Kelvin reprinted papers

—o SS 7 =o =e =e =e = ir =o =, Digitized forMicrosoft Corporation bytheInternet Archive in2008. From University ofToronto. May beused fornon-commercial, personal, research, 6educational purposes, orany fairuse. May notbeindexed inacommercial service. REPRINT OFPAPERS ON ELECTROSTATICS AND MAGNETISM BY SIRWILLIAM THOMSON, D.C.L., LLD., F.R.S., F.Il.S.E.,rLo^d Kelvi FELLOW OFSTPETEK's COLLEGE, CAMBKIDGE, AND PROFESSOR OFNATURAL PHILOSOPHY INTHEUNIVERSITY OFGLASGOW. SECOND EDITION. HLotttron : MACMILLAN &CO. 1884 [The riplits oftranslation andreproduction arcreserved.] Ac ee 1Bey Cop,2 PS AL—7e]ae4har Zz PEEFACE TOTHEFIRST EDITION. *^'statics andmathematicallyalliedsubjects, whichoriginally appearedatdifferent times duringthe lastthirty years,inthe CambridgeMathematical Journal, theCambridge andDublin Mathematical Journal, Liouville's Journal de3Iathematiques, thePhilosophical Magazine,Nichol'sCyclopcedia,theReports of theBritish Association, theTransactions orProceedings ofthe RoyalSocieties ofLondon andEdinburgh,theRoyalInstitution ofGreat Britain, andthePhilosophicalSocietiesofManchester andGlasgow. Theremainder, constitutingabout aquarterofthe whole,isnowprintedforthe firsttimefrommanuscript, which, exceptasmallpartabout twenty years old,entitled"Electro- magnets,"hasbeen written forthepresent publication,tofill uproughly gapsinthecollection. Theoriginal dates ofthe republished articles, thedates ofallnewmatterappearingas insertions ornotes inthecourse ofthose articles, andthedates ofthefresh articles have allbeencarefullyindicated. The article onAtmospheric Electricity, extracted from Nichol'sCyclopcedia^ wasoriginallywritten attherequestof mylatefriend andcolleaguetheEditor;andforthepermission toreprintitIamindebted tohisson,mycolleague,Professor John Nichol, and totheMessrs.Griffin, thepublishersofthe Cyclopcedia. vi PREFACE. Thepresent volume includes asnearlyasmaybeallthat I have hitherto written onelectrostatics andmagnetism.Ihave excluded from itelectricalpapersinwhich either thermo- dynamicsorthekinetics ofelectricityisprominent.Iintend that, assoon aspossible,itshall befollowed byacollected re- printofallmyotherpapershithertopublished. Itake thisopportunityofthankingProfessors Clerk Max- wellandTait formuch valuable assistance which theyhave givenmeinthecourse ofthiswork. WILLIAM THOMSON. Yacht "Lalla Kookh," Lamlash, Oct. 12,1872, PREFACE TOTHESECOND EDITION. This Second Edition issubstantiallyareprintoftheFirst Edition;theonlychanges madebeingthecorrection ofafew errata which hadescapeddetection inthevolume asoriginally published. W.T. TheUniversity, Glasgow, March 12,1884. CONTENTS ITICLE I.—ONTHEUNIFOKM MOTION OFHEAT IN HOMOGENEOUS SOLID BODIES, ANDITSCONNEXION WITH THEMATHEMATICAL THEOEY OFELECTEI- CITY. SECTIONS Temperature atany point within orwithout anIsothermal Surface 1—10 Uniform Motion ofHeat inanEllipsoid 11—20 Attraction ofaHomogeneous Ellipsoid onapoint within or without it 21—24 n.—ONTHEMATHEMATICAL THEOEY OFELECTEI- CITY INEQUILIBEIUM. Division I.—OntheElementaey Laws ofStatical Electbicity. InvestigationsofCoulomb, Poisson, andGreen.... 25 Examination ofHarris's Experimental results.... 26—35 Faraday's researches onElectrostatical Induction . . .36—50 m.—ONTHE ELECTEOSTATICAL CAPACITY OFA LEYDEN PHIAL AND OFATELEGEAPH WIEE INSULATED INTHEAXISOFACYLINDEICAL CON- DUCTING SHEATH. ApplicationofthePrinciples brought forward inthepreceding Articles 51—56 IV.—ONTHEMATHEMATICAL THEOEY OFELECTEICITY INEQUILIBEIUM. Division H.—AStatement ofthePrinciples onwhich the Mathematical Theory isfounded. Object oftheArticle 57 Thetwokinds ofElectricity 58—60 Electrical Quantity 61—62 Superposition ofElectric Forces 63 viii Contents. SECTIONS TheLaw ofForce between Electrified bodies.... 64 Definition oftheresultant Electric Force ataPoint... 65 Electrical Equilibrium66 Non-conductors ofElectricity 67 Conductors ofElectricity 68 Electrical Density atanyPoint ofacharged Surface... 69 Exclusion ofallNon-conductors except Air.... 70 Insulated Conductors 71 EecapitulationoftheFundamental Laws 72 ObjectsoftheMathematical Theory ofElectricity... 73 Actual ProgressintheMathematical Theory ofElectricity. . 74 ^v.—ONTHEMATHEMATICAL THEOKY OFELECTEICITY INEQUILIBKIUM. Division HI.—Geometrical Investigation with reference TOthe Distribution ofElectricity onSpherical Con- ductors. ObjectoftheArticle 75 Insulated Conducting Sphere subject tonoExternal Influence . 76 Determination oftheDistribution 77 Verification ofLawIH 78 Digression ontheDivision ofSurfaces andElements— Object oftheDigression 79 Explanation andDefinition regarding Cones.... 80 TheSohd Angle ofaCone, oracomplete Conical Surface . . 81 Sum ofalltheSolid Angles round aPoint =47r.... 82 Sum oftheSolid Angles ofallthecomplete Conical Surfaces =2ir . 83 Solid Angle subtended ataPoint byaTerminated Surface , 84 Orthogonal andObHque Sections ofaSmall Cone... 85 Area oftheSegment cutfrom aSpherical Surface byaSmall Cone 86 Theorem 87 Repulsion onanElement oftheElectrified Surface... 88 Insulated Sphere subjected totheInfluence ofanElec- trical Point. Object 89 Attraction ofaSpherical Surface ofwhich theDensity varies inversely astheCube oftheDistance from agiven point.90—92 Applicationofthepreceding Theorems totheProblem of Electrical Influence 93—95 Effects ofElectrical Influence onInternal Spherical ANi>ONPlane Conducting Surfaces 96—112 Insulated Sphere subject totheInfluence ofabody of ANYFORM electrified INANYGIVEN MANNER.... 113—127 IContents. ix SECTIONS —ONTHEMUTUAL ATTRACTION ORREPULSION BETWEEN TWO ELECTRIFIED SPHERICAL CON- DUCTORS 128—142 VII.—ONTHEATTRACTIONS OFCONDUCTING ANDNON- CONDUCTING ELECTRIFIED BODIES.... 144—148 Vm.—DEMONSTRATION OFAFUNDAMENTAL PROPO- SITION INTHEMECHANICAL THEORY OFELECTRL CITY 149—155 IX.-NOTEONINDUCED MAGNETISM INAPLATE . .156—162 X.—SURUNEPR0PRI1^t6 DELACOUCHE ^LECTRIQUE ENi:QUILIBRE ALASURFACE D'UN CORPS CON- DUCTEUR. ParM.J.Liouville 163—164 Noteonthepreceding Paper 165 XI.—ONCERTAIN DEFINITE INTEGRALS SUGGESTED BYPROBLEMS INTHETHEORY OFELECTRICITY .166—186 XII.—PROPOSITIONS INTHETHEORY OFATTRACTION. Parti 187—198 Partn. . 199—205 XHL-THEOREMS WITHREFERENCE TOTHESOLUTION OFCERTAIN PARTIAL DIFFERENTIAL EQUATIONS . 206 Additions toaFrench Translation ofthepreceding . . . 207 XIV.—ELECTRICAL IMAGES. Extraite d'une lettre deM.William Thomson kM.Liouville .208—210 Extraits dedeux lettres addressees aM.Liouville. ParM. Wiinam Thomson 211—220 Noteausujet deTArticle precedent. ParM.Liouville . . .221—230 XV.—DETERMINATION OFTHEDISTRIBUTION OFELEC- TRICITY ONACIRCULAR SEGMENT OFPLANE OR SPHERICAL CONDUCTING SURFACE, UNDER ANY GIVEN INFLUENCE 231—248 XVI.—ATMOSPHERIC ELECTRICITY. Preliminary Explanations 249—251 Thewhole Surface oftheEarth electrified[generally negatively]. 252 The State ofElectrification oftheAir[unknown andcannot be inferred with certainty from observations oftheElectricdensity oftheEarth's Surface: observation from balloons wanted].253—261 Description ofpreliminary Experiments made totestaerial Elec- tricity, andoftheInstruments employed 262—266 X Contents. ROYAL INSTITUTION LECTUEE. SECTIONS Earliest Observations ofAtmospheric Electricity.... 267—268 Essential qualities oftheApparatus required fortheobservation ofAtmospheric Electricity 269 DescriptionoftheDivided Eing Eeflecting Electrometer . .270—273 DescriptionoftheCommon House Electrometer.... 274—276 DescriptionofthePortable Electrometer 277 Burning Match andWater-dropping Collectors.... 278—279 Eemarks ontheorigin, nature, andchanges ofTerrestrial Atmo- spheric Electricity 280—291 Kew Self-recording Atmospheric Electrometer,vsrith specimen of theresults 292—293 OnElectrical Fbequency 294 Onthenecessity forincessant kecobding andfor simul- taneous OBSERVATIONS INDIFFEBENT LOCALITIES TOINVESTI- GATEAtmospheric Electricity....... 295 Observations onAtmospheric Electricity 296—300 Onsome remarkable effects ofLightning observed ina FarmHouse nearMonimail 301 XVII.— SOUND PEODUCED BYTHEDISCHARGE OFA CONDENSES 302—304 XVni.— MEASUEEMENT OFTHEELECTEOSTATIC FORCE PEODUCED BYADANIELL'S BATTEEY. Preliminary Explanation 305—306 Absolute Electrometer 307 Reduction toabsolute measure, ofthereadings oftorsional Electro- meters, bymeans oftheAbsolute Electrometer.... 308—313 General results oftheWeighings 314—316 Postscript—corrected results 317—319 XIX.—MEASUEEMENT OFTHEELECTEOMOTIYE FOECE EEQUIEED TOPEODUCE ASPAEK INAIEBETWEEN PARALLEL METAL PLATES ATDIFFEEENT DIS- TANCES. Description oftheExperiments 320—321 Table I.,Measurements byAbsolute Electrometer.... 322—323 Table II.,Measurements byPortable Electrometer.... 324 Table III.,Thetwo series compared—Additional Experiments, Tables IV.andV 325 Table VI.,Summary ofresults reduced toAbsolute Measure . . 326 Appendix—Explanation ofTerms. Measurement ofquantities of electricity. Electricdensity. Eesultant electric force atany pointinaninsulating fluid. Eelation between Electric density Contents. xi onthesurface ofaconductor and electric force atpoints inthe airclose toit.Electric pressure from thesurface ofaconductor balanced byAir. Collected formulfe. Electricpotential. In- terpretationofmeasurement byElectrometer. Relation between Electrostatic forceandvariation ofElectric potential. Stratum ofAirbetween twoparallel ornearly parallel plane orcurved metallic surfaces maintained atdifferent potentials. . .327—338 Absolute Electrometer 339 Additional Experiments 340 XX.-REPORT ONELECTROMETERS ANDELECTROSTATIC MEASUREMENTS. Definition— Requisites foraccurate Electrometry.... 341—342 Classification ofElectrometers — I.Repulsion Electrometers II.Symmetrical Electrometers in. Attracted DiscElectrometer 343 Divided Ring Electrometer described, andadjustments explained.344—357 Absolute Electrometer 358—363 NewAbsolute Electrometer . .364—367 Portable Electrometer 368—378 Standard Electrometer . . 379—382 Long-range Electrometer . .383—384 Idiostatic andHeterostatic Electrometers 385 Concluding remarks regarding Electrometers.... 386—390 XXI.—ATMOSPHERIC ELECTRICITY. NewApparatus forobserving Atmospheric Electricity. . . 391 Description andresults ofsimultaneous observations made attwo stations atdifferent elevations intheUniversityofGlasgow. 392 Effect ofsudden changes ofwind . . . . . . ,393—395 Changes observed during athunder-storm 396 Effects observed intheneighbourhood ofanescapeofhigh-pressure steam 397—399 XXn.—NEWPROOF OFCONTACT ELECTRICITY ... 400 XXIII.— ELECTROPHORIC APPARATUS ANDILLUSTRATIONS OFVOLTAIC THEORY. OnaSelf-acting Apparatus formultiplying andmaintaining Elec- tricCharges, withapplications toillustrate theVoltaic Theory.401—407 OnaUniform Electric Current Accumulator.... 408—411 OnVolta-Convection byFlame 412—415 xii Contents. SECTIONS OnElectric Machines founded onInduction andConvection,— Electric Eeplenisher, Potential-Equalizer, with applications.416—426 OntheReciprocal Electrophorus427—429 XXIV.— AMATHEMATICAL THEORY OFMAGNETISM. Introduction 430—433 Part First.—OnMagnets andthemutual Force between Magnets. Chapter I.—Preliminary Definitions andExplanations. Definition ofaMagnet 434—435 Action oftheEarth onaMagnet sensibly acouple ;Directive tendency, Magnetic Axis, Dip, Polarity 436—446 Distribution ofMagnetisminaMagnet..... 447—451 Chapter II.—OntheLaws ofMagnetic Force, andonthe dis- tribution OFMagnetism inMagnetized Matter. Mutual Action between twothinuniformly andlongitudinally Magnetized Bars 452—453 Strength ofaMagnet, Unit Strength, Magnetic Moment, Inten- sityandDirection ofMagnetization..... 454—462 Chapter HI.—OntheImaginary Magnetic Matter bymeans of WHICH thePolarity ofaMagnetized body mayberepre- sented 463—475 Chapter IV.—^Determination oftheMutual Actions between ANYgiven portions OFMAGNETIZED MaTTER. Explanations 476—478 "Resultant Magnetic Force atanyPoint".... 479—480 The'•Potential" 481—484 Potential atapointPduetoagivenMagnet.... 485—501 OntheExpression ofMutual Action between twoMagnets by means oftheDifferential Coefficients ofaFunction oftheir relative Positions 502—503 Chapter V.—OnSolenoidal andLamellar Distributions of Magnetism. Explanations 504 Definitions andExplanations regarding Magnetic Solenoids . 505 Definitions andExplanations regarding Magnetic Shells . . 506 Solenoidal andLamellar Distributions ofMagnetisms. . 507 Complex Lamellar andComplex Solenoidal Distributions of Magnetism 508—509 Action ofaMagnetic Solenoid andofaComplex Solenoid . .510—511 Potential atanypoint due toaMagnetic Shell— Action of Magnetic Shells 512 Criterion ofaSolenoidal Distribution ofMagnetism. . . 513 Contents. xui SECTIONS Criterion ofaLamellar Distribution ofMagnetism. . . 514 Resultant Force, duetoalamellarly-magnetized Magnet, onany external orinternal point.515—523 Chapter VI.—OnElectromagnets. Introductory Remarks.... .... 524 Investigation oftheAction between twoGalvanic Arcs, orbe- tween aGalvanic ArcandaMagnetic Pole.... 525—530 Unit ofstrength foranElectric Current 531—533 Hypothesis ofMatter flowing 534 Division ofElectromagnets intothree Classes.... 535 Linear Electromagnets536 Superficial Electromagnets....... 537 Solid Electromagnets 638 Analytical Investigation oftheConditions towhich theDistri- bution ofGalvanism inSolidandSuperficial Electromagnets issubject 539—543 Applications 544 Asimilar Synthetic Solution indicated 545 Electromagnets andtheirrespective equivalent Polar Magnets— Rules forDirection 546—550 Remarks andAdditions 551—553 Original Investigation of§517referred toin§518 . . . 554 XXV.-ON THEPOTENTIAL OFACLOSED GALVANIC CIRCUIT OFANYFORM 555—560 XXVI.—Chapter VII.—OntheMechanical Values ofDistribu- tions OFMatter andofMagnets. Mechanical Values ofDistributions ofMatter.... 561—563 Polar Magnets 564—568 Electromagnets 569—572 XXVII.—Chapter VIII.—Hydrokinetic Analogy 573—583 XXVIII.—Chapter IX.—Inverse Problems. Definition—Divided intotwoClasses...... 684 Class I.—Force given foreverj^ point ofspace.... 585—688 Class II.—Force orcomponent offorce given through some portion ofspace>........ 589—601 XXIX.- ONTHEELECTRIC CURRENTS BYWHICH THE PHENOMENA OFTERRESTRIAL MAGNETISM MAYBE PRODUCED Chapter X.—Magnetic Induction.602—603 XXX.— ONTHETHEORY OFMAGNETIC INDUCTION IN CRYSTALLINE ANDNON-CRYSTALLINE SUBSTANCES. xiv Contents. SECTIONS Explanations andDefinitions. Force atanypoint duetoaMagnet. Total magneticforce atapoint. "AField ofmagnetic force." "Alineofmagneticforce." "Auniform field ofmagneticforce." Resultant Distribution ofMagnetism 604—605 Axioms ofMagnetic Force 606 Laws ofMagneticInduction according toPoisson's Theory. .607—609 Conclusions from theseLaws 610—619 Appendix—Quotations from Poisson regarding Magne-Crystallic action—Explanation. Demonstration 620—624 XXXI.— MAGNETIC PERMEABILITY ANDANALOGUES IN ELECTROSTATIC INDUCTION, CONDUCTION OFHEAT ANDFLUID MOTION 625-631 XXXn.— DIAGRAMS OFLINES OFFORCE;TOILLUSTRATE MAGNETIC PERI^IEABILITY 632—633 XXXni— ONTHEFORCES EXPERIENCED BYSMALL SPHERES UNDER MAGNETIC INFLUENCE; ANDON SOME OFTHEPHENOMENA PRESENTED BYDIA- MAGNETIC SUBSTANCES. Attraction ofFerromagnetics 634—642 Repulsion ofDiamagnetics. 643—646 XXXIV.— REMARKS ONTHEFORCES EXPERIENCED BY INDUCTIVELY MAGNETIZED FERROMAGNETIC OR DIAMAGNETIC NON-CRYSTALLINE SUBSTANCES. Faraday's Law ofAttractions andRepulsions.... 647—653 Experimental illustrations ofFaraday's Law.... 654—664 OntheStability ofSmall Inductively magnetized bodies inPosi- tions ofEquilibrium 665 Ontherelations ofFerromagnetic andDiamagnetic Magnetization tothemagnetizing force 666—668 XXXV.— ABSTRACT OFTWOCOMMUNICATIONS— Oncertain Magnetic Curves;with applications toProblems inthe Theories ofHeat, Electricity, andFluid Motion . . , 669 OntheEquilibrium ofelongated Masses ofFerromagnetic Sub- stances inuniform andvaried Fields ofForce.... 669 XXXVL— REMARQUE SSURLESOSCILLATIONS d'aiguilles non cristallisdes defaible pouvoir inductif paramag- n^tique oudiamagn^tique, etsurd'autres ph^nomenes mag- n^tiques produits pardescorps cristaUisds ounoncristaUis^s; from the"Comptes Rendus" oftheFrench Academy, 1854, firsthalf-year 670 XXXVII.— ELE^IENTARY DEMONSTRATION OFPROPOSI- . TIONS INTHETHEORY OFMAGNETIC FORCE . .671-JB73 IContents. xv SECTIONS Examination oftheAction experienced byaninfinitely thin, uniformly andlongitudinally Magnetized Bar,placed inaNon- uniform Field ofForce, with itslength direct along aline of Force 674—688 XXXVni.— COKKESPONDENCE WITHPKOFESSOE TYNDALL. Letter toProfessor Tyndall onthe"Magnetic Medium," andon theeffects ofCompression 689—693 Letter from Professor Tyndall toProfessor W.Thomson on Eeciprocal Molecular Induction 694 Letter from Professor W.Thomson toProfessor TyndaU, onthe Eeciprocal Action ofDiamagnetic Particles.... 695—696 XXXIX.— INDUCTIVE SUSCEPTIBILITY OFAPOLAE MAG- NET 697—699 XL.—GENEEAL PEOBLEM OFMAGNETIC INDUCTION .700—732 XLL—HYDEOKINETIC ANALOGY FOETHEMAGNETIC INFLUENCE OFANIDEAL EXTEEME DIAMAGNETIC. OnForces experienced bySolids immersed inaMoving Liquid.733—740 Extracts fromtwoLetters toProfessor Guthrie.... 741—743 Eeport ofanAddress ontheAttractions andEepulsions dueto Vibration, observed byGuthrie andSchellbach. —Hydrokinetic Analogy forExtreme Diamagnetic 744—750 XLII.— GENEEAL HYDEOEINETIO ANALOGY FOEINDUCED MAGNETISM. Permeability inHydrokinetic Analogy 751—756 Kinetic Energy aMinimum 757—758 Analogy ofForce 759—763 ^. L—ONTHEUNIFOEM MOTION OFHEAT INHOMOGENEOUS SOLID BODIES, AND ITSCONNEXION WITH THE MATHEMATICAL THEOKY OFELECTRICITY.-^ (Art.III.ofcompletelistinMathematical andPhysical Papers,Vol.i.) [From Cambridge Mathematical Journal, Feb. 1842. Eeprinted Philosophical Magazine (1854,first half-year).] [Sincethefollowingarticle waswritten,-fthewriter finds thatmost ofhisideas have beenanticipated byM.Chasles intwoM^moires intheJournal deMathematiques ;the first, invol.III.,ontheDetermination oftheValue ofacertain DefiniteIntegral,andthesecond, invol. v.,onanewMethod ofDeterminingtheAttraction ofanEllipsoidonaPoint with- out it.Inthelatter ofthese Memoires, M.Chasles refers to apaper, byhimself, inthetwenty-fifth Cahier oftheJournal deVEcole Poly technique,inwhich itisprobablethere are still furtheranticipations, thoughthewriter ofthepresentarticle *[Note added June 1854.]—This paperfirstappeared anonymouslyinthe Cambridge Mathematical Journal inFebruary 1842. The text isreprinted without alteration oraddition. Allthefootnotes areofthepresent date (March 1854). Thegeneral conclusions established initshow that thelaws ofdistribution ofelectric ormagnetic force inanycasewhatever must be identical with thelaws ofdistribution ofthelines ofmotion ofheat incertain perfectly defined circumstances. With developments andapphcations con- tained inasubsequent paper (ii.below) ontheElementary Laws ofStatical Electricity [Cambridge andDublin Mathematical Journal, Nov. 1845), they constitute afulltheoryofthecharacteristics oflines offorce, which have been soadmirably investigated experimentally byFaraday, andcomplete the analogy with thetheory oftheconduction ofheat, ofwhich suchterms as "conducting power forlines offorce"[Exp.Res. §§2797—2802)involve the idea. t\_Note added June 1854.]—Thispreliminary notice was written some months later than thetextwhich follows, andwascommunicated tothe editor ofthejournal tobeprefixed tothepaper, which hadbeen inhis hands since themonth ofSeptember 1841. Theideas inwhich theauthor hadascertained hehadbeen anticipated byM.Chasles, were those bywhich hewasledtothedetermination oftheattraction ofanellipsoid giveninthe latter part ofthepaper. Hefound soon afterwards thathewasanticipated bythesame author inanenunciation ofthegeneral theorems regarding attraction;still laterhefound thatbothanenunciation anddemonstration ofthesame general theorems hadbeen given byGauss, whose paper ap- l'^ T.E. 1 2 UniformMotion ofHeat and[i. hasnothad access tosolateavolume ofthelatterjournal. Since, however, most ofhismethods areverydifferent from those ofM.Chasles, which arenearly entirely geometrical,the followingarticle maybenotuninterestingtosome readers :— ] 1.Ifaninfinite homogeneoussolidbesubmitted totheaction ofcertain constant sources ofheat, thestationary temperature atanypointwillvary accordingtoitsposition;andthrough every pointthere willbeasurface, over thewhole extent of which thetemperatureisconstant, which istherefore called an isothermal surface. Inthispaperthecase willbeconsidered inwhich these surfaces arefinite, andconsequentlyclosed. 2.Itisobvious thatthetemperatureofanypointwithout agivenisothermal surface, depends merelyontheformand temperatureofthe surface, being independentoftheactual sources ofheatbywhich thistemperatureisproduced, provided there arenosources without the surface. Thetemperature ofanexternal pointisconsequentlythesame asifallthe sources were distributed over this surface insuch amanner astoproducethegivenconstanttemperature. Hence wemay consider thetemperatureofanypointwithout theisothermal surface, asthesum ofthetemperatures duetocertain constant sources ofheat, distributed overthat surface. peared shortly afterM.Chasles' enunciations; and afterall,hefound that these theorems hadbeen discovered andpublished inthemost complete and general manner, with richapplications tothe theories ofelectricity and magnetism, more than tenyears previously, byGreen !Itwasnot until earlyin1845 thattheauthor, after having inquired for itinvain forseveral years,inconsequence ofanobscure allusion toitinoneofMurphy's papers, wasfortunate enough tomeet with acopy oftheremarkable paper ("An Essay ontheApplication ofMathematical Analysis totheTheories of Electricity andMagnetism," byGeorge Green, Nottingham, 1828)inwhich this great advance inphysical mathematics was firstmade. Itisworth remarking, that, referring toGreen astheoriginatoroftheterm. Murphy givesamistaken definition of"potential."Itappears highly probable that hemay never havehadaccess toGreen's essay atall,andthat this isthe explanationofthefact(ofwhich anyother explanationisscarcely conceiv- able), that inhisTreatise onElectricity (Murphy's Electricity, Cambridge, 1833) hemakes noallusion whatever toGreen's discoveries, and gives a theoiyinnorespect pushed beyond what hadbeen done byPoisson. All thegeneral theorems onattraction which Green andtheother writers referred to,demonstrated byvarious purely mathematical processes,areseen as axiomatic truths inapproaching thesubject bytheway laid Aovra inthe paperwhich isnowrepubhshed. Theanalog} with theconduction ofheat onwhich these views arefounded, hasnot, sofarastheauthor isaware, been noticed byanyother writer. I.]Mathematical Theory ofElectricity. 3 3.Tofindthetemperature produced byasingle source of heat, letrbethedistance ofanypoint fromit,and letvbe thetemperatureatthatpoint. Then, since thetemperatureis thesame forallpointssituated atthesame distance from the source,itisreadily shown that visdetermined bytheequation -r^^=A.ar Dividingbothmembers byr^,andintegrating, wehave A^ r Now letussupposethatthenaturaltemperatureofthesolid, orthetemperatureataninfinite distance from thesource, is zero :thenweshallhave (7=0, andconsequently v=- (1).r^' 4.Hence thatpartofthetemperatureofapointwithout an isothermal surface which isduetothesources ofheat situated onanyelement, dw^,ofthesurface, is——~ ,wherer^isthe r^ distance from theelement tothatpoint, andp^aquantity measuringtheintensityofthesources ofheat atdifferent partsofthesurface. Hence, thesupposition beingstillmade that there arenosources ofheatwithout thesurface, ifvbe thetemperatureattheexternalpoint,wehave "//'* « theintegrals beingextended over thewhole surface. The quantity p^must bedetermined bythecondition ^= ^1...; ;(3), foranypointinthesurface, v^beingagivenconstant tem- perature. 5.Letusnow consider what willbethetemperatureofa point within thesurface, supposingallthesources ofheatby which thesurface isretained atthetemperature v^tobedistri- buted over it.Since there arenosources intheinterior ofthe surface, itfollows that asmuch heatmust flow outfrom the interior across thesurface asflows intotheinterior, from the sources ofheat atthesurface. Hence thetotal flux ofheat from theoriginalsurface toanadjacentisothermal surface in 1—2 4UniformMotionofHeatand[i. theinterior isnothing. Hence alsothefluxofheatfrom this latter surface toanadjacentisothermal surface initsinterior must benothing ;and soonthroughthewhole ofthebody within theoriginalsurface. Hence thetemperatureinthe interior isconstant, andequaltov^,andtherefore, forpointsat thesurface, orwithinit,wehave II'(4). Now,ifwesupposethesurface tobecovered withanattrac- tivemedium, whosedensityatdifferentpointsisproportional topj,~l~n— ^'^^^^^®^^®attraction, inthedirection of theaxis ofa^,onapoint whoserectangularco-ordinates are w,y,z.Hence itfollows that theattraction ofthismedium onapointwithin thesurface isnothing,andconsequently p^ isproportionaltotheintensityofelectricityinastate ofequi- librium onthesurface, theattraction ofelectricityinastate ofequilibrium being nothingonaninteriorpoint. Since, at the surface, thevalue of\\~—-isconstant, and since, on that account,itsvalue within thesurface isconstant also, it follows, that iftheattractive force onapointatthesurface isperpendiculartothesurface, theattraction onapointwithin the surface isnothing. Hence the sole condition ofequi- librium ofelectricity,distributed over thesurface ofabody, is,that itmust besodistributed thattheattraction onapoint atthesurface, oppositely electrified, maybeperpendicularto thesurface. 6.Since, atanyoftheisothermal surfaces, visconstant, it dj\) follows that—-T- ,where nisthelengthofacurve which cuts allthesurfacesperpendicularly, measured from afixedpoint tothepoint attracted, isthetotal attraction onthelatterpoint; andthat this attraction isinatangenttothecurve?i,orin anormal totheisothermal surfacepassing throughthepoint. Forthesame reason also, ifp^representafluxofheat,andnot dfoanelectricalintensity,—-pwillbethetotal fluxofheat atthe variableextremityof?i,andthedirection ofthis flux willbe I.]Mathematical Theory ofElectricity.5 along n,orperpendiculartotheisothermal surface. Hence, if asurface inaninfinite solidberetained ataconstanttempera- ture,and ifaconducting body, bounded byasimilar surface, beelectrified, theflux ofheat, atanypoint,inthe first case, willbeproportionaltothe attraction onanelectricalpoint, similarly situated, inthesecond; andthedirection oftheflux willcorrespondtothat oftheattraction. (Iv dv 7.Let—-rr^betheexternal value of—7-attheoriginal dn^an surface, ortheattraction onapointwithoutit,andindefinitely near it.Now thisattraction iscomposedoftwoparts ;onethe attraction oftheadjacentelement ofthesurface; andtheother theattraction ofalltherestofthesurface. Hence, callingthe former ofthese a,andthelatter6,wehave Now, since theadjacentelement ofthesurface maybetaken asinfinitely larger,initslinear dimensions, than thedistance from itofthepoint attracted,itsattraction willbethesame as that ofaninfiniteplane,ofthedensity p^.Hence aisinde- pendentofthedistance ofthepointfrom thesurface, and is equalto'^.irp^.Hence Now, forapointwithin thesurface, theattraction oftheadja- centelement willbethesame, butinacontrary direction, and theattraction oftherest ofthesurface willbethesame, and inthesame direction. Hence theattraction onapointwithin the surface, andindefinitelynearit,is— '^nrp^-fh;andconse- quently,since this isequaltonothing, wemust have h=27r/0j, andtherefore dv, , ,^. -4=*"P'^^> Hencep^isequaltothetotal fluxofheat, atanypointofthe surface, dividedby47r. 8.Italsofollows that iftheattraction ofmatterspreadover thesurface benothing onaninteriorpoint,theattraction on anexteriorpoint, indefinitely nearthesurface,isperpendicular tothesurface, andequaltothedensityofthematter atthe partofthesurfaceadjacenttothatpoint, multiplied by47r. 6UniformMotionofHeatand[i. 9.IfVbethetemperatureatanyisothermal surface, andpthe intensityofthesources atanypointofthis surface, which would benecessarytosustain thetemperature v,wehave,by(5), dv . whichequation holds, whatever bethemanner inwhich the actual sources ofheat arearranged,whether overanisothermal surface ornot;andthetemperature producedinanexternal pointbytheformer sources,isthesame asthatproduced by thelatter. Also, thetotal flux ofheat across theisothermal surface, whose temperatureisv,isequaltothetotal flux of heat from theactual sources. From this,andfromwhat has beenproved above,itfollows that ifasurface bedescribed round aconductingornon-conductingelectrifiedbody,sothat theattraction onpointssituated onthissurface maybeevery- whereperpendiculartoit,and iftheelectricity beremoved from theoriginal body, and distributed inequilibriumover this surface,itsintensityatanypointwillbeequaltothe attraction oftheoriginal body onthatpoint,divided by47r, and itsattraction onanypoint without itwillbeequaltothe attraction oftheoriginal bodyonthesamepoint.* IfwecallEthetotalexpenditureofheat, orthewhole flux acrossanyisothermal surface, wehave, obviously, E-Ik'-: 10.Now thisquantityshould beequaltothesum ofthe expendituresofheatfrom allthesources. Toverify this,we must, inthe firstplace,findtheexpenditureofasinglesource. Now thetemperature produced byasinglesourceis,by(1), v=—yandhence theexpenditureisobviously equalto *[Note added June 1854.—After having established this remarkable theorem inthemanner shown inthetext, theauthor attempted toproveit bydirect integration, butonlysucceeded indoing soupwards ofayear later, when heobtained thedemonstration publishedinapaper, "Propositions in theTheoryofAttraction" {Camb. Math. Jour. Nov.1842), which appeared almost contemporaneously with apaper byM.Sturm inLiouvUle's Journal, containing thesame demonstration;exactly thesame demonstration, asthe author afterwards(in1845) found, hadbeen given fourteen years earlier by Green.] I.]Mathematical Theory ofElectricity. 7 —-^x 47rrl orto^ttA. If^=p.dcoJ^, thisbecomes^iirp.dto^.dr Hence thetotalexpenditureisJj4!7rp^d(o^,or-M^d(o^,which agreeswith theexpressionfound above. Thefollowingisanexampleoftheapplicationofthese principles:— UniformMotionofHeat inanEllipsoid. 11.Theprinciplesestablished above afford aneasymethod ofdeterminingtheisothermal surfaces, andthecorresponding temperatures,inthecage inwhich theoriginalisothermal sur- face isanellipsoid. The firststepistofindp^,which isproportionaltothe quantityofmatter atanypointinthesurface ofanellipsoid, when thematter issodistributed thattheattraction onapoint within theellipsoidisnothing. Now theattraction ofashell, bounded bytwoconcentric similarellipsoids, onapointwithin it,isnothing.Iftheshellbeinfinitely thin, itsattraction will bethesame asthat ofmatter distributed over thesurface of oneoftheellipsoidsinsuch amanner that thequantityona given infinitelysmall area atanypointisproportionaltothe thickness oftheshell atthesamepoint.Leta^,6j,c^bethe semi-axes ofoneoftheellipsoids, a^+Sa^, 6j+86^, c^-fBc^those oftheother. Letalsop^betheperpendicular from thecentre tothetangent planeatanypoint onthe firstellipsoid, and Pi+^1^^®perpendicularfrom thecentre tothetangent plane atapoint similarlysituated onthesecond. ThenBp^^isthe thickness ofthe shell, since, thetwoellipsoids being similar, thetangent planesatthepoints similarlysituated ontheir surfaces areparallel. Also, onaccount oftheirsimilarity, —i=-^=—^=-^^,andconsequently thethickness ofthesheU a,K c^ p,^ J isproportionaltop^.Hence wehave,by(5), 1dv^,/\ -|^*f.='''=*'-^'^''^' wherek^isaconstant, tobedetermined bythecondition v=v^, atthesurface oftheellipsoid. 12.Tofindtheequationoftheisothermal surface atwhich thetemperatureisv^+dv^,let-dv^=C,in(a).Thenwehave 8UniformMotion ofHeatand[i. G f(!iPidn^=7—,oxp^dn^— O^,where6^isaninfinitelysmall con- stantquantity ;andtherequired equationwillbetheequation ofthesurface traced bytheextremityofthelinedn^^drawn externally perpendiculartotheellipsoid. Letx,y\/bethe co-ordinates ofanypointinthat surface, andx,y,zthose of thecorresponding pointintheellipsoid. Then, calling a^,/5i,7i theangles which anormal totheellipsoidatthepointwhose co-ordinates are x,y^zmakes with these co-ordinates, and supposingtheaxes ofx,y,ztocoincide with theaxes ofthe ellipsoid, 2^^, 26^, 20^,respectively, wehave —^dn. X—x=an,cosa.= ,.> o o.=—opMn. =—00/,, ovx'—x=—2^i>since^^isinfinitely small, andtherefore also x—x; whence x= Inasimilar manner weshould find y',_z x^ v^ z^But— 2+f^H—2=Ij^^^hencewehave ^1 ^1 ^1 x'y'^z'^ a^+2(9,^ h^^2(9,^ c/+2(9,~ ' fortheequation totheisothermal surface whose temperature isv,+6?!;,,andwhich istherefore anellipsoiddescribed from thesame fociastheoriginal isothermalellipsoid.Inexactly thesamemanner itmight beshown thattheisothermal surface whosetemperatureisv,+dv^+dv^,isanellipsoid havingthe same foci astheellipsoid whosetemperatureisv,-f-dv^,and consequently,astheoriginal ellipsoidalso.Bycontinuingthis I.]Mathematical Theory ofElectricity.9 processitmaybeprovedthat alltheisothermal surfaces are ellipsoids, havingthesame fociastheoriginalone. 