overanyclosed surface, we have byGreen sTheorem(v.Art.92). Ifwetake thecylinderofArt.123 asoursurface, andperform the integrations andsimplify theresulting equation, wefind a 1 2 (1) Hence iffitandfaaredifferent roots of orof oroffiaJ^fjio) \J(jj.a)=0, thenrJofarMfarjdr=.(2) o EXAMPLE. Obtain(1)Art.125directly from Fourier sEquation 126.Wearenowable toobtain thedevelopments called forinArt. 123. Letf(r)=A^far)+A2J(fji2r)+AJ^r)+ (1) f-u^2)Ma?&cvbeingroots ofJQ(^t)=0, orofJI(JJWL)=0, orof liaJ^tw) \J(fjia)=0. Todetermine anycoefficient ^ fcmultiply (1)byrJQ(fLtr)drandintegrate from zero toa.The firstmember willbecome CHAP.VII.] CYLINDRICAL HARMONIC SERIES. 229 Every term ofthesecond member willvanish by(2)Art.125exceptthe term 2 ok o by(10)Art. 122. Hence Ak=T(^ Thedevelopment (1)holds goodfrom r= tor=a(v.Arts. 24,25,and88). IffAlt/42,fa,&e.,areroots ofJQ(^CL) 0,(2)reduces to If/*!, /*2, /^8&amp;gt;&c.,areroots ofJ\(pa)=0, (2)reduces to If//,!, /i2,/^3,&c.,areroots ofpaJ^fjia) XJ^a) =0, (2)reduces to Fortheimportantcasewheref(r)=1000 by(9)Art. 122,and(3)reduces to2 Q (6) k~ (4)reduces toAk=exceptfor&=1whenjj,k=andwehaveAl=3. 2A. (5)reduces to ^fc=z-.- 22NT-r(8) (X2+pfafyjjfrta) 230 CYLINDRICAL HARMONICS. [ART.127. EXAMPLES. 1.Show that in(12)Art.11anycoefficient Akhasthevalue givenin(3) Art.126;andintheanswer toArt. 11,Ex.thevalue givenin(7)Art. 126. 2.Show that ifadrumhead beinitiallydistorted sothat ithascircular symmetry,itwillnotingeneral giveamusical note ;that itmaybeinitially distorted soastogiveamusical note;that inthiscase thevibration willbe asteady vibration;thatthefrequenciesofthevarious musical notes thatcanbe givenwhen thedistortion hascircular symmetryareproportionaltotheroots ofJ(x)= ;thatthepossiblenodes forsuch vibrations areconcentric circles whose radii areproportionaltotheroots ofJQ(x)=0. 3.Acylinderofradius onemeter and altitude onemeter has itsupper surface keptatthetemperature 100,and itsbaseandconvex surface atthe temperature 15,until thestationary temperatureissetup.Find thetempera ture atpointsontheaxis25cm.,50cm.,and75cm.from thebase,andalso atapoint25cm.from thebaseand50cm.from theaxis. Ans.,29.6;47.6 ;71.2;25.8. 4.Anironcylinderonemeter longandtwentycentimeters indiameter has its convex surface covered with aso-called non-conducting cement onecentimeter thick. Oneendandtheconvex surface ofthecylinderthuscoated arekeptatthe temperature zero, theother endatthetemperatureof100. Find tothenearest tenth ofadegreethetemperatureofthemiddle pointoftheaxis,andofthe pointsoftheaxistwentycentimeters from eachendafter thetemperatures have ceased tochange. Given that theconductivityofiron is0.185 andof cement 0.000162 inC.G.S.units. Find alsothetemperatureofapoint on thesurface midway between theends, andofpoints onthesurface twenty centimeters from each end. Find thetemperaturesofthethreepoints ofthe axis,supposingthecoating aperfect non-conductor, and again, supposing the coatingabsent. Neglectthecurvature ofthecoating. Ans.,15.4;40.85;72.8;15.3;40.7;72.5;0.0;0.0;1.3. 127. Ifinstead ofconsidering thecoolingofacylinderasinArt.123we have todealwith acylindricalshell whose curved surfaces areco-axial cylinders, weareobligedtousetheBessel sFunctions ofthesecond kind. Letourequationsofcondition be u=when 2=0, u=when r=a, u=f(r)&quot;z=b, u=&quot;r=c. Then(v.Art.123) =sn CHAP. VIL] CYLINDRICAL SHELL. wherefitisarootoftheequation231 willsatisfy LaplacesEquation [(1)Art.123]and alloftheequationsof condition exceptthesecond. Hencesin istherequiredsolution if /()=4.(W)-(2) (3) Thedevelopment (3)iseasilyobtained. Calltheparenthesisforthesake ofbrevityBQ(nkr).Then bythemethod ofArt.125wegetifweintegrateoverourcylindricalshell )B(iAlr)dr=Q(4) if/j,kand/A{areroots of(1);andbyaneasyextension of(10)Art.122 c a Determining thecoefficients in(3)asinArt.124andsimplifying bythe aidof(4)wehave ^ EXAMPLE. Ifamembrane bounded byconcentric circles ofradius aandradiusb,and fastened attheedges,isinitiallydistorted intoaformsymmetrical withrespect tothecentre, andthenallowed tovibrate where ^4tisobtained from(6)Art.127byreplacingcby6. 232 CYLINDRICAL HARMONICS.[ART.128. 128. Ifinthecoolingofacylinder u=when z=0,u=when z=b, andu=/(z)when r=a,theproblemiseasilysolved. Ifin(2)and(3)Art.123//-isreplaced by /JLIwecanreadilyobtain and 2=cos asparticularsolutions ofLaplacesEquation [(1)Art.123] ;and and isreal where byArt.31(7)and(8). Hence isourrequiredsolution.dz = &amp;gt; .At.sin b /kirai &amp;lt;J((2) (3) EXAMPLES. 1.Ifthecylinderishollow andwehaveu=when z=0,u=when =b,u=when r=c,and^=/()when r=a ;then -^r/k7rri\ k7rci\-^-/kirci )M~r whereJAhasthevalue givenin(2)Art. 128,and KO(XI)=K(xi)-J(xi)logt =J(xi)logx- + )&quot; 22,42^2k7rci\ -==/kfrci\ b) \~b~) ft+ [v.(4)Art.120],and isreal. 2.Ahollow cylinder6feetlongwhose inner surface has-theradius 3inches, andwhose outer surface hastheradius 1foot, has itsbases andouter surface keptatthetemperature 0,and itsinner surface atthetemperature 100, until CHAP. VII.] TEMPERATURES UNSYMMETRICAL. 233 thepermanentstate oftemperaturesisestablished;findthetemperatures of twopointsinaplane paralleltothebases andhalf-way between them, oneof which is6inches andtheother 9inches from theaxis.Ans., 49.6; 20.2. 129. IfintheproblemofArt.123thetemperatures ofthepoints ofthe upperbase ofthecylinderareunsymmetricalsothat u=f(r,&amp;lt;f&amp;gt;) when z=b, wehave togetparticularsolutions ofLaplacesEquation [(1)Art.123]for thecasewhere D\uisnotequaltozero.Wereadilyfindthat u=sinh(/jiz)[Acos n&amp;lt;J&amp;gt;-f and u=cosh(pz)[Acos n&amp;lt;f&amp;gt; -\- aresuch solutions, andthat =,*=. ^-\^Asinhpkzu= &amp;gt; &amp;gt; ., ,\Anr.cosn isthesolution ofthegiven problemif /(r,&amp;lt;)=VV(/4n(A.cos w&amp;lt;^&amp;gt;+Bnksin TI where/i^.isarootoftheequation ^?= (3 &amp;gt; EXAMPLES. 1.Show that 2.Show that 234 CYLINDRICAL HARMONICS.[ART.129. 3.Show that inArt.129 2* a 2ir a fd+ftfr,*)rinn n,/fc 4.Obtain thecoefficients forthecasewhere theconvex surface ofthe cylinderisimpervioustoheat. 5.Obtain thecoefficients forthecasewhere theconvex surface ofthe cylinderisexposedtoairatthetemperaturezero. 6.Show that ifinadrumhead problemofArt.11theinitial distortion is unsymmetrical,sothatwehave tosolve theequation [xi]Art. 1subjectto theconditions *=f(r,&amp;lt;f&amp;gt;)when t=0,Dtz=when t=0,z=when r=a, thesolution is cosn whereA0tk ,B0tk ,A n&amp;lt;k,andBnkhave thevalues giveninEx. 3. 7.What modifications dothestatements made inEx.2,Art. 126,need to makethem applytotheunsymmetricalcasetreated inEx.6? Show thatanypossible nodal systeminEx.6iscomposedofconcentric circles andofradiiwhose outer extremities areequidistant,v.Kayleighs Sound, Vol.I.,Arts.(202-207). 8.Solve theproblemofArt.127andofArt. 127, Ex. fortheunsym metrical case. Suggestion: AJn(x)+BKn(x)isasolution ofBessel s Equation. 9.Solve theproblemofArt.128andofArt. 128,Ex.1,for-thecasewhere u=/(z,&amp;lt;)when r=a.Suggestion:u=sin/j,z(Acos n&amp;lt;^&amp;gt; -f-Bsin ntf&amp;gt;)J n(fj,ri) isasolution ofLaplacesEquation, and/(,&amp;lt;f&amp;gt;)canbedeveloped intoadouble Fourier sSeries[v.(15)Art.71]. CHAP.VII.] EXAMPLES. 235 10.Show that indealing with awedge cutfrom acylinder byplanes passed through theaxis, orwith amembrane intheform ofacircularsector, itmaybenecessarytouseBessePs Functions offractional orincommensurable orders. 11.BernouilWs Problem(v.Chapter IX).Inconsidering small transverse vibrations ofauniform, heavy, flexible, inelasticstring fastened atoneend andinitiallydistorted intosome given curve, wehave tosolve theequation D?y=&(xDly -\-Dxy),subjecttotheconditions Dty=Qwhen=0, y=f(x)when t=0,y=when x=a ;theorigin being taken atthe distance abelow thepointofsuspension andtheaxis ofXtaken vertical. Show that y=VAkcos^ikct where 1 andjjikisarootoftheequation f(X)J(2s \s and A& or--*, T,\-,n 12.Asasimplecaseunder Example10consider thevibrations ofacircular membrane fastened attheperimeter andalsoalong aradius andtheninitially distorted(v.KayleighsSound, Art.207).Inthis casewemust modifythe formula giveninEx.6bydropping outthetermsinvolving cosn&amp;lt;j&amp;gt; andby taking n=&amp;gt;Therequiredsolution is where isarootof sn 7T CYLINDRICAL HARMONICS.[Anx.T29. Fortheterms inwhich raisodd, Jm(x) canbereadily obtained from(13) 2 Art. 122,which willbecome afinite sum. Forexample, (13)Art.122givesthevalues sinx ;J$(x)=^f-sinxcosxJ; 13.Thequestionoftheflowofheat inthree dimensions involves aproblem notunlike thelast. Supposetheinitial temperaturesofallpointsinasphereofradius cgiven, indletthesurface bekeptatthetemperaturezero. Thenwehave tosolve ;heequation (1) ([iv]Art.1)subjecttotheconditions u=when r=c, u=f(r, 0, &amp;lt;)when t=0. Ifweassume u=T.R.VwhereTisafunction oftonly,Rofronly,andF ofand&amp;lt;only, (1)canbebroken upinto f+vr=o_(2) and HenceT=e~a2a2t ,T=Fro(/Lt,&amp;lt;/&amp;gt;)[v.Art.102(2)],andRisstilltobefound. Ifin(4)weletx=arand z=R^far itbecomes which issatisfied by*=Jm+^x). (v.Art.17.) Therefore R=~j=Jm+1(ar) \ar CHAP.VII.] FLOW OFHEAT INASPHERE. 237 eiby(3)Art.114, cos^+2?m,.2^ n(r)sinW &amp;lt;#&amp;gt;]P0*). where atisarootoftheequation and where Dmat=m,a,t The final solution is 7)1=00n=m ifc=oo =4=XX\P r ifc=l cf.Riemann,Par. Dif.GL, 72and73. CHAPTER YIII. LAPLACE SEQUATION INCURVILINEAR COORDINATES. ELLIPSOIDAL HARMONICS. 130. Orthogonal Curvilinear Coordinates. IfFfa y,z)=PlF2(x,y,z)=p2(1) aretheequationsinrectangular coordinates ofthree surfaces thataremutually perpendicular nomatter what thevalues ofp1}Pz ,andp8,theparameters Pl , p2,andp3,mayberegardedasasetofcoordinates forapoint ofintersection ofthethree surfaces, inthesense thatwhenPljPz ,p3aregiven thepointin questionisdetermined, andwhen thepointisgiven thecorresponding values fpi? P2&amp;gt;p&)canbefound. Fromequations (1)x,y,andzcanbeexpressedinterms ofPl ,Pz ,andp3. Supposethis done. Ifnowx,y,zaretherectangular coordinates ofthe point pi=a,pz=b,ps=c,the rectangular coordinates ofthepoints Pl=a+dpl ,Pz=b,p3=c,areobviously x+Dpx.dPl-feuy+Dpiy.dpl-fe2, *+Dp^.dpi-f-e3,where ^,e2,and e3areinfinitesimals ofhigher order than dpi.Hence thesquareofthedistance between thepoints will differ byan infinitesimal ofhigher order than that ofdp?from dnfwhere Let (2) Then ifdn-^istheelement oflength normal tothesurface normal top2=b)anddnsnormal tops=c CURVILINEAR COORDINATES. 239 Theelement ofsurface dSionthesurfacepi=aiseasily seen tobe andtheelement ofvolume dvis = * EXAMPLE. Show that hf=(DxPiY+(Z&amp;gt;ypl)a+ Suggestion: If^hasthevaluejustgiven f^-S ^&amp;gt;_^Piare^.^ AI iii hi direction cosines ofthenormal atanygiven pointofpi=a.(v.Int.CaL page 161.) Then 131. LaplacesEquationinorthogonal curvilinear coordinates. Ifweapplythespecial form ofGreen sTheorem Vdxdyd*=DnVdS(v.Art.98) tothespace bounded bythesurfacespi=a,p*=b,ps=c,pi=a-{-dp lj wehave whence andLaplacesEquationinourcurvilinear systemis 240 ELLIPSOIDAL HARMONICS.[ART.132. Ifithappensthat-V^pi^O, Vpiwillsatisfy (7)andweshall have hlhzhsDp/j-j~\=().Inlikemanner ifV2 p2=wehaveDp/7-^-j=0, and ifV2 ps=wehaveDp( -j-~r j= ;andtherefore(7)reduces to whenV2 pi=0,V2p2=0,andV2 p8=0. 132.-Ifinstead ofhaving thevalue ofthePotential Function Vgiven on thesurface ofasphereasinourSpherical Harmonicproblem, wehave it givenatallthepoints onthesurface ofanoblatespheroid, andarerequiredto find itsvalue atanyinternal orexternalpoint, wecaneasily getasolution by methods innoessential respectdifferent from those already employed,ifonly werightlychoose oursystemofcoordinates. Ifwetakeanellipse andanhyperbola having thesame foci,andrevolve them about theminor axisoftheellipse, weshall getapairofsurfaces which aremutually perpendicular ;aplane through theaxis ofrevolution will cut both thespheroid andthehyperboloid orthogonally. Theequationsofthethree surfaces canbewritten : 1=(2) where X2 &amp;gt;62 &amp;gt;ft2 ,2bbeing thedistance between thefoci. For allvalues ofX,ft,and vconsistent with theinequality above written thesurfaces(1), (2), (3)intersect inrealpoints andcutorthogonally. X,ft,andvcanbesochosen that thesurfaces will intersect inanygiven point, andtherefore canbetaken asasetofcurvilinear coordinates, and LaplacesEquationcanbeexpressedinterms ofthembytheaidofFormula [xv]Art. 1. From(1), (2),and(3)wereadily get i=xv b\l+vz ) CHAP. VIIL] whence andSPHEROIDAL COORDINATES. 241 7,2 \2 /&!A [v.130(2)].Inlikemanner weget and and[xv]Art. 1becomesXV2(6) (8) which isLaplacesEquationinterms ofourSpheroidal CoordinatesX,/A,and p. Ifnow inplaceofX,#,and vwecanintroduce some function ofX,some function of^andsome function ofvwhich, therefore, willrepresent the same setoforthogonal surfaces, and ifwecanchoose these functionsa, /?, andy,which ofcourse arefunctions ofx,y,andz,sothatVaa=0,V2 /?=0,andV2y=0,equation (8)must reduce tothesimple andsym metrical form givenin[xvi]Art. 1. These functionsa,@,andyareeasily found. Equation (8)isV2F=0 expressedinterms ofX, /JL,and v.Assume thatVisafunction ofXonly ; thenD^VQ,andDVV=0, and(8)reduces to whence and and isafunction ofXwhich satisfies LaplacesEquation. 242 ELLIPSOIDAL HARMONICS.[ART.132. Take thisasaleaving c-iatpresent undetermined, sothat CidX Inthesamewayweget /J11 andB= (v.Int. Gal.Art. 46,Ex.) 2andy=c3tan&quot;1 !/. Substitutingthese values in(8)andtakingcl= cz=b,and c3=l, (8)reduces atonce to X2 /-t2X2 /x2 y orsince X=bseca,fibsechft,and v=tany, (10) to cos2aDa2T+cosh2ftDgV+(cosh2 ft-cos2a)D*V= (11) which isLaplacesEquationinterms ofwhatwemaycallNormal Oblate SpheroidalCoordinates. Inusing (11)itistobenoted thatthepointwhose coordinates are(a, ft,y) isthepointofintersection ofanoblate spheroid whose, semi-axes are bseca and btana,animparted hyperboloidofrevolution whose semi-axes are bsechftanditanhjS, andaplane containingtheaxis ofthesystem and makingtheangle ywith afixedplane;andthat iftheaxisofrevolution is theaxisofYandthefixed planeistheplaneofXY,therectangularcoordi nates of(a, ft,y)are x=bsecasechftcosy, y=btanatanhft,z=bseca sechftsiny(12) [v.(4)]. Ifnowweletarange from to ,ftfrom ootooo,andyfrom to2?r, weshall beable torepresentallpointsinspace ;and ifweagree thatnegative values offtshall belongtopointsbelow aplane through theorigin and perpendiculartotheaxis ofrevolution andpositivevalues offttopoints above that plane,notonlyshallwehavenoambiguity,butalsotherectangular coordinates ofanypointasgivenin(12)willhave their proper signs. CHAP. VIIL] SPHEROIDAL COORDINATES. 243 EXAMPLES. 1.Ifthespheroidisaprolate spheroid, theellipse andconfocalhyperbola must berevolved about themajoraxisoftheellipse, andtheplane must con tain that axis. Inplaceofequations (1), (2),and(3)ofArt.132wehave, then, ,-1= A.2A2b2A2 where LaplacesEquation becomes L-^A[(X*- **)AF] (1)reduoesto _.g+_2+ (A2_-^^ dv ,where a=ctnh-1-?^=tanh~1yjand v^tan&quot;1^. o b Since X=bctnh a, fji=btanh/?,and v=tany (2)canbereduced to sinh^D^F+cosh^D/F^ (sinh2a+cosh2 /8)Z&amp;gt;y2r=0.(3) Inusing (3)itistobenoted that thepoint (a, ft,y)isthepointofinter section ofaprolate spheroid whose semi-axes are bctnhaand bcsch a,a biparted hyperboloidofrevolution whose semi-axes arebtanh(3and bsech/?, andaplane containing theaxisofrevolution andmaking theangle ywith a fixedplane. 244 ELLIPSOIDAL HARMONICS.[ART.133. Ifthefixedplaneisthat of(XY)therectangular coordinates ofanypoint (a, ft,y)are x=bctnha tanh/2, y=bcschasechftcosy,z=bcschasech/3siny, andamay range from ooto0, /3from ootooo,andyfrom to2?r. Negative values of(3aretobetaken forpoints lyingtotheleftofaplane through theorigin perpendiculartotheaxis ofrevolution. 2.Transform LaplacesEquationinSpherical Coordinates [xm]Art. 1 tothesymmetricalform 1 fi where a=-&amp;gt;(3=logtan -&amp;gt;andy=&amp;lt;. 3.Transform LaplacesEquationinCylindrical Coordinates[xiv]Art. 1 tothesymmetricalform D*V+DV+e2D2V= where a=logr, /3= &amp;lt;,andy=. 133. Ineach ofthecaseswehave considered,ithasbeeneasytopass from LaplacesEquationinterms ofthechosen coordinatesrepresenting an orthogonal systemofsurfaces tothesymmetrical form[xvi]Art. 1 ;and itis evident that ournew coordinate aisavalue ofVcorrespondingtosuch a distribution that thesurfaces obtained bygiving particular values toptare eqnipotentialsurfaces;thatftisavalue ofVcorrespondingtosuch a distribution that thesurfaces obtained bygiving particular values top2are equipotentialsurfaces;andthatyisavalue ofVcorrespondingtosucha distribution that thesurfaces obtained bygiving particular values top3are equipotentialsurfaces. a, /?;andyarecalled byLame&quot;thermometric parameters.&quot; Thecondition thatthese values should exist, foragiven systemofsurfaces, thatis,that thedistribution described above should bepossible,isreadily obtained. Weshallwork itoutfor a.Itismerelythecondition thatVin LaplacesEquation maybeafunction ofpialone. IfVisafunction ofptalone CHAP. VIII.] THERMOMETRIC PARAMETERS. 245 Therefore [(Z&amp;gt;IPI)2+(D,pi)s+(Ap,)2 ] whence+AV]= where^\(pi) maybeanyfunction ofplalone. Ourrequired conditions are then andwhen theyarefulfilled theoriginal curvilinear coordinatesp1?p2,p3, correspondtopossible equipotentialorisothermal surfaces, thermometric parameters a,($,andyexist, andthereduction ofLaplacesEquationtothe symmetricalform[xvi]Art. 1ispossible. 134. ReturningtoourOblate Spheroid problem ofArt.132wecanproceed asusual tobreak upourequation (11)Art. 132. Assume thatV=L.M.N, whereLisafunction ofaonly,Mof.(3only, andA7&quot; ofyonly. (11)Art.132becomes cosg q&amp;lt;M cosaM d*L[cosh2 /?cos2aN cosh2 /?d?M Lcosh2 /3cos2ada2JWcosh2 (3cos2adft21d*NNdy* The firstmember isindependentofy,andthesecond member isindependent ofaand/?,andthetwomembers areidentically equal. Thesecond member isthenindependentofa,(3,andyandmust beconstant;call itri*.Wehave, then, (1) 246 ELLIPSOIDAL HARMONICS.[ART.134. , cos*ad2L . (1)gives us N=A cosny-\-Bsinny. (3) (2)canbewritten whence cos2a+ [&amp;gt;2cos2am(m+!)]=(4) and cosh2 (3^jji+[m(m+1)-n2cosh20]Jf=0.(5) Ifweintroduce x=tanh/?in(5)itbecomes where since x=tanh/?andftmayhaveanyvalue from ootooo,xmay haveanyvalue between 1and 1.(6)isafamiliar equation havingfora particularsolution (7) (v.Arts. 101and102) Ifweintroduce in(4)x=tanaitreduces to (8)isanunfamiliar equation, but itcanbetreated as(6)wastreated ifwe takethepainstogoback tothebeginning andfollow thestepsofthetreat ment ofLegendresEquation. This labor canbesaved, however, bynotingthat ifweletx=-.(8)becomes and isidentical informwith(6).Hence L=Pl(y)andi=(l_/)i^(v.Art.101), where y=itana,areparticularsolutions of(4). Wecanavoid imaginariesifweusethevalues =-0 Piy) andL^P+^l-itf*. (9) CHAP. VIII.] SPHEROIDAL HARMONICS. 247 Sinceweassumed F=L.M.Nwehave F=(Acosny-f-Bsinny)P,^(tanh ft)( i)m~nP^(itana) andr=(.4cosny+Bsinrcy)P*(tanh fflt**+*sec-a(1} asparticularsolutions of(11)Art. 132. Iftheproblemissymmetricalwith respecttotheaxis ofthespheroid ^=0, n2andourparticularsolutions(10)reduce to V==(-i)mPrn(itana)Pm(tanl,, and F= im+lQm(itana)Pm(tanh ft). If,then,Fisgiven onthesurface ofaspheroidasafunction offtandy, wemust expressitasafunction oftanhftandy,and shall beobligedto developitinterms ofSphericalHarmonics oftanhftandybytheformulas of Chapter VII,usingthefirstequationin(10)forthevalue ofFataninternal point,andthesecond forthevalue ofFatanexternalpoint.Iftheproblem issymmetrical,wemust developinZonal Harmonics oftanhftbytheformulas ofChapterVI. Aconvenient form forQm(itana)isobtained from(2)Art.100;itis Qm(itana)=-iPm(itana)J+^(12) tana oo Hence Q(itana)= ij^.^= i\^a J-(13) tana EXAMPLES. 1.Aconductor intheform ofanoblate spheroid whose semi-axes are bsecaand btanaischarged withelectricity and isfound tobeatpotential F;findthevalue ofthepotentialfunction atanyinternal orexternalpoint. HereF=FP(tanh ft).Hence ataninternal point andatanexternal point SinceVin(2)involves aonly,theequipotentialsurfaces arealvspheroids confocalwith theconductor. 248 ELLIPSOIDAL HARMONICS.[Airr.135. 2.Theupperhalf ofanoblate spheroid whose semi-axes arebsecaand btanaiskeptatthetemperature unity, andthelower half atthetempera ture zero. Find thepermanent temperatureatanyinternalpoint. 1.3P^itana)_ 7! (v.Art.93).umaybeexpressedinterms ofx,y,and zwithout serious difficulty [v.(12)Art.132]. U 2&quot;*~4c 82*2 5c8+362c if2c=2btana=minor axisofspheroid. 135. Letusnow findthepotentialfunction atanexternal point dueto theattraction ofasolid homogeneousoblatespheroid, using themethod em ployedinArts. 98and99. Consider firstthepotentialfunction duetoashellbounded bythespheroids forwhich a= &amp;lt;f&amp;gt;anda= &amp;lt;f&amp;gt;-f~^ By(1)Art.98wehave 47rpK=\_DnF!DnF2]a_4,, (1) wherepisthedensity and Kthethickness oftheshell, Fithevalue., ofthe potentialfunction ataninternalpoint, andF2thevalue ofthepotential function atanexternal point. Let Fj= and Fjr=^jBmim+lQm(itana)Pm(tanh ft) [v.(11)Art.134]. SinceVlandF2must have thesame valuewhen a= &amp;lt;j&amp;gt; A-7?-** 4-1Qm(itanft)_/1^7?T /2^ Am--o*p^^tan^(A;.DmjX1_L^rp^-,2 [v.(12)Art.134]. 00 Hence FI=VtmjgmPm(tanh ft)Pm(itana)|- ^V %/ ( and r,=^(tanh ft)Pm(itanf)(3) CHAP. VIII.] ATTRACTION OFASPHEROID. 249 [DnV,-DnF2]a_,=[D.Fx-DaF2]tt.^.a). =* sec2a (itana) c?Pm(itana) taut (Pm(ttau$) v.Art.130(3),andArt.132(5)and(10). [Dna]a=*= Hence [D.V,-DnF2]a.,- K=[dn] a=^=bsec by(4),and(1)maybewritten sec2 Since tanh2p=$P(tanh /8)+P2(tan by(5)Art. 95,tosatisfy (5)wemust givemthevalues and2and =|,rp&2sec2 &amp;lt;(3tan24+ and ^2=irpb2sec2 &amp;lt;#&amp;gt;(3tan2(4) /K\ 250 ELLIPSOIDAL HARMONICS. [Aiu.1J5. Sothatby(3) Fi=|irpb*sec2 &amp;lt;(3tan2 &amp;lt;-f tan &amp;lt;/&amp;gt; P2(tanh ft)P z(itana) tan(/&amp;gt; andF2=f?rp&2sec2 &amp;lt;(3tan2 &amp;lt;f&amp;gt;-fl)rf^[i^ (itana) -f*3P2(tanh ft)Q2(itana)].(7) Thepotentialfunction atanexternal point due tothesolidspheroidfor which a=ais F=fF2=|Trpi2sec2atana[t# (*tana)-f-t3 P,(tanh ^8)Q2(itana)]. (8) &amp;lt;= If2aisthemajoraxisand2ctheminor axisofthespheroid C $7rp&2sec2atana=* -= whereMisthemass ofthespheroid.Therefore MV=-[iQ (itana)+tP8(tanh ft)Q2(itana)] (9) istherequiredvalue.(9)canbereduced to EXAMPLES. 1.Break uptheequation (3)Ex.1,Art. 132, fortheprolate spheroid, and obtain particularsolutions oftheterm V=(Acosny+Bsinwy)P^(tanh /3)P^(ctnh a), V=(Acosny+Bsinny)Pj(tanh /8)(- I)2 csch&quot;^ 2.Break upandsolve theequationsofExs.2and3,Art. 132,andshow thattheylead tofamiliar forms. 3.IfinEx.1,Art. 132,theconductor isaprolate spheroid whose semi- axesare&ctnha and6cscha show that F= FOataninternal point. V=F atanexternalpoint.a CHAP. VIIL] ELLIPSOIDAL COORDINATES. 251 4.Show thatthepotentialfunction atanexternal point duetotheattrac tionofahomogeneoussolidprolate spheroidis =[(ctnha)-Pa)]. Ellipsoidal Harmonics. 136. Ifweafedealing withanellipsoid instead ofaspheroid, wecantake ourorthogonal systemofsurfaces asetofconfocal quadric$ ; x_-,_jr A2^A2-^A2c (1) where X3 &amp;gt;c2 &amp;gt;p?&amp;gt;&2 &amp;gt;vz .Here the first surface isanellipsoid, the second animparted hyperboloid, andthethird abiparted hyperboloid. Each ofthethreeprincipalsections ofthesystemconsists ofconfocal conies, and it iswellknown and iseasily shown that thesurfaces cutorthogonally. A,ft, and vwillbeourcurvilinear coordinates, and areknown asEllipsoidal Coordinates. Wefindwithoutdifficulty that ar=y*=&amp;gt;z*= C2 (C2 It*)(2) == /2 2\/\2 2~\ ^32==\2 2\x 2^(3) Toavoid ambiguity, weshall supposethat ofthenine semi-axes in(1) Vc2p2istobetaken with thepositive signforapoint onthehalf ofthe imparted hyperboloid onwhich zispositive, andwith thenegative signfora point onthehalfonwhich zisnegative ;V^2v2istobetaken with the positive signforapoint onthehalfofthebiparted hyperboloid onwhich yis positive, andwith thenegative signforapoint onthehalfonwhich yis negative ;vistobetakenpositiveforapoint onthehalf ofthebiparted hyperboloid onwhich xispositive,andnegative forapoint onthehalfon which xisnegative, andthattheremainingsixaretobealways positive.It follows thatourEllipsoidal Coordinates have thedisadvantage that tofully fixapointweneed toknow notmerely thevalues ofitscoordinatesA,p,and v,butthesignsofV/c2 /*2 ,and \Jb* v2aswell. 252 ELLIPSOIDAL HARMONICS.[ART.136. Weshall seelater, Art.139,whenwecome tointroduce whatwemaycall theNormalEllipsoidal Coordinatesa,ft,andythattheyarefreefrom this disadvantage. Itistobeobserved that A.mayrange from ctooo,/*from btoc,andvfrom btob. Theelement oflength perpendicular totheEllipsoidis Theelement ofEllipsoidal surface is andtheelement ofvolume is__at;=. . .= .N N^dXdudv. (6) V(A2-&)(A2-c2 )(^2- IF)(c2 i#)(l&amp;gt;*- it)(c2-1/2 ) The surfaceintegralofanygiven function ofpand vtaken over the ellipsoidis ft C where/i(/i,v), f*(n,v)t f*(p&amp;gt;v)andfi(p,v)arethevalues ofthegiven function onthefourquartersoftheellipsoidintowhich itisdivided bytheplanes of (-XT)and(XZ). LaplacesEquation proves reducible to where-v*)DlV+(\2- v?)I&amp;gt;}r+ (A2-p*)Dir= (8) d\ C d\ c =cI . ?B=cI .J\/(A2-62 )(X*~C2 ) J\/(C2_ /42 )(/42b2 ) =cCdv /g)Jv^2 -i&amp;gt;2)fc2-i/2 )v CHAP. VIIL] NORMAL ELLIPSOIDAL COORDINATES. 253 a,/3,andycanbeexpressedasElliptic Integralsofthe first classandare -f), &amp;gt; C&quot; (10) dna (11) (v.Int.Cal.Arts. 179,192,and196). 137. Ifin(8)Art.136weassumeF=L.M.N whereLinvolves aonly,Minvolvesftonly,andNinvolvesyonly, (8)canbewritten _~LdaM (1)istoocomplicatedtobebroken upbyourusual method. If,however, welet 1&amp;lt;PL substitute in(1)andmake useofthefactthat theresult must beidentically zero,wefindthatthecoefficients arezero forallvalues ofkexcept k=a&quot;nd k=2,andthat a= b=c,anda2= b2=c2. Therefore(1)canbebroken upintothethreeequations _=(o-fa.,ft1 )M 254 ELLIPSOIDAL HARMONICS.[ART.137. WeshallAnd itconvenient totake2asm(m -f1)and as whence (2) =0. Ifnow in(2)wereplace a,/?,andybytheir values interms ofX,/u,and v,weget [m(m+1)X2 (b*-fc2)p]L= dfj, [m(m-}-l)/u-2 (ft2-(3) [m(m+l}v- (b2+c2 )jp]JV=0. Whence ifZ=^*(X),itfollows thatM=E^)andN=E(v),andthat ) (4) isasolution ofLaplacesEquation, (8)Art. 136. Theequation =(5) isknown asLame sEquation, and$%(x)asaLame sFunction oran.?&&amp;gt;- soidal Harmonic. Weshall supposemapositive integer. Togetaparticularsolution of(5)letz=2ajxk .Substitute in(5)and reduce andweget -m(m+I)]o 4-(#+c2 )[(A:+2)2p]a t+9 A;+3)(A+4)a,+4=0.(6) Wehavenowonlytochoose asequenceofcoefficients satisfying (6),andwe maytakeanytwoconsecutive coefficients arbitrarily. CHAP. VIII.] LAMP SFUNCTIONS. 255 (6)which isordinarilyarelation connecting three consecutive coefficients reduces toarelation between twowhen k=m,when k=3,andwhen k= 4.Ifwetake am+2=0,am+4 ,aw+6&amp;gt;&c.,willvanish. Letam=l. Ifraiseven thecoefficient ofain(6)willbezero;ifphassuchavalue thata_2iszero, a_4,a_6,&c.,willbezero,andthere willbenoterms in thesolution involving -negative powersofx. Ifwewrite thevalues ofam_2&amp;gt;m-4&c -&amp;gt;^7thea^ f(6)weseethat a,B_2isofthefirstdegree in^,am_4ofthesecond degreeinp,&c.,anda_2 ofthedegree -f-1in7?.There arethen+1values of ^&amp;gt;which weshall callpi,p2,ps,&c.,forwhich a_2willvanish, andforwhich oursolutions will beoftheform ifmiseven. Ifmisodd,thecoefficient ofc^in(6)willvanish andwecanchoose pso thata_ishall bezero,andthen allcoefficients oflower order will vanish. m4-1 . .,, ,m-\-1 . a_ lisofthedegree-inp,andthere willbe-values pltp2,ps, &c.,ofpforwhich Following Heine weshall callthesolutionjustobtainedKl^(x) sothat Kp(x)=xm+am_2xm-*+am_4x&amp;gt;-*+--(7) terminating with aQifmiseven, andwith a^x ifmisodd. Ifmiseven, there are77+!ofthese functions K(x), K%*(x)&amp;gt; &c.,andthere are- _ 4 ofthem ifmisodd. The coefficients canbecomputed bytheaidof(6). IfinLame sEquation (5)weletz=v^/x2b2wegettheequation -[(m+2)(m-l)z2+c-- (b2+c*)p\v=0.(8) Lettingv=*#*weobtain therelation \k(k+3)-(m+2)(m-l)]a,- {(62+c^)[(A:+2)-rf+c*(2A+5)}w +4=0.(9) 256 ELLIPSOIDAL HARMONICS.[ART.13?! Proceeding exactlyasbefore, wefindthatthere are^values ql}q2,q9,&c., ofpforwhich v=xm~l+am_3xm~s -\----- \-a^x ifmiseven, and^i^ values forwhich v=xm~l-fam_3zm-3H----- \-aifwisodd. Calling v*Jx~b*L*(x)sothat L*(x)=V*2-P[x-1+am_sx*-*+am_sxm~s+ ], (10) terminating with 040;ifmisevenandwith aifmisodd,wehave 2i values of-S*(ar), namely L%(x), Lg(x), &c., oftheform(10)ifmiseven and-values ifmisodd. Byinterchangingband cin(8), (9),and(10)wemayshow that if 33r-*+&quot;*-***-*+ ] (11) there are^values ofE(x), namely M(x), M(x), M(x), &c.,oftheform I &quot;I (11)ifmisevenand-values ifmisodd. FinallyifinLame sEquation (5)weletz=vV(x? b*)(x2c2 )weget -[(m+3)(m-2)x*- (b*+C2 )(p- 1)&amp;gt;=0.(12) Ifnowweletv=^akxkweobtain therelation \k(k+5)-(m-2)(m+3)] 4 2)(A;+4)+1-p-]a k+2+W(*+3)(A+4)a/t+4=0.(13) Proceedingasbefore wefindthat there are values sl}s2&amp;gt;s s&amp;gt;&c-jofp forwhich v=xm~*-\-am_xm~* -\-am_6xm~6 -{----- \-aifm iseven, and m values forwhich v=xm~2 -\-am_4xm~4+---- \-a-^x ifmisodd. Calling v\(x2 b-)(x2c2 )N*(x)sothat terminating with aifmisevenandwith a^x ifmisodd,wehave values x),namely N%(x),2 values ifmisodd.ofE(x), namely N(x), -#(#), N%(x), &c.,oftheform(14)ifmisevenand -rw-&quot;1 CHAP. VIIL] TABLES OFELLIPSOIDAL HARMONICS. 257 Summing upourresults weseethatthere are2m+1EllipsoidalHarmonics JE(x)each ofwhich isafinitesumofthemthdegreeinx,orinxand \lxzbz , orinxand \jxzc2 ,orinxandY#2b2andY^2 &amp;lt;?. Itwasproved byLame thatthe2m+1values ofp,namely 7^,pt,ps,&c., qi ,j8,q&c.,rr,,rs,&c.,*,s2,8,&c.,were allreal,andbyLiouville that theywere alldifferent. Wegivetables oftheEllipsoidalHarmonics form=0,m=1,m=2,and m=3.The coefficients were obtained bytheaidofformulas(6), (9), and(13). L,(x)=0 Jf(aj)=0 c2+V(62+c2 )2- c2 ) 258 ELLIPSOIDAL HARMONICS.[ART,138. Itistobenoted that since inthesolution(4)ofLaplacesEquation, wehave thesamemand^?ineach ofthethreefactors, weshall have todeal merely withproducts made upoffactors ofthesame form, forexample, K\X)K*\n)K*\V),L\\)L&amp;lt;*(tiL?(v), &c.; andthatinasolution oftheform weshallhave foragivenmjust2m+1terms. 138.From theparticularsolution ofLame sEquation [(5)Art.137] z=25*(jc),wecangetbyformula(5),Art.18,thegeneralsolution. Itis z=AE*(x)+BEXx)I .=-- (1) &amp;gt; Making A=andB=2m+1wegetasecond form ofparticularsolution of Lame sEquation,z=F&(x)where oo/nor_,_ Weshall callF*(x)aLame sFunction ofthesecond kind. Itiseasilyseen toapproachthevalue zero asxisindefinitelyincreased. EXAMPLES. 1.Ifanellipsoidalconductor ischarged withelectricity, and isfound to beatpotential VQ,show that sinceFJ&amp;gt;=JVf (A), V= ataninternalpoint, and tdx KWJ-a- rrr ** LJV(*-^(-o2 ) -,sin- c CHAP. VIII.] NORMAL ELLIPSOIDAL COORDINATES. 259 whence v.(10)Art. 136. 2.Find thevalue ofthepotentialfunction atanexternal point duetothe attraction ofasolidhomogeneous ellipsoid (v.Art.135). Observe that (P- andthat-H2 )2- whereMisthemass oftheellipsoid. dx Ans. V-*{fjg= r ax f. 1}. *2-^2^2-c.Jaj J&amp;gt; 139. Ifforthesake ofbrevity werepresent-byk,and(l- 2)bykin c \ c theformulas(11)Art.136wehave dna, ,, bT ,,x^, andfrom thesewegetwithoutdifficulty (v.Int.Cal.Art.192) r-z r ck./s TObksn8 ena(mod k) i?=beny(mod k), =cnav -^=- (modA )&amp;gt;c2-i^=cdny(mod A). 260 ELLIPSOIDAL HAKMONICS. [ART.140. Ifweletarange from toK,andftfrom to2K,andyfrom to47T, whereKandKarethecomplete Elliptic Integrals F\k^\and respectively, (a,fty)mayrepresent anypointinspace, andthere willbeno ambiguityinsign (v.Art.136). Wemaynote that if0&amp;lt;fi&amp;lt;K ,zispositive;ifK&amp;lt;(3&amp;lt;2K ,zis negative;if0&amp;lt;y &amp;lt;K, xandyarebothpositive;ifK&amp;lt;y&amp;lt;2K,xis positiveandynegative ;if2K&amp;lt;y&amp;lt;3K,xandyareboth negative ;and if 3K&amp;lt; y&amp;lt;4 A&quot;,xisnegative andypositive (v.Art.136). Wecanwrite thevalues in(4), (5), (6),and(7),Art.136,more neatly by bringingina,ftandy.Weget (3) (4) dv=- 8(A2- /*2 )(X2-v2 )O2- v&amp;gt;)dadftdy. (5) Tortheintegralofanyfunction ofa,fi,andyovertheellipsoida=a,we shallhave 2A&quot; 4K JV(a,Ay)dS= fd(] j&amp;gt;(a,fty)&amp;lt;&amp;gt;2-v2)^2- /,2 )(X2-v2 )^.- (6) o o 140. Ifwemake useoftheformula(2)Art.92 V-VDnU)dS=0 (1) andtake asourclosed surface anygiven ellipsoid, wecangetaveryimportant result. If U=E\}EP ltiE(v)andV= and J^n*^a^n~ ~HVrv~n\-/J^^2_x ,^2_^2\ UDnV-VDnU CHAP. VIII.] DEVELOPMENT INLAME SFUNCTIONS. 261 Integrating UDnVFDn/7over thewholeellipsoid, andwriting theresult equaltozero,wehave IK 4K HencedpE*(fiE*(v)E*(p)E!(v) (p?-S)dy=(2) )-^) =0-(3) Butasourellipsoid maybetaken atpleasure,A.andaareunrestricted, and if(3)istrue itmust betrueidentically. Ifwedivide(3)by[^(X)]2itbecomes andthisobviously cannot betrueunless n=mandq=p. EXAMPLES. 1.Show that itfollows from(2)Art.140that XX IdftIE^(/Ji^E^(i/)E^(fjC)E^(v)(iJ,2v2)dy=0. KK Suggestion: 2X K 2K - v*)dp. Ifinthelastintegral wereplace ftby (3+2^C itbecomes X v.Arts. 136and139and Int. Cal.Art. 196. 2.Show that ZK 4X X1K V?)dy=8I ( 262 ELLIPSOIDAL HARMONICS.[ART.141. 141.Wecannow solve theproblemoffinding thevalue ofFatanypoint inspacewhen itisgivenatallthepointsonthesurface oftheellipsoid a=a . Wehave first todevelopin.Ellipsoidal Harmonics afunction of/xand vor rather ofaand(3givenatallpoints onthesurface oftheellipsoidinquestion; and this isnoweasily accomplished byourusual method, which leads usto theresult ,A where 4^=-^^- --(2) Our final solution is OT=0 ataninternalpoint; atanexternal point. Lame* hasprovedrather ingeniouslythat K- canalwaysbefound andthat itisequalto multiplied byarational integral /b\2 function ofthecoefficients ofJCt(x)andofc2and(-1\c/ Ofcourse thelabor ofobtaining even afewterms ofthedevelopmentofa function that isintheleast complicatedisenormous. 142. IfinLaplacesEquation (8)Art.136weletV=El(X)U supposing Utobeafunction offtandyonly,wegetafter replacing-j^ byitsvaluem(m+1)X2- (b*+c?)p [v.(2)Art.137] (X2-it)DlU+ (X2-n*)D*U+(fJL*- i?)[m(m+l)Xf-(&2+c2)^]?7=;(1) CHAP. VIII.] CONICAL COORDINATES. 263 andsincebyhypothesis UisindependentofX,thecoefficient ofX2in(1) must vanish. Hence D}U+!&amp;gt;;&+ (I*- &quot;X Ofcourse U=^E^E^v)willsatisfy (2). EXAMPLES. 1.Substitute U=E&(n)E*(v)in(2)Art.142andbytheaidof(2)Art.137 show thattheequation (2)Art.142 issatisfied. 2.Obtain(2)Art.140directly from(2)Art. 142. 3.Conical Coordinates. Consider thesystemofcoordinates defined bythe equations -(?= (1) where c2 &amp;gt;^&amp;gt;b2 &amp;gt; i/2 . Show that ,_ &quot; LaplacesEquationis (2) ^ ,&quot;, where a= IfV=U.R(2)breaks upinto m(m+1)(^2-z/2 )?7= 0.(4) (3)gives R=Arm+Br~m~\ (4)gives U=EP(fJL)fip(v) (v.Art.142). Sothatasolution of(2)is But since(2)isLaplacesEquation, V=ArmYm(p, &amp;lt;f&amp;gt;),ifexpressedin Conical Coordinates, mustsatisfy it,consequently E&(i*)]S*(v)must besimply aSpherical Harmonic oftherathdegree. 264 ELLIPSOIDAL HARMONICS. [ART.143. Toroidal Coordinates. 143.Anypairofcircles belongingtotheorthogonal system obtained and figuredinArt.46canberepresented bytheequations 2ax= sinha 2ay= sinficosha cosj3 ifwetake2ainstead of2asthedistance between thepoints common tothe second setofcircles. Ifwerotate thesystemabout theaxisofywegetasetofspheres anda setofanchor ringswhich cutorthogonally.These andasetofplanes through theaxis ofrevolution willform anorthogonal systemofsurfaces, andthe parameters correspondingtothemmaybetaken asasetofcurvilinear coordinates andmaybecalled Toroidal Coordinates. IfwetaketheaxisofthesystemastheaxisofZ,theequationsofasetof thesurfaces maybewritten ) I&quot;*2+if+*2+a2 sinh2a cosh2a 2az x*+ sin(3cos(2) yxtany a,ft,andybeing regardedasthecoordinates ofapointofintersection ofthe three surfaces. Finding LaplacesEquationintheusualmanner weget asinhacosyX= coshaifcosasinhasiny &quot; asinha^ &quot;coshaqicos/sasinft coshaipcos(3 acosha cosharccos coshaq:cosft*!=-f-A2- a andLaplacesEquationbecomes asinhacoshaqrcosff asinha CHAP. VIII.] TOROIDAL HAKMONICS. 265 A(ri&amp;gt;. V)+D#D tV)+rrDSV=0.(2) Wecannot proceedfurther byourusual method, fortheassumption thatV isafunction ofaalone, orthatVisafunction offtalone, proves tobe inadmissible. Indeed, notonlyarea, ft,andynotthermometric parameters (v.Art. 133),butnothermometric parameters exist, andnopossible distribu tioncanmake ouranchor ringsorourspheresasetofequipotential surfaces. Wecan,however, simplify (2).Itcanbewritten l\fr+D$lr proves equalto .* 5hence if7=FY^*(3)becomes sinh2a(DlU+D\U)+D*U+$U=0, (4) forwhich particularsolutions canreadily befound byourusualprocess. (4)canbebroken upintothethree equations o(5) (6) sinh2a^-[m(m+1)+rc2sinh2a]=0. (7) N=Acos(m+)y+-Bsin(m+$)y M=AIcosn/8+J?isin?i^. Ifweintroduce into(7)x=etnhaitbecomes solution ofwhich is i=P;(a!)=(1-x2 )1^^(T-Art. 102). Itistobenoted that since ctnhaisgreaterthan 1 266 ELLIPSOIDAL HARMONICS. Theconstant coefficient i2canberejected andweget U=[Acos(m+fr)y+Bsin(m+fr)y](A,cosnft+Bvsinn asaparticularsolution of(4). hasbeen called aToroidal Harmonic.(d EXAMPLES. 1.Given thevalue ofthepotentialfunction atallpoints onthesurface of ananchor ring ;find itsvalue atanypointwithin thering. Suggestion:IfF=/(/?, y)when a=a,thefunction tobedevelopedis andthedevelopmentwillbeinadouble Fourier sSeries(v.Art.71). 2.Show that ifweletarange from tooo, ftfrom TTtoTT,andyfrom to2-Tr,each ofthedouble signs onpage264maybereplaced bytheminus signwithout lossofgenerality. CHAPTER IX.