13.From theform oftheequation found above fortheiso- thermalellipsoidwhosetemperatureisv^+dv^^itfollows that6^ orp^drij^is=a^da^,whereda^istheincrement ofa^,correspond- ingtotheincrementdn^ofn^. Hence,ifabeoneofthe semi-axes ofanellipsoid,a-\-dathecorrespondingsemi-axis ofanotherellipsoid havingthesamefoci,dnthethickness at anypointofthe shellbounded bythetwoellipsoids,andp theperpendicular from thecentre totheplane touchingeither ellipsoidatthesamepoint,wehave dn_a,,V da^p^^* 14.Allthatremains tobedone istofindthetemperatureat thesurface ofanygiven ellipsoid, havingthesame fociasthe original ellipsoid. Forthispurpose,letusfirst findthevalue of—-7-atanypointinthesurface oftheisothermalellipsoid whose semi-axes are a,h,c.Nowwehave, from(a), dv .J where hisconstant foranypointinthesurface oftheisothermal ellipsoid under consideration, anddetermined bythecondition that thewhole flux ofheat across this surface must beequal tothewhole flux across thesurface oftheoriginal ellipsoid. Now the first ofthesequantitiesisequalto^irkjjpdco^ (dco^ beinganelement ofthesurface),orto47rk—fJSpdco^,since —=— .ButJJSpdco^isequaltothevolume ofashell bounded bytwosimilarellipsoids, whose semi-axes are a,h,c, anda-f8(2,6+36,c+8c,and isthereforereadily shown tobe equalto47r—ahc.Hence 47r^fJSpdco^,or4i7rkJJpdco^,is equalto4^Vkahc. Inasimilar manner wehave,fortheflux ofheat across theoriginalisothermal surface, 4iVk^aJ)jC^,and therefore 4^Vkabc=4iVk^aJ)^c^, which gives k=k,-V-^.° *ahc 10 UniformMotionofHeatand[i. Hence wehave dn~^'"'.^ ahc^ ^'^• 15.Thevalue ofvmaybefound byintegratingthisequation. Toeffect this, since a,6,carethesemi-axes ofanelHpsoid passing throughthevariableextremityofw,andhavingthe same fociastheoriginal ellipsoid, whose axes area^,6^,c^,we have a'-a,'=5'-V=c'-c^\ whichgives6^=a^—/^1 ^"=0"-g'\{d), where f^a,'-h,\ g'=a,'-c,' ] Hence(c)becomes ^^^4_^^A^iP dn"^^ a^/{a'-f) ^{a'-g')' Now, by{b),dn= ,andhence ^^__4_jNaj)^c,da Integrating this,wehave da 16.Thetwoconstants, k^and C,must bedetermined bythe conditions v=v^when a=a^yand v==-0 when a=oo;the latter ofwhich must befulfilled, inorder thattheexpression found forvmaybeequalto 1j-J^^-^. 17.Toreduce theexpressionforvtoanelliptic function, let usassume a=fcoseG(j> 1... dj=/cosec (/)jJ-^^' which wemaydowithproprietyif/bethegreaterofthetwo quantities /and^,since aisalways greaterthan either of them, asweseefrom(d).Onthisassumption, equation (e) becomes / 7o\^(l~c'sm'(/)) /''^ where.c'=^(g).v=^-Mflfi,c,j+C (6). I.]Mathematical Theory ofElectricity.11 18.Determiningfrom thisthevalues ofCandk^bythe conditions mentioned above, wefind(7=0, and ^^- 4^7rafi,c,F,4^^""^^ hence theexpressionforvbecomes --.ff.«• 19.The results which havebeen obtained maybestated as follows :— If,inaninfinite solid, thesurface ofanellipsoid beretained ataconstanttemperature,thetemperatureofanypointinthe solid willbethesame asthat ofanyotherpointinthesurface ofanellipsoiddescribed fromthesame foci,andpassing through thatpoint ;andtheflux ofheat atanypointinthesurface of thisellipsoidwillbeproportionaltotheperpendicularfrom the centre toaplane touchingitatthepoint, andinversely pro- portionaltothevolume oftheellipsoid. 20.This case oftheuniform motion ofheatwas firstsolved byLam^, inhisMemoire onIsothermal Surfaces, inLiouville's Journal deMathematiqiies,vol. ii.p.147,byshowingthat a series ofisothermal surfaces ofthesecond order willsatisfythe equationd^v d^vd%_ d^''^'df'^d?~' provided theyare alldescribed from thesame foci. Thevalue which hefinds forvagreeswith(e),andhefinds, fortheflux ofheat atanypoint,theexpression KA or,accordingtothenotation which wehaveemployed, 4!7rk^aJ)^c^ ^J{a"-v')W(a'-p')' where visthegreaterreal semi-axis ofthehyperboloidof onesheet, andptherealsemi-axis ofthehyperboloidoftwo sheets, described from thesame fociastheoriginal ellipsoid, andpassing throughthepointconsidered. Hence a^,v^,p^are thethree roots oftheequation x^ y^ z^ uu-f u-g" 12UniformMotion ofBeatand[i. or Hence a'v'p^=fYx\ andaV+a'p'+vY=/y+(/^+g")x'+^'^3/^+/V. Therefore, (a^- i.^){a'-p')=a'- a'v'-aY-vY+^'*''''^' a' a? .2=«*-{/y+(/'+/)^^+/2/'+/V}+2/y^. =«4_ (c^2_j2^(^2_ ^2)_^(2a^-6^- o^)x''-{a'-0^2/' -{a'-¥)z'+2(a'-b'')(a'-c') -, =a*-{a'-¥){a^- c')- {b'+c')x'-(a'-c')y' =a*-{a'-6^(a'- c')- {b'+c'')cc^-(a'+ c^)^^'^- (a^+5')^+26V =a^6^+aV+6V- {(6^+c^)^+{a'+&)f+(a^+6^)/}; which isreadily shown, bysubstitutingforg^W+aV+ 6'^c''^ itsequal (o^W+aV+&V)(— 2+^+-2)>tobeequalto—2~• Hence theexpressionfor—-7- ,given above, becomes dv_.Jafi^Cj^ dn~ ^abc^' whichagreeswith(c). AttractionofaHomogeneous Ellipsoid onaPoint within or without it. 21.If,in(c),weput\=— ^ ,thevalue of—-7-atanypoint (Xj an willbetheattraction onthatpointofashellbounded bytwo similar concentricellipsoids, whose semi-axes are «!,ay(1- e^)ya^\J(l— e'^), anddj+da^, (a^+da^ s/{l— e^),{a^+da^ \/(l— e'^), where a^-6'=a^- b^'=ay) and a^_c'^=a/-c;=a,V^J^^^' thedensityoftheshellbeing unity. Now thisattraction isin I.]Mathematical Theory ofElectricity. 13 anormal drawn throughthepointattracted tothesurface of theellipsoid,whose semi-axes are a,h,c.Ifwecalla,/8,7 theangleswhich thisnormal makes with theco-ordinates X,y,zofthepoint attracted, w^ehave X a^ px andsimilarly,cosyS=-^f,cos7=^. Hence, calling dA,dB,dGthecomponentsoftheattraction paralleltotheaxes ofco-ordinates, wehave, from(c), dA=^iirx—i^p^da, dB= ^'rry^^fda,\(2). 22.Theintegralsoftheseexpressions,between thelimits ttj=andttj= ttj',arethecomponentsoftheattraction ofan ellipsoidwhose semi-axes area/, 6/,c/,ora/,a/^(1— e^)> ali^iX—e'"^),onthepoint {x,y,z).Now, by (1),wemay expresseach ofthequantities 6,c,6^,c^,interms ofaanda^, andtheequation enables ustoexpresseither ofthequantities a,a^interms of theother. Thesimplest way, however, tointegrate equations (2),willbetoexpresseach interms ofathirdquantity, w=^ (4).a^' Eliminatingafrom(3),bymeans ofthisquantity, wehave du Hencea,da,= ^ux^+^^^^,+^r'Z:^^ = (^4+ fs+ ^4)a^'if'du=a^p-'u-'du. Also, from(4),wehave a=— ;fromwhich wefind,by(1), 14UniformMotionofHeat.[i. 6=?!V(l-eV),c=-^V(l-eV).By(1)also, h,=a^V(l" ^')> c^=a^sJQi— e'2).Makingthese substitutions in(2),and inte- grating, wehave, callinga'thevalue ofa,whena^=a/, H ^=47r^VCl- e')^/(l-e')(''''''^'' JV(l-eV)V(l-e'V) ^0(1-eV)^ (1-eV)^ a/ =^zV(l-e')V(l-e'Of"' ^^Jo 1—tfu'(5). (l_eV)*(l-eV)^. 23.Ifthepointattracted bewithin theellipsoid,theattraction ofallthesimilar concentric shells without thepointwillbe nothing ;andhence thesuperiorlimit ofuwillbethevalue of —atthesurface ofanellipsoid,similar tothegiven one,and passing throughthepointattracted. Now, inthis case, a^=a,since aisoneofthesemi-axes of anellipsoid passing throughthepoint attracted, andhaving thesame fociasanotherellipsoid (passing throughthesame point)whosecorresponding semi-axis isa^.Hence,foran interiorpoint,wehave A=^TTxV(l- e')V(l-e')1^-^^'^"^ V(l-6V)V(l-eV) •^^^^^'o(i_eV)^(i-eV)^ C=47r^V(l~ e^)V(l- e'^)f--^^.(6). 24.These aretheknownexpressionsfortheattraction ofan ellipsoidonapointwithin it.Equations (5)agree with the expressions givenintheSupplementtoLiv. v.ofPontdcoulant's TheorieAnalytique duSystemeduMonde, wheretheyarefound bydirectintegration, byamethod discoveredbyPoisson. Theymayalsobereadily deduced fromequations (6)byIvory's Theorem. Or,ontheother hand, byacomparison ofthem, after reducingthelimits oftheintegralstoand1,bysubsti- tuting -^v foru,withequation (6),Ivory's Theorem maybe readilydemonstrated. II.—ONTHEMATHEMATICAL THEORY OFELECTRICITY IN EQUILIBRIUM. (Art.xvin. ofcompletelistinMathematical andPhysical Papers, Vol.i.) I.—ONTHEELEMENTARY LAWS OFSTATICAL ELECTRICITY.* [From Cambridge andDublin Mathematical Journal, Nov. 1845. Reprinted Philosophical Magazine, 1854, second half-year, with additional Notes ofdateMarch 1854.] 25.Theelementarylawswhichregulatethedistribution of electricityonconductingbodies have been determined by means ofdirectexperiments, byCoulomb, and intheform hehasgiven them, which isindependentofanyhypothesis,-^ theyhavelongbeen considered asrigorouslyestablished. The problemofthedistribution ofelectricityinequilibriumona conductor ofanyformwasthusbroughtwithin theprovince ofmathematicalanalysis ;butthesolution, even inthesimplest cases, presentedsomuchdifficultythatCoulomb, afterhaving investigateditexperimentallyforbodies ofvarious forms, could onlycomparehismeasurements with theresults ofhistheory byveryrudeprocessesofapproximation. Without, however, giving rigoroussolutions inparticular cases, heexamined the general problemwithgreat care,and leftnothing indefinite in theconditions tobesatisfied, sothat itwasentirely byana- lyticaldifficulties thathewasstopped.Asanexampleofthe *This paperisatranslation(withconsiderableadditions)ofonewhich appearedinLiouville's Journal deMathematiques, 1845, p.209. tCoulomb hasexpressed histheory insuch amanner that itcanonlybe attacked intheway ofproving hisexperimental results tobeinaccurate. This isshown inthefollowing remarkable passageinhissixth memoir, which follows ashort discussion ofsome ofthephysical ideas then com- monly held with reference toelectricity."J^epreviens pour mettre latheorie quivasuivre aVabri detoute dispute systematique, quedans lasupposition des deux Jiuides electriques, jen''ai d'autre intention quedepresenter avec lemoins d'elemenspossible,lesresultats decalcul etdeVexperience,etnondHndiquer lesviritables causes deVelectricite. Jerenverrai, alaJindemon travail sur Vilectriciti, Vexamen desprincipaux systemes auxquelslesphenomenes electriques outdonne naissance.''—Histoire deI'Acad^mie, 1788, p.673. 16 OntheMathematical Theory ofElectricity, [n. success ofhistheoretical investigations, wemayrefer tothe well-known demonstration ofthetheorem(usuallyattributed toLaplace)relative totherepulsionexercised byacharged conductor onapointnear itssurface.* Thememoirs ofPoisson, onthemathematicaltheory,con- taintheanalyticaldetermination ofthedistribution ofelec- tricityontwoconducting spheres placednear one another, thesolution beingworked outinnumbers inthecase oftwo equal spheresincontact, which hadbeeninvestigated experi- mentally byCoulomb(aswell asinanother case,notexamined byCoulomb, which isgivenasaspecimenofthenumerical results thatmaybededuced from theformulae). The calcu- lated ratios oftheintensities atdifferentpointsofthesurface heistherefore enabled tocomparewith Coulomb's measure- ments, andhefinds anagreementwhich isquiteasclose as could beexpected, when weconsider theexcessivelydifficult andprecariousnature ofquantitative experimentsinelectricity: butthemost remarkable confirmation ofthetheory from these researches istheentireagreementoftheprincipal features, *Thistheorem may bestated asfollows:—Let4beaclosed surface of anyform, and letmatter, attracting inversely asthesquare ofthedistance, be sodistributed over itthat theresultant attraction onaninterior point is nothing: theresultant attraction onanexterior point, indefinitely nearany partofthesurface, willbeperpendiculartothesurface and equal to4irp, ifpwbethequantityofmatter onanelement wofthesurface intheneigh- bourhood ofthepoint. Coulomb's demonstration ofthistheorem may be found inapreceding paperintheMathematical Journal, Vol. iii.p.74(above, I.7).Hegivesithimself, inhissixthmemoir onElectricity {Histoire de VAcademie, 1788, p.677), inconnexion withaninvestigation ofthetheory oftheproof planeinwhich, byanerror that isreadily rectified, hearrives at theresult thatasmall insulated conducting disc,putincontact withanelec- trified conductor atanypoint, andthenremoved, carries with itasmuch elec- tricity asliesonanelement oftheconductor atthatpoint equal inarea tothe twofaces ofthedisc; thequantity actually removed being only half ofthis. This result, however, does not atallaffect theexperimental usewhich he makes oftheproof plane, which ismerely tofindtheratios oftheintensities atdifferent pointsofacharged conductor. Asthecomplete theory ofthis valuable instrum^t hasnot, sofarasIamaware, been given inanyEnghsh work, Iannex thefollowing remarkably clear account ofit,which isex- tracted from Pouniet's Traite dePhysique:—"Quand leplan d'epreuve est tangent aune surface,ilseconfond avec I'^lementqu'il touche,ilprend en quelquesorte saplace relativement h,I'electricitd, ouplutotildevient lui- m^me r61^ment surlequel lafluide serepand ;ainsi, quand onretire ce plan, onfait lameme chose quesiTon avait ddcoupi? sur lasurface im Element dememe 6paisseuretdememe 6tendue que lui, etqu'on Vett enlevd pourleporter dans labalance sans qu'il perdit rien de 1'Electricity qui le '"eElementary LawsofStaticalElectricity. 17 eninsomevery singular phenomena,oftheexperimental results with thetheoretical deductions. Foracompleteac- count oftheexperiments wemust refer toCoulomb's fifth memoir (Histoire deVAcademie, 1787), and forthemathe- maticalinvestigationstothe first andsecond memoirs of Poisson (Memoir esdeVInstitut, 1811),ortothe treatise on ElectricityintheEncyclopcedia Metropolitana, where thesub- stance ofPoisson's firstmemoir isgiven. Themathematicaltheoryreceived byfarthemostcomplete developmentwhich ithashitherto obtained, inGreen'sEssay ontheApplication ofMathematicalAnalysistotheTheories ofElectricity andMagnetism*inwhich aseries ofgeneral theorems were demonstrated, andmany interesting applications made toactualproblems, "f* Oflateyearssomedistinguished experimentalists havebegun todoubt thetruth ofthelaws establishedbyCoulomb, and havemade extensive researches with aview todiscover the laws ofcertain phenomena whichtheyconsideredincompatible with histheory. Themost remarkable works ofthiskind couvre; une foissepar6 delasurface, cet^l^ment n'aurait plusdans sesdif- f^rents points qu'une (^paisseur ^lectrique moitid moindre, puisque lafluide devrait serdpandre pour encouvrir lesdeux faces. Ceprincipe posd,I'ex- p^rience n'exige plusquedeI'habitude etdeladext^rit^: aprfes avoir touchd unpoint delasurface avec leplan d'^preuve, onI'apporte dans labalance, oii ilpartage son Electricity avec ledisque deI'aiguille quiluiestEgale, et Tonobserve laforce detorsion aunedistance connue. Onr^p^te lamdme experience entouchant unautre point,etlerapport desforces detorsion est lerapport desrepulsions dlectriques ;onenprendlaracine carr^e pour avoir lerapport desdpaisseurs. Ainsi leg^nie deCoulomb adonnE enmeme temps auxmath^maticiens laloifondamentale suivant laquelle lamatiere dleetrique s'attire etserepousse ;etauxphysiciens unebalance nouvelle, etdesprincipes d'expErience aumoyen desquelsilspeuvent enquelque sorte sender I'Epaisseur der^lectricitd surtous lescorps,etdeterminer lespressions qu'elle exerce sur lesobstacles quiI'arr^tent." Tothisexplanationitshould beadded that,when theproof planeisstill verynear thebody towhich ithasbeen applied, theeffect ofmutual influence issuch astomake theintensity beinsensible atevery point ofthediscon thesidenext theconductor, andateach point oftheconductor which isunder thedisc. Itisonlywhen thedisc isremoved toaconsiderable distance that theelectricity spreadsitself symmetrically onitstwo faces, and that the intensity atthepoint oftheconductor towhich itwas applied, recovers its original value. Itwastheomission ofthisconsideration thatcaused Coulomb tofallintotheerror alluded toabove. *Nottingham, 1828. tThismemoir ofGreen's hasbeen unfortunately verylittleknown, either inthiscountry orontheContinent. Some oftheprincipal theorems init T.E. 2 18 OntheMathematical Theory ofElectricity. [ii. havebeenundertaken independently byMrSnow Harris and MrFaraday,and intheir memoirs, publishedinthePhilo- sophical Transactions, wefind detailed accounts oftheir re- searches. Alltheexperiments, however, which they have made, havingdirect reference tothedistribution ofelectricity inequilibrium, are, Ithink, infullaccordance with thelaws ofCoulomb, andmust therefore, instead ofobjectionstohis theory,beconsidered asconfirmingit.As,however, many have believed Coulomb's theorytobeoverturned bythese investigations,and asothers have atleastbeen ledtoentertain doubts astoitscertaintyoraccuracy,thefollowing attempt toexplaintheapparentdifficulties ismade thesubjectofthe first ofaseries ofpapersinwhich variouspartsofthemathe- matical theoryofelectricity, andcorresponding problemsin thetheories ofmagnetismandheat, willbeconsidered. 26.Wemaycommence byexamining someexperimental resultspublishedinMrHarris's firstmemoir OntheElemen- taryLawsofElectricity.^Afterdescribingtheinstruments employedinhisresearches, MrHarris*givesthe details of someexperimentswith reference totheattraction exercised byaninsulated electrified bodyonanuninsulated conductor placedinitsneighbourhood.The first result which hean- havebeen re-discovered within thelastfewyears, andpublished inthefollowing works :— Gomptes Eendus forFeb. 11th, 1839, where part oftheseries oftheorems is announced without demonstration, byChasles. Gauss's memoir on"General Theorems relating toAttractive andEe- pulsive Forces, varying inversely asthesquare ofthedistance," inthe Besultate ausdenBeobachtungen desmagnetischen Vei-eins imJahre 1839, Leipsic, 1840. (Translationsofthispaper have been pubHshed inTaylor's Scientific Memoirs forApril 1842, andintheNumbers ofLiouville's Journal forJulyandAugust 1842.) Mathematical Journal, vol.iii,,Feb. 1842, inapaper"OntheUniform Motion ofHeat, etc."(i.above). Additions totheConnaissance desTerns for1845 (published June1842), where Chasles suppHes demonstrations ofthetheorems which hehadpreviously announced. Ishould addthat itwasnot tillthebeginning ofthepresent year (1845) that Isucceeded inmeeting with Green's Essay. The allusion made tohis name with reference totheword "potential" {Mathematical Journal, vol. iii. p.190), wastaken from amemoir ofMurphy'-s,"Ondefinite Integrals with Physical Apphcations," intheCambridge Transactions, where amistaken definition ofthatterm, asusedbyGreen,isgiven.*Philosophical Transactions, 1834. iElementary LawsofStaticalElectricity. 19 nounces isthat,when other circumstances remain thesame, theattraction varies asthesquareofthequantityofelectricity withwhich theinsulated bodyischarged.Itisreadily seen, aswas firstremarked byDrWhewell inhisReport onthe TheoriesofElectricity, etc.,^ that this isarigorous deduction from themathematicaltheory, followingfrom thefactthatthe quantityofelectricityinduced upontheuninsulatedbodyis proportionaltothechargeonthe electrifiedbodybywhich it isattracted. 27.Theremainingresults have reference tothe force of attraction atdifferent distances, andwith bodies ofdifferent formsopposed. Asthese aregenerally very irregular (suchas " planecircular areas backed bysmall cones"),weshould not, accordingtoCoulomb'stheory, expect anyvery simple laws, such asMrHarris discovers, toberigorouslytrue. Accord- ingly, though theyareannounced byhimwithout restriction, wemust examine whether theexperiments from whichthey havebeendeduced areofasufficiently comprehensive character tolead toanygeneralconclusions withrespecttoelectrical action. Now, inthe firstplace, wefindthat inallofthem the attraction is"independentoftheform oftheunopposed parts" ofthebodies, which willbethecaseonlywhen theintensity oftheinducedelectricity ontheunopposed partsoftheun- insulated bodyisinsensible. Accordingtothemathematical theory, andaccordingtoMrFaraday'sresearches"oninduction incurved lines," which willbereferred tobelow, theintensity neverabsolutelyvanishes atany pointoftheuninsulated body:but "itisreadilyseen that inthecase ofMrHarris's experiments,itwillbesoslight ontheunopposed portions that itcould notbeperceivedwithoutexperimentsofavery refined nature, such asmightbemadebytheproof planeof Coulomb, which isinfact, with aslight modification, the instrument employed byMrFaradayintheinvestigation. Now tothedegreeofapproximationtowhich theintensity on theunopposed partsmay beneglected,thelaws observed by MrHarris when theopposedsurfaces areplanemaybereadily deduced from themathematicaltheory. Thus letvbethe *British Association Reportfor1837. 2—2 20 OntheMathematical Theory ofElectricity. [ii. potentialintheinterior ofthecharged body,A;aquantity which willdepend solely onthestate oftheinteriorcoating ofthebatterywith which inMrHarris'sexperiments Ais connected, and willtherefore besensiblyconstant fordifferent' positionsofArelative totheuninsulatedopposed body,B. Letabethedistance between theplane opposedfaces ofAand B,and letSbethearea oftheopposed partsofthese faces, which will ingeneralbethearea ofthesmaller,iftheybe unequal. When thedistance aissosmall thatwemayen- tirely neglecttheintensityonalltheunopposed partsofthe bodies,itisreadily shown from themathematicaltheorythat (sincethedifference ofthepotentialsatthesurfaces ofAand Bisv)theintensityoftheelectricity produced byinduction at anypointoftheportionofthesurface ofBwhich isopposed V toA,is7— .Hence theattraction onanysmall element «, oftheportion 8ofthesurface ofB,willbeinadirection perpendiculartotheplaneandequalto27r(7— ].*Hence thewhole attraction onBis 87ra'* This formulaexpressesallthelaws stated byMrHarris asresults ofhisexperimentsinthecasewhen theopposed surfaces areplane. 28.When theopposedsurfaces arecurved, forinstance when AandBareequal spheres, wecanmake noapproximation analogoustothatwhich hasledustososimple anexpression inthecase ofopposed planes;andwefindaccordinglythat nosuch simplelaw fortheattraction inthis casehasbeen announced byMrHarris. Hehas,however, found that itis expressedwith tolerable accuracy bytheformula c{c-2ay where cisthedistance between thecentres ofthespheres, atheradius ofeach, kaconstant, which willdepend onaand onthechargeofthebattery withwhichAisincommunica- *See VII.below. I II.] Elmnentary LawsofStaticalElectricity.21 tion. Though, however, thisformula may giveresults which doDot differverymuch from observation within alimited rangeofdistances,itcannot, accordingtoanytheory, becon- sidered asexpressingthephysicallawofthephenomenon. For,accordingtoit,when theballs arevery distant,Fulti- matelyvaries as-^.Now itisclear thatthelawofforce mustultimately become theinverse cube ofthedistance, since thequantityofelectricityinduced uponBwillbeultimately intheinverse ratio ofthedistance, andtheattraction between theballs astheproductofthequantitiesofelectricity directly, and asthesquareofthedistanceinversely, andhence the formulagiven byMrHarris cannotexpressthelaw offorce when theballs areverydistant. Intheexperiments bywhich hisformula istested, theforce ofattraction ismeasuredby means ofanordinarybalance andweights:theonlycom- parisonofresults which hepubHshesistranscribed inthe followingtable :— Diat ofCentres.Measured Force inGrains.Values of15ci(ci-2) c(c-2) c,=2-815 8-25+ 4-6+ 3-5-15 8-28 4-62 3-45 29.From thistableweseethattheformula isverified inthree cases totheextent ofaccuracyoftheexperiments. Comparisons extended toamuch widerrangeofdistances would berequired toestablishit,and itwould benecessarytotakeprecautions toprevent theexperimentalresults frombeinginfluenced by disturbingcauses. Intheexperiments made byMrHarris, wefindthatnoprecautions havebeen taken toavoid thedis- turbinginfluence ofextraneous conductors, which, according tothedescriptions anddrawings hegivesofhisinstruments, seem toexistvery abundantlyintheneighbourhoodofthe bodiesoperated upon, being partlymetal inconnexion with theinsulatedsystem with which thebodyAcommunicates, andpartly uninsulated metal, inthefixedpartsoftheelectro- 22 OntheMathematical Theory ofElectricity. [ii. meter, and inthemovableparts bywbichBissupported. Thegeneraleffect produced bythepresenceofsuch bodies indisturbingtheobserved lawofforce, must betomake it diminish lessrapidlywith thedistance whenAandBare separated byaconsiderable interval :and itisprobably owing, atleast inpart,tosuchdisturbingcauses thatMrHarris's resultsnearly agree,asfarasthey go,with aformula which wouldultimately giveforthelawofforce theinversesquareof thedistance between AandB,instead oftheinverse cube. 30.Thedetermination bythemathematicaltheoryofthe attraction orrepulsion between two electrifiedconducting spheres hasnothitherto, sofarasIamaware, beenattempted, andwouldpresentconsiderabledifficulty bymeans ofthe formulaeordinarily givenforsuchproblems.Itmay, however, very readilybeeffected bymeans ofageneral theorem onthe attraction between electrified conductors, which willbegiven inasubsequent paper.* Thus,ifF{c)betheforce ofattraction, correspondingtothedistance cbetween thecentres, inthe particular casewhen thetwospheresareequal (theradius of eachbeing unity), andthepotentialintheinterior ofoneof them isnothing (aswillbethecasewhen thebodyisun- insulated),thepotentialintheinterior oftheotherbeing v, Ihave found thefollowing formulae, whichexpress F[g)bya convergingseries :— ^(^)=^Kt^-^l+l+^*°-)(^^' whereQ,=c*-l "ja=(c^-2)(3.-l (B), e»«=(<=^-2)e^.-Qj P,=^c'-^ \(C). *[Note added March 1854.—The enunciation ofthe"general theorem" alluded to,theinvestigation founded on it,by.which theauthor first arrived attheconclusion made useofhere, andanother demonstration ofthesame conclusion, founded onthemethod ofelectrical images, and strictly sjTithe- tical initscharacter, arepublished, with comprehensive numerical results, inthePhilosophical Magazine forApril 1858.] II.] ElementaryLaiusofStaticalElectricity. 28 f31.These formulae enable ustocalculateQ^,Q^,Q^,Q^, .,andthenP^,P^,P^,P^, etc., successively, byasimple imiform arithmeticalprocess,foranyparticularvalue ofc, avethus calculated thevalues ofF(c)infive cases, the firstfour ofwhich arethose examined byMrHarris, andhave obtained thefollowing results, each ofwhich istrue tofive placesofdecimals :— c. 24 OntheMathematical Theory ofElectricity. [ii. 34.Inasubsequent memoir, bythesame author,* wefind additionalexperimentsontheelementary principlesofthe theoryofelectricity. The first series which isdescribed, was made forthepurposeoftestingthetruth ofCoulomb's law, thattherepulsionoftwosimilarly charged pointsisinversely asthesquareofthedistance, anddirectlyastheproductof themasses. Inexperimentsofthiskind inwhich accurate quantitativeresults areaimedat,many precautionsareneces- sary. Thus allconducting bodies, exceptthoseoperated upon, must beplaced beyondthereach ofinfluence, andthedistance between therepellingbodies must beconsiderable with refer- ence totheir linear dimensions, sothat thedistribution of electricityoneachmaybeuninfluenced bythepresenceofthe other. Also thebodies should bespheres,sothat theattrac- tionmaybethesame asifthewholeelectricityofeachwere collected atitscentre; andthedistance tobemeasured will then bethedistance between thecentres. These conditions have beenexpressly mentioned byCoulomb, andthey have been fulfilled, asfaraspossible,inhisresearches, asweseeby thedescriptionsoftheexperiments made, which wefind inhis memoirs. Hehasthus arrived bydirect measurement atthe law,whichweknow byamathematical demonstration, "f*founded upon independent experiments,tobetherigorous lawofnature, forelectrical action. None oftheseprecautions, however, have been taken intheexperimentsdescribed inMr Harris's *Philosophical Transactions, 1836.' +SeeMurphy's Electricity, p.41,orPratt's Mechanics, Art. 154. [Note added March 1854,—Cavendish demonstrates mathematically that ifthelawofforce beanyother than theinverse square ofthedistance, electricity could notrestinequilibrium onthesurface ofaconductor. But experiment hasshown thatelectricity does rest atthesurface ofaconductor. Hence thelaw offorcemust betheinverse square ofthedistance. Caven- dish considered thesecond proposition ashighly probable, buthadnotex- perimentalevidence tosupportthis opinion,inhispublished work (An attempttoexplain thephenomenaofElectricity bymeans ofanElastic Fluid). Since histime, themost perfect experimental evidence hasbeen obtained that electricity resides atthesurface ofaconductor; insuchfacts, forinstance, astheperfect equivalenceinallelectro-statical relations ofa hollow metallic conductor ofever sothin substance, orofagiltnon-con- ductor (possessing aconducting film ofnotmore than^^-^^f^ofaninch thick) andasolid conductor ofthesame external formanddimensions;the minor premiseofhissyllogismisthus demonstrated, andtheconclusion is therefore established.] II.] Elementary LawsofStaticalElectricity, 25 memoir, andthe results areaccordingly unavailable forthe accuratequantitativeverification ofanylaw,onaccount ofthe numerous unknowndisturbingcircumstancesbywhichthey are affected. Thephenomena which heobserves, however, affordqualitativeillustrations ofthemathematicaltheoryof §veryinteresting nature, asmaybeseenfrom thefollowing amplesofhisresults :— (a)When thedistance between thebodies isgreat with reference totheir linear dimensions, therepulsionisinversely asthesquareofthedistance, anddirectlyastheproductofthe masses. (6)When thedistance issmall, theaction becomesap- parently irregular. Thus ifthequantitiesofelectricity onthe twobodies beequal,the force, which isalwaysofrepulsion, doesnotincrease sorapidly when thebodiesapproach,asifit followed thelawoftheinversesquareofthedistance. (c)Ifthechargesbeunequal,therepulsionceases ata certain distance, andatallsmaller distances there isattraction between thebodies. 35.These results are,with alltheirpeculiarities,infullac- cordance with thetheoryofCoulomb, which indicates that,if thequantitiesofelectricitybeequal,andthebodiesequal and similar, there willberepulsioninevery position:but ifthere beany difference, however small, between thecharges,the repulsionwillnecessarily cease, and attraction commence, before contact takesplace,when onebodyismade toapproach theother. Unless, however, thedifference ofthecharges be sufficiently considerable, asparkmaypassbetween thebodies, andrender thecharges equal,before attraction commences. InMrHarris'sexperiments,inwhich thebodies seem tohave beennearlyoblatespheroids,theattraction isgenerallysensible before thedistance issmallenoughtoallow asparktopass,if thecharge ononebedouble ofthatontheother. MrHarris nextproceedstoinvestigatethetheoryofthe proof plane,andtoexamine whether itcanbeconsidered as indicatingwithcertaintytheintensityofelectricityatany partofacharged body, and, principallyfromanexperiment made onachargednon-conductor (ahollowsphereofglass), cumes toanegativeconclusion. Itshould beremembered, 26 OntheMathematical Theory ofElectricity. [ii. however, that, theproof plane havingnever beenappliedto determine theintensityatpointsofthesurface ofacharged non-conductor, such conclusions innowayinterfere with adoptedideas. Since there canbenomanner ofdoubt asto thetheoryofthisvaluable instrument, aswefind itexplained byM.Pouillet,* norastotheexperimentaluseofitmade by Coulomb,itisunnecessarytoenter more atlength onthe subjecthere. 36.MrFaraday'sresearches onelectrostatical induction, which arepublishedinamemoir formingtheeleventh series ofhisExperimentalResearches inElectricity, were under- taken with aview totestanideawhich hehadlong possessed, that theforces ofattraction andrepulsionexercised byfree electricity,arenottheresultant ofactions exercised atadis- tance, butarepropagated bymeans ofmolecular action among thecontiguous particlesoftheinsulating mediumsurrounding the electrified bodies, which hetherefore calls the dielectric. Bythisideahehasbeen ledtosomeveryremarkable views upon induction, or,infact,uponelectrical action ingeneral As itisimpossiblethat thephenomenaobserved byFaraday canbeincompatiblewith the results ofexperiment which constitute Coulomb'stheory,itistobeexpectedthat the difference ofhisideas from those ofCoulomb must arisesolely from adifferent method ofstating,andinterpreting physically, thesame laws :and farther,itmay,Ithink, beshown that either method ofviewingthesubject, when carriedsufficiently far,maybemade thefoundation ofamathematicaltheory which would lead totheelementary principlesoftheother as consequences.Thistheorywouldaccordingly betheexpres- sion oftheultimate lawofthephenomena, independentlyof anyphysical hypothesis wemight,from other circumstances, beled toadopt.That there arenecessarilytwo distinct elementary waysofviewingthetheoryofelectricity, may beseen from thefollowing considerations, founded onthe principles developedinaprevious paperinthisJournal.