* HISTORICAL SUMMARY. Themethod ofdevelopmentinseries which hasenabled usinthepreceding chapterstosolve problemsinvarious branches ofmathematicalphysics, had itsorigin,asmight havebeenexpected,inthetheoryofthemusical vibrations ofastretched string.Itwas intheyear 1753lthat Daniel Bernoulli enunciated theprincipleofthecoexistence ofsmalloscillations, which, in connection with TaylorsandJohn Bernoulli stheoryofthevibrating string, ledhim tobelieve that thegeneralsolution ofthisproblem could beputin theform ofatrigonometricseries. ThisprinciplealsoledhimandEuler to treat inasimilar manner theproblemsofthevibration ofacolumn ofairand ofanelastic rod.Theproblemofthevibration ofaheavy string suspended from oneendwas alsotreated inthesamemanner bythese mathematicians anddeserves special mention here asinitBessel sfunctions ofthezeroth order appearforthe first time.2Innone ofthese cases, however, wasany method givenfordetermining thecoefficients oftheseries. This lastremark alsoappliestothemorecomplicated problems ofthe vibration ofrectangular and circular membranes, which were discussed by Euler8in1764, and inthelastofwhich thegeneral Bessel sfunctions of integralorders occur. Itisinproblems connected withastronomy that the firstcompletely successful applicationofthemethod here considered occurs. Legendre ina paper publishedintheMemoires desSavants Etrangers for1785,first introduced thezonal harmonics Pmandapplied them tothedetermination of theattraction ofsolids ofrevolution. Hewasfollowed byLaplace, who in oneofthemost remarkable memoirs everwritten4determined thepotential ofasolid differing but little from asphere bymeans ofthedevelopment accordingtothespherical harmonics Ym. 1Seetwo articles byBernoulli andonebyEuler intheMemoirs oftheAcademy of Berlin forthisyear. 2SeetheTransactions oftheAcademy ofSt.Petersburg for1732-33, 1734and1781. 8Transactions oftheAcademy ofSt.Petersburg. 4&quot;Th^orie desattractions dessphe&quot;roides etdelafigure des Planetes&quot; Memoires de 1academic dessciences 1782. This article, although bearing anearlier datethan that of Legendre, was really inspired byit. Itishere that&quot;Laplace sequation&quot; first appears, occurring, however, only inpolar coordinates. *Seepreface. 270 HISTORICAL SUMMARY. published simultaneously twopapersinwhich they arrivedindependentlyof each other atabout thesame results. Ineach ofthesepapers attention is called tothefactthat theproductoftwoLame sfunctions isaspherical harmonic, and this fact ismade useoftothrow Lamp ssolution ofthe problemofthepermanentstate oftemperatures ofanellipsoid intoamore elementaryform. Besides thisthesecond solution ofLame sequationis introduced forthesake ofsolving thepotential problemfortheexterior of theellipsoid. Inthus following upthetheoryofheatandtherelatedpotential problems, wehave lostsightofthequestionofsmall vibrations, towhich during the early partofthecenturyagreat deal ofattention hadbeen devoted by Poisson, who frequently made useofthemethod ofdevelopmentinseries. Inhismemoirslmost oftheproblemsleftunfinished byBernoulli andEuler arethoroughly treated, aswell asvariousslight modifications ofthem. When, however, heattacked theproblemofthevibration ofanelasticplate hewasunable tomakemuch progress, owinginparttotheerroneous form of hisboundaryconditions. Hewas, nevertheless, able tosolve theproblem of thesymmetricalvibration ofafreecircularplate. Thecomplete theoryofthe vibration ofafreecircular platewas firstgiven byKirchhoff.2 Passing now toanewsubject,thetheoryoftheequilibriumofanelastic spherical shell,wefindasolution byLams inLiouville sJournal for1854, andbySirWilliam Thomson(1862)inthePhilosophical Transactions for 1863. Both ofthese papersconsist ofanapplicationofthespherical-harmonic analysistothis rather complicated problem. Thomson, however, considers besides Lame sproblemcertain related questions andtheform ofhisanalysis isverydifferent fromLames,being ofthesame nature asthatused inthe Appendix BofhisNatural Philosophyofwhich weshall have tospeak presently.These investigations form thestarting pointforanumber of recent memoirs among which those ofG.H.Darwin oncosmographical questionsdeserve specialmention. Closelyrelated tothis lastmentioned problemisthetheoryofthesmall vibrations ofanelastic sphere. While thesimplestcase ofthisproblem was treated byPoisson inthememoir referred toa.bove, thegeneral solution has been only recentlyobtained byJaerisch(1879)3andLamb(1882).4The functions involved arethesame asthose which occur intheproblemofthe non-stationaryflowofheat inasphereassolved byLaplace. TheAppendix BofThomson andTait sNatural Philosophy,6towhich we have already referred, deserves toberegardedasoneofthemost important 1Seeespeciallytheoneinthe Me&quot;moires de1academiedessciences, Vol. VIII., 1829. 2Crelle sJournal, Vol. 40,1850.8Crelle sJournal, Vol. 88. *Proc. Lond. Math. Soc.6First edition, 1867. Thisappendix wasevidently written asearly as1862, asThomson refers toitinthememoir quoted above. TOROIDAL ANDCONAL HARMONICS. 271 contributions tothegeneral theory. Thewayinwhichspherical harmonics areintroduced(ashomogeneousfunctions oftherectangular coordinates) was then new,1andthesolution ofthepotential problemforavarietyofnew solids wasindicated;viz., forsolids whose boundaries consist ofconcentric spheres,cones ofrevolution, andplanes. Weshallhavemore tosaypresently concerning themethod employedforthesolution oftheseproblems. Although connected only indirectlywith thetheory wearediscussing,it willbewell tomention atthispointthemethod ofelectrical images which is alsoduetoSirWilliam Thomson(1845).Thismethod enables ustosolve many potential problemsfortheinverse ofanysolidwhen oncewehave solved itforthesolid itself. Bymeans ofthismethod most ofthesolutions ofpotential problemsobtained byourmethod maybeappliedatoncewith verylittle modification tosystemsofcurvilinear coordinates derived by inversion from thosewehave used. Itwill notbenecessarytomention separately problemsofthis sort, asitisclearly immaterial whetherthey be solved directlyorbymeans ofthemethod ofinversion.2 Returning now totheContinent, wefind asthenextimportant question taken uptheproblemofthepotentialofananchorring. The firstpublication onthissubjectisamonograph byC.Neumann8 (1864),butinRiemann s posthumous paperswhich were notpublished until 1876, tenyears after his death, willbefound ashort fragment onthissubject, which(cf.thelastpage ofHattendorf sedition ofRiemann slectures :&quot;PartielleDifferentialglei- chungen &quot;)would appeartodateback tothewinter 1860-61. This fragment isofpeculiar interest, astheopening paragraphs clearly show thatRiemann hadinmind anextended article onthefundamentalprinciples ofoursubject. Wewillnextmention twopapers byMehler inwhich thefunctions known as&quot;conal harmonics,&quot; which hadalready been introduced byThomson inthe Appendix Babove mentioned, were appliedtothesolution oftwoproblems in electrostatics. The firstofthese papers4 (1868)deals with thesolidbounded bytwointersecting spheres,while inthesecond5 (1870)theinfinite cone of revolution istreated. Both ofthese problemsareessentially different from those discussed inthe&quot;Appendix B,&quot;inasmuch astheinfinite series which weusuallyhave degenerateinthese cases into definiteintegrals, justasthey doinsome simplercases treated byFourier. The later ofthetwopapers justquotedalso contains valuable informationconcerning thenature ofthe 1Thesamemethod wasused atabout thesame timebyClebsch. 2Acase inpointwould bethepotential problem fortheshellbetween twonon-intersecting eccentric spheres, since these spheres canbeinverted intoconcentric spheres. This problem, wastreated directly byC.Neumann inamonograph published inHalle in1862. 8&quot;Theorie derElektricitats- undWanne-Vertheilung ineinemHinge.&quot; Halle. *Crelle sJournal, Vol. 68,1868. 5Jahresbericht desGymnasiums zuElbing. 272 HISTORICAL SUMMARY. solution ofsimilar problemsforthehyperboloids andparaboloids ofrevolu tion. Thesolutions ofthese problems arenot,however, given. Itremains, inorder toclose thehistoryofthispartofthesubject,tomention anumber ofmemoirs which although treating entirely newproblems areoffar lessimportancethanmost ofthose considered uptothispoint, partly because thesolution isnotbroughttoapoint where itcanbeofmuch immediateuse, andpartly because most ofthemethods employedaresuch ascould notfail topresent themselves toanyoneattacking these problems. Ofthese the first isapaper byMathieu1onthevibration ofanelliptic membrane(1868),inwhich thefunctions oftheelliptic cylinder occur forthe firsttime. Thiswasfollowed inthesame yearbyapaper oncloselyalliedsubjects by H.Weber,2inwhich notmerelythecase ofthecomplete ellipseisbriefly considered, but also that inwhich theboundary consists oftwo arcs of confocalellipses andtwoarcsofhyperbolas confocal withthem. Thespecial case inwhich theellipses andhyperbolas become confocalparabolasisalso considered, wherebythefunctions oftheparabolic cylinder areforthe first time introduced. InMathieu s&quot;Cours dephysique mathematique&quot; (1873)theproblem of thenon-stationaryflow ofheat inanellipsoidistouchedupon, andan elaborate though notvery satisfactory treatment ofthespecialcaseswhere wehaveellipsoidsofrevolution isgiven. New functions appearinallof these problems. OflateyearsC.Baer hassuppliedanumber ofmissing links inthechain ofproblemshere considered bytreatinginsuccession thepotential problem fortheparaboloidofrevolution,3theparabolic cylinder4andthegeneral paraboloid.5Inthe first ofthese problemsBessel sfunctionsoccur, ashad alreadybeen stated byMehler, while inthelastwefindthefunctions ofthe elliptic cylinder.Foreach ofthethreesystemsofcoordinatesemployed the same author alsotouches uponthemore general problemofthenon-stationary flow ofheat, inwhich newfunctions occur. Exceptinthecase oftheanchor ringwehavefound sofaronlysuch solids treated byourmethod asarebounded bysurfaces ofthe first orsecond iLiou ville sJournal, Vol. XIII. 2&quot;Ueber dieIntegrationder partiellen Differentialgleichung -f^+Tchi=0.&quot; Math. Ann., Vol. I.Nophysical problemismentioned inthispaper. 3&quot;Ueber dasGleichgewicht und dieBewegung derWarme ineinem Rotationspara- boloid.&quot; Dissertation, Halle, 1881. 4&quot;Die Funktion desparabolischen Cylinders,&quot; Gymnasialprogramm Custrin, 1883. 5&quot;Parabolische Coordinate!!,&quot; Frankfurt, 1888. Seealsoapaper byGreenhill inthe Proc. Lond. Math. Soc., Vol.XIX., 1889 (readDec. 8,1887). Alsoaposthumous paper by Lam6 inLiouville sJournal for1874, Vol.XIX. CYCLIDIC COORDINATES. 273 degree. Wangerin1 (187&-76)considered inconnection with thetheoryof thepotential, more general systemsofcurvilinear coordinates than had previously beenused inphysical questions, namely, cyclidiccoordinates.2He showed, however, merely how tobreak upLaplacesequationinto three ordinarydifferentialequations.8 Animportant branch ofourtheory which wehave notyettouched upon dates back totheyear 1836,when Sturm published aseries offundamentally important papersinthe firsttwovolumes ofLiouville sJournal. The physical question which liesatthebasis ofthese papersistheproblemofthe flow ofheat inaheterogeneousbar.4Themethod hereemployed depends uponthefactthatthefunctions which occur arecharacterized bythenumber oftimes theyvanish inacertain interval. Thissame idea reappearsin Thomson andTait sAppendix Balreadyreferredto,but first finds itsfull expressioninthismore generalfield ofthethree dimensionalpotentialinan article byKlein :&quot;Ueber Korper welche vonconfocalen Flachen zweiten Grades begrenztsind&quot;5 (1881).Stillmorerecently (1889-90)Klein hasin hislectures extended thistheorytothetreatment ofsolids bounded bysix confocaleyelids, andhasindicated how allthepotential problems heretofore treated byourmethod arespecial cases ofthisone.6 Oflateyears, especiallysince theyear 1880, theyounger English mathe maticians have done avastamount ofwork inthetheory wearehere considering. Although much ofthiswork isofgreat value, hardly anyofit canberegardedasbeing arealdevelopmentofthemethod;itisrather an applicationofittoagreat varietyofproblems. Wemust therefore content ourselves with giving amere listofafewofthemore important ofthese papers. Niven: OntheConduction ofHeat inEllipsoidsofRevolution. Phil. Trans., 1880. Niven: OntheInduction ofElectric Currents inInfinite Plates and SphericalShells. Phil. Trans., 1881. 1Preisschriften derJablanowski schen Gesellschaft, No.XVIII., and Crelle sJournal, Vol. 82.See also, concerning astill further extension, theBerliner Monatsberichten for1878. 2Cyclids areakind ofsurface ofthefourth order(seeSalmon sGeom. ofthreeDimen sions, p.527).Inhisfirstmemoir Wangerin considers only eyelids ofrevolution. 8Seealsoapaper bythisauthor inGriinert sArchiv for1873, where theproblem ofthe equilibriumofelastic solids ofrevolution istreated. 4The similar problemofthevibration ofaheterogeneous string under theaction ofan external forcewastreated byMaggi (Giornale diMatematiche, 1880). Several special cases arealsoconsidered here indetail. 5Math. Ann., 18. 6Foranexposition ofthistheory seethetreatise :Ueber dieKeihenentwickelungen der Potentialtheorie, Leipsic, Teubner, 1894,bythewriter ofthepresent chapter. 274 HISTORICAL SUMMARY. Hicks :OnToroidal Functions. Phil. Trans., 1881. Hicks :OntheSteady Motion andSmall Vibrations ofaHollow Vortex. Phil. Trans., 1884, 1885. Lamb: OnEllipsoidal Current Sheets. Phil. Trans., 1887. Chree: TheEquationsofanIsotropicElastic Solid inPolar andCylin drical Coordinates, their Solution andApplication. Camb. Phil. Soc.Trans., XIV., 1889. Hobson: OnaClass ofSpherical Harmonics ofComplex Degree with ApplicationstoPhysicalProblems. Camb. Phil. Soc.Trans., XIV., 1889. Chree: OnsomeCompound Vibrating Systems. Camb. Phil. Soc.Trans., XV., 1891. Niven: OnEllipsoidalHarmonics. Phil. Trans., 1892. The historical sketch wehavejustgiven wouldnaturally requireasa supplement some account ofthework thathasbeendoneonthequestion of theconvergenceofthevarious series which occur. This, however, would carryustoo far,andwewill content ourselves witn mentioning thetwo fundamental memoirs byDirichlet inCrelle sJournal, one in1829 on Fourier sseries, andone,which hasbeen criticised tosome extent bysubse quent mathematicians,in1837onLaplacesspherical harmonic development. Another subjectwhich naturally presentsitself here isthetheoryofthe various new functions wehave met. Thosepropertiesofthese functions, however, which thephysicistneeds have usually been investigated bythe physiciststhemselves inthepapers mentioned above;while anythorough account ofthedevelopmentofthetheoryofthese functions would lead us intothevastregionofthemodern theoryoflinear differentialequations. Wewill therefore closebymerely giving alistofbooks which willbe found useful bythose wishingtocontinue their studyofthesubjectfurther. Webegin with thebooks relating directlytophysical questions: Fourier :Theorie AnalytiquedelaChaleur, 1822. Lame :LeqonssurlesFonctions inverses desTranscendantes etlesSurfaces isothermes, 1857. Lame: LemonssurlesCoordonnees Curvilignesetleurs diverses Applica tions, 1859. Mathieu :Cours dePhysique Mathematique, 1873. Riemann: Partielle Differentialgleichungen,und deren Anwendung auf physikalische Fragen (edited byHattendorf),third edition, 1882. F.Neumann :Theorie desPotentials undderKugelfunktionen (edited by C.Neumann),1887. Thomson andTait :Natural Philosophy, second edition, 1879. Raijleigh:TheoryofSound, 1877. Basset: Hydrodynamics,1888. Love :TheoryofElasticity,1892. BOOKS OFREFERENCE. 275 Heine :Handbuch derKugelfunktionen (second edition), 1878-81. Ferrers: Spherical Harmonics, 1881. .ffaentzschel :Reduction derPotentialgleichung aufgewohnliche Differential- gleichungen,1893. These lastthree books would alsobelonginthefollowinglistofbooks relatingtothetheoryofthevarious functions weuse : Todhunter :TheFunctions ofLaplace, Lame andBessel, 1875. Lommel: Studien liber dieBessel schen Funktionen, 1868. F.Neumann: BeitragezurTheorie derKugelfunktionen,1878. Andfinally concerning thequestionofconvergence: C.Neumann: Uber dienach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden Entwickelungen, 1881. APPENDIX. TABLES. TableI.,atable ofSurface Zonal Harmonics(Legendrians), gives thevaluec ofthe firstseven Harmonics Pl(cos 0),P2(cos ff),P7(cos 0)fortheargument indegrees.Itistaken from thePhilosophical MagazineforDecember, 1891, andwascomputed byMessrs. C.E.Holland, V.R.James, and C.G. Lamb, under thedirection ofProfessor John Perry. TableII.,atable ofSurface Zonal Harmonics(Legendrians), gives the values ofthe firstseven Harmonics Pl(x),P2(#),-P~(x)fortheargumentx. Itisreduced from theTables ofLegendrian Functions computed under the direction ofDr.J.W.L.Glaisher, andpublishedintheReportoftheBritish Association fortheAdvancement ofScience fortheyear 1879. Table III.,thetable ofHyperbolic Functions, gives thevalues ofex ,e~x , smhx, coshcc, andgdx(Gudermannianofx)forvalues ofxfrom 0.00 to1.00; andthevalues oflogsinhxandlogcosh#forvalues ofxfrom 1.00 to10.0. Thevalues ofgdx,logsinhx,andlogcoshxaretaken from theMathematical Tables prepared byProfessor J.M.Peirce(Boston:Ginn&Co.). Thelogsinhxandlogcoshxforvalues ofxbetween 0.00and1.00canbe obtained from thevalues givenfortheGudermannian ofxinthetablebythe aidoftherelations logsinhx=logtan(gdx} logcoshx=logsec(gdx). Table IV.gives the firsttwelve roots ofJQ(x)and J^(x)=each divided byIT.The table istaken from Lord RayleighsSound, Vol. L, page 274,and isdue toProfessor Stokes, Camb. Phil. Trans., Vol. IX., page186. Table V.gives the firstnine roots ofJ(x)=0,J(x)=0,J5(x)=0. The table istaken from RayleighsSound, Vol.L,page 274,and isdueto Professor J.Bourget, Ann. dePEcole Normale, T.III., 1866, page82. Table VI.,thetable ofBessel sFunctions, gives thevalues oftheBessel s Functions JQ(x)andJ^cc)fortheargumentxfrom x= tox=l5. Itis taken fromRayleighsSound, Vol.L,page 265,andfromLommeFs Bessel sche Functionen. 278 APPENDIX. TABLE I.SURFACE ZONAL HARMONICS. APPENDIX. TABLE I.SURFACE ZOXAL HARMONICS.279 278 APPENDIX. TABLE I.SURFACE ZONAL HARMONICS. APPENDIX. TABLE I.SURFACE ZOXAL HARMONICS.279 280 APPENDIX. TABLE II.SURFACE ZONAL HARMONICS. APPENDIX. TABLE II.SURFACE ZONAL HARMONICS.281 282 APPENDIX. TABLE III. HYPERBOLIC FUNCTIONS. APPENDIX. TABLE III. HYPERBOLIC FUNCTIONS.283 284 APPENDIX. TABLE III. HYPERBOLIC FUNCTIONS. APPENDIX. TABLE III. HYPERBOLIC FUNCTIONS.285 286 APPENDIX. TABLE IV. EOOTS OFBESSEL SFUNCTIONS. TABLE V.KOOTS OFJn(x)=Q. APPENDIX. TABLE VI. BESSEL SFUNCTIONS.287 CATALOGUE OFDOVER BOOKS Catalogue ofDover Books MATHEMATICS-INTERMEDIATE TOADVANCED General INTRODUCTION TOAPPLIED MATHEMATICS, Francis D.Murnaghan. Apractical andthoroughly sound introduction toanumber ofadvanced branches ofhigher mathematics. Among the selected topics covered indetail are: vector and matrix analysis, partial and differential equations, integral equations, calculus ofvariations, Laplace transform theory, thevector triple product, linear vector functions, quadratic and bilinear forms, Fourier series, spherical harmonics, Bessel functions, theHeayiside expansion formula, andmany others. Extremely useful book forgraduate students inphysics, engineering, chemistry, and mathematics. Index. Illstudy exercises with answers. 41illustrations, ix+389pp. 53/8x8Vi. S1042 Paperbound $2.25 OPERATIONAL METHODS INAPPLIED MATHEMATICS, H.S.Carslaw and J.C.Jaeger. Explana tion oftheapplication oftheLaplace Transformation todifferential equations, asimple and effective substitute formore difficult andobscure operational methods. Ofgreat practical value toengineers and toallworkers inapplied mathematics. Chapters on:Ordinary Linear Differential Equations with Constant Coefficients;; Electric Circuit Theory; Dynamical Appli cations; The Inversion Theorem fortheLaplace Transformation; Conduction ofHeat; Vibra tions ofContinuous Mechanical Systems; Hydrodynamics; Impulsive Functions; Chains of Differential Equations; andother related matters. 3appendices. 153problems, many with answers. 22figures, xvi+359pp.5%x8V2. S1011 Paperbound $2.25 APPLIED MATHEMATICS FORRADIO ANDCOMMUNICATIONS ENGINEERS, C.E.Smith. No extraneous material here! only the theories, equations, and operations essential andim mediately useful forradio work. Canbeused asrefresher, ashandbook ofapplications and tables, orasfullhome-study course. Ranges from simplest arithmetic through calculus, series, andwave forms, hyperbolic trigonometry, simultaneous equations inmesh circuits, etc. Supplies applications right along with each math topic discussed. 22useful tables offunc tions, formulas, logs, etc. Index. 166exercises, 140examples, allwith answers. 95diagrams. Bibliography, x+336pp. 53/8 x8. S141 Paperbound $1.75 Algebra, group theory, determinants, sets, matrix theory ALGEBRAS ANDTHEIR ARITHMETICS, L.E.Dickson. Provides thefoundation andbackground necessary toanyadvanced undergraduate orgraduate student studying abstract algebra. Begins with elementary introduction tolinear transformations, matrices, field ofcomplex numbers; proceeds toorder, basal units, modulus, quaternions, etc.; develops calculus of linears sets, describes various examples ofalgebras including invariant, difference, nilpotent, semi-simple.&quot;Makes thereader marvel athisgenius forclear andprofound analysis,&quot; Amer. Mathematical Monthly. Index, xii+241pp. 53/8x8. S616 Paperbound $1.50 THETHEORY OFEQUATIONS WITH ANINTRODUCTION TOTHETHEORY OFBINARY ALGEBRAIC FORMS, W.S.Burnside and A.W.Panton. Extremely thorough andconcrete discussion ofthe theory ofequations, with extensive detailed treatment ofmany topics curtailed inlater texts. Covers theory ofalgebraic equations, properties ofpolynomials, symmetric functions, derived functions, Horner sprocess, complex numbers and thecomplex variable, determinants and methods ofelimination, invariant theory (nearly 100 pages), transformations, introduction to Galois theory, Abelian equations, andmuch more. Invaluable supplementary work formodern students andteachers. 759examples and exercises. Index ineach volume. Twovolume set. Total ofxxiv+604pp. 53/8x8. S714 Vol IPaperbound $1.85 S715 Vol I!Paperbound $1.85 Theset$3.70 COMPUTATIONAL METHODS OFLINEAR ALGEBRA, V.N.Faddeeva, translated byC.D.Benster. 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Proofs rigorous, detailed; topics developed lucidly, inclose connection with their most frequent mathematical applications. Formerly &quot;Modern AlgebraicTheories.&quot; 155problems. Bibliography. 2indexes. 285pp.5% x8. S547 Paperbound $1.50 LECTURES ONTHEICOSAHEDRON ANDTHESOLUTION OFEQUATIONS OFTHEFIFTH DEGREE, Felix Klein. Thesolution ofquintics interms ofrotation ofaregular icosahedron around its axes ofsymmetry. Aclassic &indispensable source forthose interested inhigher algebra, geometry, crystallography. Considerable explanatory material included. 230 footnotes, mostly bibliographic. 2nd edition, xvi+289pp. 53/8x8. S314 Paperbound $2.25 LINEAR GROUPS, WITH ANEXPOSITION OFTHEGALOIS FIELD THEORY, L.E.Dickson. The classic exposition ofthetheory ofgroups, well within therange ofthegraduate student. Part Icontains themost extensive andthorough presentation ofthetheory ofGalois Fields available, with awealth ofexamples andtheorems. 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Partial contents: permutations, groups independent ofrepresentation, composition series ofagroup, isomorphism ofagroup with itself, Abelian groups, prime power groups, permutation groups, invariants ofgroups oflinear substitution, graphical representation, etc.45pp. ofnotes. Indexes, xxiv+512pp.5% x8. S38Paperbound $2.75 CONTINUOUS GROUPS OFTRANSFORMATIONS, L.P.Eisenhart. Intensive study ofthetheory and geometrical applications ofcontinuous groups oftransformations; astandard work onthe subject, called forth bythe revolution inphysicsinthe1920 s.Covers tensor analysis, Riemannian geometry, canonical parameters, transitivity, imprimitivity, differential invariants, thealgebra ofconstants ofstructure, differential geometry, contact transformations, etc. &quot;Likely toremain one ofthestandard works onthesubject formany years. . .principal theorems areproved clearly and concisely, andthearrangement ofthewhole iscoherent,&quot; MATHEMATICAL GAZETTE. Index. 72-item bibliography. 185exercises, ix+301pp. 53/8x8. S781 Paperbound $2.00 THETHEORY OFGROUPS ANDQUANTUM MECHANICS, H.Weyl. Discussions ofSchroedinger s wave equation, deBroglie swaves ofaparticle, Jordan-Hoelder theorem, Liescontinuous groups oftransformations, Pauli exclusion principle, quantization ofMaxwell-Dirac field equations, etc. Unitary geometry, quantum theory, groups, application ofgroups toquantum mechanics, symmetry permutation group, algebra ofsymmetric transformation, etc. 2nd revised edition. Bibliography. Index, xxii+422pp. 53/sx8. S269 Paperbound $2.35 APPLIED GROUP-THEORETIC AND MATRIX METHODS, Bryan Higman. The first systematic treatment ofgroup andmatrix theory forthephysical scientist. 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Covers thetheory ofgroup integration (asdeveloped bySchur and Weyl), the theory of2-valued orspin representations, therepresentations ofthesymmetric group, the crystallographic groups, theLorentz group, reducibility (Schur slemma, Burnside sTheorem, etc.), thealternating group, linear groups, theorthogonal group, etc. Index. List ofrefer ences, xi+369pp. 53/8 x8V2. S1112 Paperbound $2.35 THEORY OFSETS, E.Kamke. Clearest, amplest introduction inEnglish, well suited forinde pendent study. Subdivision ofmain theory, such astheory ofsets ofpoints, arediscussed, butemphasis isongeneral theory. Partial contents: rudiments ofsettheory, arbitrary sets and their cardinal numbers, ordered setsand their order types, well-ordered setsand their cardinal numbers. Bibliography. Key tosymbols. Index, vii+144pp.5% x8. S141 Paperbound $1.35 Catalogue ofDover Books THEORY ANDAPPLICATIONS OFFINITE GROUPS, G.A.Miller, H.F.Blichfeldt, LE.Dickspn. Unusually accurate and authoritative work, each section prepared byaleading specialist: Miller onsubstitution andabstract groups, Blichfeldt onfinite groups oflinear homogeneous transformations, Dickson onapplications offinite groups. Unlike more modern works, thisgives theconcrete basis from which abstract group theory arose. Includes Abelian groups, prime- power groups, isomorphisms, matrix forms oflinear transformations, Sylow groups, Galois theory ofalgebraic equations, duplication ofacube, trisection ofanangle, etc.2Indexes. 267problems, xvii+390pp. 53/8x8. S216 Paperbound $2.00 THETHEORY OFDETERMINANTS, MATRICES, AND INVARIANTS, H.W.Turnbull. Important study includes allsalient features andmajor theories. 7chapters ondeterminants and matrices cover fundamental properties, Laplace identities, multiplication, linear equations, rankand differentiation, etc. Sections oninvariants gives general properties, symbolic and direct methods ofreduction, binary and polar forms, general linear transformation, first fundamental theorem, multilinear forms. Following chapters study development and proof ofHilbert sBasis Theorem, Gordan-Hilbert Finiteness Theorem, Clebsch sTheorem, and include discussions ofapolarity, canonical forms, geometrical interpretations ofalgebraic forms, complete system ofthegeneral quadric, etc.New preface andappendix. Bibliography, xviii+374pp. 53/s x8. S699 Paperbound $2.25 ANINTRODUCTION TOTHETHEORY OFCANONICAL MATRICES, H.W.Turnbull and A.C.Aitken. Allprincipal aspects ofthetheory ofcanonical matrices, from definitions andfundamental properties ofmatrices tothe practical applications oftheir reduction tocanonical form. Beginning with matrix multiplications, reciprocals, and partitioned matrices, theauthors go ontoelementary transformations and bilinear andquadratic forms. Also covers such topics asarational canonical form forthecollineatory group, congruent andconjunctive transfor mation forquadratic andhermitian forms, unitary andorthogonal transformations, canonical reduction ofpencils ofmatrices, etc. Index. Appendix. Historical notes atchapter ends. Bibliographies. 275problems, xiv+200pp.5% x8. S177 Paperbound $1.55 ATREATISE ONTHETHEORY OFDETERMINANTS, T.Muir. Unequalled asanexhaustive compila tion ofnearly alltheknown facts about determinants uptotheearly 1930 s.Covers notation and general properties, rowandcolumn transformation, symmetry, compound determinants, adjugates, rectangular arrays andmatrices, linear dependence, gradients, Jacobians, Hessians, Wronskians, andmuch more. Invaluable forlibraries ofindustrial andresearch organizations aswell asforstudent, teacher, andmathematician; very useful inthe field ofcomputing machines. Revised andenlarged byW.H.Metzler. Index. 485problems andscores ofnumeri calexamples, iv+766pp. 53/8 x8. S670 Paperbound $3.00 THEORY OFDETERMINANTS INTHEHISTORICAL ORDER OFDEVELOPMENT, SirThomas Muir. Unabridged reprinting ofthiscomplete study of1,859 papers ondeterminant theory written between 1693 and 1900. Most important and original sections reproduced, valuable com mentary oneach. Noother work isnecessary fordeterminant research: alltypes arecovered each subdivision ofthetheory treated separately; allpapers dealing with each type are covered; youaretold exactly what each paper isabout andhowimportant itscontribution is. Each result, theory, extension, ormodification isassigned itsown identifying numeral sothat the full history may bemore easily followed. Includes papers ondeterminants ingeneral, determinants and linear equations, symmetric determinants, alternants, recurrents, determi nants having invariant factors, and allother major types.&quot;Amodel ofwhat such histories ought tobe,&quot;NATURE. &quot;Mathematicians must ever begrateful toSirThomas forhismonu mental work,&quot; AMERICAN MATH MONTHLY. Four volumes bound astwo. Indices. Bibliog raphies. Total ofIxxxiv+1977pp. 53/8x8. S672-3 The set,Clothbound $12.50 Calculus andfunction theory, Fourier theory,infinite series, calculus of variations, realandcomplex functions FIVEVOLUME &quot;THEORY OFFUNCTIONS SETBYKONRAD KNOPP This five-volume set,prepared byKonrad Knopp, provides acomplete and readily followed account oftheory offunctions. Proofs aregiven concisely, yetwithout sacrifice ofcomplete ness orrigor. These volumes areused astexts bysuch universities asM.I.T., University of Chicago, N.Y.City College, andmany others. &quot;Excellent introduction . . .remarkably readable, concise, clear, rigorous,&quot; JOURNAL OFTHEAMERICAN STATISTICAL ASSOCIATION. ELEMENTS OFTHETHEORY OFFUNCTIONS, Konrad Knopp. This book provides thestudent withbackground forfurther volumes inthis set, ortexts onasimilar level. Partial contents: foundations, system ofcomplex numbers and theGaussian plane ofnumbers, Riemann sphere ofnumbers, mapping bylinear functions, normal forms, thelogarithm, thecyclometric functions andbinomial series. &quot;Not only fortheyoung student, butalso forthestudent whoknows allabout what isinit,&quot;MATHEMATICAL JOURNAL. Bibliography. Index. 140pp. 53/8x8. S154 Paperbound $1.50 THEORY OFFUNCTIONS, PARTI,Konrad Knopp. With volume II,thisbook provides coverage ofbasic concepts andtheorems. Partial contents: numbers and points, functions ofacom plex variable, integral ofacontinuous function, Cauchy sintegral theorem, Cauchy sintegral formulae, series with variable terms, expansion ofanalytic functions inpower series, analytic continuation andcomplete definition ofanalytic functions, entire transcendental functions, Laurent expansion, types ofsingularities. Bibliography. Index, vii+146pp. 53/8x8. S156 Paperbound $1.35 Catalogue ofDover Books THEORY OFFUNCTIONS, PART II,Konrad Knopp. Application and further development of general theory, special topics. Single valued functions, entire, Weierstrass, Meromorphic functions. Riemann surfaces. Algebraic functions. Analytical configuration, Riemann surface. Bibliography. Index, x+150pp. 53/a x8. S157 Paperbound $1.35 PROBLEM BOOK INTHETHEORY OFFUNCTIONS, VOLUME1,Konrad Knopp. Problems inele mentary theory, forusewith Knopp sTHEORY OFFUNCTIONS, oranyother text, arranged according toincreasing difficulty. Fundamental concepts, sequences ofnumbers and infinite series, complex variable, integral theorems, development inseries, conformal mapping. 182 problems. Answers, viii+126pp. 53/a x8. S158 Paperbound $1.35 PROBLEM BOOK INTHETHEORY OFFUNCTIONS, VOLUME 2,Konrad Knopp. Advanced theory offunctions, tobeused either with Knopp sTHEORY OFFUNCTIONS, oranyother com parable text. Singularities, entire &meromorphic functions, periodic, analytic, continuation, multiple-valued functions, Riemann surfaces, conformal mapping. Includes asection ofaddi tional elementary problems. &quot;The difficult task ofselecting from theimmense material ofthe modern theory offunctions theproblems just within thereach ofthebeginner ishere masterfully accomplished,&quot; AM.MATH. SOC. Answers. 138pp.5%x8.S159 Paperbound $1.35 ACOURSE INMATHEMATICAL ANALYSIS, Edouard Goursat. Trans, byE.R.Hedrick, 0.Dunkel. Classic study offundamental material thoroughly treated. Exceptionally lucid exposition of wide range ofsubject matter forstudent with 1year ofcalculus. Vol. 1:Derivatives and Differentials, Definite Integrals, Expansion inSeries, Applications toGeometry. Problems. Index. 52 illus. 556pp. Vol. 2,Part I:Functions ofaComplex Variable, Conformal Repre sentations, Doubly Periodic Functions, Natural Boundaries, etc. Problems. Index. 38 illus. 269pp. Vol. 2,Part 2:Differential Equations, Cauchy-Lipschitz Method, Non-linear Differential Equations, Simultaneous Equations, etc. Problems. Index. 308pp.5% x8. Vol. 1S554 Paperbound $2.50 Vol.2part 1S555 Paperbound $1.85 Vol.2part2S556 Paperbound $1.85 3vol.set$6.20 MODERN THEORIES OFINTEGRATION, H.Kestelman. Connected andconcrete coverage, with fully-worked-out proofs forevery step. Ranges from elementary definitions through theory ofaggregates, sets ofpoints, Riemann andLebesgue integration, andmuch more. Thisnew revised andenlarged edition contains anewchapter onRiemann-Stieltjes integration, aswell asasupplementary section of186exercises. Ideal forthemathematician, student, teacher, orself-studier. Index ofDefinitions andSymbols. General Index. Bibliography, x+310pp. 55/8x83/8. S572 Paperbound $2.25 THEORY OFMAXIMA ANDMINIMA, H.Hancock. Fullest treatment ever written; onlywork in English withextended discussion ofmaxima andminima forfunctions of1,2,ornvariables, problems with subsidiary constraints, and relevant quadratic forms. Detailed proof ofeach important theorem. Covers theScheeffer andvonDantscher theories, homogeneous quadratic forms, reversion ofseries, fallacious establishment ofmaxima andminima, etc.Unsurpassed treatise foradvanced students ofcalculus, mathematicians, economists, statisticians. Index. 24diagrams. 39problems, many examples. 193pp.5% x8. S665 Paperbound $1.50 ANELEMENTARY TREATISE ONELLIPTIC FUNCTIONS, A.Caylcy. Still thefullest and clearest text onthetheories ofJacobi andLegendre fortheadvanced student (and anexcellent supplement forthebeginner). Amasterpiece ofexposition bythegreat 19th century British mathematician (creator ofthetheory ofmatrices and abstract geometry), itcovers the addition-theory, Landen stheorem, the 3kinds ofelliptic integrals, transformations, the q-functions, reduction ofadifferential expression, andmuch more. Index, xii+386pp.5%x8. S728 Paperbound $2.00 THEAPPLICATIONS OFELLIPTIC FUNCTIONS, A.G.Greenhill. Modern books forego detail for sake ofbrevity thisbook offers complete exposition necessary forproper understanding, use ofelliptic integrals. Formulas developed from definite physical, geometric problems; examples representative enough tooffer basic information inwidely useable form. Elliptic integrals, addition theorem, algebraical form ofaddition theorem, elliptic integrals of2nd, 3rd kind, double periodicity, resolution into factors, series, transformation, etc. Introduction. Index. 25illus .xi+357pp. 53/8x8. S603 Paperbound $1.75 THETHEORY OFFUNCTIONS OFREAL VARIABLES, James Pierpont. A2-volume authoritative exposition, byone oftheforemost mathematicians ofhistime. Each theorem stated with allconditions, then followed byproof. Noneed togothrough complicated reasoning todis cover conditions added without specific mention. Includes aparticularly complete, rigorous presentation oftheory ofmeasure; and Pierpont sown work onatheory ofLebesgue integrals, andtreatment ofarea ofacurved surface. Partial contents, Vol. 1:rational numbers, exponentials, logarithms, point aggregates, maxima, minima, proper integrals, improper integrals, multiple proper integrals, continuity, discontinuity, indeterminate forms. Vol. 2:point sets, proper integrals, series, power series, aggregates, ordinal numbers, discontinuous functions, sub-, infra-uniform convergence, much more. Index. 