-f- *Seefoot-note on§25. +OntheUniform Motion ofHeat, and itsConnexion with theMathe- matical Theory ofElectricity (i.above). II.] Elementary Laws ofStaticalElectricity, 27 37.Correspondingtoevery problemrelative tothedistribu- tion ofelectricityonconductors, ortoforces ofattraction and repulsionexercised byelectrified bodies, there isaproblemin tlieuniform motion ofheatwhichpresentsthesameanalytical conditions, andwhich, therefore, consideredmathematically,is thesameproblem. Thus, letaconductor A,chargedwith agiven quantityofelectricity,beinsulated inahollow con- ducting shell, B^which wemaysupposetobeuninsulated. Accordingtothemathematicaltheory,anequal quantityof electricityofthecontrarykind willbeattracted totheinterior surface ofB(orthesurface ofB,aswemaycall ittoavoid circumlocution), andthedistribution ofthischarge, andofthe chargeonAywilltakeplacesothattheresultant attraction at anypointofeach surface maybeinthedirection ofthenormal. This conditionbeing satisfied, itwillfollow that there isno attraction onanypointwithin A,orwithout thesurface ofB, thatis,onanypointwithin either oftheconductingbodies. Themost convenient mathematicalexpressionforthecondition ofequilibrium,isthat thepotentialatanypointP*must have aconstant value whenPisonthesurface ofA,andthe value nothing whenPisonthesurface ofB]and itwill follow from thisthatthepotentialwillhave thesame constant value foranypointwithin A,and willbeequaltonothingfor anypointwithout thesurface ofB. IfAbesubjecttotheinfluence ofanyuninsulated con- ductors, wemust consider such bodies asbelongingtothe shell inwhichAiscontained, and their surfaces asforming partofthesurface oiB:insuch cases thissurface willgene- rallybetheinterior surface ofthewalls oftheroom inwhich Aiscontained, andofalluninsulated conductors intheroom. If,however, w^ehave toconsider thecase inwhichAissubject tonoexternalinfluence, wemustsuppose every partofthe surface ofBtobeveryfarfromA.Themostgeneral problem wecancontemplateinelectricity (exclusivelyofthecase in which theinsulating medium isheterogeneous,andexercises a special action, which willbealluded tobelow),istodetermine *Theterm usedbyGreen forthesum ofthequotients obtained bydivid- ingtheproduct ofeach element ofthesurfaces ofAandB,and itselectrical intensity, byitodistance from F. 28 OntheMatliematical Theory ofElectricity. [ii. thepotentialatanypointwhen A,instead ofbeingasingle conductor, isagroupofseparateinsulated conductors charged todifferentdegrees,andwhen there arenon-conductors elec- trified inagiven manner, placedintheinsulating medium, in theneighbourhood. Theconditions ofequilibriumwill stillbe thatthepotentialateach surface duetoallthefreeelectricity must beconstant, andthetheorems stated above will stillbe true :thus the attraction willbenothingintheinterior of eachportionofA,andwithout the surface ofB\andthe wholequantityofinducedelectricityonthelatter surface will bethealgebraic sum ofthechargesofalltheinterior bodies with itssignchanged. When thepotentialdue tosuch a systemisdetermined forevery point,thecomponentofthe resultant force atanypoint P,inanydirection PL,maybe found bydifferentiation, beingthe limit ofthe difference between thevalues ofthepotentialatP,andatapoint Q,in PX,divided byPQ,whenQmoves uptowards andultimately coincides withP,andthedirection ofthe force, onanegative particle, beingthat inwhich thepotentialincreases. By Coulomb's theorem, theintensityatanypointinoneofthe conductingsurfaces isequaltotheattraction(onanegative unit)atthatpoint,divided by47r. 38.Now ifwewish toconsider thecorresponding problem inthetheoryofheat,wemustsupposethespace betweenA andByinstead ofbeingfilled withadielectric medium(thatis anon-conductor forelectricity),tobeoccupied byanyhomo- geneoussolidbody,andsources ofheat orcold tobesodis- tributed over theterminating surfaces, ortheinterior surface ofBandthesurface ofA,thatthepermanent temperature atthe firstsurface maybezero,andatthesecond shallhave a certain constant value, thesame asthatofthepotentialinthe case ofelectricity.IfAconsist ofdifferent isolatedportions, thetemperatureatthesurface ofeach willhave aconstant value, which isnotnecessarilythesame forthedifferentpor- tions. Theproblemofdistributingsourcesofheat, accordingto these conditions, ismathematicallyidentical with theproblem ofdistributing electricityinequilihriumonthesurfaces ofA andB.Inthecase ofheat, thepermanent temperatureatany point replacesthepotentialatthecorresponding pointinthe II.] Elementary LawsofStaticalElectricity. 29 electrical system, andconsequentlytheresultantflux ofheat replacesthe resultant attractio7i ofthe electrified bodies, in direction andmagnitude. Theproblemineach case isdeter- minate, andwemaytherefore employtheelementary principles ofonetheory,astheorems, relative totheother. Thus, inthe paperinwhich these considerations aredeveloped, Coulomb's fundamental theorem relative toelectricityisappliedtothe theoryofheat;and self-evidentpropositionsinthe latter theoryaremade thefoundation ofGreen's theorems inelec- tricity.* Now thelaws ofmotion forheatwhich Fourierlays down inhisTheorie AnalytiquedelaChaleur, areofthat simple elementarykindwhich constitute amathematicaltheory properlysocalled;and therefore, whenwefindcorresponding laws tobetrue forthephenomena presented byelectrified bodies, wemaymakethem thefoundation ofthemathematical theoryofelectricity:and thismay bedone ifweconsider themmerelyasactual truths, withoutadopting anyphysical hypothesis, althoughtheideathey naturally suggestisthat ofthepropagationofsome effect bymeans ofthemutual action ofcontiguous particles; justasCoulomb, althoughhis lawsnaturally suggesttheidea ofmaterialparticles attracting orrepellingoneanother atadistance, mostcarefullyavoids makingthisaphysical hypothesis,andconfines himself tothe consideration ofthemechanical effects which heobserves and theirnecessary consequences.-f* 39. Alltheviews which Faraday hasbrought forward, and illustrated ordemonstrated byexperiment,lead tothismethod ofestablishingthemathematicaltheory, and, asfarasthe analysisisconcerned,itwould, inmostgeneral propositions, beevenmoresimple,ifpossible,than that ofCoulomb. (Of course theanalysisofparticular problems would beidentical inthetwomethods.)Itisthus thatFaradayarrives ata knowledgeofsome ofthemostimportantofthegeneral *Itwasnotuntil some time after thatpaper waspublished, that Iwas able toaddthedirect analytical demonstrations ofthetheorems, which are given inthepapers on"General Propositions intheTheory ofAttraction," Gamb. Math. Jour.,vol. iii.pp.180,201(xii. below), andwhich Ihave since found arethesame asthoseoriginally given byGreen. tSee first footnoteon§25. 30 OntheMathematical Theory ofElectricity. [ii. theorems, which, from their nature, seemed destined never tobeperceived exceptasmathematical truths. Thus, inhis theory,thefollowing propositionisanelementary principle:— Letanyportionaofthesurface ofAbeprojectedonB,by means oflines (whichwillbeingeneral curved) possessingthe propertythattheresultant electrical force atanypointofeach ofthem isinthedirection ofthetangent:thequantityof electricity produced byinduction onthisprojectionisequal tothequantityoftheoppositekind ofelectricityona.*The lines thus defined arewhat Faradaycalls the"curved lines of inductive action." Foradetailed account oftheexperiments bywhich thesephenomenaareinvestigated,reference must be made toMrFaraday's ownmemoirs, publishedinthePhilo- sophical Transactio7is, and inaseparateform inhisEccpem- mental Researches. 40.Thehypothesis adopted byFaraday,ofthepropagation ofinductive action, naturallyledhim totheideathat itseffects maybeinsomedegree dependent upon thenature ofthe insulating medium ordielectric, bywhich, accordingtothis view, itistransmitted. Inthesecondpartofhismemoir he describes aseries ofresearches instituted toputthistothetest ofexperiment,andarrives atthefollowingconclusions :-^ *Thistheorem maybeprovedasfollows :— LetSheanyclosed surface, containing nopart ofthe electrified bodies within it,which wemay conceive tobedescribed between AandB;letP bethecomponentinthedirection ofthenormal, oftheresultant force at anypointofthesurface S,and letdsbeanelement ofthesurface atthe same point. Then itmaybeeasily proved (seeCamb. Math. Jour., vol. iii. p.204)that ffJ'ds^O (a), the integrations being extended over theentire surface. Now letSbe supposed toconsist ofthree parts; theportion a,ofthesurface ofA;its projection /3,ontheinterior surface ofB;andthesurface generated bythe curved lines ofprojection. Thevalue ofPateach point ofthe latter portionofSwillbenothing, since thetangent atanypoint ofalineofpro- jectionisthedirection ofthe force. Hence, if{ffPd)^'] and{ffPds)denote thevalues oiffPds,fortheportions aand/3ofS,theequation (a)becomes [ffPds] +{ffPds)=0. But ifpbetheintensityofthedistribution onthesurface AorB,atany point, wehave, byCoulomb's theorem, P Hence Ufpds] +{ffpds)=0, which isthetheorem quotedmthetext. kElementary LawsofStaticalElectricity. 31 1.Ifthedielectric beair,theinductive action isquiteinde- pendentofitsdensityortemperature (which,asMrFaraday remarks, agrees perfectlywithpreviousresults obtainedby MrHarris) ;andingeneral,ifthe dielectric beanygasor vapour capableofinsulatingacharge,theinductive action is invariable. Hence heconcludes that"allgases have thesame power of,orcapacity for,sustaininginductionthrough them (which mighthavebeenexpected when itwasfound .thatno variation ofdensityorpressure produced any effect)." When thedielectric issolid, theinduction isgreaterthan through air,and variesaccordingtothenature ofthesub- stance. Numbers which measure the"specificinductive capacities"ofthe dielectricsemployed (sulphur,shell lac, glass, etc.)arededuced from theexperiments. 42.Toexpressthese results inthelanguageofthemathe- maticaltheory,letusrecur tothesuppositionofabod}^. A, chargedwithagiven quantityofelectricity,andinsulated inthe interior ofaclosed conducting shell, B.Thepotentialofthe systemattheinterior surface ofB,andatevery pointwithout thissurface, willbenothing ;atthesurface andintheinterior of Aitwillhave aconstant value, which willdependontheform, magnitude, and relativepositionofthesurfacesAandB,on thequantityofelectricityonA,and, accordingtoFaraday's discovery,onthedielectric poweroftheinsulating medium which fillsthespace between AandB. Ifthisbegaseous,neither itsnature nor itsstate astotemperature, pressure,ordensity wdll affect thevalue ofthepotentialinA;but ifitbeasolid substance, such assulphurorshelllac,thevalue ofthepotential willbelessthanwhen thespaceisoccupied byair,and will varywith thenature oftheinsulatingsolid. 43.The result inthecase ofagaseousdielectric iswhat would follow fromCoulomb'stheory,ifweconsidergasestobe quite impermeabletoelectricity,andtobeentirely unaffected byelectrical influence. Thephenomenaobserved with solid dielectrics, whichagreewith thecircumstance observed by Nicholson, that thedissimulating powerofaLeydenphial depends onthenature oftheglassofwhich itismade, as well asonitsthickness, have beenbysome attributed toa slight degreeofconducting power,orofpenetrability, pos- 32 OntheMathematical Theory ofElectricity. [it. sessed bysolid insulators. Thisexplanation, however, seems tobeveryinsufficient;andbesides, Faradayhasestimated the nature ofthe effects ofimperfectinsulation byindependent experiments, andhasestablished, inwhat seems tobeavery satisfactory manner, theexistence ofapeculiaraction inthe interior ofsolid insulators whensubjectedtoelectrical influ- ence. Asfarascanbegatheredfrom theexperiments which haveyetbeen made,itseemsprobablethatadielectric, sub- jectedtoelectrical influence, becomes excited insuchamanner that every portionof it,however small, possesses polarity exactly analogoustothemagnetic polarityinduced inthesub- stance ofapieceofsoftironunder theinfluence ofamagnet. Bymeans ofacertainhypothesis regardingthenature ofmag- netic action,* Poisson hasinvestigatedthemathematical laws ofthedistribution ofmagnetism,and ofmagneticattractions andrepulsions.These lawsseem torepresentinthemost generalmanner thestate ofabody polarized byinfluence, and therefore, without adopting any particularmechanical hypo- thesis, wemaymake useofthem toform amathematical theoryofelectrical influence indielectrics, thetruth ofwhich canonlybeestablished byarigorous comparisonofitsresults withexperiment. 44.Letustherefore consider whatwould betheeffect, accord- ingtothistheory, which would beproduced bythepresence ofasolid dielectric, (7,placedinthespace between AandB, therest ofwhich isoccupied byair.The action of(7,when excited bytheinfluence ofthe electricities onAandB,may (asPoisson hasshown formagnetism)berepresented, whether *Faraday adopts thecorresponding hypothesis toexplain theaction ofa solid dielectric, which hestates thus:—"Ifthespace round acharged globe were filled with amixture ofaninsulating dielectric, asoilofturpentine or air,andsmall globular conductors, asshot, thelatter being atalittle dis- tance from each other, soastobeinsulated, then these intheir condition , andaction exactly resemble what Iconsider tobethecondition and action oftheparticlesoftheinsulating dielectric itself. Iftheglobe were charged, these little conductors would allbepolar;iftheglobe were discharged, they would allreturn totheir normal state, tobepolarized again upon the re- chargingoftheglobe."—{Experimental Researches, §1679.) The results of themathematical analysis ofsuchanaction aregiven inthe text. Itmay beadded that thevalue ofthecoefficient kwill differ sensibly from unityif thevolume occupied bythesmall conducting balls bear afinite ratio tothat occupied bytheinsulating medium. II.] Elementary LawsofStaticalElectricity. 33 onpointswithin orwithout C,byacertain distribution of positive electricity ononeportionofthesurface of0,andof anequal quantityofnegative electricityontheremainder. The conditionnecessary and sufficient fordeterminingthis distribution may (ascanbeshown from Poisson'sanalysis) be expressedasfollows. LetRbetheresultant force ona pointPwithout(7,andi^onapoint P'without G,due to theelectrified surfaces AandB,and totheimagineddistribu- tiononG.IfPandP'betakeninfinitelynear oneanother, andconsequentlyeachinfinitely near thesurface of(7,the componentofR'inthedirection ofthenormal must bear to thecomponentofRinthesame direction aconstant ratio (t)dependingonthecapacityfordielectric induction ofthe matter of(7.*ThecomponentsofRandR'inthetangent planewillofcourse beequal and inthesamedirection, and, ifpbetheintensityoftheimagineddistribution onthesurface ofG,intheneighbourhoodofPandP',thedifference ofthe normalcomponentswillbeWp,asisevident fromCoulomb's theorem, referred toabove. 45.Letusnowsuppose(7tobeashellsurrounding A,and letSand B'yitsinterior andexterior surfaces, besurfaces of equilibriuminthesystemofforces duetotheaction ofAand B,andofthepolarityofG. Itmaybeshown that thesame surfaces S,S\wouldnecessarilybesurfaces ofequilibrium, ifGwereremoved andthewholespace were filled withair; andconsequently,that thewhole series ofsurfaces ofequi- *From this itfollowsthat, inthecase ofheat,Gmust bereplaced bya bodywhose conducting powerisktimes asgreat asthat ofthematter oc- cupying theremainder ofthespace between AandB. [Note added March 1854.—Thesame demonstration, ofcourse, isapplic- able totheinfluence ofapiece ofsoft iron, orother "paramagnetic" (i.e., substance offerro-magnetic inductivecapacity),ortothereverse influence ofadiamagnetic onthemagnetic force inanylocality nearamagnetinwhich itcanbeplaced, andshows that thelines ofmagnetic force willbealtered byitprecisely asthelines ofmotion ofheat incorresponding thermal circum- stances would bealtered byintroducing abody ofgreateroroflessconduct- ingpower forheat. Hence weseehow strict isthefoundation foran analogy onwhich theconducting power ofamagnetic medium forlines offorce may bespoken of,andwehave aperfect explanationofthecondensing action ofaparamagnetic, andtherepulsiveeffect ofadiamagnetic, upon the lines offorce ofamagnetic field, which have been described byFaraday.— {Exp. Researches, §§2807, 2808.)] T.E. 3 84 OntheMathematical Theory ofElectricity. [ii. librium, commencingwithAandendingwith B,willbethe same inthetwo cases. Hence theresultant forceduetothe excitation ofthedielectric G(ortotheimagineddistributions ofelectricityon >SfandS'whichproduce it),onpointswithin Borwithout>Si',must besuch asnottoalter thedistributions onAandBwhen thequantityonAisgiven;and istherefore nothing. Accordingly,letQbethe total force onapoint indefinitelynear 8,andwithin it;Q'thetotal forceonapoint without 8',butindefinitelynear it.Since theforces onpoints without )S'andwithin 8'indefinitelynear theformerpoints Q Q' are,accordingtothelawstated above,-j-and-j^,itfollows* thattheintensities oftheimagineddistributions on8and8\ intheneighbourhoodofthepoints considered, are -r>-f)"^i(«-f)- Hence,ifU,U'bethepotentialsat>Si,8\due toAandB alone, and vthepotentialatanypoint P,itfollows thatthe potentialatP,duetothepolarityofthedielectric, is or or -{l-t\v-\- (l--Avythatis,0, accordingasPiswithin 8,within >S^'andwithout 8^orwithout 8'.Hence thetotalpotentialwillbe,accordingtotheposition or V. Hence the sole effect ofthedielectric 0,onthestate ofA andB,istodiminish thepotentialintheinterior oftheformer bythequantity *SeeGreen'sEssay,Art.12;orabove,i.§8. II.] Elementary LawsofStaticalElectricity. 35 Ifthewholespace between AandBbeoccupied bythesolid dielectric, thesurfaces SandAwill coincide, asalso,S'and. B,andtherefore U=V,Z7'=0.Hence thepotentialinthe V interior of^willbe-y^, orthefraction tofthepotential,with thesamechargeonA, andwithagaseousdielectric. From this itfollows that,when the dielectric issolid,itwouldrequire,toproduceagiven potentialintheinterior oiAjhtimes thecharge which would benecessarytoproducethesamepotential when thedielectric isgaseous, andtherefore thebodyAinagiven state, defined bythepotentialinitsinterior, producesontheinterior surface ofB,byinduction, throughthe solid dielectric, aquantityof electricity ktimes asgreatasthroughagaseousdielectric. On thisaccountFaradaycallsthepropertyofadielectric measured byh,its" specific inductivecapacity." 46.InFaraday's experiments anapparatus (whichisinfact aLeyden phial,inwhich anysolid orfluidmaybesubstituted, fortheglassdielectric ofanordinary Leyden phial)isused, correspondingtothecasewehave been considering,inwhich ^isaconducting sphere (2"33inches indiameter), andBa concentricsphericalshellsurroundingit(thedistance between thesurfaces ofAandBbeing'62ofaninch). IntheshellB there isanapertureintowhich ashell-lac stem isfixed; a wire, attached toA,passes throughthecentre ofthisstem to theoutside oftheshelly andsupportsaballofmetal, M,which isthus insulated andconnected withA. Itmaybeshown that insuchanapparatusthestate oftheballAandofthe shellBwillapproximatelybenotaftected bytheaperturein the latter, orbythewiresupporting M,andthatthedistribu- tion ofelectricityonMwillbeapproximatelythesame asif- thewiresupportingitandtheconductors AandBwere re- moved. Hence the sole relation between AandMwillbe thatthepotentialsintheir interiors arethesame ;andthere- forethelatter, which isaccessible, maybetaken asanindex of thestate oftheformer. 47.Todetermine thespecificinductivecapacityofanydi- electric, Faradayusestwoapparatusofthekindjust described, 3—2 36 OntheMathematical Theory ofElectricity. [ii. precisely equalandsimilar, inoneofwhich thespace between AandBisfilled with air,andintheother with thedielectric tobeexamined. One ofthese apparatusischarged,andthe intensitymeasured :theballsM,M'inthetwoarethenmade totouch andseparate again,andtheremaining intensityon the first (whichisequaltotheintensity impartedtothe second)ismeasured. Ifthisbefound todiffer from halfthe original intensity,itwill follow that thespecificinductive capacityofthesubstance examined differs from that ofair, which isunity, and itsvaluemaybedetermined bymeans of asimple expressionfrom theexperimentaldata. Toinvesti- gate this, letusfirstsupposeeachapparatustobecharged, and letitberequiredtofindtheintensity ontheballs afterthey aremade totouch, andthenremoved frommutual influence; and letthedielectrics beanytwosubstances, whose inductive capacitiesarek,k'.Letp,pbetheintensities before, and a- thecommonintensityafter contact. Then, denoting byQ,Q thequantitiesofelectricity constitutingthecharges before, andq,qafter contact, weshall have, bytheprinciples already developed,|=|^,, ^= |,^,=J. Also Q+Q'=q-^q. Hence wededuce cr=-^,-7-. Intheexperiment described, one ofthe dielectrics isair. Hence, toobtain therequired formula, wemayputk'=l,in thisequation, andthen resolve fork. Thuswefind k=^ . p-a- Ifonlyoneoftheapparatus beoriginally charged, according asitisthe firstorthesecond, weshallhave or. k=— 48. Ifthesubstance examined(thedielectric ofthe first apparatus) beanygas,orairinadifferent state astopressure ortemperaturefrom the airofthesecondapparatus, Faraday IElementary LawsofStaticalElectricity, 37 alwaysfinds theintensityafter contact tobehalftheoriginal intensity, andhence forevery gaseous bodyA;=1. 49. Ifthedielectric ofthe firstapparatusbesolid, thein- tensityafter contact isfound tobegreater than halftheoriginal intensity when the first,and lessthan halfwhen thesecond is theapparatus originally charged. Hence forasolid dielectric, A;>1.Forsulphur Faradayfinds thevalue toberather more than 2*2;forshell-lac, about 2;andforflint-glass, greaterthan 176. 50.Thecommonlyreceived ideas ofattraction andrepul- sion exercised atadistance, independentlyofanyintervening medium, arequiteconsistent with allthephenomenaofelec- trical action which have been here adduced. Thuswemay consider theparticlesofairintheneighbourhoodofelectrified bodies tobeentirely uninfluenced, andtherefore toproduceno eff"ect intheresultant action onanypoint:buttheparticles ofasolid non-conductor must beconsidered asassuminga polarizedstatewhen under theinfluence offreeelectricity,so astoexercise attractions orrepulsions onpointsatadistance, which, with theaction duetothecharged surfaces, producethe resultant force atany point.Itis,nodoubt, possiblethat such forces atadistance maybediscovered tobeproduced entirely bytheaction ofcontiguous particlesofsome inter- vening medium, andwehave ananalogyforthis inthecase ofheat,where certain effects which follow thesame laws are undoubtedly propagatedfromparticletoparticle.Itmight alsobefound thatmagneticforces arepropagated bymeans of asecond medium, andtheforce ofgravitation bymeans ofa third.Weknownothing, however, ofthemolecular action by which such effects could beproduced, andinthepresentstate ofphysicalscience itisnecessarytoadmit theknown facts in eachtheoryasthefoundation oftheultimate laws ofaction at adistance. StPeter's College, N(yv, 22.1845. III.ONTHEELECTKO-STATICAL CAPACITY OFALEYDEN PHIALANDOFATELEGRAPH WIREINSULATED INTHE AXISOFACYLINDRICAL CONDUCTING SHEATH* [FromthePhilosophical Magazine, 1855, first half-year.] 51.Theprinciples broughtforward intheprecedingarticles OntheUniform Motion ofHeat, etc.,enable uswithgreatease toinvestigatethe" capacity "fofaLeyden phialwith either air, oranyliquidorsolid dielectric, andofotheranalogous arrange- ments, such asthecopperwires ingutta-perchatubes under water, withwhich Faradayhasrecently performedsuch re- markableexperiments. ;[: 52.Thus, foraLeyden phial,letussupposeaportion Softhe surface ofaconductor Atobeeverywheresonear thesurface ofaconductor A\thatthedistance between them atanypoint isasmall fraction oftheradii ofcurvature ofeach surface in theneighbourhood ;and letzbethedistance between them at aparticular position,P.Then, bytheanalogywith heat,itis clear that ifthetwo surfaces bekeptatdifferent electrical potentials, VandV\thepotentialsatequidistant pointsin anyline across from onetotheother willbeinarithmetical V-V . progression.Hence willbetherate ofvariation ofthez potential perpendicularlyacross inthepositionP.If,inthe firstplace,the dielectric beair,the electric force inthe air *Communicated asanAdditional Note totwopapers (i.and ii.above) «'OntheUniform Motion ofHeat inHomogeneous Solid Bodies, and its connexion with theMathematical Theory ofElectricity," and"On the Mathematical TheoryofElectricity inEquilibrium ;"only not intime tobeappended tothereprints ofthose papers which appeared inthe Philosophical Magazine, JuneandJuly1854(1854,i,andii.). tDefined {Philosophical Magazine, June 1853) foranyconductor (subject ornottotheinfluence ofotherconductors), asthequantityofelectricity which ittakes tochargeittounit potential. XDescribed inalecture attheRoyal Institution, Jan. 20,1854, and subsequently published inthePhilosophical Magazine (1854,i.p.197). III.]Electro- StaticalCapacity ofaLeyden Phial, etc. 39 between thetwoabout tlieposition Pwillconsequently beV-V ,andtherefore the electricaldensity (accordingtothe orem provedinthe firstarticle) ononesurface must be 1V—V 1V—v , ,andontheother— .Thequantity of lectricityintheposition P,onanareadsofthesurface S,is 1V-V therefore^ds,andtherefore thewholequantityon >Sfis V-r [ds(^eorempre [lV-V Stt z ectricity ii /' 47r j^' which isGreen's general expressionforthe electrification of eithercoatingofaLeyden phial.Ifthethickness ofthe dielectric beconstant andequaltor,itbecomesv-r s 47rT* 53.Now ifA'beuninsulated, wehave F'= ;andthen, tocharge Stothepotential F,ittakes thequantity Fx-— . Hence the" capacity"of8is S_ 47rT* Ifinstead ofairthere beasolid orliquiddielectric ofinductive capacity, k,occupyingthespace between thetwo surfaces, the quantityofheat conducted across, intheanalogous thermal circumstances, would bektimes asgreatasinthecase cor- respondingtothe airdielectric, with thesame difference of temperatures ;and intheactual electrical arrangement,the quantityofelectricityoneach oftheconductingsurfaces would bektimes asgreataswith airfordielectric andthesame dif- ference ofpotentials. Theexpressionforthecapacityofan actual Leyden phialistherefore kB_ 47rT' kbeingtheinductivecapacityofthesolid non-conductor of which itisformed, ritsthickness, andSthearea ofitwhich iscoated oneach side. 54.Toinvestigatethecapacityofacopperwire inthecir- cumstancesexperimentedonbyFaraday,letusfirstconsider the analogous circumstancesregardingtheconduction ofheat;that is,letusconsider theconduction ofheat thatwould takeplace 40 OntheElectro- Statical Capacity ofa[in, across thegutta-percha,ifthecopperwire initsinterior were kept continuallyatatemperaturealittle above that ofthe water which surrounds it.Here thequantityofheatflowing outwards from anylengthofthecopper wire, thequantities flowingacross different surfacessurroundingitinthegutta- percha,andthequantity flowingintothewater from thesame lengthofgutta-percha tube, inthesame time,must beequal. Buttheareas ofthesamelengthofdifferentcylindricalsurfaces areproportionaltotheir radii, andtherefore theflow ofheat acrossequalareas ofdifferentcylindricalsurfaces inthegutta- percha,coaxial with thewire,must beinverselyastheir radii. Hence, inthecorrespondingelectricalproblem, with airasthe dielectric instead ofgutta-percha,ifEdenote theresultant electrical force atanypointPintheairbetween aninsulated, electrified, infinitely long cylindrical conductor, andanun- insulated, coaxial, hollowcylindrical conductorsurrounding it, and ifa?bethedistance ofPfrom theaxis,wehave X whereAdenotes aconstant. But ifvbethepotentialatP; bythedefinition of" potential"wehave —=-Pdx Hence dx X^ and,byintegration, v=—A\ogx +G. Assigningtheconstants AandGsothat thepotential may have thevalueVatthesurface ofthewire, andmayvanish atthehollowconductingsurface roundit,ifrandrdenote the radii ofthesecylinders respectively, wehave and —-7-dx 55.Taking x=r,wefindbythistheelectric force intheair III.] LeydenPhial andofaTelegraph Wire. 41 infinitelynear theinner electrified conductor;anddividing the value found, by47r(accordingtothegeneral theorem), wehave 1V 47r ,r' rlog- forthe electricaldensityonthe surface oftheconductor. Multiplyingthisby^irrl, thearea ofalengthIofthesurface, wefind^VI forthewholequantityofelectricityonthatlength. Hence,if hbethespecificinductivecapacityofgutta-percha,the electri- cityrestingonalengthIofthewire intheactual circumstances willamount toi^^v OrifSdenote thesurface ofthewire,wehave, forthequantity ofelectricity which itholds, y_hS_.r 4i7rrlog— and therefore itscapacityisthesame asthat ofaLeyden phial withanequalarea ofcoatedglassofthicknessequalto // . Trlog- ,if/denote thespecificinductivecapacityofthe glass. 56.Inthecaseexperimented onbyMrFaraday,thediameter ofthewirewas-j^g-thofaninch,andtheexterior diameter ofthe gutta-percha covering wasabout fourtimes asgreat. Hence thethickness oftheequivalent Leyden phialmust havebeen Asthesurface ofthewireamounted to8300square feet,we mayinfer that ifthegutta-perchahadonlythesame induc- tivecapacityasglass (anditprobablyhasalittlegreater),the insulated wire,when theouter surface ofthegutta-perchawas uninsulated, would havehadanelectricalcapacity equaltothat ofanordinary Leyden batteryof8300squarefeet ofcoated glass^dofaninch thick. Invebcloy, Abban, June, 1854. IV.ONTHEMATHEMATICAL THEORY OFELECTRICITY INEQUILIBRIUM. (Art. XXXVIII. ofcompletelistinMathematical andPhysical Papers, Vol.i.) n.—ASTATEMENT OFTHEPRINCIPLES ONWHICH THEMATHE- MATICAL THEORY OFELECTRICITY ISFOUNDED. [Cambridge andDublin Mathematical Journal, March, 1848.] 57.Thispapermayberegardedasintroductorytosome others which will follow, containingvariousinvestigationsin theTheoryofElectricity. Thefundamental mathematicalprin- ciplesofthephenomenaofElectricityinEquilibriumarestated andexplainedinasconcise amanner asseems consistent with clearness. Toavoidlengtheningthepaperandunnecessarily distractingtheattention ofthereader, nodetails aregivenwith reference totheexperimentswhich have been, orwhich might be,made forestablishingthevariouspropositions asserted; and, forthesame reasons, scarcely anyallusion ismade tothehis- toryofthesubject. Withregardtothenature oftheevidence onwhich themathematicaltheoryofelectricity rests, thereader isreferred tothepreceding paper"OntheElementary Laws ofStaticalElectricity," where, besides somegeneralex- planationsonthesubject,theworkscontainingaccounts ofthe actual experimentalresearches ofprincipal importanceare indicated. Thatpaperismarked asthe first ofaseries which itwasmyintention topublishinthisJournal, andofwhich thesecond nowappears.Inthisseries itwillnotbeattempted toadhere toasystematiccourse ofinvestigations such asmight constitute acompletetreatise onthesubject; andmyonly reason forpublishingthisintroductoryarticle isforthesake ofreference inotherpapers, therebeingnopublished work in which theprinciplesarestated inasufficientlyconcise and correct form, independentlyofanyhypothesis,tobealtogether satisfactoryinthepresentstate ofscience. TheTwoKindsofElectricity. 58.Ifapieceofglassandapieceofresin arerubbedtogether andthenseparated,itisfound thattheyattract oneanother IV.]Fundamental Laws a^idPrinciples. 43 mutually. Thetermelectricity*hasbeenappliedtotheagency developedinthisoperation ;theexcitation ofthebodies, to which theattractive force isdue, iscalled electrical, andthe bodies soexcited aresaid tobeelectrified,ortobechargedwith electricity. Ifsecondpiecesofglassand resin berubbedtogether and thenseparated,andplacedintheneighbourhoodofthe firstpair ofelectrified bodies, itmaybeobserved— (1)That thetwopieces ofglass repeloneanother. (2)That eachpiece ofglassattracts eachpiece ofresin. (3)That thetwopieces ofresinrepeloneanother. Hence itisinferred that thetwopiecesofglass possesselec- tricalpropertieswhich differ intheir characteristics from those ofthe resin.;andthetwokinds ofelectricity thus indicated are called vitreous andresinous, after thesubstances onwhichthey aredeveloped.Bodies mayinvariousways bemade electric; butthe characteristicspresentedarealwaysthose ofeither vitreouselectricityorresinouselectricity. 59.Anelectrified bodyexerts noforce, whether ofattraction orofrepulsion, uponanynon-electric matter. When inanycase bodies notpreviouslyelectrified areobserved tobeattracted, or urgedinanydirection, byanelectrical mass, itisbecause the bodies havebecomeelectricallyexcitedbyinfluence. 60. Ifasmallpieceofglassandasmallpieceofresin, which havebeen electrified bymutual friction, beplaced successively inthesamepositionintheneighbourhoodofanelectrifiedbody, theywillbeacted upon byequal forces, inthesame line, butincontrarydirections. Hence thetwobodies aresaidtobe equally chargedwith thetwokinds ofelectricity respectively. ElectricalQuantity. 61.The forcebetween two electrified bodiesdepends,ceteris paribus, ontheamounts oftheircharges,oronthequantities ofelectricitywhichthey possess. Ifasmallpieceofglassandasmallpieceofresinbeelectrified bymutual friction tosuchanextent that,whenseparatedand placedataunit ofdistance, theyattract oneanother with a unit offorce, thequantityofelectricity possessed bytheformer *FromijXeKTpou, amber, onaccount ofsuchphenomena having been first observed withamber asoneofthesubstances rubbed together. 