95illustrations. 1229pp. 53/8 x8. S558-9, 2volume set,paperbound $5.20 Catalogue ofDover Books FUNCTIONS OFACOMPLEX VARIABLE, James Pierpont. Long one ofbest inthe field. A thorough treatment offundamental elements, concepts, theorems. Acomplete study, rigor ous, detailed, with carefully selected problems worked out toillustrate each topic. Partial contents: arithmetical operations, realterm series, positive term series, exponential functions, integration, analytic functions, asymptotic expansions, functions ofWeierstrass, Legendre, etc. Index. List ofsymbols. 122 illus. 597pp.5% x8. S560 Paperbound $2.45 MODERN OPERATIONAL CALCULUS: WITH APPLICATIONS INTECHNICAL MATHEMATICS, N.W. McLachlan. Anintroduction tomodern operational calculus based upon theLaplace trans form, applying ittothesolution ofordinary and partial differential equations. Forphysi cists, engineers, andapplied mathematicians. Partial contents: Laplace transform, theorems orrules ofthe operational calculus, solution ofordinary ana partial linear differential equations with constant coefficients, evaluation ofintegrals and establishment ofmathe matical relationships, derivation ofLaplace transforms ofvarious functions, etc. Sixappen dices deal with Heaviside sunit function, etc. Revised edition. Index. Bibliography xiv+ 218pp. 53/8 x8V2. S192 Paperbound $1.75 ADVANCED CALCULUS, E.B.Wilson. Anunabridged reprinting ofthework which continues toberecognized asone ofthemost comprehensive and useful texts inthe field. Itcontains animmense amount ofwell-presented, fundamental material, including chapters onvector functions, ordinary differential equations, special functions, calculus ofvariations, etc., which areexcellent introductions tothese areas. Forstudents with only oneyear ofcal culus,nwe than 1300 exercises cover both pure math and applications toengineering and physical problems. Forengineers, physicists, etc., this work, with its54page intro ductory review, isthe ideal reference and refresher. Index, ix+566pp.5%x8. S504 Paperbound $2.45 ASYMPTOTIC EXPANSIONS, A.Erdelyi. The onlymodern work available inEnglish, this isan unabridged reproduction ofamonograph prepared forthe Office ofNaval Research. Itdis cusses various procedures forasymptotic evaluation ofintegrals containing alarge parameter andsolutions ofordinary linear differential equations. Bibliography of71items, vi+108pp. 53/8 x8. S318 Paperbound $1.35 INTRODUCTION TOELLIPTIC FUNCTIONS: with applications, F.Bowman. Concise, practical introduction toelliptic integrals and functions. Beginning with the familiar trigonometric functions, itrequires nothing more from thereader than aknowledge ofbasic principles ofdifferentiation and integration. Discussion confined totheJacobian functions. Enlarged bibliography. Index. 173problems andexamples. 56figures, 4tables. 115pp. 53/a x8. S922 Paperbound $1.50 ONRIEMANN STHEORY OFALGEBRAIC FUNCTIONS AND THEIR INTEGRALS: ASUPPLEMENT TOTHEUSUAL TREATISES, Felix Klein. Klein demonstrates how themathematical ideas in Riemann ?work onAbelian integrals canbearrived atbythinking interms ofthe flow ofelectric current onsurfaces. Intuitive explanations, not detailed proofs given inan extremely clear exposition, concentrating onthekinds offunctions which canbedefined onRiemann surfaces. Also useful asanintroduction totheorigins oftopological problems. Complete and unabridged. Approved translation byFrances Hardcastle. New introduction. 43figures. Glossary, xii+76pp.5% x8V2.. S1072 Paperbound $1.25 COLLECTED WORKS OFBERNHARD RIEMANN. This important source book isthe first tocon tain thecomplete text ofboth 1892Werke andthe1902 supplement, unabridged. Itcontains 31monographs, 3complete lecture courses, 15miscellaneous papers, which have been of enormous importance inrelativity, topology, theory ofcomplex variables, and other areas ofmathematics. Edited byR.Dedekind, H.Weber, M.Noether, W.Wirtinger. German text. English introduction byHans Lewy. 690pp. 53/sx8. S226 Paperbound $3.75 THETAYLOR SERIES, ANINTRODUCTION TOTHETHEORY OFFUNCTIONS OFACOMPLEX VARIABLE, P.Dienes. This book investigates the entire realm ofanalytic functions. Only ordinary calculus isneeded, except inthe lasttwo chapters. Starting with anintroduction toreal variables andcomplex algebra, theproperties ofinfinite series, elementary func tions, complex differentiation and integration arecarefully derived. Also biuniform mapping, athorough two part discussion ofrepresentation and singularities ofanalytic functions, overconvergence andgaptheorems, divergent series, Taylor series on itscircle ofcon vergence, divergence and singularities, etc. Unabridged, corrected reissue offirst edition. Preface and index. 186examples, many fully worked out.67figures, xii+555pp. 53/ax8. S391 Paperbound $2.75 INTRODUCTION TOBESSEL FUNCTIONS, Frank Bowman. Arigorous self-contained exposition providing allnecessary material during thedevelopment, which requires onlysome knowl edge ofcalculus and acquaintance with differential equations. Abalanced presentation including applications and practical use. Discusses Bessel Functions ofZero Order, ofAny Real Order; Modified Bessel Functions ofZero Order; Definite Integrals; Asymptotic Expan sions; Bessel sSolution toKepler sProblem; Circular Membranes; much more. &quot;Clear and straightforward. . .useful notonly tostudents ofphysics andengineering, buttomathe matical students in general,&quot; Nature. 226 problems. Short tables ofBessel functions. 27 figures. Index, x+135pp.5% x8. S462 Paperbound $1.50 Catalogue ofDover Books ELEMENTS OFTHETHEORY OFREAL FUNCTIONS,J.E.Littlewood. Based onlectures given at Trinity College, Cambridge, thisbook hasproved tobeextremely successful inintroducing graduate students tothemodern theory offunctions. Itoffers afullandconcise coverage ofclasses and cardinal numbers, well-ordered series, other types ofseries, andelements ofthetheory ofsets ofpoints. 3rdrevised edition, vii+71pp.5% x8. 5171 Clothbound $2.85 5172 Paperbound $1.25 TRANSCENDENTAL ANDALGEBRAIC NUMBERS, A.0.Geifond. First English translation ofwork byleading Soviet mathematician. Thue-Siegel theorem, itsp-adic analogue, onapproximation ofalgebraic numbers bynumbers infixed algebraic field; Hermite-Lindemann theorem on transcendency ofBessel functions, solutions ofother differential equations; Gelfond-Schneider theorem ontranscendency ofalpha topower beta; Schneider swork onelliptic functions, withmethod developed byGeifond. Translated by L.F.Boron. Index. Bibliography. 200pp. 53/8 x8. S615 Paperbound $1.75 ELLIPTIC INTEGRALS, H.Hancock. Invaluable inwork involving differential equations contain ingcubics orquartics under theroot sign, where elementary calculus methods areinade quate. Practical solutions toproblems that occur inmathematics, engineering, physics: differential equations requiring integration ofLame s,Briot s,orBouquet sequations; deter mination ofarcofellipse, hyperbola, lemniscate; solutions ofproblems inelastica; motion ofaprojectile under resistance varying asthecube ofthe velocity; pendulums; many others. Exposition isinaccordance with Legendre-Jacobi theory and includes rigorous dis cussion ofLegendre transformations. 20figures. 5place table. Index. 104pp. 5Vs x8. S484 Paperbound $1.25 LECTURES ONTHETHEORY OFELLIPTIC FUNCTIONS, H.Hancock. Reissue oftheonlybook inEnglish with soextensive acoverage, especially ofAbel, Jacobi, Legendre, Weierstrasse, Hermite, Liouville, andRiemann. Unusual fullness oftreatment, plus applications aswell as theory, indiscussing elliptic function (the universe ofelliptic integrals originating inworks ofAbel and Jacobi), their existence, and ultimate meaning. Use ismade ofRiemann to provide themost general theory. 40page table offormulas. 76figures, xxiii+498pp. S483 Paperbound $2.55 THETHEORY ANDFUNCTIONS OFAREAL VARIABLE ANDTHETHEORY OFFOURIER SSERIES, E.W.Hobson. One ofthebest introductions tosettheory andvarious aspects offunctions and Fourier sseries. Requires only agood background incalculus. Provides anexhaustive coverage of:metric and descriptive properties ofsets ofpoints; transfinite numbers and order types; functions ofareal variable; theRiemann andLebesgue integrals; sequences andseries ofnumbers; power-series; functions representable byseries sequences ofcontinuous functions; trigonometrical series; representation offunctions byFourier sseries; complete exposition (200pp.) onsettheory; andmuch more. &quot;The best possible guide,&quot; Nature. Vol. I: 88detailed examples, 10figures. Index, xv+736pp. Vol. II:117 detailed examples, 13 figures. Index, x+780pp. 6V8x9V4. Vol. I:S387 Paperbound $3.50 Vol. II:S388 Paperbound $3.00 ALMOST PERIODIC FUNCTIONS, A.S.Besicovitch. This unique andimportant summary bya well-known mathematician covers indetail thetwostages ofdevelopment inBohr stheory of almost periodic functions: (1)asageneralization ofpure periodicity, with results and proofs; (2)thework done byStepanoff, Wiener, Weyl, andBohr ingeneralizing thetheory. Bibliography, xi+180pp. 53/8x8. S18Paperbound $1.75 THEANALYTICAL THEORY OFHEAT, Joseph Fourier. This book, which revolutionized mathe matical physics, islisted intheGreat Books program, andmany other listings ofgreat books. Ithasbeen used with profit bygenerations ofmathematicians and physicists whoare interested ineither heat orintheapplication oftheFourier integral. Covers cause and reflection ofrays ofheat, radiant heating, heating ofclosed spaces, use oftrigonometric series inthetheory ofheat, Fourier integral, etc. Translated byAlexander Freeman. 20 figures, xxii+466pp. 53/8x8. S93Paperbound $2.50 ANINTRODUCTION TOFOURIER METHODS ANDTHELAPLACE TRANSFORMATION, Philip Franklin. Concentrates upon essentials, enabling thereader with only aworking knowledge ofcalculus togain anunderstanding ofFourier methods inabroad sense, suitable formost applica tions. Thiswork covers complex qualities withmethods ofcomputing elementary functions forcomplex values oftheargument and finding approximations bytheuse ofcharts; Fourier series and integrals with half-range andcomplex Fourier series; harmonic analysis; Fourier andLaplace transformations, etc.; partial differential equations with applications to transmission ofelectricity; etc.Themethods developed arerelated tophysical problems of heat flow, vibrations, electrical transmission, electromagnetic radiation, etc.828problems with answers. Formerly entitled &quot;Fourier Methods.&quot; Bibliography. Index, x+289pp.5% x8. S452 Paperbound $2.00 THEFOURIER INTEGRAL ANDCERTAIN OFITSAPPLICATIONS, Norbert Wiener. Theonly book- length study oftheFourier integral aslinkbetween pure andapplied math. Anexpansion oflectures given atCambridge. Partial contents: Plancherel stheorem, general Tauberian theorem, special Tauberian theorems, generalized harmonic analysis. Bibliography, viii+ 201pp. 53/8x8. S272 Paperbound $1.50 Catalogue ofDover Books Differential equations, ordinary andpartial; integral equations INTRODUCTION TOTHEDIFFERENTIAL EQUATIONS OFPHYSICS, LHopf. Especially valuable totheengineer with nomath beyond elementary calculus. Emphasizing intuitive rather than formal aspects ofconcepts, theauthor covers anextensive territory. Partial contents- Law ofcausality, energy theorem, damped oscillations, coupling byfriction, cylindrical and spherical coordinates, heat source, etc. Index. 48figures. 160pp.5% x8. S120 Paperbound $1.35 INTRODUCTION TOTHETHEORY OFLINEAR DIFFERENTIAL EQUATIONS, E.G.Poole. Authorita tivediscussions ofimportant topics, withmethods ofsolution more detailed than usual, for students with background ofelementary course indifferential equations. Studies existence theorems, linearly independent solutions; equations with constant coefficients; with uniform analytic coefficients; regular singularities; thehypergeometric equation; conformal repre sentation; etc. Exercises. Index. 210pp.5% x8. S629 Paperbound $1.65 DIFFERENTIAL EQUATIONS FORENGINEERS, P.Franklin. Outgrowth ofacourse given 10 years atM. I.T.Makes most useful branch ofpure math accessible forpractical work. Theoretical basis ofD.E. s;solution ofordinary D.E. sand partial derivatives arising from heat flow, steady-state temperature ofaplate, wave equations; analytic functions; con vergence ofFourier Series. 400problems onelectricity, vibratory systems, other topics. Formerly &quot;Differential Equations forElectrical Engineers.&quot; Index 41 illus. 307pp. 5y& x8. S601 Paperbound $1.65 DIFFERENTIAL EQUATIONS, F.R.Moulton. Adetailed, rigorous exposition ofallthenon- elementary processes ofsolving ordinary differential equations. Several chapters devoted to thetreatment ofpractical problems, especially those ofaphysical nature, which are far more advanced than problems usually given asillustrations. Includes analytic differential equations; variations ofaparameter; integrals ofdifferential equations; analytic implicit functions; problems ofelliptic motion; sine-amplitude functions; deviation offormal bodies; Cauchy-Lipschitz process; linear differential equations with periodic coefficients; differential equations ininfinitely many variations; much more. Historical notes. 10figures. 222prob lems. Index, xv+395pp. 53/8x8. S451 Paperbound $2.00 DIFFERENTIAL ANDINTEGRAL EQUATIONS OFMECHANICS ANDPHYSICS (DIE DIFFERENTIAL- UNOINTEGRALGLEICHUNGEN DERMECHANIK UNO PHYSIK), edited byP.Frank and R.yon Mises. Most comprehensive and authoritative work onthemathematics ofmathematical physics available today intheUnited States: thestandard, definitive reference forteachers, physicists, engineers, andmathematicians now published (intheoriginal German) atarela tively inexpensive price forthe first time! Every chapterinthis 2,000-page set isbyan expert inhis field: Carathe&quot;odory, Courant, Frank, Mises, and adozen others. VolI,on mathematics, gives concise butcomplete coverages ofadvanced calculus, differential equa tions, integral equations, and potential, and partial differential equations. Index, xxiii+ 916pp. Vol. II(physics): classical mechanics, optics, continuous mechanics, heat conduction and diffusion, the stationary and quasi-stationary electromagnetic field, electromagnetic oscillations, andwave mechanics. Index, xxiv+1106pp. Twovolume set.Each volume avail- ab,e separately. 5y8x,*. S7.7 v.1 I Clotjbo^ |TjD Theset$15.00 LECTURES ONCAUCHY SPROBLEM,J.Hadamard. Based onlectures given atColumbia, Rome, this discusses work ofRiemann, Kirchhoff, Volterra, and theauthor/s own research onthe hyperbolic case inlinear partial differential equations.Itextends spherical and cylindrical waves toapply to all(normal) hyperbolic equations. Partial contents: Cauchy sproblem, fundamental formula, equations with oddnumber, with even number ofindependent var iables; method ofdescent. 32figures. Index, iii+316pp. 53/ax8. S105 Paperbound $1.75 THEORY OFDIFFERENTIAL EQUATIONS, A.R.Forsyth. Out ofprint forover adecade, the complete 6volumes (now bound as3)ofthismonumental work represent themost com prehensive treatment ofdifferential equations ever written. Historical presentation includes in2500 pages every substantial development.Vol. 1,2:EXACT EQUATIONS, PFAFF S PROBLEM; ORDINARY EQUATIONS, NOT LINEAR: methods ofGrassmann, Clebsch, Lie, Dar- boux; Cauchy stheorem; branch points; etc. Vol. 3,4:ORDINARY EQUATIONS, NOT LINEAR; ORDINARY LINEAR EQUATIONS: Zeta Fuchsian functions, general theorems onalgebraic integrals Brun stheorem, equations with uniform periodic coffiecients, etc. Vol. 4,5: PARTIAL DIFFERENTIAL EQUATIONS: 2existence-theorems, equations oftheoretical dynamics, Laplace transformations, general transformation ofequations ofthe2nd order, much more. Indexes. Total of2766pp. 53/8x8. S576-7-8 Clothbound: theset$15.00 PARTIAL DIFFERENTIAL EQUATIONS OFMATHEMATICAL PHYSICS, A.G.Webster. Akeystone work inthe library ofevery mature physicist, engineer, researcher. Valuable sections on elasticity compression theory, potential theory, theory ofsound, heat conduction, wave propagation, vibration theory. Contents include: deduction ofdifferential equations, vibra tions, normal functions, Fourier sseries, Cauchy smethod, boundary problems, method of Riemann-Volterra. Spherical, cylindrical, ellipsoidal harmonics, applications,etc.97figures, vii+440pp. 53/8 x8. S263 Paperbound $2.25 Catalogue ofDover Books ORDINARY DIFFERENTIAL EQUATIONS, E.L.Ince. Amost compendious analysis inrealand complex domains. Existence andnature ofsolutions, continuous transformation groups, solu tions inaninfinite form, definite integrals, algebraic theory, Sturmian theory, boundary prob lems, existence theorems, 1storder, higher order, etc. &quot;Deserves thehighest praise, anotable addition tomathematical literature,&quot; BULLETIN, AM.MATH. SOC. Historical appendix. Bib liography. 18figures, viii+558pp. 53/8x8. S349 Paperbound $2.75 INTRODUCTION TONONLINEAR DIFFERENTIAL AND INTEGRAL EQUATIONS, Harold T.Davis. Athorough introduction tothisimportant area, ofincreasing interest tomathematicians and scientists. First published bytheUnited States Atomic Energy Commission, itincludes chap tersonthedifferential equation ofthe first order, theRiccati equation (asabridge between linear andnonlinear equations), existence theorems, second order equations, elliptic integrals, elliptic functions, and theta functions, second order differential equations ofpolynomial class, continuous analytic continuation, thephase plane and itsphenomena, nonlinear me chanics, thecalculus ofvariations, etc.Appendices onPainlev6 transcendents andVander Poland Volterra equations. Bibliography of350 items. 137problems. Index, xv+566pp.5% x8V2. S971 Paperbound $2.00 THEORY OFFUNCTIONALS AND OFINTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS, VitO Volterra. Unabridged republication ofthe only English translation. Anexposition ofthe general theory ofthefunctions depending onacontinuous setofvalues ofanother function, based ontheauthor sfundamental notion ofthetransition from afinite number ofvariables toacontinually infinite number. Though dealing primarily with integral equations, much material oncalculus ofvariations isincluded. Thework makes noassumption ofprevious knowledge onthepart ofthereader. Itbegins with fundamental material andproceeds to Generalization ofAnalytic Functions, Integra-Differential Equations, Functional Derivative Equations, Applications, Other Directions ofTheory ofFunctionals, etc.New introduction by G.C.Evans. Bibliography and criticism ofVolterra swork byE.Whittaker. Bibliography. Index ofauthors cited. Index ofsubjects, xxxx+226pp.5%x8. S502 Paperbound $1.75 LINEAR INTEGRAL EQUATIONS, W.V.Lovitt. Systematic survey ofgeneral theory, withsome application todifferential equations, calculus ofvariations, problems ofmath, physics. Partial contents: integral equation of2nd kind bysuccessive substitutions; Fredholm sequa tion asratio of2integral series inlambda, applications oftheFredholrr *heory, Hilbert- Schmidt theory ofsymmetric kernels, application, etc.Neumann, Dirichlet, vibratory prob lems. Index, ix+253pp. 53/8x8. S176 Paperbound $2.00 Foundations ofmathematics THECONTINUUM ANDOTHER TYPES OFSERIAL ORDER, E.V.Huntington. Thisfamous book gives asystematic elementary account ofthemodern theory ofthecontinuum asatype of serial order. Based ontheCantor-Dedekind ordinal theory, which requires notechnical knowledge ofhigher mathematics, itoffers aneasily followed analysis ofordered classes, discrete anddense series, continuous series, Cantor stransfinite numbers. 2nd edition. Index, viii+82pp. S^/s x8. S130 Paperbound $1.00 CONTRIBUTIONS TOTHEFOUNDING OFTHETHEORY OFTRANSFINITE NUMBERS, Georg Cantor. These papers founded anew branch ofmathematics. Thefamous articles of1895-7 are translated, with an82-page introduction byP.E.B.Jourdain dealing with Cantor, theback ground ofhisdiscoveries, their results, future possibilities. Bibliography. Index. Notes, ix+211 pp. 53/8x8. S45Paperbound $1.35 ELEMENTARY MATHEMATICS FROM ANADVANCED STANDPOINT, Felix Klein. This classic text isanoutgrowth ofKlein sfamous integration andsurvey course atGottingen. Using one field ofmathematics tointerpret, adjust, illuminate another, itcovers basic topics ineach area, illustrating itsdiscussion with extensive analysis.Itisespecially valuable inconsidering areas ofmodern mathematics. &quot;Makes thereader feel theinspiration of... agreat mathematician, inspiring teacher . . .with deep insight into thefounda tions and interrelations,&quot; BULLETIN, AMERICAN MATHEMATICAL SOCIETY. Vol. 1.ARITHMETIC, ALGEBRA, ANALYSIS. Introducing theconcept offunction immediately, itenlivens abstract discussion with graphical andgeometrically perceptual methods. Partial contents: natural numbers, extension ofthenotion ofnumber, special properties, complex numbers. Real equations with realunknowns, complex quantities. Logarithmic, exponential functions, goniometric functions, infinitesimal calculus. Transcendence ofeand pi,theory ofassemblages. Index. 125 figures, ix+274pp.5%x8. S150 Paperbound $1.85 Vol. 2.GEOMETRY. Acomprehensive view which accompanies thespace perception inherent ingeometry with analytic formulas which facilitate precise formulation. Partial contents: Simplest geometric manifolds: linesegment, Grassmann determinant principles, classification ofconfigurations ofspace, derivative manifolds. Geometric transformations: affine transforma tions, projective, higher point transformations, theory oftheimaginary. Systematic discussion ofgeometry and itsfoundations. Indexes. 141 illustrations, ix+214pp.5%x8. S151 Paperbound $1.75 Catalogue ofDover Books ESSAYS ONTHETHEORY OFNUMBERS: 1.CONTINUITY ANDIRRATIONAL NUMBERS; 2.THE NATURE ANDMEANING OFNUMBERS, Richard Dedekind. Thetwomost important essays on the logical foundations ofthenumber system bythefamous German mathematician. The first provides apurely arithmetic andperfectly rigorous foundation forirrational numbers and thereby arigorous meaning tocontinuity inanalysis. Thesecond essay isanattempt to give alogical basis fortransfinite numbers andproperties ofthenatural numbers. Discusses the logical validity ofmathematical induction. Authorized English translations byW.W. Deman of &quot;Stetigkeit und irrationale Zahlen&quot; and &quot;Was sind undwas sollen die Zahlen?&quot; vii+115pp. 53/8 x8. T1010 Paperbound $1.00 Geometry THEFOUNDATIONS OFEUCLIDEAN GEOMETRY, H.G.Forder. The first rigorous account of Euclidean geometry, establishing propositions without recourse toempiricism, andwithout multiplying hypotheses. Corrects many traditional weaknesses ofEuclidean proofs, and investigates theproblems imposed ontheaxiom system bythediscoveries ofBolyai and Lobachevsky. Some topics discussed areClasses and Relations; Axioms forMagnitudes; Congruence and Similarity; Algebra ofPoints; Hessenberg sTheorem; Continuity; Existence ofParallels; Reflections; Rotations; Isometries; etc. Invaluable forthe light itthrows on foundations ofmath. Lists: Axioms employed, Symbols, Constructions. 295pp. 53/ax8. S481 Paperbound $2.00 ADVANCED EUCLIDEAN GEOMETRY, R.A.Johnson. Foryears thestandard textbook onadvanced Euclidean geometry, requires only high school geometry andtrigonometry. Explores inunusual detail and gives proofs ofhundreds ofrelatively recent theorems and corollaries, many formerly available only inwidely scattered journals. Covers tangent circles, thetheorem of Miquel, symmedian point, pedal triangles and circles, theBrocard configuration, andmuch more. Formerly &quot;Modern Geometry.&quot; Index. 107diagrams, xiii+319pp. 53/8x8. S669 Paperbound $1.65 HIGHER GEOMETRY: ANINTRODUCTION TOADVANCED METHODS INANALYTIC GEOMETRY, F.S. Woods. Exceptionally thorough study ofconcepts andmethods ofadvanced algebraic geometry (asdistinguished from differential geometry). Exhaustive treatment of 1-, 2-, 3-,and 4- dimensional coordinate systems, leading ton-dimensional geometry inanabstract sense. Covers projectivity, tetracyclical coordinates, contact transformation, pentaspherical coordi nates, much more. Based onM.I.T. lectures, requires sound preparation inanalytic geometry andsome knowledge ofdeterminants. Index. Over 350 exercises. References. 60figures, x+423pp. 53/8x8. S737 Paperbound $2.00 CONTEMPORARY GEOMETRY, Andr6 Delachet. Translated byHoward G.Bergmann. Therecent developments ingeometry covered inuncomplicated fashion. Clear discussions ofmodern thinking about thetheory ofgroups, theconcept ofabstract geometry, projective ge9metry, algebraic geometry, vector spaces, new kinds ofmetric spaces, developmentsindifferen tialgeometry, etc.Alarge part ofthebook isdevoted toproblems, developments, and applications oftopology. Foradvanced undergraduates and graduate students aswell as mathematicians inother fields whowant abrief introduction tocurrent work ingeometry. 39figures. Index, xix+94pp. 53/8x8V2. S988 Paperbound $1.00 ELEMENTS OFPROJECTIVE GEOMETRY, L.Cremona. Outstanding complete treatment ofprojec tivegeometry byone oftheforemost 19th century geometers. Detailed proofs ofallfunda mental principles, stress placed ontheconstructive aspects. Coverspomology,lawofduality, anharmonic ratios, theorems ofPascal andBrianchon, foci, polar reciprocal figures, etc.Only ordinary geometry necessary tounderstand thishonored classic. Index. Over 150 fullyworked outexamples andproblems. 252diagrams, xx+302pp. 53/8x8. S668 Paperbound $1.75 ANINTRODUCTION TOPROJECTIVE GEOMETRY, R.M.Winger. One ofthebest introductory texts toanimportant area inmodern mathematics. Contains fulldevelopment ofelementary concepts often omitted inother books. Employing theanalytic method tocapitalize onthe student scollegiate training inalgebra, analytic geometry and calculus, theauthor deals with such topics asEssential Constants, Duality, The Line atInfinity, Projective Properties andDouble Ratio, Projective Coordinates, The Conic, Collineations and Involutions inOne Dimension, Binary Forms, Algebraic Invariants, Analytic Treatment ofthe Conic, Collinea tions inthePlane, Cubic Involutions andtheRational Cubic Curve, and aclear discussion ofNon-Euclidean Geometry. Forsenior-college students and graduates.&quot;An excellent text book . . .very, clearly written . . .propositions stated concisely,&quot; A.Emch, Am. Math. Monthly. Corrected reprinting. 928problems. Index. 116 figures,xii+443pp. 53/8x8. S949 Paperbound $2.00 ALGEBRAIC CURVES, Robert J.Walker, Professor ofMathematics, Cornell University. Fine introduction toalgebraic geometry. Presents some oftherecently developed algebraic meth ods ofhandling problems inalgebraic geometry, shows how these methods are related to theolder analytic andgeometric problems, andapplies them tothose same geometric prob lems. Limited tothetheory ofcurves, concentrating onbirational transformations. Contents: Algebraic Preliminaries, Projective Spaces, Plane Algebraic Curves, Formal Power Series, Transformations ofaCurve, Linear Series. 25illustrations. Numerous exercises atends of sections. Index, x+201pp. 53/8x8V2. S336 Paperbound $2.00 Catalogue ofDover Books THEADVANCED GEOMETRY OFPLANE CURVES ANDTHEIR APPLICATIONS, C.Zwikker. Anun usual study ofmany important curves, their geometrical properties and their applications, including discussions ofmany less well-known curves notoften treated intextbooks on synthetic andanalytic Euclidean geometry. Includes both algebraic andtranscendental curves such astheconic sections, kinked curves, spirals, lemniscates, cycloids, etc.andcurves generated asinvolutes, evolutes, anticaustics, pedals, envelopes andorthogonal trajectories. Dr.Zwikker represents thepoints ofthecurves bycomplex numbers instead oftwo real Cartesian coordinates, allowing direct andeven elegant proofs. Formerly: &quot;Advanced Plane Geometry.&quot; 273 figures, xii+299pp.5% x8Va. S1078 Paperbound $2.00 ATREATISE ONTHEDIFFERENTIAL GEOMETRY OFCURVES ANDSURFACES, L.P.Eisenhart. Introductory treatise especially forthegraduate student, foryears ahighly successful text book. More detailed andconcrete inapproach than most more recent books. Covers space curves, osculating planes, moving axes, Gauss method, themoving trihedral, geodesies, conformal representation, etc. Last section deals with deformation ofsurfaces, rectilinear congruences, cyclic systems, etc. Index. 683problems. 30diagrams, xii+474pp.5% x8. S667 Paperbound $2.75 ATREATISE ONALGEBRAIC PLANE CURVES, J.L.Coolidge. Unabridged reprinting ofone of few fullcoverages inEnglish, offering detailed introduction totheory ofalgebraic plane curves and their relations togeometry and analysis. Treats topological properties, Riemann- Roch theorem, allaspects ofwide variety ofcurves including real, covariant, polar, contain ingseries ofagiven sort, elliptic, polygonal, rational, the pencil, twoparameter nets, etc. Thisvolume will enable thereader toappreciate thesymbolic notation ofAronhold and Clebsch. Bibliography. Index. 17illustrations, xxiv+513pp.5% x8.S543 Paperbound $2.75 ANINTRODUCTION TOTHEGEOMETRY OFNDIMENSIONS, D.M.Y.Sommerville. Anintroduc tionpresupposing noprior knowledge ofthe field, theonlybook inEnglish devoted exclu sively tohigher dimensional geometry. Discusses fundamental ideas ofincidence, parallelism, perpendicularity, angles between linear space; enumerative geometry; analytical geometry from projective andmetric points ofview; polytopes; elementary ideas inanalysis situs; content ofhyper-spacial figures. Bibliography. Index. 60diagrams. 196pp.5% x8. S494 Paperbound $1.50 GEOMETRY OFFOUR DIMENSIONS, H.P.Manning. Unique inEnglish asaclear, concise intro duction. Treatment issynthetic, andmostly Euclidean, although inhyperplanes and hyper- spheres atinfinity, non-Euclidean geometry isused. Historical introduction. Foundations of 4-dimensional geometry. Perpendicularity, simple angles. Angles ofplanes, higher order. Symmetry, order, motion; hyperpyramids, hypercones, hyperspheres; figures with parallel elements; volume, hypervolume inspace; regular polyhedroids. Glossary. 78figures, ix+ 348pp. 53/8x8. S182 Paperbound $2.00 CONVEX FIGURES ANDPOLYHEDRA, L.A.Lyusternik. Anexcellent elementary discussion by aleading Russian mathematician. Beginning with thebasic concepts ofconvex figures and bodies and their supporting lines and planes, theauthor covers such matters ascentrally symmetric convex figures, theorems ofEuler, Cauchy, Steinitz andAlexandrov onconvex polyhedra, linear systems ofconvex bodies, planar sections ofconvex bodies, theBrunn- Minkowski inequality and itsconsequences, andmany other related topics. Nomore than a high school background inmathematics needed forcomplete understanding. First English translation byT.J.Smith. 182 illustrations. Index, x+176pp. 53/8x8V2. S1021 Paperbound $1.50 NON-EUCLIDEAN GEOMETRY, Roberto Bonola. Thestandard coverage ofnon-Euclidean geom etry. Itexamines from both ahistorical andmathematical point ofview thegeometries which have arisen from astudy ofEuclid s5thpostulate upon parallel lines. Also included arecomplete texts, translated, ofBolyai sSCIENCE OFABSOLUTE SPACE. Lobachevsky s THEORY OFPARALLELS. 180diagrams. 431pp. 53/a x8. S27Paperbound $2.00 ELEMENTS OFNON-EUCLIDEAN GEOMETRY, D.M.Y.Sommerville. Unique inproceeding step- by-step, inthemanner oftraditional geometry. Enables thestudent with only agood knowledge ofhigh school algebra andgeometry tograsp elementary hyperbolic, elliptic, analytic non-Euclidean geometries; space curvature and itsphilosophical implications; theory ofradical axes; homothetic centres andsystems ofcircles; parataxy and parallelism; absolute measure; Gauss proof ofthedefect area theorem; geodesic representation; much more, allwith exceptional clarity. 126problems atchapter endings provide progressive practice and familiarity. 133figures. Index, xvi+274pp. 53/8x8. S460 Paperbound $1.75 INTRODUCTORY NON-EUCLIDEAN GEOMETRY, H.P.Manning. Sound elementary introduction to non-Euclidean geometry. The first two thirds (Pangeometry and theHyperbolic Geometry) require agrasp ofplane and solid geometry and trigonometry. The last sections (the Elliptic Geometry and Analytic Non-Euclidean Geometry) necessitate also basic college cal culus forunderstanding the text. Thebook does notpropose toinvestigate thefoundations ofgeometry, butrather begins with thetheorems common toEuclidean andnon-Euclidean geometry and then takesupthe specific differences between them. Asimple and direct account ofthebases ofthis important branch ofmathematics forteachers and students. 94figures, vii+95pp. 53/8x8. S310 Paperbound $1.00 Catalogue ofDover Books ELEMENTARY CONCEPTS OFTOPOLOGY, P.Alexandroff. First English translation ofthefamous brief introduction totopology forthebeginner orforthemathematician notundertaking extensive study. This unusually useful intuitive approach deals primarily with theconcepts of complex, cycle, andhomology, and iswholly consistent with current investigations. Ranges from basic concepts ofset-theoretic topology totheconcept ofBetti groups. &quot;Glowing example ofharmony between intuition andthought,&quot; David Hilbert. Translated byA.E.Farley. Introduction byD.Hilbert. Index. 25figures. 73pp. 53/8x8. S747 Paperbound $1.00 Number theory INTRODUCTION TOTHETHEORY OFNUMBERS, L.E.Dickson. Thorough, comprehensive ap proach with adequate coverage ofclassical literature, anintroductory volume beginners canfollow. Chapters ondivisibility, congruences, quadratic residues &reciprocity, Dioptiantine equations, etc. Fulltreatment ofbinary quadratic forms without usual restriction tointegral coefficients. Covers infinitude ofprimes, least residues, Fermat stheorem, Euler sphi function, Legendre ssymbol, Gauss slemma, automorphs, reduced forms, recent theorems ofThue &Siegel, many more. Much material not readily available elsewhere. 239 prob lems. Index. Ifigure, viii+183pp.5% x8. S342 Paperbound $1.75 ELEMENTS OFNUMBER THEORY,I.M.Vinogradov. Detailed 1stcourse forpersons without advanced mathematics; 95% ofthisbook canbeunderstood byreaders whohave gone no farther than high school algebra. Partial contents: divisibility theory, important number theoretical functions, congruences, primitive roots and indices, etc. Solutions toboth problems andexercises. Tables ofprimes, indices, etc.Covers almost every essential formula inelementary number theory! Translated from Russian. 233problems, 104 exercises, viii+ 227pp.5% x8. S259 Paperbound $1.75 THEORY OFNUMBERS andDIOPHANTINE ANALYSIS, R.D.Carmichael. These twocomplete works inonevolume form oneofthemost lucid introductions tonumber theory, requiring only afirm foundation inhigh school mathematics. &quot;Theory of Numbers,&quot; partial contents: Eratosthenes sieve, Euclid sfundamental theorem, G.C.F. andL.C.M. oftwo ormore integers, linear congruences, etc &quot;Diophantine Analysis&quot;: rational triangles, Pythagorean triangles, equations Ofthird, fourth, higher degrees, method offunctional equations, much more. &quot;Theory ofNumbers&quot;: 76problems. Index. 94pp. &quot;Diophantine Analysis&quot;: 222problems. Index. 118pp. 5%x8. S529 Paperbound $1.35 Numerical analysis,tables MATHEMATICAL TABLES ANDFORMULAS, Compiled byRobert D.Carmichael andEdwin R. Smith. Valuable collection forstudents, etc.Contains alltables necessary incollege algebra andtrigonometry, such asfive-place common logarithms, logarithmic sines andtangents of small angles, logarithmic trigonometric functions, natural trigonometric Tunctions, four-place antilogarithms, tables forchanging from sexagesimal tocircular andfrom circular tosexa gesimal measure ofangles, etc. Alsomany tables andformulasnpt ordinarily accessible, including powers, roots, and reciprocals, exponential and hyperbolic functions, ten-place logarithms ofprime numbers, andformulas andtheorems from analytical andelementary geometry andfrom calculus. Explanatory introduction, viii+269pp.5% x8V2. Sill Paperbound $1.25 MATHEMATICAL TABLES, H.B.Dwight. Unique for itscoverage inonevolume ofalmost every function ofimportance inapplied mathematics, engineering, and the physical sciences. Three extremely fine tables ofthethree trig functions and their inverse functions to thousandths ofradians; natural andcommon logarithms; squares, cubes; hyperbolic functions andtheinverse hyperbolic functions; (a2+b2 )exp. i/2a;complete elliptic integrals ofthe 1stand2nd kind; sineandcosine integrals; exponential integrals Ei(x)and Ei(-x); binomial coefficients; factorials to250; surface zonal harmonics and first derivatives; Bernoulli and Euler numbers and their logs tobase of10;Gamma function; normal probability integral; over 60pages ofBessel functions; theRiemann Zeta function. Each table with formulae generally used, sources ofmore extensive tables, interpolation data, etc.Over half have columns ofdifferences, tofacilitate interpolation.Introduction. Index, viii+231pp.5% x8. S445 Paperbound y2.orj TABLES OFFUNCTIONS WITHFORMULAE ANDCURVES, E.Jahnke &F.Emde. Theworld smost comprehensive 1-volume English-text collection oftables, formulae, curves oftranscendent functions. 