44 OntheMathematical Theory ofElectricity. [ly. issaid tobeunity;thelatterpossesses whatmaybecalled a unit ofresinouselectricity. Ifmbodies, eachpossessingaunit ofvitreouselectricity, be incorporated together,thesingle bodythuscomposedischarged withmunits ofthesame kind ofelectricity:Itissaidtopossess aquantityofelectricity equaltom,oritselectrical mass ism. Asimilar definition isapplicablewith reference tothemeasure- ment ofresinouselectricity. 62. Iftwobodiespossessing equal quantitiesofvitreous and resinouselectricitybeincorporated,thesingle bodythuscom- posedwillbefound either tobenon-electric, ortobeinsuch a state that, without theremoval ofanyelectricityofeither kind fromit,itmay, merely byanalteration inthedistribution of what italready possesses,bedeprivedofallelectrical symptoms. Thus itappearsthat abodyeithervitreouslyorresinously electrified, maybedeprivedofitscharge merely bysupplying itwithanequal quantityoftheother kind ofelectricity. Inconsequenceofthis fact,wemayestablish acomplete systemofalgebraicnotation with reference toelectricalquantity, whether ofvitreous orresinouselectricity, byadoptingas universal thelawthatthetotalquantityofelectricity possessed bytwobodies, orthequantity possessed byonebodymade up oftwo,isequaltothesum ofthequantitieswithwhich they areseparately charged. Thus letmbethequantityofelec- tricitywithwhich avitreouslyelectrifiedbodyischarged, and letm'bethequantitycontainedbyabody equally charged with resinouselectricity. Wemust have m4-m'=0, andtherefore m'isequalto—m.Now itisusual toregard vitreouselectricityaspositive ;andwemust thereforeregard theother kind asnegative ;sothatabody possessing munits ofresinouselectricityistobeconsidered ascharged witha quantity—mofelectricity. TheSuperposition ofElectrical Forces. 63. Ifabody,electrified inagiveninvariable manner, be placedintheneighbourhoodofanynumber ofelectrified bodies, itwillexperienceaforcewhich istheresultant oftheforces thatwould beseparately exerted uponitbythedifferent bodies IV.] Fundamental Laws andPrinciples. 45 iftheywereplacedinsuccession inthepositions whichthey actually occupy, withoutanyalteration intheir electrical con- ditions. Thislaw istrueeven ifanynumber ofthebodies considered bemerelydifferentpartsofonecontinuous mass. Cor. 1.The total mechanical action between two electrified bodies, whetherpartsofonecontinuous mass orisolated bodies, istheresultant oftheforces duetothemutual actions between allpartsofeitherbody and allpartsoftheother, ifweconceive thetwobodies tobearbitrarily divided each intopartsinanymanner whatever. Cor. 2.Wemay,inanyelectricalproblem, imagine the charge possessed byabodytobedivided intotwo ormore parts,each distributedarbitrarilywith thesolecondition that thesum ofthequantitiesofelectricityinanyverysmall spaceofthebody duetothedifferent distributions shall be equaltothegiven quantityofelectricityinthatspace, accordingtotheactual distribution ofelectricityinthebody ; andwemayconsider theforceactuallyexerted upon anyother electrified bodyasequivalenttotheresultant oftheforces due tothesepartialdistributions. TheLawofForce betweenElectrifiedBodies. 64.The force between twosmall electrified bodies varies inverselyasthesquareofthedistance between them. Cor. Iftwo small bodies becharged respectivelywith quantities mandmofelectricity, theywillmutually repel withaforceequalto—rj-; (anaction which willbereallyattractive whenmandmhave unlikesigns,aswould bethecasewere thebodiesdissimilarly electrified). Fortwo units, placedatadistanceunity, repel withaforce equaltounity,andtherefore ifplacedatadistance A,theywillrepelwith aforce—^ ;andtheexpressionforthe repulsion between munits andmunits isdeduced from this, accordingtotheprincipleofthesuperpositionofforces, by multiplying bymm. 46 OntheMathematical Theory ofElectricity. [iv. Definition oftheResultant Electrical Force ataPoint, 65.Letaunit ofnegative electricitybeconceived tobecon- centrated atapointPintheneighbourhoodofanelectrified bodyorgroupofbodies, withoutproducing anyalteration in thepreviously existingelectrical distribution. The force exerted uponthis electricalpointiswhatweshallthroughoutunder- stand astheresultantforceatPdue totheelectricityofthe bodyorbodies considered. Cor. IfRhetheresultantforceatPinany case, then theforceactuallyexerted upon anelectrical mass m,concen- trated atP,willbeequalto—mR. ElectricalEquilibrium. 66.When abodyheld atrest iselectrified, andwhen, being eithersubjecttoelectrical action from other bodies, orentirely isolated, thedistribution ofitschargeremainspermanently unaltered, theelectricity uponitissaid tobeinequilibrium. Electricalequilibrium maybedisturbed invariousways. Thus ifabody chargedwithelectricityinequilibriumbe touched, orevenapproached byanother electrifiedbody,the equilibrium maybebroken, andcanonlyberestored after a different distribution hasbeen effected, byamotion ofelectricity throughthebodyoralongitssurface :orifabodybeinitially electrified inanyarbitrary manner, whether byfriction orother- wise, itmaybethat, assoon astheexcitingcause isremoved, theelectricitywill eithergradually become altered from its initial distribution, bymoving slowly throughthebody,orwill suddenly assume acertain definite distribution. The laws whichregulatethedistribution ofelectricityin equilibriumonbodies invarious circumstances havebeen the subjectofmostimportant experimentalresearches;andhaving been established withperfect precision byCoulomb, andplaced beyondalldoubt byverifications afforded insubsequentex- periments, theyconstitute thefoundation ofanextremelyin- teresting branch oftheMathematical TheoryofElectricity. In connexion with these laws,andbeforestating them,itwillbe convenient toexplainthenature ofthedistinction which is drawn between thetwogreatclasses ofbodies innature, called- Conductors ofElectricity, andNon-Conductors ofElectricity. I^^^Fundamental Laws andPrinciples. 47 Non-ConductorsofElectricity. 67.Abodywhich affords sucharesistance tothetransmis- sion ofelectricity through it,oralongitssurface, that, ifitbe once electrified inanyway,itretainspermanently, without anychangeofdistribution, thecharge which ithasreceived, is called aNon-Conductor ofElectricity. Nobodyexists innature which fulfilsstrictlytheterms ofthis definition; butglassandresin, besides manyother substances, aresuch thattheymay,within certain limits and subjecttocertain restrictions, beconsidered asnon-conductors. Conductors ofElectricity. 68.Averyextensive class ofbodies innature, includingall themetals, many liquids, etc., arefound topossesstheproperty that, inallconceivable circumstances ofelectrical excitation, the resultant force atanypointwithin their substance vanishes. Such bodies arecalled Conductors ofElectricity,sincetheyare destitute oftheproperty, possessed bynon-conductors, of retaining permanently, byaresistance toevery change, any distribution ofelectricity arbitrarily imposed ;theonlykind of distribution which can existunchangedforaninstant ona conductor beingsuch assatisfies thecondition thattheresultant forcemust vanish intheinterior. Itisfound byexperimentthat theelectricityofacharged conductor restsentirelyonitssurface, andthat the electrical circumstances arenot atallaffected bythenature ofthe interior, butdepend solely upontheform oftheexternal conductingsurface. Thus the electricalpropertiesofasolid conductor, ofahollowconducting shell, orofanon-conductor enclosed inanenvelop, however thin(thefinestgold leaf, for instance), areidentical, providedtheexternal forms bethe same.Ahollow conductor never shows symptomsofelectricity onitsinterior surface, unless anelectrified bodybeinsulated within it;inwhich casetheinterior surface willbecome elec- trifiedbyinfluenceorbyinduction, insuch awayastomake thetotal resultant force atanypointintheconductingmatter vanish, bybalancing,foranysuchpoint,theforce duetothe electricityoftheinsulatedbody. 48 OntheMathematical Theory ofElectricity,•[iv. Ithasbeen frequentlyassumed thatelectricity penetrates to afinite depthbelow thesurface ofconductors;and, inaccord- ance with certain hypotheticalideasregardingthenature of electricity,the"thickness ofthestratum" atdifferentpointsof thesurface ofaconductor hasbeen considered asasuitable term with reference tothevaryingoruniform distribution of electricityover thebody.Alltheconclusion with reference to thisdelicatesubjectwhich canasyetbedrawn fromexperiment, isthatthe"thickness,"ifitexist atall,must belessthan that ofthefinestgoldleaf;andinthepresentstate ofscience we mustregarditasimmeasurablysmall. Itmaybeconceived thattheactual thickness oftheexcited stratum atthesurface ofanelectrified conductor isofthesame order asthespace throughwhich thephysical propertiesofthepervadingmatter change continuously from those ofthe solids tothose which characterize thesurroundingair. Electrical DensityatanyPoint ofaCharged Surface. 69.Inthis,andinallthepapers which willfollow, instead oftheexpression"the thickness ofthestratum," Coulomb's farmorephilosophical term, ElectricalDensity,willbeemployed with reference tothedistribution ofelectricity onthesurface ofabody;aterm which istobeunderstoodstrictlyin accordance tothefollowing definitions, withoutinvolvingeven theidea ofahypothesis regardingthenature ofelectricity. The electrical densityofauniformly chargedsurface isthe quantityofelectricitydistributed overaunit ofsurface. The electrical densityatanypointofasurface, whether the distribution beuniform ornot,isthequotientobtained by dividingthequantityofelectricitydistributed overaninfinitely small element atthispoint, bythearea oftheelement. ExclusionofallNon-Conductorsexcept Air. 70.Inthepresent paper,and insome others which will follow, nobodies willbeconsideredexcept conductors; and theairsurrounding them, which willbeconsidered asoffering aresistance tothetransference ofelectricity between two detached conductors, butasotherwise destitute ofelectrical properties. Afulldevelopmentofthemathematicaltheory, oftheinternal electricalpolarizationofsolid orliquid non-con- IV.]Fundamental Laws andPrinciples. 49 ductors, subjecttotheinfluence ofelectrified bodies, discovered byFaraday (inhisExperimentalKesearches onthespecificin- ductivecapacitiesofnon-conducting media), must bereserved foralatercommunication.* Insulated Conductors. 71.Aconductor separated from theground, andtouchedonly byair,issaid tobeinsulated. Insulation maybepractically effected bymeans ofsolidpropsofmatter, such asglass,shell- lac,orgutta percha;-f-and ifthepropsbesufficiently thin, itisfound that theirpresencedoes notinanywayalter or affect the electrical circumstances, and that theirresisting power,asnon-conductors ofelectricity, prevents anyalteration inthequantityofelectricity possessed bytheinsulated body ; sothat however the distribution maybeaffected bythe influence ofsurrounding bodies, itisonlybyatemporary breakingoftheinsulation that theabsolute chargecanbein- creased ordiminished. Ifaninsulated unchargedconductor beplacedintheneigh- bourhood ofbodies chargedwithelectricity,itwillbecome "electrified byinfluence," insuch amanner that itsresultant electrical force ateveryinternalpointshall counterbalance the force due totheexteriorchargedbodies: but, inaccordance withwhat hasbeen stated inthepreceding paragraph,the totalquantityofelectricitywillremainequaltonothing; that istosay,thetwokinds ofelectricity produced uponitby influence willbeequaltooneanother inamount. Recapitulation oftheFundamental Laws. 72.Thelaws ofelectricityinequilibriuminrelation with conductors may—-ifwetacitlytake intoaccount suchprinciples *Theresults ofthisTheory wereexplained briefly inapaper entitled"Note sur lesLois Elementaires deI'Electricit^ Statique" (published,in1845, in Liouville's Journal), andmorefully inthe firstpaperofthepresent series, on the"Mathematical TheoryofElectricity" (ii.above). Asimilar view ofthis subject hasbeen taken byMossotti, whose investigations arepublished ina paper entitled "Discussione Analitica sull'Influenza cheI'Azione diunMezzo Dielettrico hasulla Distributione dell' Elettricita aliaSuperflcie dipiuCorpi Elettrici Disseminati inEsso"(Vol.xxiv. oftheMemorie della Societd Italiana delle Scienze Residente inModena, dated1846). tIthasbeenrecently discovered byFaraday thatgutta perchaisoneofthe bestinsulators among known substances {Phil. Mag., March, 1848). T.E. 4 50 OntheMathematical Theory ofElectricity, [iv. asthesuperpositionofelectrical forces, andtheinvariableness ofthequantityofelectricity onabody, except byaddition or subtraction(intheextendedalgebraicsense ofthese terms)— beconsidered asfully expressedinthethreefollowing pro- positions:— I.Therepulsionbetween two electricalpointsisinversely proportionaltothesquareoftheir distance. II. Electricityresides attheboundaryofachargedconductor. III.The resultant force atanypointinthesubstance of aconductor, duetoallexistingelectrified bodies, vanishes. Ithasbeenproved byGreen thatthesecond ofthese laws isamathematicalconsequenceofthe firstandthird;and ithas beendemonstrated byLaPlace* thatthe firstlawmaybein- ferred from thetruth, inacertainparticular case, ofthesecond and third. Thethree laws were, however,firstannounced by Coulomb, astheresult ofhisexperimentalresearches onthe subject. Objects oftheMathematical Theory ofElectricity. 73.The variedproblems which occur inthemathematical theoryofelectricityinequilibrium maybedivided into the twogreatclasses ofSyntheticalandAnalytical investigations. Inproblemsoftheformer class, theobjectisineach casethe determination either ofaresultant force orofanaggregate electrical mass, accordingtospecialdataregardingdistributions ofelectricity:inthelatter class, inverseproblems,such asthe determination ofthe electricaldensityateachpointofthe surface ofaconductor inanycircumstances, accordingtothe laws stated above, aretheobjects proposed. Ithasbeen proved (byGreen andGauss)that there isa determinate uniquesolution ofeveryactualanalytical problem oftheTheoryofElectricityinrelation with conductors. The demonstration ofthiswith reference tothecomplete Theoryof Electricity (including theaction ofsolidnon-conducting media discoveredbyFaraday),aswell aswith reference totheTheories ofHeat, Magnetism, andHydrodynamics, maybededuced from twotheoremsprovedintheCambridge andDublin Mathemati- calJournal for1847, "RegardingtheSolution ofcertain Partial *[Originally byCavendish, asIlearned after the firet publication ofthis paper. Seefootnote ofMarch 1854on§34above.] IV.]Fundamental Laws andPrincijyles. 51 Differential Equations" (xiii. below, orThomson and Tait's Natural Philosophy, App. A.). The fullinvestigationofanyactual case ofelectricalequi- librium willgenerallyinvolve bothanalyticalandsynthetical problems;as itmaybedesirable, besidesdeterminingthe distribution, tofindtheresultingelectrical force atpointsnot intheinterior ofanyconductor, ortofindthetotalmechanical action duetotheattractions orrepulsionsoftheelements of twoconductors, oroftwoportionsofoneconductor;and besides, itisfrequently interestingtoverify syntheticallythe solutions obtained foranalytical problems. Actual ProgressintheMathematical Theory ofElectricity. 74.InPoisson's valuable memoirs onthissubject,thedis- tribution ofelectricity ontwo electrifiedspheres,uninfluenced byother electric matter,isconsidered;acompletesolution of theanalytical problemisarrived at;andvariousspecialcases ofinterest areexamined indetail withgreat rigor.Inavery elaborate memoir byPlana*, thesolutiongiven byPoisson is worked outmuchmorefully,theexcessive mathematical difficul- tiesinthewayofmanyactual numericalapplicationsofinterest beingsuch astorender awork ofthiskindextremely important. The distribution ofelectricityonanellipsoid (includingthe extreme cases ofellipticand circular discs, and ofastraight rod),andtheresults ofconsequent synthetical investigationsare wellknown. Theanalytical problem regarding anellipsoid subjecttothe influence ofgivenelectrical masses, hasbeen solved byM. Liouville, bythe aidofaveryrefined mathematical method suggested bysomeinvestigationsofM.Lam^ with reference to corresponding problemsintheTheoryofHeat. Green'sEssayonElectricityand hisotherpapers onallied subjects contain, besides thesolution ofseveralspecial problems ofinterest, most valuable discoveries with reference tothe general TheoryofAttraction, andopenthewaytomuch more extendedinvestigationsintheTheoryofElectricitythan any thathaveyetbeenpublished. Glasgow College, March 4,1848. *Tiiriyi Academy ofSciences, tome vii.Scrie ii.published separatelyina quarto volume of333pages:Turin, 1845. 4—2 v.—ONTHEMATHEMATICAL THEORY OFELECTRICITY INEQUILIBRIUM. (Art.XXXVIII. ofcompletelistinMathematical andPhysical Papers, Vol.i.) in.—GEOMETRICAL INVESTIGATIONS WITHREFERENCE TOTHE DISTRIBUTION OFELECTRICITY ONSPHERICAL CONDUCTORS.* [Cambridge andDublin Mathematical Journal, March, May, andNov.1848, Nov. 1849, Feb.1850.] .75.There isnobranch ofphysicalscience which affords a surer foundation, ormore definiteobjectsfortheapplicationof Vmathematical reasoning,than thetheoryofelectricity. The small amount ofattention which thismost attractivesubject hasobtained isnodoubt owingtotheextremedifficultyofthe analysis bywhich even averylimitedprogresshasasyetbeen made; andnoother circumstance could havetotally excluded fromanelementarycourse ofreading,asubject which, besides itsgreat physical importance, abounds somuch inbeautiful illustrations ofordinarymechanicalprinciples.This character ofdifficultyandimpracticabilityisnothoweverinseparable from themathematicaltheoryofelectricity:byveryelemen- tarygeometrical investigations wemayarrive atthesolution *Theinvestigations given inthispaper (§§75—127)form thesubject ofthe firstpartofaseries oflectures ontheMathematical Theory ofElectricity given int]ieUniversityofGlasgow during thepresent session [1847—8].They are adaptations ofcertain methods ofproof which firstoccurred tomeasappli- cations oftheprinciple ofelectncal images, made withaview toinvestigating thesolutions ofvarious problems regarding spherical conductors, without the explicit useofthe differential orintegral calculus. Thespirit,ifnotthe notation, ofthedifferential calculus must enter intoany investigations with reference toGreen's theory ofthepotential, and therefore amore extended view ofthesubjectisreserved forasecond part ofthecourse oflectures. Acomplete expositionoftheprinciple ofelectrical images (ofwhich ashort account wasread atthelatemeeting oftheBritish Association atOxford) has notyetbeen published; butanouthne ofitwascommunicated bymeto M.Liouville inthree letters, ofwhich extracts arepublished intheJournal de Mathematiques (1845 and1847, vols, x., xii.). [See xiv.below.] Afulland elegant expositionofthemethod indicated, together withsome highly interesting applicationstoproblems ingeometry notcontemplated byme,aregiven by M.Liouville himself, inanarticle written with reference tothose letters, and published along with thelastofthem. Icannot neglect thepresent opportunity ofexpressing mythanks forthehonour which hasthusbeen conferred uponme bysodistinguished amathematician, aswellasforthekindmanner inwhich hereceived those communications, imperfectasthey were, and forthefavour- ablemention made ofthem inhisownvaluable memoir. v.]Geometrical Investigations regarding SphericalConductors. 53 ofagreat varietyofinteresting problemswith reference tothe distribution ofelectricityonspherical conductors, including Poisson's celebrated problemofthetwospheres,andothers which mightatfirstsightberegardedaspresentingdifficulties ofafar higherorder. Theobjectofthefollowing paperistopresent, inassimpleaform aspossible, someinvestigationsofthiskind. Themethods followed, beingforthemostpart synthetical,were suggested byaknowledgeofresults founded onalessrestricted view ofthetheoryofelectricity;and itmust notbeconsidered either thattheyconstitute thebest ortheeasiest wayofad- vancingtowards acomplete knowledgeofthesubject,orthat theywould besuitable asinstruments ofresearch inendeavour- ingtoarrive atthesolutions ofnewproblems. Insulated Conducting Sphere subjecttonoExternalInfluence. 76.Wemaycommence with thesimplest possible case, that ofaspherical conductor, chargedwithelectricity andinsulated inaposition removed from allother bodies which could influence thedistribution ofitscharge. Inthis, asintheother cases which willbeconsidered, thevariousproblems,oftheanalytical andsynthetical classes, alluded toinaprevious paper (iv. §73),willbesuccessively subjectsofinvestigation. Thus let usfirstdetermine thedensityatanypointofthesurface, and then, afterverifyingtheresult byshowingthatthelaws(§72) aresatisfied, letusinvestigatetheresultant force atanexternal point. DeterminationoftheDistribution. 77.Letabetheradius ofthesphere, andEtheamount of thecharge. AccordingtoLaw IL,thewholechargewill reside onthe surface, and,onaccount ofthesymmetry,itmust beuniformly distributed. Hence, ifpbetherequired densityatanypoint, wehave E ^~ 47ra'' Verification ofLaw III. 78.Thewell-known theorem, thattheresultant forceduetoa uniformsphericalshell vanishes foranyinteriorpoint,consti- tutes theverificationrequiredinthis case. Thistheorem was 54*OntheMathematical Theory ofElectricity [v. firstgiven byNewton, and istobefound inthePrincipia; butashisdemonstration isthefoundation ofevery synthetical investigationwhich willbegiveninthispaper,itmaynotbe superfluoustoinsert ithere; andaccordinglythepassageof thePrincipiainwhich itoccurs, translatedliterally,isgiven here. Newton, First Book, Twelfth Section, Prop.LXX. Theorem XXX. Ifthedifferentpointsofasphericalsurface attractequally with forcesvarying inverselyasthesquaresofthedistances, aparticle placedwithin the surface isnotattracted inany direction. LetHIKL bethespherical surface, andPtheparticlewithin it.Lettwo linesHK, IL,intercepting verysmall arcsHI, KL,bedrawnthrough P;then onaccount ofthe similartriangles HPI,KPL(Cor. 3,Lemma YII.Newton), those arcs willbe proportionaltothedistances HP,LP] and anysmall elements ofthesphericalsurface at HIandKL,eachbounded allround bystraight linespassing throughP[andverynearly coincidingwithHIC], willbeintheduplicateratio ofthose lines. Hence theforces exercised bythematter ofthese elements ontheparticlePare equal ;fortheyareasthequantitiesofmatterdirectly,andthe squaresofthedistances, inversely ;andthese two ratios com- pounded givethatofequality. Theattractions therefore, being equalandopposite, destroyoneanother: andasimilarproof shows that alltheattractions duetothewholesphericalsur- facearedestroyed bycontraryattractions. Hence theparticlePisnoturgedinanydirection bythese attractions. Q.E.D. DigressionontheDivision ofSurfacesintoEletnents. 79.The division ofasphericalsurface intoinfinitely small elements willfrequentlyoccur intheinvestigations which follow: andNewton's method, described intheprecedingde- monstration, inwhich thedivision isefiected insuchamanner that allthepartsmaybetakentogetherinpairs ofopposite elements withreferencetoaninternalpoint;besides other v.]Oeometrical Investigations regarding SphericalConductors. 55 methods deduced fromit,suitable tothespecial problemstobe examined; willberepeatedly employed. Thepresent digression, inwhich some definitions andelementary geometrical pro- positions regardingthissubjectarelaiddown, willsimplify thesubsequent demonstrations, bothbyenabling us,through theuseofconvenient terms, toavoid circumlocution, andby affordingusconvenient means ofreference forelementary principles, regarding whichrepeated explanations mightother- wisebenecessary. Explanations andDefinitions regardingCones. 80. Ifastraightlinewhichconstantly passes througha fixedpointbemoved inanymanner, itissaid todescribe, or generate,aconical surfaceofwhich thefixedpointisthe vertex. Ifthegeneratinglinebecarried from agiven positioncon- tinuously through anyseries ofpositions, notwo ofwhich coincide, till itisbroughtback tothefirst,theentire lineonthe two sides ofthefixedpointwillgenerateacompleteconical surface, consistingoftwo sheets, which arecalled vertical or oppositecones. Thus theelements HIandKL, described inNewton's demonstrationgiven above, maybeconsidered asbeingcutfrom thesphericalsurface bytwooppositecones havingPfortheircommon vertex. TheSolidAngle ofaCone, orofacompleteConicalSurface. 81. Ifanynumber ofspheres bedescribed from thevertex ofacone ascentre, thesegments cutfrom theconcentric sphericalsurfaces willbesimilar, andtheir areas willbeasthe squaresofthe radii. Thequotient obtained bydividingthe area ofoneofthese segments bythesquareoftheradius of thesphericalsurface from which itiscut, istaken asthe measure ofthesolidangle ofthecone. Thesegmentsofthe samesphericalsurfaces made bytheopposite cone, arere- spectively equal andsimilar totheformer. Hence the solid anglesoftwo vertical oroppositecones areequal:eithermay betaken asthesolidangleofthecompleteconical surface, of which theoppositecones arethetwosheets. 56 OntheMathematical Theory ofElectricity. [v. SumofalltheSolidAngles round aPoint=47r. 82.Since thearea ofasphericalsurface isequaltothe squareofitsradiusmultiplied by 4f7r, itfollows thatthesum ofthesolid anglesofallthedistinct cones which canbede- scribed with agiven pointasvertex,isequalto47r. SumoftheSolid Angles ofallthecomplete ConicalSurfaces=27r. 83.The solidanglesofvertical oroppositeconesbeing equal, wemayinfer fromwhatprecedesthat thesum ofthe solidanglesofallthecompleteconical surfaces which canbe described without mutual intersection, with agiven pointas vertex, isequalto27r. Solid Anglesubtended ataPointbyaTerminatedSurface. 84.The solidangle subtended atapoint byasuperficial area ofanykind,isthesolidangleoftheconegenerated bya straightlinepassing throughthepoint, and carriedentirely round theboundaryofthearea. Orthogonal andObliqueSectionsofaSmall Cone. 85.Averysmall cone, thatis,acone such thatanytwo positionsofthegeneratinglinecontain butaverysmallangle, issaid tobecutatright angles,ororthogonally, byaspherical surface described from itsvertex ascentre, orbyany surface, whether planeorcurved, which touches thesphericalsurface atthepartwhere thecone iscutbyit. Averysmall cone issaid tobecutobliquely, when the section isinclined atanyfiniteangletoanorthogonalsection; and thisangleofinclination iscalled theobliquity ofthe section. Thearea ofanorthogonalsection ofaverysmall cone is equaltothearea ofanobliquesection inthesameposition, multiplied bythecosine oftheobliquity. Hence thearea ofanobliquesection ofasmall cone isequal tothequotientobtained bydividingtheproductofthesquare ofitsdistance from thevertex, intothesolidangle, bythe cosine oftheobliquity. Area oftheSegmentcutfromaSpherical Surface byaSmall Cone. 86.LetEdenote thearea ofaverysmall element ofa v.]Geometrical Investigations regarding Spherical Conductors. 57 sphericalsurface atthepointE(thatistosay,anelement every partofwhich isvery near thepoint E)ylet o)denote thesolidanglesubtended byEatanypoint P,and letPE, producedifnecessary, meet thesurfaceagaininE' :then, a denotingtheradius ofthespherical surface, wehave „2a.co.PE^ ^=—EE^-' For,theobliquityoftheelement E,considered asasection ofthecone ofwhichFisthevertex andthe element E,asection;beingtheangle between thegiven sphericalsurface andanother de- scribed fromPascentre, withPEasradius; isequaltotheangle between the radii,EP andEC, ofthetwospheres. Hence, bycon- sideringtheisoscelestriangle ECW, wefindthatthecosine of theobliquityisequalto^ j^p,orto-^— ,andwearrive at thepreceding expressionforE, 87.Theorem,^ Theattraction ofauniformsphericalsurface onanexternalpointisthesame asifthewhole mass were collected atthecentre. LetPbetheexternalpoint,Cthecentre ofthesphere, andCAP astraightlinecuttingthe sphericalsurface inA.TakeIin/^ \^ CP, sothatOP,GA,CImaybe continualproportionals, and letthe wholesphericalsurface bedivided intopairs ofoppositeelements with referencetothepointI. LetHandH'denote themagnitudesofapairofsuch *This theorem, which ismore comprehensive than that ofNewton inhis firstproposition regarding attraction onanexternal point (Prop, lxxi.),is fully established asacorollary toasubsequent proposition (Prop,lxxiii. Cor.2).Ifwehadconsidered theproportion oftheforces exerted upon two external points atdifferent distances, instead of,asinthetext, investigating theabsolute force ononepoint, and ifbesides wehadtaken togetherallthe pairs ofelements which would constitute twonarrow annular portions ofthe surface, inplanes perpendicular toPC7,thetheorem and itsdemonstration would have coincided precisely with Prop. lxxi. ofthePrincipia. 58 OntheMathematical Theory ofElectricity. [v. elements, situatedrespectivelyattheextremities ofachord HH';and let o)denote themagnitudeofthesolidanglesub- tended byeither ofthese elements atthepoint/. Wehave(§85) H=——TTTTTy andH cosciir coscur Hence, ifpdenote thedensityofthesurface(§69),theattrac- tions ofthetwoelementsHandH'onParerespectively CDIW WIW ^COScm'FH'' PcosCH'I'PH"' Now thetwotriangles PCH,HGIhave acommonangleat(7, and, sincePG :CH ::CH :CI,thesides about thisangleare proportional.Hence thetrianglesaresimilar;sothat the anglesCPHandCHI areequal, and IH_CH_a* 'HP~CP~CP' Inthesamewayitmaybeproved, byconsideringthetriangles PCH', H'CI, thattheangles CPH' andCHI areequal, and that IH'^CH'_aHPCP~CP' Hence theexpressionsfortheattractions oftheelements // andHonPbecome ct) a'* , ft) a^ ^cosCHI'^P' ^cosCH'I'CP' which areequal,since thetriangle HCH' isisosceles; and, for thesame reason, theangles CPH, CPH', which have been provedtoberespectively equaltotheangles CHI,CHI, are equal. Weinfer that theresultant oftheforces due tothe twoelements isinthedirection PC,and isequalto 2ft>./3.^p. Tofindthetotal force onP,wemust take thesum ofall *From thisweinfer thattheratio ofIHtoHP isconstant, -whatever bethe position ofifonthespherical surface, awell-known proposition. —(Thomson's Euclid, VI.Prop. G.) 7.]GeometricalTnvestigations regarding Spherical Conductors. 59 theforcesalongFGdue tothepairsofopposite elements; and, since themultiplierof(oisthesame foreachpair,we must add allthevalues of«,andwetherefore obtain(§83), fortherequired resultant, 4f'7Tpa^ Thenumerator ofthisexpression; beingtheproductofthe densityintothearea ofthesphericalsurface;isequaltothe mass oftheentirecharge ;andtherefore theforceonPisthe same asifthewhole masswere collected atG. Q.E.D. Cor.The force onanexternalpoint, infinitelynear the surface, isequalto47rp,and isinthedirection ofanormal at thepoint. The force onaninternalpoint, however near the surface, is,byapreceding proposition, equaltonothing. Repulsiononanelement oftheElectrified Surface. 88.Let (Tbethearea ofaninfinitelysmall element ofthe surface atanypoint P,andatanyotherpointHofthesurface letasmall element subtend- ingasolidangle co,atP,betaken. Thearea ofthiselement willbeequalto cosGHP' andtherefore therepulsion alongHP,which itexerts onthe element a-atP,willbeequalto pco,per(o2 cosCHP'^^ cosGEPP"^' Now thetotalrepulsion ontheelement atPisinthedirection GP; thecomponent inthis direction oftherepulsiondue to theelement H,is o).pV; and, since alltheconescorrespondingtothedifferent elements ofthesphericalsurface lieonthesame side ofthetangent planeatP,wededuce, fortheresultantrepulsiononthe elementcr, 27r/)V. From thecorollarytothepreceding proposition,itfollows that 60 OntheMathematical Theory ofElectmcity. [v. thisrepulsionishalftheforcewhich would beexerted onan externalpoint, possessingthesamequantityofelectricityas theelement a,andplaced infinitelynear thesurface. Glasgow College, March 14,1848. INSULATED SPHEEE SUBJECTED TOTHEINFLUENCE OFAN ELECTRICAL POINT—(§§89—95). 89.Aconducting sphere placedintheneighbourhoodofan electrified bodymustnecessarily become itself electric, even if itwerepreviously uncharged;since(Law ill.)theentire resul- tant force atanypointwithin itmust vanish, andconsequently there must beadistribution ofelectricityonitssurface which will forinternalpointsbalance theforceresultingfrom theex- ternal electrifiedbody.Ifthesphere, being insulated, bepre- viously chargedwith agiven quantityofelectricity,thewhole amount will(§71)remain unaltered bytheelectrical influence, but itsdistribution cannot beuniform, since inthat case,it would exert noforceonaninternalpoint, andthere would re- main theunbalanced resultant duetotheexternalbody. In what follows,itwillbeproved thattheconditions aresatisfied byacertain assumed distribution ofelectricityineachinstance; butthepropositionthatnoother distribution cansatisfythe conditions, which ismerelyacaseofageneral theorem referred toabove(§73),willnotbespeciallydemonstrated with re- ference totheparticular problems ;although weshallhave to assume itstruthwhen acertain distribution which isproved syntheticallytosatisfythe conditions isasserted tobethe uniquesolution oftheproblem. AttractionofaSpherical Surface ofwhich thedensity vaHes inverselyasthecubeofthedistance fromagiven point 90.Letusfirstconsider thecase inwhich thegiven pointS andtheattractedpointPareseparated bythesphericalsur- face. Thetwofigures representthevarieties ofthis case in which thepoint>Sibeing without thesphere,Piswithin;and, Sbeing within, theattractedpointisexternal. Thesame de- monstration isapplicable literallywith reference tothetwo I*^^Geometrical Investigations regarding SphericalConductors. 