4thcorrected edition, new76-page section giving tables, formulae forelementary functions not inother English editions. Partial contents: sine, cosine, logarithmic integral; factorial function; error integral; theta functions; elliptic integrals, functions; Legendre, Bessel, Riemann, Mathieu, hypergeometric functions, etc.Supplementary books Bibliography. Indexed. &quot;Out ofthewayfunctions forwhich weknow noother source SCIENT FICCOM PUTING SERVICE, Ltd.212 figures. 400pp. 5*/ax8. S133 Paperbound $2.00 Catalogue ofDover Books JACOBIAN ELLIPTIC FUNCTION TABLES, L.M.Milne-Thomson. Aneasy tofollow, practical book which gives notonly useful numerical tables, butalso acomplete elementary sketch oftheapplication ofelliptic functions. Itcovers Jacobian elliptic functions and adescription oftheir principal properties; complete elliptic integrals; Fourier series andpower series expansions; periods, zeros, poles, residues, formulas forspecial values oftheargument; transformations, approximations, elliptic integrals, conformal mapping, factorization ofcubic and quartic polynomials; application tothependulum problem; etc.Tables and*graphs form thebody ofthebook: Graph, 5figure table ofthe elliptic function sn(um);en(um); dn(um).8figure table ofcomplete elliptic integrals K,K,E,E,andthenome q.7figure table oftheJacobian zeta-function Z(u). 3figures, xi+123pp.5% x8. S194 Paperbound $1.35 TABLES OFINDEFINITE INTEGRALS, G.PetitBpis. Comprehensive andaccurate, this orderly grouping ofover2500 ofthemost useful indefinite integrals willsave youhours oflaborious mathematical groundwork. After alistof49common transformations ofintegral expressions, with awide variety ofexamples, thebook takes upalgebraic functions, irrational monomials, products and quotients ofbinomials, transcendental functions, natural logs, etc.You will rarely ornever encounter anintegral ofanalgebraic ortranscendental function notincluded here; anymore comprehensive setoftables costs atleast $12or$15. Index. 2544 integrals, xii+154pp. 6Vs x9V4. S225 Paperbound $2.00 SUMMATION OFSERIES, Collected byL.B.W.Jolley. Over 1100common series collected, summed, andgrouped foreasy reference formathematicians, physicists, computer techni cians, engineers, and students. Arranged forconvenience into categories, such asarith metical andgeometrical progressions, powers andproducts ofnatural numbers, figurate and polygonal numbers, inverse natural numbers, exponential and logarithmic series, binomial expansions, simple inverse products, factorials, andtrigonometric andhyperbolic expansions. Also included areseries representing various Bessel functions, elliptic integrals; discussions ofspecial series involving Legendre polynomials, the zeta function, Bernoulli sfunction, and similar expressions. Revised, enlarged second edition. New preface, xii+251pp.5% x8V2. S23Paperbound $2.25 ATABLE OFTHEINCOMPLETE ELLIPTIC INTEGRAL OFTHETHIRD KIND, R.G.Selfridge, J.E. Maxfield. The firstcomplete 6-place tables ofvalues oftheincomplete integral ofthethird kind, prepared under theauspices oftheResearch Department ofthe U.S. Naval Ordnance Test Station. Calculated onanIBM type 704 calculator andthoroughly verified byecho- checking and acheck integral atthecompletion ofeach value ofa.Ofinestimable value inproblems where thesurface area ofgeometrical bodies canonly beexpressed interms oftheincomplete integral ofthe third andlower kinds; problems inaero-, fluid-, and thermodynamics involving processes where nonsymmetrical repetitive volumes must be determined; various types ofseismological problems; problems ofmagnetic potentials due to circular current; etc.Foreword. Acknowledgment. Introduction. Use oftable, xiv+805pp.5% x83/a. S501 Clothbound $7.50 PRACTICAL ANALYSIS, GRAPHICAL ANDNUMERICAL METHODS,F.A.Willers. Translated by R.T.Beyer. Immensely practical handbook forengineers, showing how tointerpolate, use various methods ofnumerical differentiation and integration, determine theroots ofasingle algebraic equation, system oflinear equations, useempirical formulas, integrate differential equations, etc.Hundreds ofshortcuts forarriving atnumerical solutions. Special section on American calculating machines, byT.W.Simpson. 132 illustrations. 422pp.5% x8. S273 Paperbound $2.75 NUMERICAL INTEGRATION OFDIFFERENTIAL EQUATIONS, A.A.Bennett, W. E.Milne, H. Bateman. Replication oforiginal monograph prepared forNational Research Council. New methods ofintegtation ofdifferential equations developed by3leading mathematicians: THE INTERPOLATIONAL POLYNOMIAL andSUCCESSIVE APPROXIMATIONS byA.A.Bennett; STEP-BY- STEP METHODS OFINTEGRATION byW.W.Milne; METHODS FORPARTIAL DIFFERENTIAL EQUATIONS byH.Bateman. Methods forpartial differential equations, transition from differ ence equations todifferential equations, solution ofdifferential equations tonon-integral values ofaparameter will interest mathematicians and physicists. 288 footnotes, mostly bibliographic; 235-item classified bibliography. 108pp.5%x8. S305 Paperbound $1.35 INTRODUCTION TORELAXATION METHODS, F.S.Shaw. Fluid mechanics, design ofelectrical networks, forces instructural frameworks, stress distribution, buckling, etc. Solve linear simultaneous equations, linear ordinary differential equations, partial differential equations, Eigen-value problems byrelaxation methods. Detailed examples throughout. Special tables fordealing with awkwardly-shaped boundaries. Indexes. 253 diagrams. 72tables. 400pp.5%x8. S244 Paperbound $2.45 NUMERICAL SOLUTIONS OFDIFFERENTIAL EQUATIONS, H.Levy&E.A.Baggott. Comprehensive collection ofmethods forsolving ordinary differential equations offirstand higher order. Allmust pass 2requirements: easy tograsp and practical, more rapid than school methods. Partial contents: graphical integration ofdifferential equations, graphical methods forde tailed solution. Numerical solution. Simultaneous equations andequations of2ndandhigher orders. &quot;Should be inthehands ofall inresearch inapplied mathematics, teaching,&quot; NATURE. 21figures, viii+238pp. 53/8x8. S168 Paperbound $1.85 Catalogue ofDover Books Probability theory andinformation theory ANELEMENTARY INTRODUCTION TOTHETHEORY OFPROBABILITY, B.V.Gnedenko and A.Ya. Khinchin. Translated byLeo F.Boron. Aclear, compact introduction designed toequip the reader with afundamental grasp ofthetheory ofprobability.Itisthorough and authori tative within itspurposely restricted range, yetthelayman with abackground inelementary mathematics willbeable tofollow itwithout difficulty. Covers such topics astheprocesses involved inthecalculation ofprobabilities, conditional probabilities and themultiplication rule, Bayes sformula, Bernoulli sscheme andtheorem, random variables and distribution laws, and dispersion andmean deviations. New translation offifth (revised) Russian edi tion (1960) theonly translation checked andcorrected byGnedenko. New preface forDover edition byB.V.Gnedenko. Index. Bibliography. Appendix: Table ofvalues offunction&amp;lt;(a). xii+130pp. 53/8 x8V2. T155 Paperbound $1.50 ANINTRODUCTION TOMATHEMATICAL PROBABILITY, Julian Lowell Coolidge. Athorough intro duction which presents themathematical foundation ofthetheory ofprobability. Asub stantial body ofmaterial, yetcanbeunderstood with aknowledge ofonly elementary cal culus. Contains: TheScope andMeaning ofMathematical Probability; Elementary Principles ofProbability; Bernoulli sTheorem; Mean Value and Dispersion; Geometrical Probability; Probability ofCauses; Errors ofObservation; Errors inMany Variables; Indirect Observations; The Statistical Theory ofGases; andThe Principles ofLife Insurance. Sixpages oflogarithm tables. 4diagrams. Subject andauthor indices, xii+214pp. 53/8x8V2. S258 Paperbound $1.50 AGUIDE TOOPERATIONS RESEARCH, W.E.Duckworth. Abrief nontechnical exposition of techniques and theories ofoperational research. Agood introduction forthelayman; also canprovide the initiate withnew understandings. Nomathematical training needed, yetnot anoversimplification. Covers game theory, mathematical analysis, information theory, linear programming, cybernetics, decision theory, etc. Also includes adiscussion ofthe actual organization ofanoperational research program andanaccount oftheuses ofsuch pro grams inthe oil, chemical, paper, and metallurgical industries, etc. Bibliographies at chapter ends. Appendices. 36figures. 145pp. 5V4 x8Va. T1129 Clothbound $3.50 MATHEMATICAL FOUNDATIONS OFINFORMATION THEORY, A. I.Khinchin. Forthe first time mathematicians, statisticians, physicists, cyberneticists, andcommunications engineers are offered acomplete andexact introduction tothis relatively -new field. Entropy asameasure of afinite scheme, applications tocoding theory, study ofsources, channels and codes, detailed proofs ofbothShannon theorems foranyergodic source andanystationary channel with finite memory, andmuch more arecovered. Bibliography, vu+^4P pape8 rbound $135 SELECTED PAPERS ONNOISE ANDSTOCHASTIC PROCESS, edited byProf. Nelson Wax, U.of Illinois 6basic papers fornewcomers inthe field, forthose whose work involves noise characteristJcs. Chandrasekhar, Uhlenbeck &Ornstein, Uhlenbeck &Ming, Rice, Doob. In cluded isKacsChauvenet-Prize winning Random Walk. Extensive bibliographyists200 articles, upthrough 1953. 21figures. 337pp. 6Vsx9V4. S262 Paperbound $2.75 THEORY OFPROBABILITY, William Burnside. Synthesis, expansionofindividual papers pre sents numerous problemsinclassical probability, offering many original views succinctly effectively Game theory, cards, selections from groups; geometrical probabilityinsuch aTeas assuppositions astoprobability ofpositionofpoint ona.line, points onsurface ofsphere, etc. Includes methods ofapproximation, theory oferrors, direct calculation of probabilities,etc. Index. 136pp. 53/8x8. S567 Paperbound $1.00 Statistics ANALYSIS &DESIGN OFEXPERIMENTS, H.B.Mann. Offers amethod forgrasping theanalysis Sfvariance andvahance design within ashort time Partial contente: Chi-square distrib irtion and analysis ofvariance distribution, matrices, quadratic forms, likelihood ration Jestsand tests oflinear hypotheses, power ofanalysis, Galois fields, n &quot;- ^&quot;^&quot;3^?^ &quot;JfJJJ estimates etc.15pp. ofuseful tables, x+195pp. 5x7%. S180 Paperbound $1.45 Catalogue ofDover Books METHODS OFSTATISTICS, L.H.C.Tippett. Aclassic initsfield, thisunusually complete sys tematic introduction tostatistical methods begins atbeginner slevel and progresses to advanced levels forexperimenters and poll-takersinallfields ofstatistical research. Sup plies fundamental knowledge ofvirtually allelementary methods inusetoday bysociologists, psychologists, biologists, engineers, mathematicians, etc. Explains logical andmathematical basis ofeach method described, with examples foreach section. Covers frequency distribu tions andmeasures, inference from random samples, errors inlarge samples, simple analysis ofvariance multiple and partial regression and correlation, etc. 4threvised (.1952) edition. 16charts. 5significance tables. 152-item bibliography. 96tables. 22figures. 395pp.6x9. S228 Clothbound $7.50 STATISTICS MANUAL, E.L.Crow, F.A.Davis, M.W.Maxfield. Comprehensive collection of classical, modern statistics methods, prepared under auspices ofU.S.Naval Ordnance Test Station, China Lake, Calif. Many examples from ordnance willbevaluable toworkers in allfields. Emphasis isonuse, with information onfiducial limits, sign tests, Chi-square runs sensitivity, quality control, much more. &quot;Well written . . .excellent reference work,&quot; Operations Research. Corrected edition ofNAVORD Report 3360 NOTS 948. Introduction. Appendix of32tables, charts. Index. Bibliography. 95illustrations. 306pp. 53/8x8. S599 Paperbound $1.75 Symbolic logic ANINTRODUCTION TOSYMBOLIC LOGIC, Susanne K.Langer. Probably theclearest book ever written onsymbolic logic forthephilosopher, general scientist andlayman.Itwillbepar ticularly appreciated bythose who have been rebuffed byother introductory works because ofinsufficient mathematical training. Nospecial knowledge ofmathematics isrequired. Starting with thesimplest symbols andconventions, youare ledtoaremarkable grasp of theBoole-Schroeder andRussell-Whitehead systems clearly and quickly. PARTIAL CONTENTS: Study offorms, Essentials oflogical structure, Generalization, Classes, Thedeductive system ofclasses, The algebra oflogic, Abstraction ofinterpretation, Calculuspfpropositions, Assumptions ofPRINCIPIA MATHEMATICA, Logistics, Logic ofthe syllogism, Proofs of theorems. &quot;One oftheclearest andsimplest introductions toasubject which isverymuch alive The style iseasy, symbolism isintroduced gradually, and the intelligent non-mathe matician should have nodifficulty infollowing the argument,&quot; MATHEMATICS GAZETTE. Revised, expanded second edition. Truth-value tables. 368pp. 53/8 x8. S164 Paperbound $1.85 ASURVEY OFSYMBOLIC LOGIC: THECLASSIC ALGEBRA OFLOGIC, C. I.Lewis. Classic survey ofthe field, comprehensive andthorough. Indicates content ofmajor systems, alternative methods ofprocedure, and relation ofthese totheBoole-Schroeder algebra and toone another. Contains historical summary, aswell asfullproofs andapplications oftheclassic, or Boole-Schroeder, algebra oflogic. Discusses diagrams forthelogical relations ofclasses, the two-valued algebra, propositional functions oftwo ormore variables, etc.Chapters 5and 6 oftheoriginal edition, which contained material notdirectly pertinent, have been omitted in this edition attheauthor srequest. Appendix. Bibliography. Index, viii+352pp.5% x8%. S643 Paperbound $2.35 INTRODUCTION TOSYMBOLIC LOGIC AND ITSAPPLICATIONS, R.Carnap. One oftheclearest, most comprehensive, and rigorous introductions tomodern symbolic logic byperhaps its greatest living master. Symbolic languages areanalyzed andone constructed. Applications tomath (symbolic representation ofaxiom systems forsettheory, natural numbers, real numbers, topology, Dedekind and Cantor explanations ofcontinuity), physics (the general analysis ofconcepts ofdetermination, causality, space-time-topology, based onEinstein,), biology (symbolic representation ofanaxiom system forbasic concepts).&quot;A masterpiece,&quot; Zentralblatt fiirMathematik und ihre Grenzgebiete. Over 300 exercises. 5figures. Bibliog raphy. Index, xvi+241pp. 53/8x8. S453 Paperbound $1.85 Clothbound $4.00 SYMBOLIC LOGIC, C. I.Lewis, C.H.Langford. Probably themost cited book insymbolic logic, this isone ofthe fullest treatments ofparadoxes. Awide coverage ofthe entire field ofsymbolic logic, plus considerable material that hasnotappeared elsewhere. Basic totheentire volume isthedistinction between the logic ofextensions and ofintensions. Considerable emphasis isplaced onconverse substitution, while thematrix system presents thesupposition ofavariety ofnon-Aristotelian logics.Ithas especially valuable sections onstrict limitations, existence ofterms, 2-valued algebra and itsextension topropositional functions, truth value systems, thematrix method, implication and deducibility, general theory ofpropositions, propositions ofordinary discourse, and similar topics. &quot;Authoritative, most valuable,&quot; TIMES, London. Bibliography. 506pp. 53/8x8. S170 Paperbound $2.35 THEELEMENTS OFMATHEMATICAL LOGIC, Paul Rosenbloom. First publication inany language. Thisbopkisintended forreaders who aremature mathematically buthave no previous training insymbolic logic. Itdoes not limit itself toasingle system, butcovers the field asawhole. Itisadevelopment oflectures given atLund University, Sweden, in 1948. Partial contents: Logic ofclasses, fundamental theorems, Boolean algebra, logic of propositions, logic ofpropositional functions, expressive languages, combinatory logics, development ofmathematics within anobject language, paradoxes, theorems ofPost and Goedel, Church stheorem, andsimilar topics, iv+214pp. 53/8x8. S227 Paperbound $1.45 Catalogue ofDover Books PHILOSOPHY OFSCIENCE ANDMATHEMATICS FOUNDATIONS OFSCIENCE: THEPHILOSOPHY OFTHEORY ANDEXPERIMENT, N.R.CampbellAcritique ofthemost fundamental concepts ofscience ingeneral andphysics inparticularExamines why certain propositions areaccepted without question, demarcates science from philosophy, clarifies theunderstanding ofthe tools ofscience. PartOneanalyzes thepresuppositions ofscientific thought: existence ofthe material world, nature ofscientific laws, multiplication ofprobabilities, etc.: PartTwocovers thenature ofexperiment andthe application ofmathematics: conditions formeasurement, relations between numerical lawsand theories, laws oferror, etc.Anappendix covers problems arising from relativity, force motion, space, andtime. Aclassic initsfield. Index, xiii+565pp. SS/B x83/8. S372 Paperbound $2.95 THENATURE OFPHYSICAL THEORY, P.W.Bridgman. Here ishowmodern physics looks toa highly unorthodox physicist aNobel laureate. Pointing outmany absurdities ofscience, and demonstrating theinadequacies ofvarious physical theories, Dr.Bridgman weighs andana lyzes thecontributions ofEinstein, Bohr, Newton, Heisenberg, andmany others. This isanon-technical consideration ofthe correlation ofscience and reality. Index, xi+138pp5% x8-S33Paperbound $1.25 THEVALUE OFSCIENCE, Henri Poincare&quot;. Many ofthemost mature ideas ofthe &quot;last scientific universalist&quot; covered withcharm andvigor forboth thebeginning student andtheadvancedworker. Discusses thenature ofscientific truth, whether order isinnate intheuniverse orimposed upon itbyman, logical thought versus intuition (relating tomath, through theworks ofWeierstrass, Lie, Klein, Riemann), time andspace (relativity, psychological time simultaneity), Hertz sconcept offorce, interrelationship ofmathematical physics topure math, values within disciplines ofMaxwell, Carnot, Mayer, Newton, Lorentz, etc. Index, ni+147pp. 53/ex8. S469 Paperbound $1.35 SCIENCE ANDHYPOTHESIS, Henri PoincarS. Creative psychology inscience. How such concepts asnumber, magnitude, space, force, classical mechanics were developed, andhow themodern scientist uses them inhisthought. Hypothesis inphysics, theories ofmodern physics. Introduction bySirJames Larmor. &quot;Few mathematicians have had thebreadth of yilonSLPomcar6 &amp;gt;ar &quot;dnone ishissuperior inthe gift ofclearexposition,&quot; E.T.Bell.Index. 