61 figures ;but, foravoidingtheconsideration ofnegative quanti- ties,some oftheexpressions maybeconvenientlymodified to suitthesecondfigure.Insuch instances thetwoexpressions aregiveninadouble line, theupper beingthatwhich ismost convenient forthe firstfigure,andthelower forthesecond. Lettheradius ofthespherebedenoted bya,andlet/bethe distance ofSfrom G,thecentre ofthesphere (notrepresented inthefigures). JoinSFandtakeTinthis line(oritscontinuation)sothat (fig.l) SP.ST=f-a^\ ^ (fig. 2)8P.TS =a'-f}^^• Through Tdrawanylinecuttingthesphericalsurface atK,K'. JoinSK,SK\and letthelines sodrawn cutthesphericalsur- faceagaininEE\ Letthewholesphericalsurface bedivided intopairsofop- positeelements with reference tothepointT.LetKandK' beapairofsuch elements situated attheextremities ofthe chordKK\andsubtendingthesolidangle«atthepoint T; and letelements EandE'betakensubtendingatSthesame solidangles respectivelyastheelementsKandK\Bythis means wemaydivide thewholesphericalsurface intopairsof conjugate elements, E,E\since itiseasilyseenthatwhenwe have taken every pairofelements, K,K\thewhole surface will * If,ingeometrical investigations inwhich diagrams arereferredto,the distinction ofpositive andnegative quantities beobserved, theorder ofthe letters expressing astraightline willdetermine thealgebraic sign ofthe quantitydenoted: thusweshould have, universally,ifA,Bbetheextremities ofastraight line,AB=-BA, eachmember ofthisequation being positive ornegative according totheconventional direction inwhich positive quantities areestimated. Inthepi-esent instance, lengths measured along thelineSPin thedirection fromStowards P,orincorresponding directions inthecontinua- tion ofthis lineoneither side, are,inboth figures, considered aspositive. Hence, inthe first figureSTwillbepositive; butwhen/islessthan a,STmust benegative onaccount oftheequation SP .ST=f'^-a^.Hence the second figure representsthiscase;and,ifwewish toexpress thecircumstances without theuseofnegative quantities, wemust change thesigns ofboth members oftheequation, and substitute forthe positive quantity-ST its equivalent TS,sothatwehaveSP .TS=a^-p, asthemost convenient form oftheexpression, when reference ismade tothesecondfigure. Seeabove (Symbolical Geometry, §4),involume oftheCambridge andDtiblin Mathe- vwtical Journal for1848, where theprinciples ofinterpretation ofthesign- in geometry arelaiddown bySirWilliam K.Hamilton[orTait's Quaternions, §20,1868]. 62 OntheMathematical Theory ofElectricity. [Vi have been exhausted, withoutrepetition, bythededuced ele- FlG. 1. Fig. 2. ments, E,E'.Hence theattraction onPwillbethefinal re- sultant oftheattractions ofallthepairsofelements, EE' . Now ifpbetheelectrical densityatE,and ifFdenote the attraction oftheelementEonP,wehave Accordingtothegivenlawofdensity weshallhave _J^P- SE'' where Xisaconstant. Again,sinceSEK isequallyinclined tothesphericalsurface atthetwopointsofintersection, we have(§§85,86) SE'_SE'2aco.TK'E=^SK'KSK' KK' andhence X8E'2a<D.TK^ SE''SK'' Kir=\.2a TK' 0).EF''KK''SE .SK\EP' Now,byconsideringthegreatcircle inwhich thesphereiscut byaplane throughthelineSK,wefindthat (fig. 1)SK.SE=f'~a' (fig.2)KS.SE=a'-f' andhence8K.SE =SP.ST,from which weinfer that the triangles ^OT,PSE Sivesimilar; sothatTK:SK::PE:SP. Hence(2), TK' SK\PE'' andtheexpressionforFbecomes 2a1 SP''' F=^X.KK"SE.SP^,(3). IGeometrical Investigations regarding SphericalConductors. 63 (4).Modifyingthisby(2)wehave (fig.l)F=X.j~,.^,_l,^^^.SK (fig. 2)F=X.^,. ^^^_J.^^^.KS Similarly,ifF'denote theattraction ofF'onP,wehave (fig.l)F'= ^j^'-^f_l^^sF^-^^'' (fig. 2)r=X^.^^^-^.K'S. Now inthetriangles which havebeenshown tobesimilar, the angles TKS,EPS areequal ;andthesamemaybeprovedof theangles TK'S, E'PS. Hence thetwosidesBK,SK'ofthe triangle KSK' areinclined tothethird atthesameanglesas those between thelinePSanddirections PE,PE' ofthetwo forces onthepoint P;andthe sidesSK,SK' aretoone another asthe forces, FyF,inthedirections PE,PE'. It follows, by"thetriangleofforces," thattheresultant ofFand F'isalong PS,andthat itbears tothecomponentforces the same ratios asthesideKK' ofthetrianglebears totheother two sides. Hence theresultant force duetothetwoelements EandE',onthepoint P,istowards S,and isequalto ^2a ft) TTTTt X.2a.ft)X .^>i.w.r-7^^ KT—FT^..KK,orKK''(P- a').SP'''if~a'OSP'' The total resultant force willconsequently betowards S; andwefind,bysummation(§83)foritsmagnitude, X.4!7ra if-d'jSP-'- Hence weinfer that theresultant force atanypoint P, separated fromSbythespherical surface,isthesame asif aquantityofmatterequalto-^ ^were concentrated atthe pointS. 91.Tofindtheattraction whenSandPareeither both without orbothwithin thesphericalsurface. Take in C'>Sf(fig. 3),orinCSproduced through S(fig. 4),a point S^,such that CS.C8.=a\ 64 OntheMathematicalTheory ofElectricity. [v. Then, byawell-knowngeometrical theorem(seenoteon§87), ifEbeanypoint onthespherical surface, wehave SE_f S,E~a'"i I xlence wenaveSE^f.S^E'' Hence, pbeingtheelectricaldensityatE,wehave^i" ^ IS^E' S,E'' I ./. ^\a^ .~' Hence, bytheinvestigationinthepreceding paragraph,the attraction onFistowardsS^,and isthesame asifaquantity Fm. 3. ofmatterequalto->^^ ^were concentrated atthatpoint ;f /i~« beingtaken todenoteCS^.Iffor/^andX^wesubstitute their values, -J.and-^,wehave themodifiedexpression X^.47ra forthequantityofmatter which v/emust conceive tobe collected atS^. 92.Pkop. Ifasphericalsurface beelectrified insuch away that theelectricaldensityvariesinverselyasthecube ofthe distance from aninternalpoint8(fig. 4),orfrom thecorre- spondingexternalpoint 8^,itwill attract anyexternalpoint, asifitswhole mass were concentrated atS,andanyinternal i^Pjieometrical Investigations regarding Spherical Conductors, 65 pointasifaquantityofmattergreater than thewhole mass in theratio ofato/were concentrated atS^. LetthedensityatEbedenoted, asbefore, by-^^.Then, ifweconsider twooppositeelements atEandE'which sub- tend asolidangle(oatthepoint 8,theareas ofthese elements bemg (§96)w^—and-^r^,thequantityofelec- tricity which they possesswillbe X,.2a.oj/l 1\ X.2a.ft) E'E[W^WSJ^"^SE.E'S' NowSE.E'Sis constant (Euc.ill.35),and itsvalue isc^—p. Hence, bysummation, wefind forthetotalquantityofelec- tricityonthesphericalsurface XAira Hence,ifthisbedenoted bym,theexpressionsinthepreced- ingparagraphs,forthequantitiesofelectricity which wemust supposetobeconcentrated atthepointSotS^yaccordingasP iswithout orwithin thespherical surface, becomerespectively m,and^m. Q.E.D. Application ofthepreceding Theorems totheProblem ofElectrical Influence. 93.Prob. Tofindtheelectricaldensityatanypointofan insulated conducting sphere (radius a)chargedwith aquantity Q(either positive,ornegative,orzero)ofelectricity, andplaced with itscentre atagivendistance /fromanelectricalpointM possessing munits ofelectricity. IftheexpressionfortheelectricaldensityatanypointEof thesurface be p= We'-^''<^*)' Xandkbeing constants; theforce exerted bythe electrified surface onanyinternalpointwillbethesame asifthecon- stant distribution k,which(§78)exerts noforce onan internalpoint, were removed; and therefore(§90)willbe T.E. 5 dQ OiltheMathematical Theory ofElectricity. [v. thesame asifaquantityofmatterequalto-'^were collected atthepointM.Hence,ifthecondition X.47ra f^a'^~'^^^ besatisfied, thetotal attraction onaninternalpoint, due to theelectrified surface andtotheinfluencing point,willvanish. Hence thisdistribution satisfies the condition ofequilibrium (§72) ;and tocompletethesolution ofthepro- posed problemitonlyremains tode- termine thequantity k,sothat the totalquantityofelectricity onthe surface mayhave thegivenvalueQ.Now(§92)the total mass ofthedistribution, depending ontheterm^^^3inthe expressionforthedensity,sinceMisanexternalpoint,is equalto aX .^ira Hence, adding ^ira^k, thequantity depending ontheconstant term k,weobtain theentirequantity, which must beequalto Q;andwetherefore have theequation aX.47ra ,a^ r\ /\ f'j2zr^+^™^^Q W- Fromequations {h)and(c)wededuce 47ra 47ra Hence, bysubstitutingin(a),wehave ^~47ra 'ME''^ 47ra^^^^' astheexpressionoftherequireddistribution ofelectricity. Thisagrees with theresult obtained byPoisson, bymeans ofaninvestigationinwhich theanalysis known asthat of " Laplace's coefiicients,"isemployed. 94.Tofindtheattraction exerted bytheelectrified conductor onanyexternalpoint. v.]Geometrical Investigations regarding Spherical Conductors. G7 Wemayconsiderseparatelythedistributionscorresponding totheconstant andthevariable term intheexpressionforthe electrical densityatanypointofthesurface. The attraction ofthe first ofthese onanexternalpointis(§87)thesame asifitswhole masswere collected atthecentre ofthesphere: theattraction ofthesecond onanexternalpointis(§92) thesame asifitswhole mass were collected ataninterior point /,taken inMG sothatMI.MC= a^Hence, according totheinvestigationinthepreceding paragraph, weinfer that theconductor attracts anyexternalpoint with thesame force aswould beproduced byquantities Q+^m,and—^mof electricity,concentrated atthepointsGand/respectively. Cor. Theresultant force atanexternalpoint infinitelynear thesurface isinthedirection ofthenormal, and isequalto 47r/9,ifpbetheelectricaldensityofthesurface, intheneigh- bourhood. 95.Tofindthemutual attraction orrepulsion between the influencing point, M,andtheconducting sphere. Accordingtowhatprecedes,therequiredattraction orrepul- sion willbetheentire force exerted uponmunits ofelectricity atthepoint M,byQ+-:^matGand—^matapoint /, taken inGM, atadistance-^from G,Hence, iftherequired attraction bedenoted byF(aquantity which willbenegative iftheactual forcebeofrepulsion), wehaveMa 4-^+/vA. {B), f\r-aj ^/'(r-«T^^^- Cor. 1,If§bezero ornegative,thevalue ofFisneces- sarily positive, sincefmust begreaterthan a;andtherefore there isaforce ofattraction between theinfluencing point 5—2 68 OntheMathematical Theory ofElectricity. [v. andtheconducting sphere,whatever bothedistance between them. Cob. 2.IfQbepositive,then forsufficiently largevalues of/,Fisnegative,while forvaluesnearly equaltoa,Fist positive.Hence ifanelectricalpointbebroughtinto thai neighbourhoodofasimilarly chargedinsulatedsphere, anal ifitbeheld atagreat distance, themutual action willbe repulsive;ifitthen begraduallymoved towards thesphere, therepulsion,which willatfirst increase, will, afterattaining amaximum value, begintodiminish tilltheelectricalpoint ismoved uptoacertain distance where there willbenoforce either ofattraction orrepulsion ;ifitbebroughtstillnearer totheconductor, theaction willbecome attractive and will continually augmentasthedistance isdiminished. Ifthevalue ofQbepositive, andsufficiently great,aspark willbeproducedbetween thenearestpartoftheconductor andtheinfluencing point,before theforcebecomes changed fromrepulsiontoattraction. StPKTIBB*fl COLLBOE, July 7,1848. EFFECTS OFELECTRICAL INFLUENCE ONINTERNAL SPHERICAL, ANDONPLANE CONDUCTINQ SURFACES. 96.Intheprecedingarticles ofthis series certain problems with reference toconductors boundedexternally byspherical surfaces have been considered. Itisnowproposedtoexhibit thesolutions ofsimilarproblems with reference tothe dis- tribution ofelectricityonconcavespherical surfaces, andon planes. Theobjectofthefollowingshortdigressionistodefine and explaintheprecise significationofcertain technical terms and expressionswhich willboused inthisandinsubsequent papers ontheMathematical TheoryofElectricity. External andInternalConducting Surfaces. 97.Def. 1.Aclosed surfaceseparating conducting matter ^^Geometrical InvestiyaticMS regarding SphericalConductors. 69 within itfrom air*without it,iscalled anexternalconducting surface. Def. 2.Aclosed surfaceseparatingairwithin itfromconduct- ingmatter without itiscalled aninternal conducting surface. Thus, accordingtothese definitions, asolid conductor has onlyone"conducting surface," andthat"anexternal conduct- ingsurface." Aconductor containingwithin itoneormore hollowspaces filled with air,possessestwoormore"conducting surfaces;" namely,one"external conducting surface," andoneormore "internal conductingsurfaces." Acomplex arrangement, consistingofahollow conductor and other conductors insulated withinit,presentsseveral external andinternalconducting surfaces; namely,an"external conductingsurface" foreach individual conductor, andasmany "internal conductingsurfaces" asthere arehollowspacesinthe different conductors. 98.Inanyarrangementsuch asthis, there aredifferent masses ofairwhich arecompletely separated from oneanother byconductingmatter. NowamongtheGeneral Theorems alluded toin§73, itwillbeprovedthat theboundingsur- face orsurfaces ofanysuchmass ofaircannotexperience anyelectrical influence from thesurfaces oftheother masses ofair,orfromanyelectrified bodies within them. Henceany statical phenomenaofelectricitywhich maybeproducedina hollowspace surrounded continuously byconducting matter,— whether thisconducting envelopebeasheet even asthin as gold leaf, oramassive conductor ofanyexternal form and dimensions,—willdepend solely ontheform ofthe internal conductingsurface. 99.Prop. Aninternalconducting surfacecannot receive a charge ofelectricity independently oftheinjtu£nce ofelectrified bodies within it. 100.Thedemonstration ofthisproposition dependsonwhat precedes, andononeoftheGeneral Theorems, alreadyalluded ^^(§73),bywhich itappearsthat itisimpossibletodistribute achargeofelectricityonaclosed surface insuchamanner that *See§70,excludingallnon-condactors except air,orgases. 70 OntheMathematical Theory ofElectricity. [v. theremaybenoresultant force exerted onexternalpoints, and consequently impossible,withmerelyadistribution ofelectricity onaninternal conducting surface, tosatisfythecondition of electrical equilibriumwith reference tothQconductingmatter which surrounds it. Thepreceding proposition (§99)isfullyconfirmed byex- periment (Faraday's Experimental Researches, §§1173, 1174). Infact,thecertaintywithwhich itstruth hasbeenpractically demonstrated inavastvarietyofcases, byallelectrical experimenters, mayberegardedasavery strong partofthe evidence onwhich theElementary Laws asstated above (§72)rest. 101. Itmightbefurther stated that thetotalquantityof electricity produced byinfluence onaninternalconducting surface isnecessarily equalineverycase tothetotalquantity ofelectricity ontheinfluencingelectrified bodies insulated within it.This will alsobedemonstrated amongtheGeneral Theorems;but itstruth inthespecialcasewhich wearenow toconsider, will, asweshall see,beestablishedbyaspecial demonstration. ElectricalInfluenceonanInternalSpherical Conducting Surface. 102. Ininvestigatingtheeffects ofelectrical influence upon anexternal, orconvex, spherical conductingsurface(§§93, 94,95),wehave considered theconductor tobeinsulated and initially chargedwith agiven amount ofelectricity.Inthe present investigationnosuch considerations arenecessary, since, accordingtothestatements inthepreceding paragraphs, itisofnoconsequence,inthecasenowcontemplated, whether theconductorcontainingtheinternalconductingsurface be insulated ornot;and itisimpossibletochargethis internal surfaceinitially,ortochargeitatall,independentlyofthe influence ofelectrified bodies within it.With themodifications andomissionsnecessaryonthisaccount, theprecedinginvesti- gationsareapplicabletothecasenow tobeconsidered. 103. Prob. Tofindtheelectricaldensityatanypointofan internalspherical conductingsurface with anelectricalpoint insulated within it. v.]Geometrical Investigations regarding Spherical Conductor's. 71 Letmdenote thequantityofelectricityinthe electrical pointM;fitsdistance fromGthe centre ofthesphere, andatheradius ofthesphere. Iftheexpressionfortheelectrical densityatanypointEoftheinternal surface be PME'' (\aconstant) ;theforce exerted bytheelectrifiedspherical surface onanypointwithout itwill(§90)bethesame asif aquantityofmatterequalto—^— 7^2were collected atthepoint 31.Hence ifwetake\such tha.t X .4^7ra_ thetotal resultant force, duetothegivenelectricalpointand totheelectrified surface, willvanish atevery pointexternal to thespherical surface, andconsequentlyatevery pointwithin thesubstance oftheconductor;sothatthecondition ofelectrical equilibrium (§72),intheprescribed circumstances, issatisfied. Weconclude, therefore, thattherequired densityatanypoint E,oftheinternalsphericalsurface isgiven bytheequation ^47ra 'ME'^^ This solution oftheproblemiscomplete,since itsatisfies alltheconditions thatcan_possiblybeprescribed,and itis unique,asfollows from thegeneral Theorem referred toin§73.f *Wecannothere, asin(a)of§93,annex aconstant term, since inthis case there would result aforce duetoacorresponding quantity ofelectricity, concentrated atthecentre ofthesphere onallpoints oftheconducting mass. tFor ifthere weretwodistinct solutions there would betwo different dis- tributions onthespherical surface, eachbalancing onexternal points theaction oftheinternalinfluencing body, andtherefore eachproducingthesame force at externalpoints. Hence adistribution, inwhich theelectrical density ateach pointisequal tothedifference oftheelectrical densities inthose two,would produce noforce atexternal points. But,bythetheorem alluded to,nodis- tribution onaclosed surface ofanyformcanhave thepropertyofproducing no force onexternal points; and therefore thehypothesisthat there aretwo distinct solutions isimpossible.Thetheorem made useofinthisreasoningissusceptible ofspecial analytical 72 OntheMathematical Theory ofElectricity. [v. Cor. The totalquantityofelectricity produced bythe in- fluence ofanelectrical pointwithin aninternalsphericalcon- ductingsurface isequal,butoftheoppositekind tothat of theinfluencing point. This follows atoncefrom theinvestigationof§92;from which wealsodeduce theconclusion stated below inthenext section. 104. Theentire electrical force, which vanishes forallpoints external totheconducting surface, may,forpointswithinit,be found bycompoundingtheforceduetothegiven influencing pointM(charged, byhypothesis,with aquantity mofelec- tricity)with thatdue toanimaginary point /,taken inCM produced,atsuch adistance fromCthatCM .CI=a^,and chargedwithaquantityofelectricity equalto—^m. Cor. Theresultant force ataninternalpoint infinitelynear thesurface,isinthedirection ofthenormal, and isequalto 47rp,ifpbetheelectricaldensityofthesurface intheneigh- bourhood. 105.Themutual attraction between theinfluencing point ilf,andthesurfaceinductivelyelectrified willbefound asin §95,providedtheuniform supplementarydistribution which was there introduced beomitted. Hence, omittingtheterm of[B) which depends onthissupplementary distribution; orsimply, without reference to{B),consideringthemutual forcebetween matifand—^mat/,aforcewhich isnecessarilyattractive asthetwo electricalpointsMandIpossess oppositekinds of electricity; weobtain a F=l ^_afm_ &-/) astheexpressionfortherequiredattraction. demonstration (with theaidofthemethod inwhich''Laplace'scoefficients" areemployed)forthecase ofaspherical surface; butsuchaninvestigation would beinconsistent with thesynthetical character ofthepresent series of papers, andItherefore donomore atpresent than allude tothegeneral theorem. Geometrical Investigations regardiiig Spherical Conductors. 73 ''JlectricalInfluenceonaPlane Conducting Surface ofinfinite extent. 106. If,ineither thecase ofanexternal orthecase ofan internalspherical conducting surface, theradius ofthesphere betakeninfinitely great,theresults willbeapplicabletothe presentcaseofaninfiniteplane ;and itisclear thatfrom either ?wemaydeduce thecompletesolution oftheproblemofdeter- miningthedistribution ofelectricity, produced uponacon- ducting plane, bytheinfluence ofanelectricalpoint. The "supplementary distribution," which, inthecase ofaconvex spherical conducting surface, must ingeneralbetaken into account, will, inthecase ofasphereofinfinite radius, be afinitequantityofelectricity uniformlydistributed over a surface ofinfinite extent, andwilltherefore produce noeffect; andthesame results willbeobtained whether wededuce them from thecase of-anexternal orofaninternalspherical surface. 107. LetMbeanelectricalpoint possessingaquantitymof electricity placedintheneighbourhoodofaconductor bounded onthesidenext ifbyaplane LL' which wemust conceive tobeindefi- nitely extended ineverydirection;it isrequiredtodetermine theelectrical densityatanypointEoftheconduct- ingsurface. DrawMAperpendiculartothe plane, and let itslength bedenoted byp.Wemay,inthe firstplace, conceive that instead oftheplanesur- facewehave aspherical conducting surfaceentirely enclosingthe airin whichMisinsulated; and, suppos- ingtheshortest linefrom 31tothe spherical surface tobeequaltop,weshould have, accordingto thenotation of§103, /=a—p. Hence theexpression (A)becomes P=-2ap 47rapm V27r 4!7raJME' 74 OntheMathematical Theory ofElectricity. [v, Inthis, letabesupposedtobeinfinitely great;thesecond term within thevinculum willvanish, andweshallhavesimply p^-^^^^) fortherequiredelectricaldensityatthepointEofthe in- finite planeelectrifiedinductively throughtheinfluence ofthe pointM. Cor. The totalamount oftheelectricity produced byin- duction isequalinquantity, butoppositeinkind, tothat of theinfluencing pointM.Wehave seenalreadythatthesame propositionistrue ingeneralforinternalsphericalsurfaces inductively electrified; but itdoes nothold foranexternal spherical surface, even ifweneglectthe"supplementary distribution," asitappears from thedemonstration of§92, that theamount ofthedistributionexpressed bythe first term(that which variesinverselyasthecube ofthedistance from theinfluencing point)ofthevalue ofpinequation {A) of§93,isequalto—^m.The infiniteplane may,aswe have seen,beregardedasanextreme case ofeither anexternal oraninternalsphericalsurface;andtheproposition which is ingeneraltrue forinternal, butnottrue forexternalspherical surfaces, holds inthislimitingintermediate case. 108.Todetermine theresultant force atanypointintheair, before theconducting plane,itwillbeonly necessary,asin §104, tocompoundtheaction ofthegivenelectricalpoint with that ofanimaginary pointI. Tofind thispoint, wemustproduceMAbeyondJ.toa distance AI,determined bytheequation CM.CI—a^\ which, ifwedenoteAIbyp\becomes(a—p)(a+p)=a^. From thiswededuce ,_op_p a andthence, inthecaseofa=oo,wededuce p=2). Again,forthequantityofelectricitytobeconcentrated at7, wehave theexpression a ,, . 771=m,or,whena—co,m=—??i.a—p v.]GeometricalInvestigations regarding SphericalConductors, 75 Hence theforce atanypointbefore theplanewillbeob- tained bycompoundingthatduetothegivenelectricalpoint Mywith aforce due toanimaginary point I,possessingan equal quantityoftheother hand ofelectricity, andplacedat anequaldistance behind theplaneintheperpendicular MA .produced. 109. Ifreference bemade tothegeneraldemonstration(§90) lonwhich allthespecialconclusions with reference totheeffects tofelectrical influence onconvex, concave, orplane conducting surfacesdepend, weseethat thegeometricalconstruction em- ployedfails inthecase ofasphereofinfinite radius, becoming nugatoryinalmostevery step: wehave however deduced conclusions which arenotnugatory, but,onthecontrary, assume aremarkably simpleform forthiscase;andwemayregardas rigorouslyestablished thesolution oftheproblemofelectrical influence onaninfiniteplane which hasbeen thus obtained. 110. Itisinterestingtoexamine thenugatoryforms which occur inattemptingtoapplythedemonstrations of§§90and 92,tothecase ofaninfiniteplane ;and itisnot difficult to derive aspecial demonstration, freefrom allnugatory steps,of thefollowing proposition. LetLL' beaninfinite "materialplane,"ofwhich the "density"indifferentpositionsvaries in- verselyasthecube ofthedistance from a point 8,orfromanequidistant point >S^^,on theother sideoftheplane. The resultant *-^E. force atanypointPisthesame asifthe whole matter oftheplane were concentrated atS\andtheresultant force atanypoint"^ Pj,ontheother side oftheplane,isthe same asifthewhole matter were collected atS^. 111. Inthecourse ofthedemonstration^ (inthatpartwhichcorrespondstothe in- vestigationin§93)itwouldappear that, ifthedensityatany pointEoftheplaneisgiven bytheexpression _\ 76 OntheMathematical Theory ofElectricity. [v. theentirequantityofmatter distributed overtheinfinite extent oftheplaneisgiven bytheexpression 27r\—m=.P Thispropositionand thatwhichprecedesit*contain the simplest expressionofthemathematical truths onwhich the solution oftheproblemofelectrical influence onaninfinite plane depends, andwemightatonce obtain fromthem the resultsgivenabove. Foranisolatedinvestigationofthiscase ofelectricalequilibrium,thiswould beabetter form ofsolu- tion :butIhavepreferredthemethod given above, since the solution ofthemoregeneral problem,ofwhich itisaparticular case,hadbeenpreviously given. 112.Thecaseofelectrical influence which hasbeen considered *Thetwopropositions maybeanalytically expressed asfollows :— Let0,thepoint inwhich SS^cutstheplane, beorigin ofco-ordinates, and letthis linebeaxis ofz.Then, taking OX,OF intheplane,lettheco- ordinates ofPbe(x,y,z).Letalsothose ofEbe(^, ri,0);sothatwehave \ Hence theproposition stated inthetext(§111), that theentire quantity ofmatter distributed over theinfinite extent oftheplaneisequal to,is thusexpressed:— J-ccJ-00(^2+^2 -J-p2)f p Thisequation maybevery easily verified, andsoanextremely simple analytical demonstration ofoneofthetheorems enunciated above isobtained. Again, theproposition with reference totheattraction oftheplane may, accordingtothewell-known method, beexpressed most simply bymeans of thepotential. This must, invirtue oftheenunciation in§110, beequal to thepotential duetothesame quantity ofmatter, collected atthepoint S,or thepoint Si,according astheattracted pointisseparated from theformer orfrom thelatter bytheplane. Hence wemusthave 27rX ("^r^ M^ T" ]-coJ-^ (^•2+^2+^2)||(^_^).2+ (y_^)2+^2|i ^^2+y2^.(i^+^)2j^' thepositive ornegative signbeing attached tozinthedenominator ofthe second member, according aszisgiven with apositiveornegative value. Thisequation (ofwhich ageometrical demonstration isincluded in§§107and 108, inconnexion with §90)isincluded inaresult(the evaluation ofa certain multiple integral),ofwhich three different analytical demonstrations were given inapaper Oncertain Definite Integrals suggested byProblems in theTheory ofElectricity, publishedinMarch 1847 inthisJournal, vol. ii. p.109(ix.below). v.]Geometrical Investigations regarding Sphe7ical Conductors. 77 niightatfirstsight appeartobeofasingularly unpractical nature, since aconductorpresentingononesideaplanesurface ofinfinite extent ineverydirection would berequiredforfully realizingtheprescribedcircumstances.If,however, wehave aplanetable ofconducting matter, orcovered with asheet of tinfoil, orifwehave awallpresenting anuninterrupted plane surface ofsome extent, theimagined circumstancesare,aswe readily see,approximatelyrealized with reference tothein- fluence ofanyelectrical pointintheneighbourhood ofsucha conducting plane, providedthedistance oftheinfluencing point from theplanebesmall comparedwith itsdistance from the nearestpartwhere thecontinuityoftheplane surface isinany waybroken. FORTBREDA, BELFAST, Oct. 17,1849. INSULATED SPHERE SUBJECT TOTHEINFLTJENCE OFABODY OF ANYFORM ELECTRIFIED INANYGIVEN MANNER. 113. Theproblemofdeterminingthedistribution ofelec- tricity uponasphere,oruponinternal orplane spherical conducting surfaces, under theinfluence ofanelectricalpoint, wasfullysolved in§§89...112 ofthis series ofpapers. On theprincipleofthesuperpositionofelectrical forces(§63) wemayapplythesamemethod tothesolution ofcorresponding problemswith reference totheinfluence ofanynumber ofgiven electricalpoints. 114. Thus letif,M',M"beanynumber ofelectricalpoints possessing respectively m,m\munits ofelectricity,atdis- tances/,/',/"fromGthecentre ofa sphereinsulated andchargedwith a quantity Qofelectricity. The actual distribution ofelectricity onthespheri- calsurface must besuch thattheforce duetoitatanyinternalpointshall beequal andoppositetotheforce due totheelectricityat 3/,M\M".Now ifthere were adistribution ofelectricity on thesphericalsurface such that thedensityatanypointE would beTrrra, the force due tothis atanyinternalpoint 78 OntheMathematical Theory ofElectricity. [v would(§90)bethesame asthatdue toaquantity~ r^J—a concentrated atthepointM\andtherefore ifwetake ^_(/'-a')m 47ra' theforce atinternalpointsdue tothis distribution would be equal andoppositetotheforce duetotheactualelectricity ofM.Wemight similarly expressdistributions which would respectivelybalance theactions ofif',M'\ etc.,upon points within thesphere;andthence, bysupposingallthose distri- butions tocoexist onthesurface, weinfer that asingledis- tribution such thatthedensityatEisequalto ({f-a')m1{r-a')m1{f"^-a')m"1 ) I4™ME'^ 4i7ra M'E''^ 47ra M''E'] would balance thejointaction ofallthe electricalpoints if,if',M'\onpointswithin thesphere. Again, from§92, weinfer that thetotalquantityofelectricityinsuch adis- tribution is fa a,,a, Hence, unless thedatachance tobesuch thatQisequalto thisquantity,asupplementarydistribution willbenecessary toconstitute theactual distribution which itisrequiredto find. Theamount ofthissupplementarydistribution willbe ^a a,a„Q+jm+^m +^,m; which must besodistributed astoproduce noforceoninternal points. 115. Takingthen the distribution found above, which balances theaction oftheelectricityatM,M', etc.,onpoints within thesphere, andauniformsupplementary distribution; andsuperimposingoneonthe other, weobtain aresultant electrical distribution inwhich thedensityatanypointEof thesurface ofthesphereisgiven bytheequation ^ \UirME'^ 47ra M'E'^^^^'] Q-\-^m+-,ni+etc.'^ \-^w•••••«' andwedraw thefollowingconclusions :— .]Geometrical Investigations regarding Spherical Conductors. 70 (1)The total force atanyinternalpoint, due tothis distri- aution andtotheelectricityofM^M', etc.,vanishes. (2)Theentirequantityofelectricity onthesphericalsur- face isequaltoQ. Hence this distribution ofthegiven charge onthesphere satisfies thecondition ofelectricalequilibrium under thein- fluence ofthegivenelectricalpoints M^M\ etc.;and(§73) itistherefore thedistribution whichactuallyexistsuponthe sphericalconductor intheprescribed circumstances. 116. Theresultant force atanyexternalpointmaybefound asintheparticularcase treated in§94.Thus, ifwejoin MG,M'CyM"C,andtake inthelines sodrawn, points /,T,I" respectively,atdistances fromCsuch that CI.CM= Cr .CM'=CI" .CM"=a^ theresultant action duetotheactualelectricityofthespherical surface will, atanyexternalpoint,bethesame asifthesphere wereremoved, and electricalpoints J,/',etc.,substituted inits stead, besides(exceptinthecasewhen thesupplementarydis- tributionvanishes) anelectricalpointatC:andthequantities ofelectricity which must beconceived forthisrepresentation, tobeconcentrated atthesepoints,arerespectively—7.m, atI —-7r,m, SitIf .(2). and +7+2^+etc.,atC 117.Bymeans oftheseimaginaryelectricalpoints wemay giveanother form totheexpressionforthedistribution onthe spherical surface, which inmany important cases, especially thatoftwomutually influencing spherical surfaces, isextremely convenient. For(asin§94,Cor.)itisreadilyseen thatthe firstterm, intheexpressionforpmultiplied by47r,or (f-a')m1 aME" istheresultant force atE,duetoMand/,andthat thisforce isinthedirection ofanormal tothesphericalsurface through 80 OntheMathematicalTheory ofElectricity. [v. E\and that similar conclusions hold with reference tothe other similar terms of(2). Again,thelastterm, ^a a, ,Q\-^.m -\-J,m+etc. 47ra^ istheexpressionfortheforce atE,due totheimaginary electricpoint G,divided by47r;and this force also isinthe direction ofthenormal. Hence, with reference tothe total resultant action atE,due toM,M\ etc.,andthespherical surface, ortheimaginaryelectricalpoints withinit,weinfer (1)That thisforce isinthedirection ofthenormal; (2)That ifRbeitsmagnitudeconsidered aspositiveor negative accordingasitisfrom ortowards thecentre ofthe sphere,andptheelectrical densityatE,wehave "=4^^(«)• These twopropositionsconstitute theexpression,forthe case ofasphericalconductorsubjecttoanyelectricinfluence, ofCoulomb's Theorem,* 118. The total action exerted bythegiven electricalpoints, andbythespherewith itselectricitydisturbedbytheir in- fluence uponagivenelectrified body placed anywhereintheir neighbourhood, might,aswehave seen, befound bysubsti- tutinginplaceofthespherethegroupofelectricalpoints whichrepresentsitsexternal action, providedthere were no disturbance produced bytheinfluence ofthis electrifiedbody. Thishypothesis, however, cannot betrue unless thesphere, afterexperiencingasaconductor theinfluence ofMjM\ etc., were tobecome anon-conductor soastopreservewithrigidity thedistribution ofitselectricity when thenew electrified body isbroughtinto itsneighbourhood:andconsequently, when it isasserted that theresultant force atanyexternalpointPis duetothegroupofelectricalpoints determined inthepreced- ingparagraphs,wemustremember thatthedisturbinginfluence thatwould beactuallyexerted uponthedistribution onthe *Forageneral demonstration ofthistheorem, virtually thesame asthe original demonstration given byCoulomb himself, seeCamhridge Mathematical Journal (1842),vol. iii.p.75(or§§7,8,above). v.]GeometricalInvestigations regarding Spherical Conductors. 81 sphericalsurface byaunit ofelectricityatthepoint P,is excluded inthedefinition(§65)oftheexpression 'Hhe re- sidtant electricalforceatapoint'' 119. Theactual force exerted uponanyone,M,oftheinflu- encing pointsmaybedeterminedbyinvestigatingtheresultant force atM,due toalltheothers and totheconductor, and multiplyingitbythequantityofelectricity, m,situated atthis point,since inthiscasetheinfluence ofthebodyonwhich the force isrequiredhasbeenactually taken intoaccount. 120. Itfollows thattheentire mutual action between allthe givenelectricalpoints andthesphere under their influence isthesame asthemutual action between thetwosystemsof electricalpoints, m2itM^ ra ^^mSitI m'atM' and/m atr Q+:?m+:^ni+etc.,atG. This actionmaybefullydetermined withanyassigned data, bytheelementary principlesofstatics. 121. There isaremarkable characteristic ofthisresultant action whichoughtnottobepassed over, asitisrelated toa veryimportant physical principleofsymmetry,ofwhich many other illustrations occur inthetheories ofelectricityandmag- netism. Itisexpressedinthefollowing proposition:— Themutual action between asphericalconductor andanygiven electrified bodyconsists ofasingle forceinalinethroughthe centreofthesphere. Letusconceive thegivenelectrified bodyeither toconsist ofagroupofelectricalpoints,ortobedivided intoinfinitely smallparts, each .ofwhich mayberegardedasanelectrical point. Themutual action between thegiven bodyandthe conducting sphere under itsinfluence istherefore tobefound bycompoundingalltheforces between thepoints M,M',etc., ofthegiven body,andthepoints /,/',etc....and(7,ofthe imaginary systemwithin thespheredetermined bythecon- T.E. 6 82 OntheMathematicalTheory ofElectricity. [v struction andformulge ofthepreceding paragraphs. Ofthese theforces betweenMand C,between M'andG\etc.;and again, betweenMand/,between M'and/',etc.,areactually inlinespassing through C)and, therefore,ifthere were nc other forces tobetaken intoaccount theproposition would be proved. Butwehave alsoasetofforces betweenMand/ betweenMand /'', etc.,none ofwhich, exceptinparticulai cases, areinlinesthrough G,and, therefore,itremains form todetermine thenature oftheresultant action ofailthese forces. For thispurposeletusconsidei ^,anytwopoints if,M'ofthegivenin- fluencing body and thecorresponding imaginary points /,F;and letustake theforce betweenMandF,andalong with itthe force between /and M', These two forces lieintheplaneMOM', since, bythecon- structiongiven above, /andFarerespectivelyinthe lines CMandCM'; andhencetheyhave asingleresultant. Now theforce inMF isdue tomunits ofelectricityatM,and —TjVfi units atF\and(§64)itistherefore aforce ofre- pulsion equalto a, a, jf^2>^^^force ofattractionequalto"yrjrfT'• a,m .m Similarly, wefind ^ fortheattraction betweenMand I.Now since, byconstruc- tion,CM.CI=GM' .OF,thetriangles FMG, IM'G, which have acommonangleat(7,aresimilar. Hence a' FM'_GF.GM_£_£IM"GFGM'~a ,, , fromwhich wededuce a, a, j.m.m -j/m.m ^ Z^/•=L fS /'if^-^IM"^' Now ifwemultiplythe firstmember ofthisequation by sinGMI', weobtain themoment roundGoftheforcebetween '.]Geometrical Investigations regarding Spherical Conductors. 83 ^'andM]andsimilarly, bymultiplyingthesecond member )ysinCM' I,wefindthemoment oftheforcebetween M'and ^;and, since theangleatMisequaltotheangleatM\we :nferthat themoments ofthetwo forces roundCareequal. ?rom this itfollows thattheresultant oftheforces inMI'and (M'l isaforce inalinepassing throughC.Now the entire Ip:oupofforces betweenpointsofthegiven body andnon- \correspondent imaginary points,consists ofpairs such asthat I^hichwehavejustbeenconsidering ;andtherefore themutual 'iction istheresultant ofanumber offorces inlinespassing \:hroughCThis,compoundedwith theforces between if,M', 3tc.,andthecorresponding imaginary points, andthe forces between M,M', etc.,andtheimaginaryelectricalpointatC, ^ivesforthe totalmutual action afinal resultant inaline passing throughC. 122. Itfollows from thistheorem that ifasphericalcon- ductor besupportedinsuch amanner astobeable toturn freelyround itscentre, orroundanyaxispassing throughits centre, itwillremain inequilibrium whensubjectedtothe influence ofanyexternal electrified bodyorbodies. Wemay arrive atthesame conclusion bymerely consideringtheperfect symmetryofthesphere,round itscentre orroundanyline throughitscentre, without assuming any specificresults with reference tothe distribution ofelectricityonsphericalcon- ductors. For ifthere were atendencytoturn roundany diameterthroughtheinfluence ofexternal electrifled bodies, thesphere would, onaccount ofitssymmetry, experiencethe sametendency when turned intoanyotherposition,itscentre andtheinfluencingbodies remaining fixed; andthere would therefore result acontinuallyaccelerated motion ofrotation. Thisbeingaphysical impossibility, weconclude that the sphere canhavenotendencytomovewhen itscentre isfixed, whatever betheelectrical influence towhich itissubjected. 123. Itisvery interestingtotrace the different actions which, accordingtothesyntheticalsolution oftheproblemof electrical influenceinvestigated above, must balance toproduce thisequilibrium round thecentre ofasphericalconductor subjectedtotheinfluence ofagroupofelectricalpoints.Let us,forexample,consider thecase oftwoinfluencing points. Forfixingtheideas,letusconceive thespheretobecapable 6—2 84 OntheMathematicalTheory ofElectricity. [v ofturning round avertical axis,and lettheinfluencing point besituated inthehorizontalplaneofitscentre, C.Ifatfirs there beonlyone electricalpoint, J/,which wemaysuppoa tobepositive,thesphereunder itsinfluence willbeelectrifie( with adistribution symmetrical round thelineMC, but witl morenegative, or,asthecasemay be,lesspositive, electricity onthehemisphereofthesurface nextMthanontheremot* hemisphere.Ifanotherpositiveelectricalpoint,M\bebrough intotheneighbourhoodofthesphere,onalevelwith itscentre andononeside ortheother ofMG,and ifforamoment w< conceive thespheretobeaperfect non-conductor ofelectricity thissecondpoint, acting ontheelectricityasdistributed unde: theinfluence ofthe first, willmake thesphere tend totun round itsvertical axis. Thus iiAA^beadiameter ofthe sphereinthelineMACA^^thesphere would tend toturn froa itsprimitive positionsoastobringthepointAofitssurfact nearer M'. Ifnowthesphere besupposedtobecome aperfeci conductor, thedistribution ofitselectricitywillbealtered sc astobeno'longer symmetrical roundAA^.This alteratior wemayconceive toconsist ofthesuperpositionofadistribu- tion ofequal quantitiesofpositiveandnegativeelectricities symmetricallydistributed round thelineM'C, with thenega- tiveelectricity preponderatingonthehemispherenearest U M'.Toobtain thetotal action ofthetwopointsonthe elec- trifiedsphere,itwillnowbenecessarytocompoundtheaction ofM',andtheaction ofM,onthissuperimposeddistribution with theactionpreviouslyconsidered. Ofthese theformer consists ofasimpleforce ofattraction inthelineM'C', but the latter, ifreferred toGthecentre ofthesphere,willgive besides asimple force, acouple round avertical axis, tending toturn thesphereinsuch adirection astobringthepointA ofitssurface nearer M.Now, asweknow apriorithat there canbenoresultanttendencytoturnarisingfrom the entire action uponthesphere,itfollows that themoment ofthis couple must beequaltothemoment ofthecontrary couple^ which, aswehave seenpreviously,results from theaction oi M'onthesphereasprimitivelyelectrified under theinfluence ofM.This ispreciselythepropositionofwhich asynthetical demonstration wasgivenin§121,andweaccordinglyseethat that demonstration ismerelytheverification ofaproposition '.]GeometricalInvestigations regarding Spherical Conductors. 85 fwhich thetruth isrendered certain byapriori reasoning lunded ongeneral physical principles. 124.When theinfluencing body,instead ofbeing,aswe lavehitherto conceivedit,afinitegroupofisolated electrical •tunts, isacontinuous masscontinuously electrified, wemust inagineittobedivided intoaninfinite number ofelectrical joints; and then, bymeans oftheintegral calculus, theex- )ressionsinvestigated above may bemodified soastobe ipplicabletoanyconceivable case. 125. Itappearsfromtheconsiderations adduced in§§99,100, hat itisimpossibletohaveaninternalspherical conducting surface, oraninfiniteplane conducting surface, insulated and chargedwith agiven amount ofelectricity; andthat conse- .i^uently,therebeingno"uniformsupplementarydistributions" oobetaken into account, thesolutions ofordinary problems with reference tosuch surfaces aresomewhatsimpler than those inwhich itmaybeproposedtoconsider aninsulated conducting sphere possessing initiallyagivenelectricalcharge. Alltheinvestigationsofthepresent article, exceptthose which have reference tothe"supplementarydistribution"andwhich arenotrequired,areatonceapplicabletocases ofinternal or ofplane conductingsurfaces. 126. Theimportanceofconsideringtheimaginaryelectrical points /,I\etc.(and G,thecentre ofthesphereinthecase of anexternalspherical surface),whether forsolving problems with reference tothemutual forces called into action bythe electrical excitation, orfordeterminingthe distribution of electricityonthespherical surface, hasbeenshown inwhat precedes. Hence itwillbeuseful, beforegoingfurther inthe subject,toexamine thenature ofsuchgroupsofimaginary points, when theinfluencingbodies areeither finitegroupsof electricalpoints,orcontinuouslyelectrified bodies. [SeeXIV. below, orThomson andTait's Natural Philosophy, §§512... 518.] 127. Theterm ElectricalImages,which willbeappliedto theimaginaryelectricalpointsorgroupsofelectricalpoints, LSsuggested bythereceived languageofOptics;andtheclose malogyofoptical images will, itishoped,beconsidered asa mjBficientjustificationfortheintroduction ofanewandextremely convenient mode ofexpressionintotheTheoryofElectricity. Stockholm, September 20,1849. VI.-ON THEMUTUAL ATTEACTION ORREPULSION BE- TWEEN TWOELECTRIFIED SPHERICAL CONDUCTORS. (Art. Lxiv. ofMatJiematical andPhysical Papers, VoLii.) [Philosophical Magazine, AprilandAugust 1853.] 128. Inacommunication made totheBritisti Association at Cambridgein1845, Iindicated asolution adaptedfornumerical calculation, oftheproblemofdeterminingthemutual attraction between twoelectrifiedsphericalconductors. Apaper (ii.above) published inNovember ofthesameyearinthe firstNumber of theCambridge andDublin Mathematical Journal contains a formulaactually expressingthecompletesolution forthecase of aninsulatedsphere andanon-insulated sphereofequalradius (§30,above), andnumerical results calculated forfour dijBferent distances forthesake ofcomparisonwith experimentalresults which hadbeenpublished byMrSnow Harris. The investi- gation bywhich Ihad arrived atthis solution, which was equally applicabletothegeneral problemoffindingtheattrac- tionbetweenanytwo electrifiedspherical conductors, hasnot hitherto beenpublished ;but itwascommunicated inJuly 1849 toM.Liouville, along withanother verydifferent method bywhich Ihadjustsucceeded inarrivingatthesame result, inaletter thesubstance ofwhich constitutes thepresent communication. Formulae marked(8)....(18)inthat letter expressed the details ofthe solutionaccordingtothetwo methods. Theyarereproducedhere interms ofthesame nota- tion,andwith thesamenumbers affixed. Thefirst-mentioned method isexpressed bytheformulae(16), (17), (18),andthe otherby(8). . i .(15). The formulaj marked with letters (a), (6), etc., inthepresent paper, expressdetails ofwhich I hadnotpreserved exactmemoranda. 129. LetAandBdesignatethetwospherical conductors; letaand hbetheir radii, respectively;and letcbethe dis- tajice between their centres. Letthem bechargedwith such quantitiesofelectricity, that,when noother conductors andno :i.] OnElectrified Spherical Conductors, 87 'xcited electrics arenearthem, thevalues ofthepotential* vithin themmaybeuandvrespectively. 130.The distribution ofelectricityoneach surface maybe jetermined withgreat facility byapplyingthe"principle 3fsuccessive influences"suggested byMurphy (Murphy's Electricity, Cambridge, 1833, p.93),anddeterminingthe effect jfeach influence bythemethod of^'electricalimages," given inapaperentitled "GeometricalInvestigations regarding SphericalConductors, "fThefollowingstatement shows as much asisrequiredoftheresults ofthisinvestigationforour present purpose. 131.Letusimagine anelectricalpoint containingaquantity ofelectricity equaltouatobeplacedatthecentre ofA,and another vhatthecentre ofB.Theimageoftheformer inB willbe .ua, atapointinthelinejoiningthecentres, and c 52 distantby—from thecentre ofB.TheimageofthisinAwill ben--''?^<^, inthesame line, atadistance^ j^from the c c centre ofA;theimageofthispointinBwillbea ac— c b 6^-• ita,atadistance »from thecentre ofB:and c— c soon :andinasimilar manner wemayderive aseries of imaginary pointsfrom vbatthecentre ofB.Tospecify com- pletelythese two series ofimaginary points,letp^,p\,p^,p\, p^,p\, etc.,denote themasses oftheseries ofwhich the first isatthecentre ofA;andlet/^, /'j, f^,f.^, etc.,denote the distances ofthesepointsfrom thecentres ofAandBalter- *Thepotential atanypoint intheneighbourhood of,orwithin, anelectrified body,isthequantity ofwork thatwould berequiredtobring aunit ofpositive electricity from aninfinite distance tothat point,ifthegiven distribution of electricity were maintained unaltered. Since the electrical force vanishes at every point within aconductor, thepotentialisconstant throughoutitsinterior. tCambridge andDublin Mathematical Journal, Feb.1850(v.above, §127). 88 OntheMutual Attraction orRepulsion [vi. nately ;and, again,letq^,q^^q^^q^^ ...,denote themasses, and 9t> 9'i>^2'^'2' •••>^^^distances ofthesuccessivepointsofthe other series from thecentres ofBandAalternately. These quantitiesaredetermined byusingthefollowing equations,and givingnsuccessivelythevalues 1,2,3,... :— g^=0, q^=vb7.=o, VI.]between twoElectrified SphericalConductors. 89 andtherefore f^-fs-dt c-fs-i-9t+i G-fs-2-gt+2"' o-f^-g,^,_^-uq,+^1 and ;Similarly, wefind PsPt _Psq o-fs-9\—uq Ps-lP. -_PiPj±?:±.- C~fs~J tC"/«+!~/t o-f\=-Up,^s> and^4^=-»" Now—=— ;and—' and—areeach independent ofuand u V- "'U V V;hence thefollowingnotation maybeadopted conveniently: u , V Pn O y9.n .(13).U V Then, takingntodenote ^+5inthepreceding equations, we have Pn-tPt ^U' . qn-tqt ^'^ Hence wehave 2(14). (cJn-t+l yt)\^J Fn-t+iqt ^n fromwhich weconclude that t=(X) 5=00 n^riPHt 5=1 t=l\0—J,—gt) '^n=lf^nt=l and,byusingthisandtransformationssimilarly obtained for theotherpartsoftheexpressionforF,weobtain n=<xi(„.„,rt=n t=n-l n=l {i^nLt=l t=l ,/2rf=n-i t=n-lS (Q„-A) |...(15). 133.Thequantities P^,Q„,S^which occur inthisexpression, maybedeterminedsuccessivelyforsuccessive values oftiinthe following manner:—Bysubstituting,in(8),forp,,,Pn>qn>2n their values by(13),andeliminating /^,/'^,g^^,g\,wefind 90 OntheMutual Attraction orRepulsion [VI. cP=a8^, +h8, fromwhichwederive'-P„=aQ^,+ bQ„ y(a); P=-a'-b' _o'-a»-y_ ah /Sf.(6). Bygiving wthevalues 1and2in(13)and(8),wefind „2 a6' ^'=^'a6Q.+ s='a6-s:,(c). Bytheseequations wehavedirectlythevalues ofthe firsttwo terms ofeach ofthesetsofquantities P^,P^, P^,etc., Q^,Q^,Qg* etc.,andS^,S^,S^,etc.;andtheothersmaybecalculated suc- cessively bythepreceding equations. 134.Thepolynomials which constitute thenumerators ofthe successive terms ofthesecond member of(15)mayalsobe calculatedsuccessively, bymeans ofequations obtained inthe following manner. Wehaveby(c), (6),and(a), J-.Qn+-P.Q„-+-PaQ...+etc.=iQ„+(?1=^P, +1)Q„.. /c'-a V a\ =^&_.+ab"" c^-a'-b'A--Px <?«-.+etc. andsimilarly wefind ccrc^—a^—b^ ab^(^.^»-.+SA-,+ etc.)-(S,S^,+8,S^+etc.y. VI.]between twoElectrified Spherical Conductors. 91 ^.P»-.+ ^A-.+^s-P„-.+etc. and ^.«„-+ «.«„-.+«,Q»-3+etc. Hence,ifweput «=1 ^=1 and t=lie). interms ofwhich notation theexpression (15)forFbecomes wehave ^,C 0/^,fQ,Ga P',= «U.=c'-a^-¥ ah e-a^-v- ah <7»-(«;-.-^G„)(fl^)- Alsowehavedirectlyfrom{e)and(c), ^'i=i7);''^^ a61 2-'23c^-a=^ a^6'^ -P.= 0,(A). 135. Theseequationsenable ustocalculatesuccessively the values ofB\,8\,S\, etc.,P\,P\,P\, etc.,andQ\,Q\,Q\, etc., afterthevalues of^j,S^jetc., P^,P^, etc.,andQ^,Q^, etc., havebeen found. 92 OntheMutual Attraction orRepulsion [vi. 136.Thesolution of{h)asequationsoffinite differences with reference ton,andthedetermination ofthearbitraryconstants ofintegration by (c),leads togeneral expressionsforS^,P„, andQ,jandbyusingthese in(^),integratingtheequationsso obtained, anddeterminingthearbitraryconstants bymeans of(^),general expressionsforS'^, P'^,andQ^areobtained. TheexpressionforFmaytherefore beputintheform ofan' infinite series, with afiniteexpressionforthegeneralterm. Further, thevalue ofthis seriesmaybeexpressed, bymeans of analysissimilar tothatwhich Poisson hasused forsimilar purposes,interms ofadefiniteintegral.Idonot,however, inthepresent communicationgiveanyofthisanalysis, except forthecaseoftwospheresincontact which isdiscussed below, because, exceptforca^es inwhich thespheresareverynear oneanother, theseries forFisrapidly convergent, andthe terms ofitmaybesuccessivelycalculated withgreat ease,by regulararithmeticalprocesses,foranysetofvalues ofc,a,and 6,byusingfirsttheequations (c),tocalculate>Sfj,8^,P^,Pg, Q^,Qg ;then(6)with thevalues2,3,etc., successivelysubsti- tuted forn,tocalculateS^,S^,etc.,andPg,P^,etc.,andQ^,Q^, etc.;then(A)and{g)tocalculate byasimilar succession of processes,thevalues ofS\,S\,S\, etc.,F\,F\,F^, etc.,and Q'vQ\^Q\>e^c. 137.Thefollowingisthemethod, alluded toabove, bywhich Ifirstarrived atthesolution ofthisproblemintheyear 1845. 138. The"mechanical value"ofadistribution ofelectricity onagroupofinsulated conductors, maybeeasily shown tobe equaltohalfthesum oftheproductsobtained bymultiplying thequantityofelectricityoneachconductor intothepotential within it.*Hence,ifFandFdenote thequantitiesofelec-. tricityonthetwospheresinthepresent case,and ifWdenote themechanical value ofthedistribution ofelectricityonthem, wehave 1^=i(^^+^v). *This proposition occurred tomeinthinking over thedemonstration which Gauss gave ofthetheorem thatagiven quantity ofmatter mayhedistributed in oneandonlyonewayoveragiven surface soastoproduce agiven potential at every point ofthesurface, andconsidering themechanical signification ofthe function ontherendering ofwhich aminimum thatdemonstration isfounded. ItwaspubHshed,Ibeheve, byHelmholtz in1847, inhistreatise Ueber die Erhaltung derKraft, bythetranslation ofwhich, inthe lastnumber ofthe NewScientific Memoirs, agreat benefit hasbeen conferred ontheBritish scientific public. VI.]hetween twoElectrified Spherical Conductors. 93 Now ifthetwospheres, kept insulated, bepushed towards one another, soastodiminish thedistance between their centres from ctoc—dc, thequantityofwork that willhave tobespent willbeF.dCysinceFdenotes therepulsiveforceagainst which this relative motion isaffected. Butthemechanical value of thedistribution inthealtered circumstances must beincreased byanamountequaltotheworkspentinproducing noother effect butthisalteration. HenceF.dG=—dW,andtherefore ^_^MBu^)^^g^_ where uand varetobeconsidered asvaryingwithc,andD andEasconstants. Now, accordingtothenotationexpressed in(13),wehave (i.+1.+ etc.)«- (i+1 +etc.).=D ...(17). du dv Determining -7-and-j-bythe differentiation ofthese equa- tions, andusingtheresults in(16),wefind Thisexpression agrees perfectlywith(/),given above; since, bydifferentiatingtheequations (6)and(c)with reference toc, wefindthatthequantitiesdenoted above byB\,S\,S\, etc., P'j,P\,P\, etc., Q\,Q\, §'3, etc.,andexpressed bythe equations (g)and{h),areequal respectivelyto ^dS, ,dS, ,dS, ,dP, ^dP, ^dP, "do' ""ITc'^do'''do'\do'^dc'®"•' dO^ dQ^if-',etc.^dc^dc^dc' 139. The series(/)or(18)forFbecomesdivergentforthe case oftwospheresincontact, butthedoublyinfinite series fromwhich thiswasderived inthe first ofthetwoinvestiga- tionsgiven above, isconvergent when theterms areproperly grouped together;and itssummaybeexpressed bymeans ofa definiteintegralinthefollowingmanner :— 94 OntheMutual Attraction orRepulsion [VI. 140. Since thetwospheresareincontact, thepotentials within them must beequal,thatis,wemusthaveu=v. For thesake ofsimplicity,letussupposetheradii ofthetwospheres tobeequal, and leteachbetaken asunity. Then weshall havea=6=1,and c=2;andtheterms ofdoublyinfinite series (9)inthis case areeasily expressed,*inverysimple forms, by equations (8).Thuswefind F=v^x 1 2'1.2 32+A24^ 2.12^ 3'"^ 4-^1.4 1.5^-^+^-etc. 2.3 2.4 5'^"^ 6' 3.1 3.2 3.3 +-¥- 5^+etc. —etc. 4.1 4.2, -^2-+-^-etc. 5.1+ (32etc.,{k). Ifweaddtheterms inthevertical columns, wefind 1.2.3 _.... .....^etc,2.3.4 3.4.5 ). 3''¥ which isadiverging series, and isthesame asweshould have foundbyusingtheform(/)or(18). But ifweaddtheterms inthehorizontallines,wefindthefollowing convergentseries fori^:— Liog^.ede ,,iogl.^w ^logl.e^de \ (i+oy *From equations (8)wefind, inthiscase, 2w-2 Hence_,_2n-l ^_Jn~9n—2'Jn— 9r Pn-2n-2n'Pn=2n=2n-l V 2^1 Psqt _{2s-l){2t-l) {c-f.-9t? {2(s+t)-2p p\q't 2s.2t {c-f.-9\? {2(s+«)p P.p't _ q.q't _ 2t{2s-l) {c-f,-f't? ic-g.-g't) {2(s+<)-l}2' andthen,by(9),weobtain theexpression forFinthisparticular case, given in thetext. VI.]between twoElectrified Spherical Conductors. 95 Hence, since(1+6)''=1-20 +SB'-etc.,wehave .log].Odd ^=47TW®' or,byactualintegration, =^^1X(log2-J)=?;^JX(-69315--25) =i;'x-073858. Thequantityofelectricityoneachsphere being equaltothe sum ofthemasses oftheimaginaryseries ofpoints withinit, is,accordingtotheformulaeforp^, q\,p.2, q\,etc., v(1—J+i—J+etc.),orVlog2. Hence wehave thefollowing expressionfortherepulsion be- tween thetwospheres,interms ofQthequantityofelectricity oneach, r,_rt^ ix(log2-i).^^ (log2r• 141. If^denote thedistance atwhich twoelectricalpoints, containing quantities equaltothequantities onthetwospheres, must beplacedsoastorepeloneanother withaforceequalto theactual force ofrepulsion between thespheres, wehave (vAog^y „ OntheMutual Attraction orRepulsion [TI. with their centres atdistances 2'1,2'2,2"3,etc.,upto4,has been undertaken, and isnownearly complete. Glasgow College, March 21,1853. 142. Thefollowingnumerical results havebeen calculated(by means oftheformulae established above)forapplicationtothe theoryofanew electrometer which Ihaverecently hadcon- structed todetermine electricalpotentialsinabsolute measure, from therepulsionsofuninsulated balls intheinterior ofa hollow insulated and electrified conductor, bymeans ofabifilar ortorsion balancebearingavertical shaft whichpasses through asmallaperturetotheoutside oftheconductor :— Table I.—ShowingtheQuantities ofElectricity ontwoequal Spheri- calConductors, ofradiusr,and themutualforcebetween them, whenchargedtopotentials uandvrespectively. Col. 1. between twoElectrified Spherical Conductors. 97 Table II.—ShowingthePotentials intwoequal Spherical OonductorSj and themutual force betweentherriywhen charged withquantitiesDandEofelectricity respectively. Qo\.l. VII.—ONTHEATTRACTIONS OFCONDUCTING AND NON-CONDUCTING ELECTRIFIED BODIES. (Art.VII.ofcompletelistinMathematical andPhysical Papers, Vol.i.) [From theCambridge Mathematical Journal, May 1843.] 144. Inmeasuringtheaction exerted upon anelectrified body, byaquantityoffreeelectricitydistributed inanymanner overanotherbody,themethods followed inthecases inwhich theattracted bodyisconductingandnon-conductingare different. Now, theonlydifference between the state ofa conducting bodyandthat ofanon-conducting body is,that theelectricityishelduponaconducting bodybythepressure oftheatmosphere (toacertain extent atleast),while onanon- conducting bodyitisheldbythefrictionoftheparticlesofthe body. 145.Tofindtheattraction ofanelectrical massE,onanon- conductingelectrified bodyA,theobvious wayistoproceedas inordinarycases ofattraction, consideringtheelectricityonA astheattracted mass. Infindingtheaction onaconducting bodyA,themethod followed istoconsider itselectricityasexertingnopressure upontheparticlesofthebody,butdisturbingitsequilibrium, bymakingthepressureofthe airunequalatdifferentparts ofitssurface. These twomethods ofmeasuring theaction of^on^should obviouslylead tothesame result, since the action must bethesame, whether Abeconductingornon- conducting,thedistribution remaining thesame. Itisthe objectofthefollowing papertoshow thattheydolead tothe same result. 146.Wemust first findthepressureofanelement ofthe electricityof^,ontheatmosphere. Letdsbethearea oftheelement, andpdsitselectrical mass. Letdsformpartofanother element a,indefinitely largerthan dsinevery direction, but sosmall that itmaybeconsidered asplane. Now,ifpabeamaterialplane,itcanexercise no attraction onpds,inadirectionperpendiculartotheplane, and VII.] Conducting andNon-conducting ElectrifiedBodies. 99 itmaybereadily shown that this isalsotrue ifpabeaplate ofmatter ofdifferent densities, arrangedinparallel planes,the thicknessbeingeither finite orindefinitely small, andthelaw ofdensity being anywhatever. 147. Hence, theforceactingonpdsisduetotherepulsion ofallthe electrical mass, except cr;and, since theelectricity onAisinequilibrium under theinfluence ofE,therepulsion actsalongthenormalthrough ds,and isinmagnitude ^irp^ds (seeI.above, §7),which istherefore thepressureofdsonthe air.Hence,ifpbethebarometricpressureoftheatmosphere, thepressureonds,perpendiculartothesurface,is [p— '^irp^)ds. Hence, ifXbethewholepressureonA,resolvedalongafixed lineX'X, and ifvbetheanglewhich thenormalthroughds makes with this line,wehave X=-Jf {p- 27rp^)cosvds, theintegrals beingextended over thesurface ofA.Now, ffpcosvds=0, since thepressureoftheatmospheredoesnotdisturb theequi- librium ofA.Hence, wehave X=27rjfp^cosvds(a), which istheexpressionfortheattraction onaconducting body A,eitherseparate from thebodyonwhichEisdistributed, or connected with it. 148. Toshow that this isidentical with theexpressionfor theattraction ofEontheelectricityofA,letBpds andRpds bethecomponentsoftherepulsiononpds,which areduetoE, andtotheelectricityofA;and leta,a'betheangles which their directions make withXX'. Thenweshallhave 27rpcos J/=-5cosa+i?'cosa'; therefore X=//(jRcosa+R'cosa')ds. Now,jjRcosads istheattraction oftheelectricityofAon itself inthedirection XX\ and istherefore =0.Hence, X=JJBcosads.{b). ButthisexpressionforXistheattraction of^ontheelectricity ofA :[also, themoment roundOX isthesame forthediminution ofairpressureasfortheattraction ofEontheelectricityof A:]andhence thetwomethods ofmeasuringtheaction lead to thesame result. 7—2 VIII.—DEMONSTRATION OFAFUNDAMENTAL PROPOSITION INTHEMECHANICAL THEORY OFELECTRICITY. (Art.XIV.ofcompletelistinMathematical andPhysical Papers,Vol.i.) [From theCambridge Mathematical Journal, Feb. 1845.] 149. Ifamaterialpoint beinapositionofequilibrium when under theinfluence ofanynumber ofmassesattractingitor repellingitwith forces which areinversely proportionaltothe squareofthedistance, theequilibriumwillbeunstable.* The firstthingtobeproved is,that ifthematerialpoint receive aslight displacement,there willingeneral beamoving force called into action. 150. Let bethepositionofequilibrium:Panyadjacent point ;Fthepotentialoftheinfluencing masses, fi,atP,which pointwesupposenottobecontained within anyportionof/t; Uthevalue ofVat0.Now itisshown byGauss, inhis M^moire onGeneral Theorems inAttraction, (alsoinThomson andTait's NaturalPhilosophy, §497,)thatVcannot have the constant valueUthrough anyfinite volume, however small, adjacentto0,withouthavingitforevery pointexternal tofi. But this isimpossible,asmaybeshown inthefollowing manner. Let crbeaclosed surfacecontaining within itaquantityof matter, fi^,consistingofanynumber ofdetachedportionsof/z-, orofthewhole of//-,if//,beacontinuous mass. Let dcrbean element ofa-,andPtheforce due tothe total action of//,, resolved inadirectionperpendiculartoda,which maybecon- sideredpositive when directed towards thespacewithin a.Then, byatheorem demonstrated inthisJournal(seexii.below, §200), wehavejjPdcr=47r/Lt, theintegrations being extended overthewhole ofa.HenceP cannot be=forevery pointofthesurfaceo-,andtherefore V cannot beconstant forallthespaceexterior toyu,. *Thistheorem was firstgiven byMrEarnshaw, inhisMemoir onMolecular Forces, read -attheCambridge Philosophical Society, March 18,1839. See Vol. VII.oftheTransactions. VIII.] Mechanical Theory ofElectricity. 101 HenceFcannot have theconstant valueUforevery point ofanyfinite volume, however small, adjacentto0. 151.Now letasphere Sbedescribed round ascentre, with anyradius a,sufiicientlysmall thatnoportionofjishall be included, and letPbeanypointofthesurface 8,anddsan element ofthesurface atP. Intheequations (3)and(4)ofthearticlealreadyreferred to (xii.below, §199),letthesphere>S^bethesurface there con- sidered;letV=F,andv^=- ,ifOP=r. HenceP^=-gandi;^=- ,atevery pointof8;a a mfjL,^1=1,JJJvdm^=U. Also IhPds=-JJPds=0,a andJJJv^dm=0,sinceSdoes notcontain anyofthematterfi. Wehave therefore, bycomparing (3)and(4)of§199, 0=4i'n-U--JjVds. ThereforeJfVds=4<7ra' U, which shows that themean value forthesurface ofasphere, ofthepotentialofanyexternal masses,isequaltothevalue atthecentre. Let V=V+u. Thereforejjuds=0. 152. Now, ashasalready been shown, ucannot be=for every pointPadjacentto0,andtherefore ifthesphere pass throughapointP"where uisnegative,theremust alsobeapoint P'inthesurface, forwhich uispositive. But ifweassume the potentialofanattracting particletobepositive,thedirection oftheresultant force, resolvedalong anystraight line, willbe thatinwhichVincreases. Hence there willbeaforce towards 0,forpoints displaced along OP',andfrom 0,forpointsdis- placed along OP". Hence ifM,thematerialpointinequili- brium at0,bedisplaced along OP", themovingforcegenerated willtend toremove itfurther from 0,which istherefore an unstableposition. 153.Asanapplicationofthistheorem, letusconsider the 102 MechanicalTheory ofElectricity. [viii. case ofanynumber ofmaterialpoints repellingoneanother accordingtotheinversesquareofthedistance, andcontained in theinterior ofarigidclosedenvelope. Letthesystem bein equilibrium when acteduponbyattractingorrepellingmasses distributed inanymanner without theenvelope. Itwillgenerally bepossiblethattheremaybeapositionor positionsofequilibrium,inwhich atleastsome oftheparticles arenotincontact with the surface. Ifnowwesupposeall theparticlesfixedexcept one,notincontact with thesurface, theequilibriumofthisparticle is,ashasbeen shown, unstable. Hence, generally,theequilibriumofthesystemisunstable if anyoftheparticlesbenotincontact with the surface, and therefore innature theparticlescannot remain insuch aposi- tion. There must, however, besome stablepositionorpositions inwhich theparticlescan rest,butinsuch, alltheparticles mustbeincontact withthesurface oftheenvelope. The sole condition ofequilibriuminthiscase willbethat theresultant forceoneachparticleshallbeinthedirection ofanormal to the surface, anddirected towards theexteriorspace.Ifthe number ofparticlesbeinfinite, andthere beonepositionin which thewhole surface iscovered, there canbenoother in which this isthecase, asisshown inthepaperinthisJournal already quoted (xii.below, §204) ;and itisalsoreadilyseen that thispositionwillbestable, andthatnoother inwhich the surface isnotentirelycovered canbestable. Inthiscasethe particleswillbedistributedaccordingtothelawoftheintensity ofelectricityonthesurface, thespacewithin being conducting matter, andthemasses withoutbeing anyelectrified bodies. Ifamechanicaltheorybeadopted, electricitywillactually be anumber ofmaterialpointswithoutweight,whichrepelone anotheraccordingtotheinversesquareofthedistance. Thus theresult wehave arrived atis,thatthere canbepermanently nofreeelectricityintheinterior ofaconducting body under anycircumstances whatever. 154.If,asmayhappen throughtheinfluence oftheexterior masses, there cannot beapositionofequilibriumofthepar- ticles coveringthewhole surface, there willbeapermanent distribution, inwhichpartofthesurface isuncovered. This, VIII.] Mechanical Theory ofElectricity. 