272pp. 53/8x8. S221 Paperbound $1.35 PHILOSOPHY ANDTHE PHYSICISTS, L.S.Stebbing. The philosophical aspects ofmodern science examined interms ofalively critical attack ontheideas ofJeans andEddington. Discusses thetask ofscience, causality, determinism, probability, consciousness, therelation oftheworld ofphysics tothat ofeveryday experience. Probes thephilosophical significance ofthePlanck-BohrC9nceptofdiscontinuous energy levels, theinferences tobedrawn from Heisenberg sUncertainty Principle, theimplications of&quot;becoming&quot; involved inthe2ndlaw ofthermodynamics, andother problems posed bythediscarding ofLaplacean determinism. 285pp. 53/8x8. T480 Paperbound $1.65 THEPHILOSOPHICAL WRITINGS OFPEIRCE, edited byJustus Buchler. (Formerly published asTHEPHILOSOPHY OFPEIRCE.) This isacarefully balanced exposition ofPeirce scomplete system, written byPeirce himself. Itcovers such matters asscientific method, pure chance vs.law, symbolic logic, theory ofsigns, pragmatism, experiment, and other topics. Intro duction byJustus Buchler, Columbia University, xvi+368pp. 53/8k8. T217 Paperbound $2.00 LANGUAGE, TRUTH AND LOGIC, A.Ayer. Aclear introduction totheVienna andCambridge schools ofLogical Positivism. Itsets upspecific tests bywhich youcanevaluate validity of ideas, etc.Contents: Function ofphilosophy, elimination ofmetaphysics, nature ofanalysis, apriori, truth and probability, etc. 10th printing.&quot;Ishould like tohave written itmyself,&quot; Bertrand Russell. Index. 160pp. 53/8x8. T10Paperbound $1.25 MATHEMATICS ANDSCIENCE: LAST ESSAYS (DERNIERES PENSEES), Henri Poincare. Translated by J.W.Bolduc. Aposthumous volume ofarticles and lectures bythegreat French mathe matician, philosopher, scientist. Here arenine pieces, never before translated into English,onsuch subjects asThe Evolution ofLaws, Space andTime, Space and3Dimensions, The Logic ofinfinity inMathematics (discussing Russell stheory oftypes), Mathematics and Logic,TheQuantum Theory and itsModern Applications, Relationship Between Matter and Ether,Ethics andScience andTheMoral Alliance. First English translation ofDernieres Pensees.New index, viii+128pp. 53/8x8V2. S1101 Paperbound $1.25 THEPSYCHOLOGY OFINVENTION INTHEMATHEMATICAL FIELD, J.Hadamard. Where doideas come from? What roledoes theunconscious play? Areideas best developed bymathematical reasoning, word reasoning, visualization? What arethemethods used byEinstein, Poincar6, Galton, Riemann? How canthese techniques beapplied byothers? Hadamard, one ofthe world sleading mathematicians, discusses these andother questions, xiii+145pp. 53/ax8. T107 Paperbound $1.25 Catalogue ofDover Books EXPERIMENT ANDTHEORY INPHYSICS, Max Born. ANobel laureate examines thenature and value ofthecounterclaims ofexperiment andtheory inphysics. Synthetic versus analytical scientific advances areanalyzed inthework ofEinstein, Bohr, Heisenberg, Planck, Eddington, Milne, andothers byafellow participant. 44pp. 53/8x8. S308 Paperbound 75$ THEPHILOSOPHY OFSPACE ANDTIME, H.Reichenbach. Animportant landmark inthedevelop ment oftheempiricist conception ofgeometry, covering theproblem ofthefpundationsof geometry, thetheory oftime, theconsequences ofEinstein srelativity, including: relations between theory andobservations; coordinate andmetrical properties ofspace; thepsycholog icalproblem ofvisual intuition ofnon-Euclidean structures; andmany other important topics inmodern science andphilosophy. Themajority ofideas require only aknowledge ofinter mediate math. Introduction byR.Carnap. 49figures. Index, xviii+296pp.5% x8. S443 Paperbound $2.00 OBSERVATION ANDINTERPRETATION INTHEPHILOSOPHY OFPHYSICS: WITH SPECIAL REFER ENCE TOQUANTUM MECHANICS, Edited byS.Kbrner. Acollection ofpapers byphilosophers andphysicists arising outofasymposium held atBristol, England in1957 under theauspices oftheColston Research Society. One ofthemost important contributions tothephilosophy ofscience inrecent years. Thediscussions center around theadequacy orinadequacy of quantum mechanics initsorthodox formulations. Among thecontributors are A. J.Ayer, D.Bohm, K.Popper, F.Bopp, S.Korner, J.P.Vigier, M.Polanyi, P.K.Feyerabend, W.C. Kneale. W.B.Gallie, G.Ryle, SirCharles Darwin, and R.B.Braithwaite. xiv+218pp. 53/8x8V2. S131 Paperbound $1.60 SPACE ANDTIME INCONTEMPORARY PHYSICS: ANINTRODUCTION TOTHETHEORY OFRELA TIVITY AND GRAVITATION, Moritz Schlick. Exposition ofthetheory ofrelativity bythe leader ofthefamed &quot;Vienna Circle.&quot; Itsessential purpose istodescribe the physical doctrines ofspecial and general relativity with particular reference totheir philosophical significance. Explanations ofsuch topics asthegeometrical relativity ofspace,thecon nection with inertia and gravitation, themeasure-determination ofthespace-time continuum, the finite universe, etc., with their philosophical ramifications. Index, xii+89pp.5% x8V2. T1008 Paperbound $1.00 SUBSTANCE ANDFUNCTION, &EINSTEIN STHEORY OFRELATIVITY, Ernst Cassirer. Twobooks bound asone. Cassirer establishes aphilosophy oftheexact sciences that takes intocon sideration newer developments inmathematics, andalsoshows historical connections. Partial contents: Aristotelian logic, Mill sanalysis, Helmhqltz&Kronecker, Russell &cardinal num bers, Euclidean vs.non-Euclidean geometry, Einstein srelativity. Bibliography. Index, xxi+ 465pp. 53/e x8. T50Paperbound $2.25 PRINCIPLESOFMECHANICS, Heinrich Hertz. This lastwork bythe great 19th century physicist isnotonly aclassic, butofgreat interest inthelogic ofscience. Creating anew system ofmechanics based upon space, time, andmass, itreturns toaxiomatic analysis, tounderstanding oftheformal orstructural aspects ofscience, taking intoaccount logic, observation, and apriori elements. Ofgreat historical importance toPoincar6, Carnap, Ein stein, Milne. A20-page introduction byK.S.Cohen, Wesleyan University, analyzes theimpli cations ofHertz sthought and the logic ofscience. Bibliography. 13-page introduction by Helmholtz. xlii+274pp. 53/8x8. S316 Clothbound $3.50 S317 Paperbound $1.85 THEANALYSIS OFMATTER, Bertrand Russell. How dooursenses concord with thenew physics? Thisvolume covers such topics aslogical analysis ofphysics, prerelatiyity physics, causality, scientific inference, physics and perception, special andgeneral relativity, Weyl s theory, tensors, invariants and their physical interpretation, periodicity and qualitative series. &quot;The most thorough treatment ofthesubject that hasyetbeen published,&quot; THENATION. Introduction byL.E.Denonn. 422pp.5% x8. T231 Paperbound $1.95 FOUNDATIONS OFGEOMETRY, Bertrand Russell. Analyzing basic problems intheoverlap area between mathematics and philosophy, Nobel laureate Russell examines thenature ofgeo metrical knowledge, thenature ofgeometry, and theapplication ofgeometry tospace. Itcovers the history ofnon-Euclidean geometry, philosophic interpretations ofgeometry especially Kant projective and metrical geometry. This ismost interesting asthesolution offered in1897 byagreat mind toaproblem still current. New introduction byProf. Morris Kline ofN.Y.University, xii+201pp. 53/8x8. S232 Clothbound $3.25 S233 Paperbound $1.75 IDENTITY AND REALITY, Emile Meyerson. Called byEinstein a&quot;brilliant study inthetheory ofknowledge,&quot; thisbook bytherenowned Franco-German thinker isamajor treatise in thephilosophy ofscience andepistemology. Thorough, critical inquiries into causality, scien tific laws, conservation ofmatter and energy, theunity of.matter, Carnot sprinciple, the irrational, theelimination oftime. Searches outthesolutions ofepistemological questions that form thebases ofthe scientific method. Authorized translation byKate Loewenberg.Author sprefaces. Editor spreface. Appendices. Index. 495pp. 53/sx8V2. T65Paperbound $2.25 ESSAYS INEXPERIMENTAL LOGIC, John Dewey. This stimulating series ofessays touches upon the relationship between inquiry and experience, dependence ofknowledge upon thought,character oflogic; judgments ofpractice, dataandmeanings, stimuli ofthought, etc. Index, viii+444pp. 53/8 x8. T73Paperbound $2.25 Catalogue ofDover Books MATHEMATICS, HISTORIES ANDCLASSICS HISTORY OFMATHEMATICS, D.E.Smith. Most comprehensive non-technical history ofmath inEnglish. Discusses lives andworks ofover athousand major andminor figures, with footnotes supplying technical information outside thebook sscheme, and indicating dis puted matters. Vol I:Achronological examination, from primitive concepts through Egypt Babylonia, Greece, the Orient, Rome, theMiddle Ages, theRenaissance, and upto1900? Vol 2:Thedevelopment ofideas inspecific fields and problems, upthrough elementarycalculus. Twovolumes, total of510 illustrations, 1355pp. 5% x8.Setboxed inattractive container.T429, 430Paperbound, theset$6.00 ASHORT ACCOUNT OFTHEHISTORY OFMATHEMATICS, W.W.R.Ball. Most readable non technical history ofmathematics treats lives, discoveries ofevery important figure from Egyptian, Phoenician mathematicians tolate 19th century. Discusses schools ofIonia Pythagoras, Athens, Cyzicus, Alexandria, Byzantium, systems ofnumeration; primitive arith metic; Middle Ages, Renaissance, including Arabs, Bacon, Regiomontanus, Tartaglia, Cardan, Stevinus, Galileo, Kepler; modern mathematics ofDescartes, Pascal, Wallis, Huygens, Newton, Leibnitz, dAlembert, Euler, Lambert, Laplace, Legendre, Gauss, Hermite, Weierstrass,scores more. Index. 25figures. 546pp.5% x8. S630 Paperbound $2.25 AHISTORY OFGEOMETRICAL METHODS,J.L.Coolidge. Full, authoritative history ofthetech niques which men have employed indealing with geometric questions. . .from ancient times tothemodern development ofprojective geometry. Critical analyses ofthe originalworks. Contents: Synthetic Geometry theearly beginnings, Greek mathematics, non-Euclidean geometries, projective anddescriptive geometry; Algebraic Geometry extension ofthesystem oflinear coordinates, other systems ofpoint coordinates, enumerative andbirational geometry, etc.; and Differential Geometry intrinsic geometry andmoving axes, Gauss andtheclassical theory ofsurfaces, and projective andabsolute differential geometry. Thework ofscores of geometers analyzed: Pythagoras, Archimedes, Newton, Descartes, Leibniz, Lobachevski, Riemann, Hilbert, Bernoulli, Schubert, Grassman, Klein, Cauchy, andmany, many others. Extensive (24- page) bibliography. Index. 13figures, xviii+451pp.5% x8Vfe. S1006 Paperbound $2.25 THEMATHEMATICS OFGREAT AMATEURS, Julian Lowell Coolidge. Enlightening, often surprising, accounts ofwhat canresult from anon-professional preoccupation with mathematics. Chapters onPlato, Omar Khayyam and hiswork with cubic equations, Piero della Francesca, Albrecht Durer, asthetruediscoverer ofdescriptive geometry, Leonardo daVinci and hisvaried mathe matical interests, John Napier, Baron9fMerchiston, inventor oflogarithms, Pascal, Diderot, IHospital, andseven others known primarily forcontributions inother fields. Bibliography. 56figures, viii+211pp. 53/8x8V2. S1009 Paperbound $1.50 ARTANDGEOMETRY, Wm. M.Ivins, Jr.Acontroversial study which propounds theview that theideas ofGreek philosophy andculture served nottostimulate, buttostifle thedevelop ment ofWestern thought. Through anexamination ofGreek artand geometrical inquiries andRenaissance experiments, thisbook offers aconcise history oftheevolution ofmathe matical perspective and projective geometry. Discusses thework ofAlberti, Durer, Pelerin, Nicholas ofCusa, Kepler, Desargues, etc. inawholly readable text ofinterest tothe art historian, philosopher, mathematician, historian ofscience, and others, x+113pp.5% x 83/a. T941 Paperbound $1.25 ASOURCE BOOK INMATHEMATICS, D.E.Smith. Great discoveries inmath, from Renaissance toend of19th century, inEnglish translation. Read announcements byDedekind, Gauss, Delamain, Pascal, Ferrnat, Newton, Abel, Lobachevsky, Bolyai, Rjemann, DeMoivre, Legendre, Laplace, others ofdiscoveries about imaginary numbers, number congruence, slide rule, equations, symbolism, cubic algebraic equations, non-Euclidean forms ofgeometry, calculus, function theory, quaternions, etc. Succinct selections from 125 different treatises, articles, most unavailable elsewhere inEnglish. Each article preceded bybiographical, historical introduction. Vol. I:Fields ofNumber, Algebra. Index. 32 illus. 338pp.5% x8.Vol. II: Fields ofGeometry, Probability, Calculus, Functions, Quaternions. 83 illus. 432pp.5% x8. Vol. 1:S552 Paperbound $2.00 Vol. 2:S553 Paperbound $2.00 2vol. set,$4.00 ACOLLECTION OFMODERN MATHEMATICAL CLASSICS, edited byR.Bellman. 13classic papers, complete intheir original languages, byHermite, Hardy and Littlewood, Tchebychef, Fej6r, Fredholm, Fuchs, Hurwitz, Weyl, vander Pol, Birkhoff, Kellogg, vonNeumann, and Hilbert. Each ofthese papers, collected here forthe first time, triggered aburst ofmathematical activity, providing useful new generalizations orstimulating fresh investigations. Topics dis cussed include classical analysis, periodic andalmost periodic functions, analysis andnumber theory, integral equations, theory ofapproximation, non-linear differential equations, and functional analysis. Brief introductions andbibliographies toeach paper,xii+292pp.6x9. S730 Paperbound $2.00 THEWORKS OFARCHIMEDES, edited byT.L.Heath. Alltheknown works ofthegreat Greek mathematician arecontained inthisonevolume, including therecently discovered Method ofArchimedes. Contains: OnSphere &Cylinder, Measurement ofaCircle, Spirals, Conoids, Spheroids, etc. This isthe definitive edition ofthegreatest mathematical intellect ofthe ancient world. 186-page study byHeath discusses Archimedes and the history ofGreek mathematics. Bibliography. 563pp. 53/a x8. S9Paperbound $2.45 Catalogue ofDover Books THETHIRTEEN BOOKS OFEUCLID SELEMENTS, edited bySirThomas Heath. Definitive edition ofone ofthevery greatest classics ofWestern world. Complete English translation of Heiberg text, together with spurious Book XIV. Detailed 150-page introduction discussing aspects ofGreek and Medieval mathematics. Euclid, texts, commentators, etc. Paralleling thetext isanelaborate critical apparatus analyzing each definition, proposition, postulate, covering textual matters, mathematical analysis, commentators ofalltimes, refutations, sup ports, extrapolations, etc. This isthe full Euclid. Unabridged reproduction ofCambridge U. 2nd edition. 3volumes. Total of995 figures, 1426pp. 5% x8. 888,89,90, 3volume set,paperbound $7.50 ACONCISE HISTORY OFMATHEMATICS, D.Struik. Lucid study ofdevelopment ofmathematical ideas, techniques from Ancient Near East, Greece, Islamic science, Middle Ages, Renaissance, modern times. Important mathematicians aredescribed indetail. Treatment isnotanecdotal, but analytical development ofideas. &quot;Rich incontent, thoughtful ininterpretation,&quot; U.S. QUARTERLY BOOKLIST. Non-technical; nomathematical training needed. Index. 60illustra tions, including Egyptian papyri, Greek mss., portraits of31eminent mathematicians. Bib liography. 2nd edition, xix+299pp. 53/s x8. T255 Paperbound $1.75 AHISTORY OFTHECALCULUS, AND ITSCONCEPTUAL DEVELOPMENT, Carl B.Boyer. Pro vides laymen andmathematicians adetailed history ofthedevelopment, ofthe calculus, from early beginning inantiquity tofinal elaboration asmathematical abstractions. Gives asense ofmathematics notasatechnique, butasahabit ofmind, intheprogression of ideas ofZeno, Plato, Pythagoras, Eudoxus, Arabic and Scholastic mathematicians, Newton, Leibnitz, Taylor, Descartes, Euler, Lagrange, Cantor, Weierstrass, and others. This firstcom prehensive critical history ofthe calculus was originally titled &quot;The Concepts ofthe Calculus.&quot; Foreword byR.Courant. Preface. 22figures. 25-page bibliography. Index, v -f 364pp. 53/8x8. S509 Paperbound $2.00 AMANUAL OFGREEK MATHEMATICS, SirThomas L.Heath. Anon-technical survey ofGreek mathematics addressed tohigh school andcollege students andthelayman whodesires asense ofhistorical perspective inmathematics. Thorough exposition ofearly numerical notation and practical calculation, Pythagorean arithmetic andgeometry, Thales and the earliest Greek geometrical measurements andtheorems, themathematical theories ofPlato, Euclid s &quot;Ele ments&quot; and hisother works (extensive discussion), Aristarchus, Archimedes, Eratosthenes and themeasurement ofthe earth, trigonometry (Hipparchus, Menelaus, Ptolemy), Pappus and sdfrc Heron ofAlexandria, anddetailed coverage ofminor figures normally omitted from histories 36.Presented inarefreshingly interesting andreadable style. Appendix. 2Indexes, pp.53/8x8. S279 Paperbound $2.25 THEGEOMETRY OFREN DESCARTES. With thisbook Descartes founded analytical geometry. Excellent Smith-Latham translation, plus original French text with Descartes own diagrams. Contains Problems theConstruction ofWhich Requires Only Straight Lines and Circles; On theNature ofCurved Lines; OntheConstruction ofSolid orSupersolid Problems. Notes. Diagrams. 258pp. 53/8x8. S68Paperbound $2.00 APHILOSOPHICAL ESSAY ONPROBABILITIES, Marquis deLaplace. Thisfamous essay explains without recourse tomathematics the principle ofprobability, and theapplication ofprob ability togames ofchance, natural philosophy, astronomy, many other fields. Translated from the6thFrench edition byF.W.Truscott, F.L.Emory, withnew introduction forthis edition byE.T.Bell. 204pp. 53/8x8. S166 Paperbound $1.35 Prices subjecttochange without notice. Dover publishes books onart,music, philosophy, literature, languages, history, social sciences, psychology, handcrafts, orientalia, puzzles and entertainments, chess, petsandgardens, books explaining science, inter mediate andhigher mathematics, mathematical physics, engineering, biological sciences, earth sciences, classics ofscience, etc.Write to: Dept. catrr. Dover Publications, Inc. 180Varick Street, N.Y. 14,N.Y. PGS 7 ®, CLPOEOC eoe “ee, &$ Guhl% ¢ oy ¢ 8 ‘0, $ eyMy MyF >$ © "no%, o! ”, * ay FTI C “0,e, “ipy £ tu,®, ¢Berkel "ay Buckle“eyry G%,%, 4oe, e * PsCud *,rr 3°& by * %<Buchel, %, oe@ %,o onyRS “tgBe y$ oe,= “m, - a IIIS - %&, ca %¢Z on Butte,%, *Bert, *, zs mS by RS a’ o!4 * Prasat. 5 FP $ TIS * & ;*Biche, %©@. “wfss“y Oy* K BeFo, o ns>, D« raed s “2 .*, “eya € Yin U.C.BERKELEY LIBRARIES \/ r\Qb 4*^A a\X/\* \& *U *\,//\^&amp;lt;2r 7 eeejee aoeae 0soll ooooaeeaSreeeeoeaePeoni ae ae oe Sear lyeeeeLithaigck aeelieatieafA naeCeieseeCis oeoe oeoo |Lo.a ies | aeeoe| eeaeae | avee aean ieoe a. ap seiaealt