103 however, isnever thecasewithelectricity,asacertainquantity oflatentelectricityisthen decomposed,sothat thewhole surface iscovered withelectricity,eitherpositiveornegative. Alltheabovereasoning would stillapply,ifweconsidered the masses ofsomepointstobenegative,andofsomepositive,and *theforcebetween anytwotobearepulsion equaltothepro- 'duct oftheir masses dividedbythesquareoftheir distance. 155. Sinceevery particleisonthesurface, thewholemedium (ifitcanbeproperlysocalled),willbeanindefinitelythin stratum, thethickness beinginfacttheultimate breadth ofan atom ormaterialpoint.Ifwesupposethese atoms tobemerely centres offorce, thethickness will therefore beabsolutely nothing, andthusthefluidwillbeabsolutely compressible and inelastic. Anythickness which thestratum canhavemust depend onaforce ofelasticity,oronaforcegenerated bythe contact ofmaterialpoints, and ineither case willtherefore require*anultimate lawofrepulsion more intense than that of theinversesquare,-}- when thedistance isvery small, andwe therefore conclude that thiscannot betheultimate lawof repulsioninanyelastic fluid. As,however,allexperiments yetmade serve toconfirm thefactthat there isnoelectricity intheinterior ofconducting bodies, orthat thestratum has absolutely nothickness, weconclude that there isnoelasticity intheassumed electric fluid,andthusthelawofforce, deduced independently bydirectexperiments,isconfirmed. StPeter's College, Jan. 16,1845. *[Note added Jan. 1869.—Thiswaswritten without knowledge ofDavy's ^'repulsive motion," andwithout theslightest idea that elasticity ofevery kind ismostprobably aresult ofmotion. Theconclusions ofthetext are,however, notaffected bytheseviews.] tThis agrees witharesult of]MrEarnshaw.I IX.—NOTEONINDUCED MAGNETISM INAPLATE. (Art.XX.ofcompletelistinMathematical andPhysical Papers,Vol.i.) [From theCambridge andDublin Mathematical Journal, Nov. 1845.] 156. Ifaplateofsoftironbesubmitted totheaction ofa magnetofanykind,itimmediately becomes magnetized "by induction;" andtheeffects ofthisareexhibited intheattrac- tionorrepulsionitexercises uponsmallmagneticbodies inits neighbourhood. Thedetermination ofthese effects, from the elementarylaws ofmagnetic induction,isaproblemofcon- siderablepracticalinterest. Inthecase ofaplatebounded by infiniteparallel planes,Ihave succeeded inobtainingacom- pletesolution ofaverysimple nature, bymeans ofaprinciple which willbedevelopedinafuturepaper (seeabove, §§127, 107,108, 44).Theobjectofthepresentnote istocompare this solution with aformulagiven byGreen inhisEssay on Electricity andMagnetism,asanapproximate result, butAvhich appearstobeinadmissible. 157. Lettheinfluencing magnet, whichmaybeofanyform andsize,andmagnetizedinanymanner, bedenoted byQ;and letussupposeittobeheld behind theplateofsoftiron. The solution which Ihave obtained enables ustofindthe total magneticaction onapoint, P,situated inany position,either within orwithout theplate ;butatpresentIshallonlystate theresultwhenPisbeforetheplate.Inthis case theactual magneticeffect onPmaybeproduced bysupposing Qandthe platetoberemoved, andacertainimaginaryseries ofmagnets Q'iQiiQ^y^tc,tobesubstituted, thesystem being constructed thus. Each oftheimaginary magnetsisequalandsimilar to Q,andsimilarly magnetized ;Q'occupiestheplaceofQ,and theothers aresimilarly placedbehindit,alongalineperpen- dicular totheplate,thedistance betweencorresponding points ofeach consecutivepairbeing equaltotwice thethickness of IX.]Note onInduced MagnetisminaPlate. 105 theplate.The intensities ofthesuccessive magnetsdecrease inageometrical progression,ofwhich thecommon ratio ism^ (aquantity measuring (§45)theinductivecapacityformagnet- ismoftheplate), commencingwith that ofQ',which isequal to1—rn^yiftheintensityofQbeunity.Itishardly necessary topointouttheanalogy between thisandthecorresponding result inoptics,inwhich theilluminationproduced througha plateofglass, byacandle,isfound tobeduetothecandle itself, with diminishedbrightness, and toarow ofimages behindit,with intensitiesdecreasinginageometrical progres- sion,which arisefrom successive internal reflections. 158. Iftheironplatebeinfinitely thin,alltheimages, Q^, Q.^, etc., willcoincide with Q' ;and, since thesumoftheir intensities isunity,thetotal effect willbethesame asthat ofQ,which willtherefore beunaffected bytheinterpositionofthescreen. Thesame willbethecase ifthedistance ofQbeinfinitely great,andthethickness ofthescreen finite;butinthis case, atleast asfarasthepresentresult canshow us,thedimensions oftheplaneswhich bound theplate must beinfinitely great compared withthedistance ofQ. 159. The result which Ihave stated isapplicablealso to theimaginarycase inwhich, instead ofbeingamagnet, Qis amass ofpositiveornegative magnetism.* Thus, letQbea unit ofpositive magnetismcollected inapoint, which case isinvestigated byGreen. Toexpresstheactionanalytically, letQbetaken asoriginofco-ordinates, alineperpendicular totheplateasaxis ofoo,andtheplane throughthis line,and P,asplaneof(cc,y).Thendenoting byathethickness ofthe plate, andconsidering §asapositiveunit ofmatter, weshall have, forthetotalpotentialatP,duetoQandtheplate. *This expression doesnotimply anyhypothesis ofamagnetic matter orofa fluid orfluids, but itismerely used forbrevity inconsequence oftheprinciple established byCoulomb, Poisson, andAmpere, thattheaction ofamagnetized body ofanykind, orofacollection ofelectric"closed currents," mayalways be represented byanimaginary positive andnegative distribution ofmatter, of which thewhole mass isalgebraically nothing. Byanelement ofpositive or negative magnitude, wemerely mean aportionofthisimagined matter. 106 Note onInduced MagnetisminaPlate.[ix. 160. For allmagneticbodiesmisbetween and 1,the former limitbeingitsvalue when theinductivecapacityfor magnetismisnothing,andthelatterbeing never attained, though itisapproachedinsuch bodies asiron, ofwhich theinductive capacityisgreat.Intheextreme case ofm=1,thelaws of induction inamagnetic body degenerateintothose ofelectrical equilibriumonthesurface ofaconductor ofelectricity.Ifin theexpressionforFweputm=1,oneofthefactors vanishes andtheother becomes infinite, buttheultimate value ofthe productisnothing, which shows thatthe effect oftheplateis todestroyallaction behind it.Thisweknow tobethecase when aninfiniteconductingscreen ofanyform isplacedbefore anelectrified body. 161. Inthecasewhen theplateisofiron,thevalue ofmis nearly unity. Hence, astheseries ismultiplied by1—m^ it mightbeimagined that, ifwe" neglectsmallquantitiesofthe order(1—g)compared with those which areretained," (1—^ being,inGreen's notation, aquantityofthesame order as 1—m),anapproximateresult would beobtained byputting wi=1inthesuccessive terms ofthe series within thevin- culum. And itisthus thatGreen, having,intheinvestigation, neglected quantities multiplied by(1—^)^arrives attheresult, 4(l-gr) f1 1 ,1^.1 3{(^•-'+/)i"^ {(oj-f2a)^+2/^p"^ {(^+4a)*^+2/^ji'^^'''•J• As,however, this series hasaninfinite sum,itisclear thatno value ofmcanbesufficientlynear tounitytorender the approximationadmissible. Ifinstead ofQwewere tosub- stitute amagnet,oranycollection ofpositive andnegative particles,such that thesum ofthemasses iszero, theseries forthepotential, deduced from Green'sexpression,would con- verge:andthesame remark isapplicabletotheseries which would befound fortheattraction ofthesystemonapoint beyondthescreen, evenwhen^isapositive point, bydiffer- entiatingtheexpressionforF.Notwithstanding this, the approximationisstill inadmissible; since,ifweexpandthe rigorous expressionineither case inascending powers (1—w), wefind that,thoughthe firstterm isfinite, thecoefficients of alltheterms which follow itareinfinite. f^X.]Note onInduced MagnetisminaPlate. 107 162. Althoughthemethod bywhich Iobtained therigorous -solution isquitedistinct from that followed byGreen, being ndependentofanymathematicalprocess,itmaybesatis- :actorytoshow that theresult canbededuced from hisown analysis,andeven withgreatereasethan hissolution isob- ^»ined aftermaking unnecessary approximation. By*averyremarkableinvestigation,inwhich heextends fLaplace'swell-knownanalysisforsphericalco-ordinates tothe 3asewhen theradius ofthesphere becomes infinite, Green arrives{EssayonElectricity, p.64)atthefollowing expression forthetotalpotentialatP,due tothepositiveunit ofmatter Q,andtotheinterposed plate,before making anyapproxima- tion :— Letm=y-^.Thenwehave,byexpansion,andbychanging theorder oftheintegration, c^7.6-^^(1+m'e-^^+m*e-^Y«+etc.)cos{^r^y) (X mXa;+2a) m^{x+4ta) \ V+/3y'^ {x+^af+^y'^{x+ 4^af+^y*"^*^- J 2/-, 2^v f^''m^%de,=-(1—m)2,— 2.. ...„ ,where Xi=x+zia, =(1—m^)%7—z TT i whichagreeswiththeexpression given above. StPeter's College, Oct. l^th, 1845.Jo I X.—SUEUNEPROPRIETE DELACOUCHE ELECTRIQUE EN EQUILIBRE ALASURFACE D'UNCORPS CONDUCTEUR. ParM.J.LiouviLLE. (Art.XXIV. ofcompletelistinMathematical andPhysical Papers, Vol.i.) [From theCambridge andDublin Mathematical Journal, Nov. 1846.] 163.Lam^thode laplus g^n^rale queTonconnaissepour- former descouches^lectriques,enequilibrealasurface de corps conducteurs, consiste aconsiderer unemasse ilf;etle potentiel, y._ ff[f(ps,' /,z')dx'dydz' decette masse, parrapport aunpoint quelconque (a;,y,z)j dont ladistance aupoint {x\y,z),ouaI'element f{x,y\z)dx'dy'dz\ estd^sign^e parA.Prenons ensuite unesurface deniveau ou d'equilibre relativement aI'attraction delamasse if,etqui entoure cette masse, c'est adireprenons unesurface ferm^e (A),contenant lamasseMdans sonintdrieur, etpourtous les points delaquelle Vconserve unevaleur constante. Enfin dV soit-T-ds lavariation infinimentpetite queV^prouve lorsqu'on passed'unpoint decette surface aunpointexterieur infini- ment voisin situd surlanormale aunedistance ds. C'est la dV d^rivde-T-, multipli^esiTonveutparune constante, qui regleralaloidesdensites deI'electricit^ enequilibresurun corps conducteur termine parlasurface(A).Plusieursgeo- metres sontparvenus,chacun deleur cot^,acebeau tb^or^me; mais c'estGeorgeGreenqui I'a,jecrois, donn^ lepremierdans unexcellent memoirepublieen1828, sous cetitre :AnEssay ontheApplication ofMathematical AnalysistotheTheoriesof Electricity andMagnetism.Jemeproposedemontrer quela coucbeelectriqueenequilibreainsi obtenue apr^cisementle meme centre degravity quelamasse iHf. 164.Pla9ons I'originedescoordonnees x,y,z,aucentre de.' X.] Propri^tedelaCouche MectriqueenEquilihre. 109 gravitydelamasseM;etd^signons parx^unequelconquedes coordonn^es ducentre degravitydelacouche^lectrique, laquelle jsera fournieparlaformule ''//?'^"=/f?'^"' *6illesintegrations s'appliquenth,lasurface(A)dontr^ldment estrepresent^ pardad. IIs'agitdeprouver quex^—0. D'apres I'expressiondeF,ona d'V d'Vd'V,^, ^^+^+^=-^^'^(^'^'^)'"^=^^ suivant quelepoint (x,y,z)appartientounonalamasse M. Pourplusdesimplicity,ecrivonstoujours (fFd'V ^d'V,.,. enregardantlafonctionf{x, y,z)comme nulle hors dela masseM;etcombinons cetteEquationavec cette autre deforme analogue^^^_ dx''"^ dy'^ dz'~^' oilnoussupposons queUestunefonction dex,y,z,quireste finie etcontinue ainsi quesesd^riv^es dans toutTespace intdrieur a(A).Nous aurons ^ dx'^dx'^^ dfdf^^dz'^ dz'^^^j ^^^y^^^' Multiplions pardxdydz,etint^gronsdans toutTespace interieur a(A).Enconservant acZsetadwlamemesignifi- cation que ci-dessus, ontrouve, apresdestransformations bien connues : W^^^"" ~\\^^s^"^"^'^JJIWi^^ y^^)dxdydz, Maisr^quationenUestsatisfaiteparU=x;nous avons done: L'intdgrale triple dusecond membre, divisdeparM,donne I'abscisse ducentre degravitydelamasse M.Cecentre ^tant ^I'originedescoordonn^es, I'int^graledont nousparlousest aussi.110ProprietydelaCoucheElectriqueenEquilihre [x. nulle. Jevaisprouver queTint^grale jlV-j-d(oTest D'abord onpeutfaire sortirVdusigne /,puisque,surlasur- dx face(A),Vestconstant. Observons ensuite que -7-apour valeur lecosinus deTangleaquelanormale dsfaitavec I'axe des X.Notreint^graledeviendra done :VfJcos (xdo). Or I'integrale //cosadco estnulle, d'apres untheoreme connu, comme composded'^l^ments deux adeux^gauxetdesignes Cfdx contraires. Ainsi jjV-j-dco=0.IIrestedone finalement //•oj-j-da)— 0, etTonenconclutaj^=0,cequ'ilfallait ddmontrer. TouL, 4Juillet 1846. NOTEONTHEPEECEDING PAPER ByWilliam Thomson. [Extracted fromaLetter toM.Liouville.] 165. ". ..Thedemonstration which youhavegivenhasledme tothisother theorem, thatthemassM,andtheshellsurround- ing it,have thesameprincipal axes, through anypoint. Todemonstrate this, letU=yzintheformula which you havegiven. Then, since, ifwedenotebyKtheconstant value ofVattheshell,wehave //-f-^/Zf-' //'wefind r—dV Iyz-^day=^iirjjjyz .f{x, y,z)dxdydz (1), whichprovesthepropositionenunciated. IfwetakeU=a?^wefind *Seexii.below, §200, (8). X.] dlaSurface61unCorpsConducteur. Ill 'from which, observingthat .=2KJJJdxdydz; wededuce 47ryP'"?^"'= i^^-^(^"^)dxdydz+JUx^fi^, y,z)dxdydz. Let^,5,(7bethemoments ofinertia ofthemassJfround theaxes ofco-ordinates, andA^,B^,G^,those oftheshell,round thesame axes,itbeing supposedthatthequantityofmatter of theshell isthesame asthatofif;* thepreceding equation,and thetwoothers whichcorrespond relativelytotheaxes ofyand z,arewith thisnotation, A^=Q+A,B^=Q +B,C,=Q+C(2),t where §,=—- fjj[V—K) dxdydz, isaquantity which isindependentofthepositionofthe origin. Fromequations (2),wehave B-G^B^-C^, G-A=G-A^,^-5=^,- J5,...(3). Ademonstration ofyourtheorem and ofthetheorems ex- pressed bytheequations (1)and(3)maybearrived atbycom- paringtheexpressionsfortheequal potentials Jproduced bythe massM,andtheshell atverydistantpoints."|| StPeter's College, July 15,1846. *Inthiscasethe"density" ofthedistribution atanypoint oftheshell 1-dV willbeequal to7—.—-— .See i.above, §7. 'iir ds ^ tIftheorigin betaken atthecentre ofgravity, andtheaxes ofco-ordinates principal axes of31(and therefore ofthe shell, according totheproposition enunciated above), these equations show thatthe"central ellipsoid" (seenote top.202ofCambridge andDublin Mathematical Journal, 1846)fortheshell is confocal withthat forthebodyM. XAshell constructed round themassM,inthemanner described byM. Liouville, with aquantity ofmatter equal toM,exerts thesame forceupon points without theshell, aswasproved firstbyGreen (seealso i.above, §9);andsince thepotential ofeach vanishes ataninfinite distance,itfollows that thetwo bodies produce equal potentials atevery point without theshell. li[SeeThomson andTait's Natural Philosophy, §539.] XI.—ONCERTAIN DEFINITE INTEGRALS SUGGESTED BY PROBLEMS INTHETHEORY OFELECTRICITY. (Art.XXVIII. ofcompletelistinMathematical andPhysical Papers, Vol.i.) [From theCambridge andDublin Mathematical Journal, March 1847.] 166. Itfollows from thesolution oftheproblemofthe dis- tribution ofelectricityonaninfiniteplane,* subjecttothe influence ofanelectricalpoint,that thevalue ofthedouble integral, J—COJ-00 IS{(I-^f+('?-yf+^^1*{(S-^y+{^1-y'f+z'^] 27r ^[{x-x'f +{y-yy +{z+zjf Adirectanalyticalverification ofthisresult istherefore interest- inginconnexion with thephysical problem.Inthefollowing paperthemultiple integral /"<"/'°° ud^,d^,...d^, isconsidered, and itsvalue isshown tobe n(«+1){(^1-<y+(^.-<f+ ...+(w+uy}i(^-^)' aresult ofwhich theonementioned above isaparticularcase. Several distinct demonstrations ofthistheorem aregiven,and some other formulae, which have occurred tomeinconnexion withit,areadded. 167.The firstpartofthefollowing paper, which isatransla- tion,withslight alterations, ofamemoir inLiouville's Journal,f contains ademonstrationsuggestedtomebyamethod followed byGreen inprovingtheremarkable theorem inArt.(5)ofhis Essay onElectricity.Inthesecondpartsome formulae are given which, inthecase oftwo variables, aresuch aswould *Seeabove, §111, footnote. +1845, p.137,"Demonstration d'unTh6ordme d'Analyse" (April 1845). XL]Problems intheTheory ofElectricity. 113 occur intheanalysisofproblemsinheatandelectricity, with reference toabodybounded inone direction byaninfinite plane,ifthemethods indicated byFourier were followed;and fromthem thevalue ofthemultiple integral mentioned above isdeduced. In§ill.theevaluation iseffectedbyadirect processofreduction, suggested bygeometricalconsiderations*. PART I. 168. Letthevalue ofthemultiple integral, which,ifwe useaveryconvenient notationanalogoustothat offactorials, maybewritten thus, rrr ra! bedenoted byU. Letw+w'=a,itbeingunderstood thatuanduaretaken as positive. Then,ifweassume {2(f-xf+2;^}*(*-i) {S(^-xf+(2w-vf 1R'= wehave{Z{i-xy +{a-vfY^^-^)(2), -2{s-l)uU= "/:s^JlR'-p[d^Y,when v—u. Itiseasilyseen that thesecond member ofthisequation vanishes whenv=±co ,andthat itdoes notbecome infinite, evenwhen one ofthevalues 0,2w,oraisassignedto u. Hence thepreceding equation maybewritten Butwehavevehave /[rjfs«*=[/:.]'/ff*[*i- =[f:j'''f™-/[/:]-.^jHi.* When wetake theintegralwithrespecttovbetween the *See"Extrait d'une lettre aM.Liouville, etc." Liouville's Journal^ 1845, p.364(xiv. §210,below).k T.E. 114 Once7i^ainDefinite Integrals suggested hy [xi. limits—00and u,the firstterm vanishes, since ateach limit jK=0.Thus thepreceding equationisreduced to 169.Nowwehave~^-^+1--^2~= 0, forallvalues offj,fg..., providedvbenotequaltoa.Hence thisequationissatisfied for allthevalues ofthe variables between the limits oftheintegrationintheprecedingex- pression,andwemaytherefore employittoeliminate-t-t' wethusobtain -2(.- l)uU=\l [rj'(E'f+Rtf)mdv. Takingoneoftheterms ofthesecond member, andintegrating byparts,wehave -/:[f:]'-(/->f^^.)™-'* =-/:[/:]"(/:.ff<'f.)™-* rurr® j—on j_c since theintegrated partsvanish ateach limit. Byapplying asimilarprocesstoeachtermunder thesign ^,wefind -2{s-l)uU/:.[/:'fcPR^d^R\ dv. But,ifwedenote byQand Q'thetwopartsofR,inequation (1),sothatR=Q—Q\wehave d^.^d^;^ dv'"^^ d^' for allvalues ofthevariablesv,^j,etc.,within thelimits of integration ;hence there remains -2(n-l)uU=Jl [L]""'S+^f)^'^^"^- Todetermine thevalue ofthisexpressionitmayberemarked CI.]Problems intheTheory ofElectricity.115 hatthequantity under theintegral signsvanishes for all rallies ofthevariables which differsensiblyfrom those ex- pressed by mdmoreover, that ifweconsiderseparatelytheterms ofthe \second member, each isfound tobeaconverging integral:it !follows that,ifwedenote byPthevalue which R'receives when thevariables have these valuesassigned, wehave I_2(.-l)«fr=pj]J...(^+2g)«?.(Z?,...C(3)- where the limits ofintegration must besuch astoinclude thevalues 0,x^,x^, etc.,butareotherwise entirely arbitrary. Byconsidering separatelythe different terms ofthisexpres- sion,andintegratingeachwithrespecttothevariable towhich itisrelated, withoutyetassigningthelimits oftheintegration, wefind -2(s-l)uU=p[jf...^dldi,...+11^M^,+ eto)(4). 170. Letusnowassume ^i= i\+oc^,^^=v^+x^, etc., andv^-i-v^^^ ...+vf=r\ fromwhichwehave ^_J^ dQ_5-1 dQ_s-1 Theintegrationsinequation (3)maybeextended toallthe values ofthevariables whichsatisfythecondition andthelimits in(4)willthen besuch astoinclude allthe values whichsatisfytheequation ^'+<+<+...+^/=«'.orr'=a^etc. Ifintheintegrations weonlytake thepositivevalues ofthe variablesv,v^,v^,etc.,whichsatisfythelimiting condition, we mustmultiplyeachintegral by2*+^;andwemaythensimply take, inthesuccessive terms thesecond member of(4), dQ__Sj-l dQ__s-l dv a«+i^'d^^~a*+i^^'^*'^- Thuswehave 8—2 116 OncertainDefinite Integrals suggested hy [XI. uU-.2^P (//... vdv^dv^ .,dv^-\-ff... v^dvdv^ ...dv^+etc.)^ ?^Sl±^JJ...(a^-v,^-v:-...-vf)idv,dv,...dv, =(«+!) p//...(i-k-k...-i:)n-n-K..i-Hkdk...di,- inwhich lastexpressionthelimits include allpositivevalues satisfyingthecondition ?,+?,+...+Z,51. Hence, byLiouville*s theoremf, u^=(^+i)^?i//i-^)*^*'-''^*= r4(.+i whichgivestherequiredvalue oftheintegralU. 171. Ifwedenote byTTanyintegral correspondingtoZ7, inwhich thesystemofvariables u,x^,x^...andu,x\,x^... areinverted, weshall have uU=uU\sincePisafunction symmetricalwithrespecttothetwosystems; andwethere- forededuce from thepreceding result, [d^Y —uJ—CO J—CO.'[Ml {S(f-xY+uy^'+^) [t{^-xY+u^'-^) 1.(5). ~ TK^+1){t{x-xj-\-{u+u)Y'-^^ 172. Ishalladdanother demonstration ofthistheorem, as anapplicationofsome remarkableanalysis given byMrGreen inhismemoir ''Onthedetermination oftheexterior andinterior attractions ofellipsoidsofvariable densities|." u[d^YLetF= [/:]' (2(1-xj+1*^)^(^+1) {S(f-x'Y+'u'f('-^y"'^^^' anintegralwhichmayalsobeexpressedthus : n-\du\\j-^\ {t(i-xy +^^^p-i){S(f-^')' +^'2}K*-i) J *[Byputting,inthis, v=fdv ;Vj=fdvj^;etc.,wehave 2'PuU=-^1 {s+l)fff...dvdvj^dv^.-I a"^""1 whence immediately, byasimpler case ofLiouville's theorem than inthetext, orbyGreen's transformation(see §186),thesame result.] +SeeGregory's Examples (Ed. 1841), p.469. XRead attheCambridge Phil. Soc,May 6,1833. SeeTrans, ofthat date. XI.]Problems intheTheory ofElectricity. 117 From thislatter form,weseethattheequation ^^+^^=^^7) issatisfied, provided udoesnotvanish. HenceFisafunction which satisfies thisequationforallvalues of^j, iCg•••and for allthevalues ofubetween and oo .Atthese limits the value ofVmaybeeasily determined, andthegeneralvalue inferred inthefollowing manner :— 173."When u=0,thequantity under thesignsofintegration intheexpressionforVvanishes for allthevalues off^,f^... which arenotequaltocc^^ o)^...respectively. Hence itfollows that,when u=0, V.' [/:u[d^J ' dz^dz^ ...dzg t{t{oo^^W+^'¥'-'^ JoJo••• (1+h+l,+...+ ls)i^'+^^ 1 ^isr^h¥-^dh1 rr i-n-K..di,di^ {%(x-xj+w'^)K^-i) T{is)Jo (1+h)^^'+^) Ti{s+1){Z{x-xJ+uy^'-^^' Also,when w=oo,thevalue ofVisnothing. 174.ThusweseethatVhasthesame value astheexpression rj(5+1)• {X{x-xy+{u-h u'Y]^^'-^^' when u=0,andwhenu=oo;which enables ustoinfer that 7ri(«+i) 1^" ri(7+ 1)']X{x- xy+(u+uy}^^^-^)' forallpositive values ofu,providedubetaken aspositive ; forthesecond member ofthisequationsatisfies equation (7) forallpositive values ofu,and foranyvalues oftheother variables, and atthelimits u=andm=oohasthesame value asV,and therefore, byatheorem ofGreen's*, inthe memoir referredto,must beequaltoVforallpositivevalues ofw. *[IncludedinTheorem 2ofxiii. below.] il8 OncertainDefinite Integrals suggested by [xi. 175.From what hasbeenproved above wemaydeduce the solution ofthefollowing problem:— Having givenforallvalues of^j,fa•••'*^®value ofthe multiple integral pdx^dx^...dx; {(^.- 1.)^+«-i/+ ...+(^/-h?+^^'1*^^-')•••^^' where u'andpareanyunknown functions ofx^,^2'•••^«''^^^^^ berequiredtofindthevalue of Qp'dx^dx^...dxs wherex^,x^...Xsareanygiven quantities,anduagiven positive quantity. Denotingtheexpression {a)by <J>,andtheexpression (6)by 0,wehave, from thetheorem established above, ^=^ IhtX)' ^pd<dx;...dx;. [d^y[Q ffT___MSl___ Qpdx;dx^...dxs z^ri(g+i)rrT [^?? apdx;dx^...dxs ^[^flV - ,...(0). But,byhypothesis, ^isgivenforallvalues off^,fg--- f»> andtherefore thisequation expressesthesolution oftheproblem. Wemayalsodeduce from thetheorem(5)theexpression bymeans ofwhich<^maybedetermined when thevalue, ^,of -^correspondingtot^= isgiven. 176. Fortheparticularcase ofu=0,thetheorem{d)isin- cluded inatheoremgiven byGreen, inwhich thenumber nin theexponentofthedenominator maydiffer from thenumber s ofvariables, thesole conditionbeingthat71—5+1 must be positive;but itisonlyinthecase ofw=sthatageneral theorem such as[d),bymeans ofwhich thegeneralvalue ofj> isobtained from thevalue-^whenu=0, canbeestablished.du XI.] Problems intheTheory ofElectricity. 119 177. Letusnowapplythese formulae tothecase of5=2: wemayinthiscaseconveniently replace x^,x^,uhy x,y,z,and fi'?2'^yftV'Equations (c)and(d)become .^±_r r ^^dri where "^denotes thevalue of-^whenx=f,y= 1^],z=0. 178.The firstofthese theorems maybededuced from avery generaltheoremgiven byGreen inhisessay onElectricityand Magnetism [§(5)eq.(6)]. Thesecond maybedemonstrated in thefollowing manner:— LetX,y,zbeconsidered astheco-ordinates ofapoint P', where there issituated aquantityofmatterpdxdydz,inthe volume dxdydz.Then<j>willbethepotential onapoint P{x,y,z),above theplaneofx,ywhich wemayregardashori- zontal, duetoaquantityofmatter, M,{=JfJp'dx'dy'dz') situated below thisplane. Now itfollows fromatheorem, first, sofarasIamaware, given byGauss, forasurface ofanyform, that there isadeterminate distribution ofmatterupon*the plane (xy)which willproducethissamepotential onpoints above theplane.Letkbethedensityofthisdistribution ata pointn(f ,7])oftheplane,sothat whichgives d^ /'"r M^dv -'i:idz.j_j_oo{(f_^)2+(^-2/y+/}r Let^=0; thendenoting bykandf-^jthevalues ofkand -^atthepoint {xyy,0),wefind =-k.27r. 120 OncertainDefinite Integrals suggested hy [xi. since thevalue oftheintegralinthesecond member is27r, whatever bethevalue ofz.Hence weconclude that andequation (/)isestablished. 179. Itshould beremarked thatthetotalquantityofmatter distributed over theplane xymust beequaltothemassM, which itrepresents:this isreadilyverified from thepreceding formulae. 180. Thesame formulae admit ofaninteresting application inthetheoryofheat. Thus let</>bethepermanent tempera- ture ofapointPinaninfinite homogeneous solid, heatedby constant sources distributed below theplane {xy), (thecase in which some ofthesources areinthisplane beingofcourse included).Ifthetemperature ^atanypoint11intheplane (xy)begiven,theformula{e)enables ustofindthetemperature atanypointabove theplane. 181. Asanexample,letussupposethatthesources ofheat aresuch thatthetemperatureofaportionAoftheplane {xy\ between twolinesparalleltoOFand atequal distances, a,on itstwo sides, hasaconstant valuec,andthetemperatureofthe remainder oftheplanezero. Inthiscasetheformula(e)will give,forthetemperatureatapoint (x,y,z)above theplane, zcrr» d^ 'l'~27r].^]-a{(^-xy+(7)--yy+z'}^ c/\-iX+a^_iX—a\=—tan tan TTV-s ^/ c^_,2ax=-tan' -2 2 2• From thisweconclude thattheisothermal surfaces which corre- spondtothiscasearecircularcylinders, which intersect theplane (xy)inthetwoparallellinesbounding A. Theapplicationtothisexample, and allothers inwhich the isothermal surfaces arecylindrical, maybemadedirectly by puttings=1inthegeneralformulae. I XL]Problems intheTheory ofElectricity.121 [/: andPART II. 182. Inowproceedtofindthevalues, which willbedenoted byFand W,oftheintegrals "^1^[c?fl^[cosmg]^ where thesymbols [cos m^~\^, [cosmx\^denote theproducts cosm^fj.cosm2^2• •^^^'^sfs> cosm^x^.cosm^x^. .cosm^^s; andthenotation isinotherrespects thesame asbefore. Bymeans oftheformula [cosm^-fsinmf.^J(—1)Y=cos(^7n^)+sin(2m^).^/(— 1), itiseasily shown that (t^'+uy^^-^)^''^• Hence, byasuitable linear transformation, inwhich oneofthe assumptionsis2mf= 77(Sm^)^, wehave[if yu,denote(Xm^)^]V= HCOSfiT}.drj1: [/.J(^2+v^+:Ef^)K^-i) Now,bymeans ofLiouville's theorem*, wefind.(6). Hence47ri(«-i)rrf*-2cos fjuv.d^dv ijoJoF=(c). Differentiatingwithrespecttou,bywhich thefurther reduction oftheintegralwillbefacilitated, wehave NowdV_...47r^(^-i) \ .f*~2cosfjLTJ.C?^C?77 g-g^g.r" dts-1) r^r ..(d). (1.^)KH^"*Jo II+iv'+u')t]^^'+1) s-17;^+Z*^ *SeeCambridge Mathematical Journal, Feb. 1841, p.221[orGregory's Examples, Ed.1841, p.469]. 122 Oncertain Definite Integrals suggested hy [xi. Hence^F 4^«»-«ruco^Mdy ,. From this,byintegrationwithrespecttou,wededuce thevalue ofF:thuswehave theresult Vr 'V[(Zfp[cos mfp_27r^(^+^) e'^^'^^)^" [J-00J(Sr+uy^'-^)~ TKs-1)(Sm^)*^ ^' 183. Toevaluate theintegralWwemayinthe firstplace reduce ittoadoubleintegral byaprocesssimilar tothat in- dicated above, forobtainingtheexpression (c) ;andwethus find 47r^(^-^)rr<^?7icZyi.m^-^cos(y^r).e-(^+^^)*" ^-rK*-f)Joio (m^+w^)*^''^' where rdenotes(S^c^)*.Ifwetakem=pcos^,n=psin^,this becomes W=:pj^—-dOdpp'-^cos*-2^ cos(r/jsin^)6-p«...(6). Nowwehave d^ d^V dA?"^dp)^^^^^^^^^^^^'^"^^p^-^cos^/^.cos(rpcos^)e"^". ..(c). Consideringfirstthecasewhere siseven,let/= ^5—1;wethus find /d^ d^\i«-i p'-^cos*-2^ COS(rpcos^)6"^^=fj-2+T^jcos(r/0sin^)e-^", and,bysubstitution in(6),wehave ^=rK^) Joio^^^^•fe+dP)'''^'P'^^^^"" rK5-1)W^ c^rV J ^*'+r-'sin^^ ri(s- 1)U^^'"^ c^rV(i*^+ryI [dmj [(XI.]Problems intheTheory ofElectricity. 123 Inthesecond case,when sisodd,let/=|(5—1)in(c);then, makinguseoftheresult in(6),wehave mce,whether sbeoddoreven,weconclude that reosma^l^^^=2^-M('-^)ri(.-l)^^.^^^y,_„(F). 184.Theinvestigation which wehavejustgone through,of theintegrals (F),(W)constitutes theverification of"Fourier's theorem" inaparticularcase. For,bythistheorem, wehave, aF{cc^,x^...)he a.function which remains thesamewhen the signsofanyofthevariables arechanged, \j']\dmyicos mx]4 [d^y[cos m|]^i^(?„ f,...)...(e): and ifwetake -^(fu f2•••)^ (2f'+w'')M«-i)' theresult oftheintegrationswithrespecttof^,fg---*isgiven by (V),and thesecond member thus becomes amultiple integral withrespecttom^,m^...^which isshown by(W) tobeequaltothe firstmember.Conversely,ifweassume Fourier's theorem, wemaydeduce thevalue W,bymeans of it,from that ofV.Theintegrals VandWarealsocon- nected bymeans ofanother caseofFourier's theorem, found by taking,in(e), Inthisway,after thevalue ofWhasbeen found, that ofVmay bededuced. 185.Theformula3 (F)and(W)maybeappliedtoevaluate themultiple integral u,andweshall thus obtain theresult of theinvestigationin§I.inadifferent manner. 124 OncertainDefinite Integrals suggested hy [XI. Bymeans oftheequationobtained bydifferentiating (F) withrespecttom,wefind 1 u {S(f-xf+w^)K^+i)2^-1(s_1)rj,h[s-i) ri{s-1) Makingthissubstitution, foroneofthefactors oftheexpression under theintegral signs U,wehave Uu= 2^-1(5-1) 7ri(^-i)ri(5-l)-/•QO J—00[dij [/:[dmj [cosw(f-a;)]«€-(^'"'^*« [fjWKcos^(.-.o]^e-(-*[/;;;j™g^^ 2^-2(s-i){rj(5-i)p [c?m]* [cosm(a;-a?0]*€-(^)^ lt~W'^^^^'^^' whichagreeswiththevalue obtained above. PAET III. 186. Thevalue oftheintegral ZJmayalsobeobtainedbya directprocessofreduction, asfollows :— Byasuitable linear transformation, inwhichassumptions such as^^—x^=Saf aremade,wefind where/^=2(^-x'f. Letusnowassume fj=/ocos<^, ?;=/3sin</>cosl9j, ^3=/3sin (^sin^^cos ^2"-' £_i= />sin<;&sind^sin(9^cos6^^, f,= /osin (j)sin^^sin^^sin^^^, I XI.] Problems intheTheory ofElectricity.125 fromwhich wededuce* [d^]'=p"-'sin'-^<^ sin'-'6>, sin*"^!?,sine^_^[dd]'-'d<lydp; atransformation givenfirstbyGreen. Equation (a)isthus reduced to i^'^^-'.'o !o{p'+uy^'+^\p'- 2p/cos </)+/^+ uy^'-''^""^ ^' where H^_^denotes theproductIfsin'-^ede .r^m'-^ede rdo. etp=utanJ^ ;wethusget J .' sin*-i^sin«-2(jydc^d^^ {2(/'+w'+iO+2(/'+u'^-u')cos^-4?^/sin^cos</)]K*-i) andwemaynov/conveniently assume 2(/^+u'^-u^)cos^—4w/sin^cos =2[{f+u'-u'J+42/Y'l*cose=2hhcos6, and sin sin^=sin9sin6, fromwhich wededuce K'={u+uf+f\ ]^={u'-uf+f\ sin^dcpd^=sinOdcpdO; theexpressionforUbecomes Tj-inff.sin^-^^ sin^-V^96^^ Uu-,Ii,.2J^ Jo(/.^^^^2^cos"'^-F)i(*-i') ^1^r sin^-^(96?(9~*^"Vo(h'-2hkcord +A;^j^^*-^)* Let A,sin (-^/r-^)=A;sin>/r ; bymeans ofthistransformation, observingthat h>7c,we readilyfind .^7ri(«+i) 1 ri(5+1){Z{x -a!')'+(u+iO'}*^'~'^ which isthesame astheresultpreviouslyobtained. StPetee's College, Oct. 3,1846. *SeeCambridge Mathematical Journal, Nov. 1843, p.24,First Series; [or Green, "Attraction ofEllipsoids," §6,Camb. Phil. Trans., May, 1833.] and let v= j-XII.—PKOPOSITIONS INTHETHEOEY OFATTEACTION. .(Art. VI.ofcompletelistinMathematical andPhysical Papers, Vol.i.) [From theCamb. Math. Jour., Nov. 1842andFeb. 1843.] 187. Let X,y,zbetheco-ordinates ofanypointPinan attractingorrepelling bodyM\ letdmbeanelement ofthe mass, atthepoint P,which willbepositiveornegativeaccord- ingasitisattractive orrepulsive ;letx\y\zbetheco-ordinates ofanattractedpointP';let {dm theintegral includingthewhole ofM.Thisexpressionhas been called byGreen thepotential*ofthebody 3/,onthe point P,andthesamename hasbeen employed byGauss (inaM^moire on"General TheoremsrelatingtoAttractive andRepulsive Forces, intheResultate ausdenBeobachtungen desmagnetischenVereins imJahre 1839, Leipsic 1840, edited byM.Gauss andWeber)-]". Byaknown theorem, thecom- ponentsoftheattraction ofMonP\inthe directions of X,y,z,are dv dv dv ~d^" ~rf7' ~I?' and ifdybetheelement ofany line, straightorcurved, whichpasses through P',theattraction inthedirection ofthis .dv'element is-7-7.Hence itfollows that ifasurface bedrawn d'y through anypointP'forevery pointofwhich thepotential hasthesame value, theattraction onevery pointinthesurface iswhollyinthedirection ofthenormal. Surfaces forwhich thepotentialisconstant aretherefore called, byGauss, surfaces ofequilibrium.Ithasbeenshown inaformerpaper (i.above), * ["This Ifound inareference tohismemoirs, inMurphy's firstmemoir on "definite integrals. Ever since Ihavebeen trying toseeGreen's memoir, but "could nothear ofitfromanybodytillto-day, when Ihave gotacopyfrom "MrHopkins. Jan. 25,1845."(Private notewhich Ifindwritten onp.190 ofvol. iii.ofmycopy oftheCamb. Math.Jour.)] tTranslations ofthispaper have been pubHshed inTaylor's Scientific Memoirs forApril, 1842, andintheNumbers ofLiouville's Journal forJuly andAugust, 1842. sil.] PropositionsintheTheory ofAttraction. 127 bhat ifM,instead ofanattractive mass, wereagroupofsources Dfheat orcold intheinterior ofaninfinitehomogeneous solid, y'would bethepermanent temperature produced bythem at F.Inthat case, thesurfaces ofequilibrium would be{so- thermalsurfaces. 188.When theattraction of(positiveornegative) matter, asforinstanceelectricity, spreadover asurface isconsidered, thedensityofthematter atanypointismeasuredbythe quantityofmatter onanelement ofthesurface, dividedbythat element. Theprincipal objectofthispaperistoprove thefollowing theorems :— Ifupon E,oneofthesurfaces ofequilibrium enclosing an attracting mass, itsmatter bedistributed insuch amanner that itsdensityatanypointPisequaltotheattraction ofM onP;then— (1)Theattraction ofthematterspreadoverE,onanexternal point,isequaltotheattraction ofMonthesamepoint multi- pliedby47r. (2)The attraction ofthematter on^,onaninternalpoint, isnothing. 189. These theorems wereprovedinaprevious paper (i.§§ 5,9),from considerations relative totheuniform motion ofheat; butinthefollowing theyareproved bydirectintegration:— LetubethepotentialofM,onthepoint P,(ccyz)inE. Thecomponentsoftheattraction ofMonP,inthedirections ofX,y,z,are du du du ^ dx' dy^dz' andhence,ifa,^,7betheangles which anormal tojE^atP makes with these directions, thetotal attraction onPis /du du ^du \ du ifdnbeanelement ofthenormalthroughP. This istherefore theexpressionforthedensityatPofthe matter wehavesupposedtobespreadover E.Letdsbean element of^atP;let v'bethepotentialofE,onapoint P', (x'y'z), either within orwithoutE;and letAbethedistance fromPtoP.Then 128PropositionsintheTheory ofAttraction.[xn. /du du ^du \^ du^ v=-\\ —J r~i'"s"r">' thebracketsenclosingtheintegrals denotingthattheintegra- tions aretobeextended over thewhole surface E.Now for ds,wemaychoose anyoneoftheexpressions, ,_dydz,_dxdz,_dxdy V cosa'cos13' cos7* Henceanyintegraloftheform {J(Acos OL+Bcos^-j-Ccos7)ds]l. maybetransformed intothesumofthethreeintegrals, (ffAdydz), (JjBda>dz), USGdxdy), byusingthefirst, second, andthird oftheexpressionsfords inthe first, second, andthird terms oftheintegral respectively. Hence, if^=^^^.5=^t. C=#>^,dx^ dy^dz^ (IS^'^^)"' {/(2"'" '-'+f°°^^+S"^^ ^)H = {//^(S^y^'+ fy^''^'+S'^''^y)]f'^' the limits oftheintegrationsrelative toyandz,xand0, a?and3/,beingsochosen astoinclude thewhole ofthesurface considered. 190. Makinguseofthistransformation in(a)wehave , (fffdu dydz dudxdz dudxdy\].,. ,T- ffdudydz ^^,,,,fd'^u 1dud1\^°"IJd.A^^^'^y'^'Hdi^'A^d.T.A) rrrj 77/^^^ 1dud1\ 191. Hence,iftheintegralsinthesecond member include every pointinthespacecontained between E,andanother surface ofequilibrium, E^,without E,andwhich weshallsup- posetobealsowithout P',wehave {/£¥}, -{/£¥)=I© i4:Ei)"'* theaccentdenoting that, intheterm accented, theintegralsare II.] PropositionsintheTlieory ofAttraction. 129 )beextended over thesurface E.Modifyinginasimilar lanner thesecond andthird terms ofv,wehave du ,'du ^ dii , fff/d/^i^ d'^ud^udud1dud1dud1\ jjy^s Now, forallpointswithout M, d^ud%d^u_r. d^^'^dy^'^d?'' )yaknown theorem;andsuchpoints onlyareincluded inthe ntegralsinthesecond member of(c). Also,byintegration byparts, Modifying similarlythetworemainingterms ofthesecond nember of(c),wehave ~ds -[\KiW'-^lyl^''^'^ii^''^y)] -\\KdA+1-4+ilW^'^^w- Now, sinceEandE^aresurfaces ofequilibrium, uiscon- stant foreach. Again, dx^^^ dy'^'^dz'^~^' except whenPcoincides with P',atwhichpoint uhasthe value u.Hence, thevalue oftheintegrals, (n(d^1 d^1d^l\.,j isonlyaffected bythese elements, forwhich u=u\andhence umaybetaken without theintegral sign,asbeing constant andequaltou.If,therefore, forbrevity, weput T.E. 9 1/^180PropositionsintheTheory ofAttraction.[xii accordingastheintegralsrefer toE,ortoE^,and ///&i+|"4+©3'^^''2/'^^=^(')' theintegrations including every pointbetweenEandE';equa- tion(c)becomes du , '^\+v'={nl{h)-(u)(h)-u'k (c"). Now itisobvious that, atagreatdistance from if,the surfaces ofequilibriumarevery nearly spherical.LetE'be taken sofaroffthat itmaybeconsidered asspherical,without sensible error, and let7bethedistance ofanypointinE'from thecentre, afixedpointinM,or,which isthesame, theradius ofthesphere. Then~3- >^^^~3~ >isthe attraction ofM, MonapointinE\and isthereforeequalto—^ ,and therefore, bytheknownexpressionforthepotentialofauniformspherical shell, onaninteriorpoint, duJ -{/^},.»'"{/?},=?-'-'<•>.(A Itnowonlyremains todetermine theintegrals (Ti)^ {h)^^andh 1 AByputting,in(6),-^=1, </>=-r,wefind thefollowing transformation, for{h), h=(—ds =-[—- Jdn JdnA^' Now letthepoint {xi/z)bereferred tothepolar co-ordinates, 7,0,<l>.Then,ifP'bepole,7=A.Also, if-^betheangle between Aanddn,theexpressionfordsis ,A'sinedOdcb .,dA (Is=—norsince cosy*=-j- , cosyjr^dn A^sinOdOdcl) dA' n dn Hence h^—jjsinOdOdcf). XII.] PropositionsintheTheory ofAttraction. 131 IfP'bewithin thesurface towhich theintegrals refer, the limits for6areandtt,andfor<^,and 27r,and inthat case, ^=—47r;therefore, sinceP'isalways withinE^, W,=-4t '.{g). IfP'bewithout thesurface considered, then, foreach value of6,wemust take thesum oftheexpressions -sin6d6d(f), and—sin6(—dO) d<f)f ind, therefore, each element oftheintegralisdestroyed by motherequaltoit,butwith acontrary sign,andthevalue of thecomplete integralistherefore zero. Hence, accordingasP'iswithout orwithin E, [h)=0,or(h)=-47r{h). Again,tofindthevalue ofk,wehave,bydividingitinto three terms, andintegratingeach once, =(h)^- (Ji)=-4i7r-0, or=-47r+47r; iand, therefore, accordingasP'iswithout orwithin E, k=— 4!7r,ork=(k). Hence, makinguseof(/), {g), Qi), {k),in(c''),wehave v'=4:7rUj when P'iswithout E.(1), v'=4!'7r(M), when P'iswithin E.(2). From the firstoftheseequationsitfollows that theattrac- tion ofE,onapoint withoutit,isthesame asthat ofM, multiplied by47r;and since thesecond shows that the potentialofEoninternalpointsisconstant, weinfer that theattraction ofEoninternalpointsisnothing. These theorems^ alongwith some others which were also provedintheprevious paperinthisJournal, alreadyreferred to,had, Ihave since found, beengiven previously byGauss. One ofthemostimportantofthese isthefollowing:—Ifamass ifbewhollywithin orwhollywithout asurface, anequal mass maybedistributed over this surface[intheformer case, ora certain lessmassmaybedistributed over itinthelattercase] insuch amanner that itsattraction, intheformer caseon 9—2 132 PropositionsintheTheory ofAttraction.[xii. externalpoints,andinthelatter oninternal, willbeequalto theattraction ofMonthesamepoints.This theorem, which wasprovedfromphysicalconsiderations inthepaper Onthe UniformMotion ofHeat, etc., isproved analyticallyinGauss's Memoire, butthesame method isused inboth toinferfrom it thetruth ofpropositions (1)and(2). FromProp. (2)itfollows that,ifEbethesurface ofan electrified conducting body,theintensityoftheelectricityat anypointwillbeproportionaltotheattraction ofMonthe point.Hence wehave themeans offinding aninfinite number offorms forconducting bodies, onwhich thedistribution of electricitycanbedetermined. Thus,ifMconsists ofagroupofmaterialpoints, m^,m^, etc., whose co-ordinates are0?^,y^^z^,;^^,y^,z^,etc.: thegeneral equationtothesurfaces ofequilibriumis m, m^ _, i~ I/ \2 ./ \« . J \911~i~etc.—/ andtheintensityofelectricityatanypointofasolidbody, bounded byoneofthem, willbethevalue of {©-©-©T atthepoint. Totakeasimplecase :—Let there beonlytwo material points,ofequal intensity. The surface willthenbeasurface ofrevolution, and willbesymmetrical withregardtoaplane perpendicular, throughitspointofbisection, tothelinejoining thetwopoints,andwouldprobably very easily beconstructed inpractice. Weshould thushave asimple method ofverifying numericallythemathematicaltheoryofelectricity. PAET II. [FromtheCambridge Mathematical Journal, February 1843.] 199. Ishallnowproveageneral theorem, whichcomprehends thepropositionsdemonstrated inPartI.,alongwith several others ofimportanceinthetheories ofelectricityandheat. LetMandM^betwo bodies, orgroupsorattractingorre- pelling points;and letvandv^betheirpotentials onxyz ; letRandR^betheir total attractions onthesamepoint ;and mB.co.e,^,^^=jlj(i^^^.II^gj)...,.....(a).s:ii.] PropositionsintheTheory ofAttraction. 133 let6betheanglebetween thedirections ofRandR^^and 2)97, OL^P^y^ytheangleswhichtheymake withxyz.Let8be a,closed surface, dsanelement, correspondingtothe co- ordinatesxyz'jandPandP^thecomponentsofR,R^tina directionperpendiculartothesurface atds.Thenwehave iJcosa=-^, 2?cos^=-^-,iJcos7=-^, ^'•=°^"'=-d^' ^^'"'^^-d^' -^.'=°^%=-d^' COS^=cosacosa^+cos/3cos^S^+cos7cos7^ ; , dvdv, dvdv, dvdv.^^^ ^ hence, -i--T+^- -T^+-7--y^=ititicos^. aa?dxdydydzdz* Hence dvdv, .dvdv^.cZv (Z?;, dydy where weshallsupposetheintegralstoinclude every pointin theinterior ofS.Now, byintegration byparts,thesecond rmember maybeputunder theform, where thedoubleintegralsareextended over thesurface S, andthetriple integrals,asbefore, overevery pointinits interior. Ifwetransform the firstterm ofthisby(h),PartI., dvandobserve that—^-=P,itbecomesan -jjvJPds. . . dj^vd% .d^v,,,. ^«^^"'dS'+df-^d?=^('')' except whenxyzisapointoftheattractingmass. Ifthisbethecase,and ifZ?bethedensityofthematter at thepoint,wehave therefore (jTa+X^+T^jdxdydz +iirdm= i(d). 134PropositionsintheTheory ofAttraction,[xii. Hence(a)istransformed into SJjRR^cosedxdydz=4<7rjfjv^dm-jjv^Pds (3) ; similarly, byperformingtheintegrationin(a),ontheterms dvdvdv . .odv^ dv^ dv^ dx'dy'dz' dx'dy*dz^ weshould havefound JJJBE^cosedxdydz=^irjjjvdm^-jjvP^ds (4). 200. Ifthetriple integralsin(a)wereextended over allthe space without S,oroverevery point between S,andanother surface, S^,enclosing it,ataninfinite distance, itmay be shown, asinPartI.,thatthesuperiorvalues ofthedouble in- tegralsin(h),correspondingtoS^,vanish. Hence, theinferior valuesbeingthose whichcorrespondtoS,wehave, instead of (3)and(4), JfJRR^cos6dxdydz=4<irjjjv^dm +l^vj^ds (5), SJjRR^cosedxdydz=4<7rjfjvdm^ +jjvP^ds (6). Itisobvious that vandv^intheseequations maybeany functions, each ofwhichsatisfy equations (c)and(d),whether weconsider them aspotentialsortemperatures,orasmere analyticalfunctions with therestriction that, in(5)and(6),v andVjmust besuch astomakejjvJPdsandJJvP^dsvanish atS^[and (acondition thenecessityforwhich hasbeen dis- covered byHelmholtz),* that, in(3)and(4),ifSbemultiply continuous, vandv^must besingle-valued functions through- outit].Ifeach ofthemsatisfy (c)forallthepointswithin the limits ofthetriple integrals considered, dmanddm^willeach vanish;but ifthere beanypointswithin thelimits,forwhich either vorv^doesnotsatisfy (c),thevalue ofdmordm^atthose pointswillbefound from(d), 201. Thus let v^^=1,forevery point. Thenwemust have dm^=0.AlsoR^=0,P^=0. Hence(3)becomes JfPds=^4i7rfJJdm=^4^7rm (7), *[See Helmholtz; Crelle's Journal, 1858 (Wirbelbewegung),translated byTait, Phil. Mag. 1867,i.(Vortex-Motion);orThomson (Vortex- Motion, §§54... 58),Trans. Boyal Society ofEdinburgh, 1868.] XII.] PropositionsintheTheory ofAttraction. 135 ifmbethepartofMwithin S,Thisexpressionisindependent ofthequantityofmatter without S,and ifm= itbecomes HPds^O (8). Ififbeagroupofsources ofheat inasolidbody,Pwillbe thefluxacross aunit ofsurface, atthepoint xyz.Hence the total flux ofheat acrossSisequaltothesum oftheex- penditures from allthesources intheinterior;and ifthere be nosources inthe interior, thewhole flux isnothing.Both these results, thoughourphysicalideas ofheatwouldreadily leadustoanticipate them, arebynomeans axiomatic when consideredanalytically.Inexactlyasimilar manner, Poisson* provesthat the total flux ofheat outofabody duringan instant oftime isequaltothesum ofthediminutions ofheat ofeachparticleofthebody, duringthesame time. This follows atoncefrom(7).For ifwesupposethere tobeno sources ofheatwithin S,butthetemperatureofinteriorpoints tovary with thetime, onaccount ofanon-uniform initial distribution ofheat,wehave dhd\ d^v_dv •d^^'^df^dz^~'Jt^ dv Hence, by (c?),wemust use—-rrdxdydz,instead ofAiirdm, andtherefore(7)becomes llp^-lW^d^dyd- Itwastheanalysis used byPoisson, inthedemonstration ofthistheorem, thatsuggested thedemonstrationsgivenin Part I.ofpropositions (1)and(2). 202.Asanotherexampleoftheapplicationofthetheorem expressed by(3)and(4),letv^bethepotentialofaunit of mass, concentrated atafixedpoint, x'y'z. Hence, M^=land dm^=0,except whenxyz,atwhichdm^issupposedtobe situated, coincides withxyz\ and, ifAbethedistance of xyzfromx'y'z, ^i=-r . Hence, accordingasx'y'ziswithout orwithin 5, JJJvdm^=0,orfjvdm^=v'jjjdm^=v(e), *SeeTMorie delaGhaleur, p.177. 136PropositionsintheTheory ofAttraction.[xii. thetriple integrals beingextended over thespacewithin S. Now letussupposeMtobesuch, that vhasaconstant value {v)atS.ThenjjvP.ds=(v)jjP^ds, which, by (7),is=0,or to47r(v),accordingasxy/iswithout orwithin S.Hence, by comparing (3)and(4),wehave, inthetwocases, and 47r 111— r^—I(—r—=—47r{v)+4<7rv; thereforejj^=47r(i;) (10). These arethetwopropositions (1)and(2)provedinPartI., which aretherefore, aswesee,particularcases ofthegeneral theoremexpressed by(3)and(4)* 203. If i;=Vj,and ifboth arisefrom sources situated with- out/Sf,(3)becomes SJjR'dxdydz=SJvPds (11), aproposition given byGauss. Ifvhave aconstant value{v) over S,wehave jjvPds=iv)fjPds=0,by(8), hence jjjP^dxdydz=0. Therefore J?=and v={v)forinteriorpoints. Hence,if thepotential produced byanynumber ofsources have thesame value overevery pointofasurface which contains none of them, itwillhave thesame value foreveryinterior pointalso. Ifweconsider thesources tobespreadover>S^,itfollows that V=(y)atthesurface isacondition which impliesthat the attraction onaninteriorpointwillbenothing. Hence thesole condition forthedistribution ofelectricity overaconducting surface, isthat itsattraction shall beeverywhere perpendicular tothe surface, aproposition which wasproved from indirect considerations, relative toheat, inaformerpaper.*)" *Itmay behere proper tostate that these theorems, which were first demonstrated byGauss, arethesubject ofaMemoire byM.Chasles,inthe Additions totheConnaissance desTemps for1845, published inJune, 1842. InthisM6moire herefers toanannouncement ofthem, without ademonstra- tion, intheComptes Eendus desSeances deVAcademie desSciences, Feb. 11, 1839, adate earher than that ofM.Gauss's Memoire, which wasread atthe Koyal Society ofGottingen inMarch, 1840. tS3e I.above, §5. ul] PropositionsintheTheory ofAttraction. 137 204. Inexactlyasimilar manner,ifnone ofthesources be without>S,bymeans of(5)and(7),itmaybeshown that fJfR'dxdydz=-4>'n-M(v) (12); thetriple integrals beingextended over allthespace without S,Hence aquantityofmatterfjucanonlybedistributed in onewayonS,soastomake(v)beconstant. For ifthere were twodistributions of/jl,eachmaking (v)constant, there would beathird, correspondingtotheir difference, which would also make(v)constant. Thewhole mass inthethird casewould be nothing. Hence, by(12),wemust havefJfR^dwdydz=0,and therefore i?=forexternalpoints ;and, since(y)isconstant at thesurface,Rmust be=forinteriorpointsalso.Now this cannot bethecase unless thedensityateachpointofthe surface benothing,onaccount ofthetheorem ofLaplace, that, ifpbethedensityatanypointofastratum which exerts no attraction oninteriorpoints,itsattraction onaninteriorpoint close tothe surface willbe47r/3.This important theorem, which shows thatthere isonlyonedistribution ofelectricityon abodythat satisfies thecondition ofequilibrium, was first given byGauss. Itmaybereadily extended, ashasbeendone byLiouville,* tothecase ofanynumber ofelectrified bodies, influencing oneanother, bysupposing Stoconsist ofanumber ofisolatedportions, which willobviouslynotaffect thetruth of (5)and(6). Then, ifwesuppose vtohave theconstant values, (v),{v)\ etc., atthedifferent surfaces, andthequantitiesofmatter on these surfaces tobeM,M\ etc.,weshould have, instead of(11), jjjR^dxdydz=47r{M(v)+M'{v)'+etc.} (13), andfrom this itmaybeshown, asabove, thatthere isonlyone distribution ofthesamequantitiesofmatter, M,M', etc.,which satisfies theconditions ofequilibrium. 205. IfbothMandM^bewhollywithin ^,bycomparing (5)and(6),orifbothbewithout Sybycomparing (3)and(4), wehave JJPv^ds-^JfP.vds (14). *SeeNote toM.Chasles' Memoire intheConnaissance desTempsfor1845. 138PropositionsintheTheory ofAttraction.[xii. Now letShe a.sphere, and letr^<^bethepolarco-ordi- nates, from thecentre aspole,ofanypointinthesurface to which thepotentialsvandv^correspond. Thenweshall have (Li) dv F=-'-j-, Pj=--7-*, andwemayassume ds=r*sin6ddd<j). Hence(14)becomes rr\^sineded(j>=rTv^sinedOd<l> (15). Thisequationleads atonce tothefundamentalpropertyof Laplace'scoefficients. For ifvandv^beoftheformsYj^"", Yy,mandnbeing any positiveornegative integers,zero included, andY^andF„being independentofr,wehave,by substitution in(15), mrpF^F^sindded(i>=n{^TF„r„ sinOdOd^. Ifmbenot=ti,thiscannot besatisfied unless F^r„sin(9c?(9#=(16).JO This isthe*fundamentalpropertyofLaplace'scoefficients. There aresome otherapplicationsofthegeneraltheorem which hasbeen established, especiallytotheTheoryofElec- tricity,which must, however, beleftforafutureopportunity. *[Forajustification ofthisuseofthedefinite article, seeMurphy'sElec- tficity^ Chap.i.Props,i.andii.,Cambridge 1833.] XIII.THEOREMS WITHREFERENCE TOTHESOLUTION OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS.' (Art. XXXVI. ofcompletelistinMathematical andPhysical Papers,Vol.i.) [FromtheCambridge andDublin Mathematical Journal, Jan. 1848.] 206. Theorem 1.Itispossibletofindafunction V,of £c,y,z*which shallsatisfy,forallrealvalues ofthese variables, thedifferentialequation aumd(a^'^)dum dx dydzr \/> abeing anyrealcontinuous ordiscontinuous function ofx,y,z, andpafunction which vanishes forallvalues ofx,y,z,exceed- ingcertain finite limits(suchasmayberepresented geo- metrically byafinite closedsurface), within which itsvalue is finite, butentirely arbitrary. Theorem 2.There cannot betwodifferent solutions ofequa- tion(A)forallrealvalues ofthevariables. 1.(Demonstration). —Let C/'be afunction ofx,y, z,given by theequation 77_[ff pdxdydz ~ii%-^7+(3/-2/r+(3-/)i^^"^' theintegrationsinthesecond memberincludingallthespace forwhichpisfinite;sothat,ifweplease, wemayconceive thelimits ofeachintegrationtobe—ooand+oo,asthus all thevalues ofthevariables forwhichpisfinite willbeincluded, andtheamount oftheintegralwillnotbeaffected bythose values ofthevariables forwhichpvanishes, beingincluded. Again,Ybeing anyrealfunction ofx,y,z,let *The case ofthree variables, which includes theapplications tophysical problems,isalone considered here; although theanalysisisequally applicable whatever bethenumber ofvariables. 140 Theorems withreferencetotheSolutionof [xiii. J-ooJ-ooj-00[\ax (XdxJ \ayaayJ 4-("^-^'syH^^"-^^)- Itisobvious that,although Vmaybeassignedsoastomake Qasgreatasweplease,itisimpossibletomake thevalue ofQ lessthanacertain limit, sinceweseeatoncethat itcannot be negative. Hence §,considered asdependingonthearbitrary function V,issusceptibleofaminimum value;andthecalculus ofvariations willleadustotheassigningofVaccordingto thiscondition. Thuswehave ./dVldU\ dSV],,, Hence, bytheordinary processofintegration byparts,the integratedtermsvanishingateach limit,* wededuce -i»«=///'-(i('S-f)-|("-f;f^ Butbyawell-known theorem(provedinPratt's Mechanics, andinthetreatise onAttraction inEarnshaw'sDynamics),we have d'Ud'Ud'U Hence thepreceding expression becomes +B("'3+M'"!«'' Wehave, therefore, forthecondition thatQmaybeamaximum orminimum, theequation, AIol'--\ +—((i'—\ 4--^l^a^—1=-47r dx\ dxJdy\ dy)dz\ dz)^' tobesatisfied forallvalues ofthevariables. *Allthefunctions ofx,y,zcontemplatedinthispaperaresupposedto vanish forinfinite values ofthevariables. XIII.]certain PartialDifferential Equations. 141 Now itispossibletoassignVsothatQmaybeaminimum, andtherefore there exists afunction, V,which satisfiesequa- tion (A). 2.{Demonstration). —LetFbe asolution of(A),and letV^ beanydifferent function ofx,y,z,that istosay,anyfunction such thatFj—V,which wemaydenote by <^,does notvanish forallvalues ofx,y,z.Letusconsider theintegral Q^, obtained bysubstituting V^forVintheexpressionforQ.Since \dx adxJ\dx adxJ \dxadxj dx dot?' wehave Now,byintegration byparts,wefind I I I Ia-i 7—1^ 7•dxdydz J-ooJ-ooJ-ccV dx adxJdx"^ theintegrated termvanishingateach limit.Applyingthis andsimilarprocesseswith reference toyandz,wefindan expressionforthesecond term ofQ^ywhich, onaccount of equation (A),vanishes. Hence ^^^^^WKi^W^'i)"-''''^«)' which shows thatQ^isgreater than Q,Now theonly pecu- liarityofQis,that V,fromwhich itisobtained, satisfies the equation (A),andthereforeV^cannot beasolution of(A). Hence nofunction different fromVcanbeasolution of(A). Theanalysis given above, especially wheninterpretedin various cases ofabruptvariations inthevalue ofa,and of infinite orevanescent values, throughfinitespaces, possesses veryimportant applicationsinthetheories ofheat, electricity, magnetism, andhydrodynamics, whichmayform thesubjectof future communications. Edinbarnet, Dumbartonshire, Oct.9,1847. 142 Theorems withreferencetotheSolutionof [xiii. ADDITION TOAFRENCH TRANSLATION OFTHE PRECEDING. [FromLiouville's Journal deMathematiques, 1847.] 207. Dans lesapplications quipresententleplus d'inter^t, ilfautconsid^rer destransitions subites dans lavaleur de a. Parexemple,siaaunevaleur constante dans toutI'espaceex- t^rieur aunesurface fermde 8,dans I'interieur delaquellea estinfinie, notre analyseconvient aucasd'uncorpsconducteur ;Si soumis aI'influence d'une niasse^lectrique donnde(ffjpdxdr/dz), etcetteapplicationnepr^sente aucune difficulty Onentire, eneffet, lesdemonstrations donn^esparGreen, quelasolution analytique duproblemedeladistribution d'^lectricite dans ces circonstances estpossibleetqu'elleestunique. Dans uneapplicationaI'hydrodynamique, ouauncertain problemedemagn^tisme,ilfaut consid^rer unespacedans lequellavaleur deasoitz^ro.L'interpr^tation dur^sultat ne pr^senteaucune difficulte, mais il6stplusdifficile debien comprendre comment lademonstration tellequejeI'aidonnde plushaut sepreteacecas.EnessayantdeTexpliquer nettement, j'aitrouv^ unedemonstration directe dutheoreme suivant, quirenferme ler^sultat dont ils'agit: "IIestpossibledetrouver une fonction Vquis'evanouisse pourlesvaleurs infinimentgrandesdesvariables x,y,z,et satisfasse aI'^quation^^^__ dx^^df'^d^~^' pourtous lespointsext^rieurs aune surface fermde ^,avec cette condition dans laquelleFestune fonction arbitraire descoordonndes d'unpointsurlasurface ^,etdnest 1'element d'une normale exterieure alasurface encepoint." Pour ledemontrer, consideronsI'integrale uii.]certain PartialDifferential Equations. 143 -elative aI'espaceexterieur aS.Parmi toutes lesfonctions V piverifient lacondition JJVFdS==A, 3^Aestunequantity quelconque,ilyenaunepour laquelle I'int^grale Qestunminimum. Une fonction V,ainsi d^ter- minde, satisfait auxEquations dx^dy^dz^ an (oilcestuneconstante), comme ons'enassureparlecalcul desvariations. Suivant lesvaleurs de-4,caura desvaleurs proportionnelles;onpentprendre Atellequec= 1.Delaon conclut lethdoreme dnoncd IIserait faciled'ajouter une demonstration, quelasolution duprobl^medeladetermination deVsous cesconditions estunique.* *[Proyided5fisasimply continnous surface. If/Sbeamultiply continuous surface, as,forinstance, theinner boundary ofanendless tube(afinite tube with itsends united, soastoconstitute acircuit), wemayaddtoVthevelocity- potentialofaliquid moving throughitirrotationally (Thomson and Tait's Natural Philosophy, §§184—190;Thomson, Vortex Motim, §§54...58)without violating theconditions prescribed inthetext.Compare above, §200, footnote.] XIV.ELECTRIC IMAGES. EXTEAIT D'UNE LETTRE DEM.WILLIAM THOMSON AM.LIOUVILLE. (Art. XIX.ofcompletelistinMathematical andPhysical Papers, Vol.i.) [PromLiouville's Journal deMathematiques, 1845.] "Cambridge, 8Octobre 1845. 208. "...Pendant monsejouraParis, jevous aiparM du principedesimages pourlasolution dequelques problemes relatifs aladistribution deI'electricite. IIyaunefoule de problemes auxquels jenepensais pas alors, etohj'aitrouv^ plustardqu'on peat I'appliquer.Parexemple,onparvient ainsi aexprimer alg^briquementladistribution d'electricit^ surdeuxplansconducteursquisecoupentsousunangle - ,quand unpoint electriqueestposedansI'espaceentre les deuxplans. (L'ideeestanalogueacelledukaleidoscopede Brewster.) Quandilyatroisplans .quisecoupent perpen- diculairement, ouquandilyaunplan quicoupe perpendicu- lairement deuxplans quisecoupentsousunangle- ,onpeut egalementtrouver ladistribution sous I'influence dunpoint electriquedonn^. Onpeutaussiexprimertres-facilement la distribution surlesparoisintdrieures dunparallelipipederect- angulaire creux, soumis aTinfluence d'unpoint electrique pos^ endedans, enseservant desinte'gralesd^finies. "Soient Glecentre d'unesphere S;Q,Q'deuxpoints prissurunmemerayonGAetsursonprolongement,detelle maniere que GQ.GQ'=CA'^; etPunpoint quelconquesurlasurface S.On a,comme on salt, PQ' AQ'- Onpeut,kcause decethdorfeme, appeler QetQ'points r^cipro- ques relatifs alasphere 8,dont chacun estI'imagedeI'autre I I XIV.]Electric Images.145 dans lasphere.Suivant cette definition, I'imaged'uncligne ousurface sera lelieudesimagesdepoints prissurcetteligne ousurface. Ainsi, ontrouve queI'imaged'unplanoud'une sphereesttoujoursunesphere (leplan^tantcomprissous cette designation).Lesimagesdedeuxspheressecoupentsous le meme angle,r^elouimaginaire, quelessurfaces donn^es. "Soient Q,Qdeuxpoints r^ciproques,relativement kune sphere>S^,etq,q',sleursimagesetI'image delasphereBdans uneautrespheredonnee. Lespoints q,qserontr^ciproques relativement alaspheres. 209."AI'aide decesth^oremes, jeparviens facilement a determiner lesimagessuccessives d'unpoint quelconque (qui n'estpasn^cessairement dans laligne quipasse parleurs centres),dans deuxspheres quisecoupentsousunangle donn^. Quand cetangleestimaginaire, jeparviensainsi a exprimerladistribution deI'^lectricite surlesdeuxspheres, sous I'influence d'unpoint quelconque, charge d'electricite, au moyendesseries deM.Poisson(qui convergent comme des seriesgeom^triques). Quand Tangled'intersection est re'el et comprisdans Fexpression- ,onparvientainsi aexprimer algebriquementladistribution d'unequantite donnee d'^lec- tricitd surlasurface ext^rieure desspheres, quin'estsoumise a aucune influence ouquiTestacelle d'unpoint donn^. S'ily atrois surfacessph^riques quisecoupent perpendiculairement, onexprime alg^briquement, parlesmemesprincipes,ladistri- bution sur lasurface ext^rieure. Jeparviensaussi adeter- miner lestemperaturesstationnaires dans I'interieur d'une lentille dont lesdeux surfaces secoupentsousunangle- ,la temperaturedechaque point decessurfaces etant donnee. 210. "SiTonveutdeterminer ladistribution d'eiectricite sur unesurface donnee*Sf,sous I'influence d'unpoint quelconque Q, onreduit, parlesmemesprincipes,leproblemealadetermina- tiondeladistribution, sansaucune influence, surI'image de ^S' dansunespheredecrite ducentre Q,avecunrayon quelconque. Uneapplication generaledecetheoreme conduit aunedemon- strationrigoureuse dutheoreme deM.Gauss, qu'on peut pro- duire, aumoyen d'une distribution determinde dematiere sur T.E. 10 146 ElectricImages. [xiv unesurface fermeequelconque, unevaleur donnee dupotentie achaque pointdelasurface. IIyaaussibeaucoup d'applica- tionsspeciales [seebelow, §§218...220] qu'on peutfaire d( cetheoreme auxcasdanslesquels Sestunesphere, undisquf circulaire, ouunsegmentd'une surfacesph^riquefaitparur plan.J'en aiaussi d^duit unedemonstrationg^om^triquedi tb^or^me quevous avezpubliddans lenum^ro d'avril 184c devotre Journal {voir page 137), dont voiciI'expression analy- tique***" [seeabove, xi.§§167, 186]. EXTRAITS DEDEUX LETTKES ADRESSEES 1M.LIOUVILLE PARM.WILLIAM THOMSON. [FromLiouville's Journal deMath^matiques, 1847.] **Cambridge,26juin 1846. 211. "...Lesrecbercbes surlesquelles jevous ai^crit, le 8octobre 1845, m'ont conduit aI'emploid'unsystemenouveau decoordonn^es ortbogonales tres-commode dansquelques pro- blemes destbdories delacbaleur etdeI'^lectricit^. Les sur- faces coordonnees dans cesystemesont lessurfaces engendrees parlarotation, autour d'un axeconvenable, d'unsystemede coordonneescurvilignesdansunplan,etlesplansmeridiens. Eneffet, soitMunplanmeridienquelconque;lescoordonnees d'unpointPdans ceplansontdeux cerclesquisecoupent a angledroitencepoint,etdont lepremier passe pardeuxpoints fixesA,A\dans I'axederevolution X'X, tandis quelesecond estlacourbeorthogonale delaserie entiere des cerclesqui passent parlespoints A,A\Ondemontre facilement que cette courbe estuncerclequipasse pardeuxpoints imaginaires B,B',dans ladroite Y'OYperpendiculaireaX'OXyadesdis- tances auxdeux cotes de dontchacune estegalekaJ—1, aetant lavaleur desdistancesegales A'O,OA.Eneffet,la premiereserie estexprimee parFequation (1)w'+f-2uy=a\ uetantunparametre variable, etTonendeduit (2) iio'+y'-2vx=-a\ pourrequation delacourbeorthogonale. I av.]Electric Images. 147 212. "Posons u=acoiOy v=aJ-1 .coti^ ; ?seraTangle quelatangente ducercle(1),aupointAouA\ aitavec I'axeX'X, et-^/rseraTangle imaginaire quelatangente lucercle(2),aupointBouB'^faitavec Y'Y,Pour avoir la ;^rie entiere descercles(1),ilfaudrait donner autoutes les /aleurs r^elles de—co^oo,oua^toutes lesvalours deatt; )t,pourlaserie(2),ilfaudrait donner avtoutes lesvaleurs de Ia00,etde—COa—a.Onpentconsid^rer unpointPcomme l^termin^ sansambiguity parlescoordonnees 6,'\{r(enprenant ?+TTaulieude^pourTautrepointd'intersection desmemes lercles). LesEquationsdetransformation, entre lescoordon- le^s (x,y)et{6, yfr)d'unmemepoint P,sont ;3)af+y^-2aycote=a\ ;4)a)^+y^-2aa)cotfJ^=^a\ Dnend^duit sin'yfrJ—1 cos '\jr—cos6' sin •^cosY—cos6 -2 ocos >lr4-cos^ cosyjr—COS Dans lesapplications physiques,ils'agit d'exprimerladistance \,entredeuxpoints P,P'enfonction desnouvelles coordon- Q^es.Ontrouve facilement, aTaide desformules donnees si-dessus, dans lecasdePetP'dansunmemeplanm^ridien M, (cos yjr—cos6)(cos i/r'—COS6')' Pour letroiscoordonnees d'unpoint dansTespace, jeprends 0, yfrquifixent saposition dansunplan m^ridien, etTangle <^ queceplanfaitavecunplan m^ridien fixe. Jetrouve main- tenant, pourladistance entredeuxpoints quelconques P,P\ .2_92cos(-\|r— yfr')—[cos6COS6'+sin sin6'cos(j)— ((>')] (cos ^fr—COS6)(cos yjr'—COS6') Pour dviterTemploidequantit^s imaginaires, jepose 2cos'\jr=r-\--, 2cosi|r'=/+— ,I 10—2 148 Electric Images. [xiv. d'ouTond^duit 2cos(t-t')=p+ ^', etrexpression prdcddenteser^duit a -y-2rr[coscos6'+sin^sin6'cos((^~ <^^)]+r^^-^ (r'-2rcos^+1)(r'*-2/cos^+1) AI'aide decetteexpression, ontrouve ,d ^—Va+.^d{s'Vj +-1^(s-'v)_f, r,2/JJJ2 ^>dr' sin (9 c^0'sin'^df oh s=(r^-2rcos+1)^ pour I'dquationdumouvement uniforme delachaleurexprimee parlescoordonn^es r,0,<j>. "Lessurfaces representees parT^quation r=constante sont desspheres engendr^es parlarevolution d*une sdrie de cercles autour deladroitequicontient leurs centres. Sup- posons queI'espaceentre deux decesspheres (quand chaque sphereestendehors deI'autre, cetespaceseraI'espaceinfini en dehors desdeuxspheres),dont lesEquationssont soitremplid'un milieu solide homog^ne, quelestemperatures detous lespointsdechaquesurface soient donn^es, etqu'il s'agissededeterminer latemperaturestationnaire d'unpoint quelconquedans lesolide; onresoudra ceproblemeavec beaucoupdefacilite aumoyendeI'analysedeLaplace,eu employantlescoordonneesque j'aiindiquees. Dans lecas particulierd'unetemperatureconstantepourchaque sphere,on parvient, apres quelques reductions, atrouver lasolutionque Poisson adonnee pourleprobleme correspondant dedeux sphereseiectrisees. 213. "IIyaimsyst^me nouveau ettrbs-remarquablede coordonnees, qu'ontrouve enposant rcos^=f,rsin^cos^= -77,rsin^sin<^=f, r,0,</)appartenant ausysteme expliqueci-dessus. Dans ce syst^me (f, 77,f),lessurfaces coordonnees sont desspheres orthogonales quipassent parunpoint fixe, etquitouchent, pai IV.] ElectricImages. 149 ms^quent,troisplans orthogonaux menes parcepoint. Je lisparvenu aconsid^rer cessyst^mes decoordonn^es en lerchant lesimagesdesseries desurfaces dessystemes (polaire brectangulaire) ordinaires, dans desspheres convenablement ispos^es. *' L'applicationdusyst^me (f, rj,f)auxprobl^mes dephysique, our lecasdedeuxsystemes quisetouchent Fun I'autre, en .onne lessolutions avecbeaucoupdefacility;mais ilestplus impledefaire directement larecherche decescoordonn^es, ?uedelesddduire dusysteme (r,0,(j>).Eneffet, soient ^^+2/^+^^-|=0, a^+f+z^^^=0, x'+f+ z'-^^=^0, esEquationsdetroisspheres quisecoupentaunpointP Reliessecoupentaussi aTorigine 0).Jeprends f,77,fpour lescoordonn^es decepoint (ilfaudrait substituer,-,-^ ^aaa dans cesEquations,aulieude|,77,f,pourretrouver lescoordon- ndesf,7j,findiqudes ci-dessus). DecesEquationsontire a;'+2/'+^'= |2-j-^2-:j7^2> f V r "^"r+^'+r* ^~r+77^+r' ^~^-^v'+v' etI'equation d'^v d^v d^v_ d?'^dy''^d?~ devient, pourlesnouvelles coordonnees, oilp=(f^+,»+f^)i. PourexempledeI'emploi qu'on pentfaire decesystemede coordonnees, supposons quelatemperatured'unpoint (a, 7},f) estunefonction donn^eF{rj,f)descoordonnees97,fdesa 150 Electric Images. [xiv. positionsur lasphere a,etquelatemperature d'unpoint (a^, 7},f)estFj^(rj,f),etqu'il s'agitdedeterminer latemperature permanented'unpoint quelconque P(f, rj,f)dansI'espace entre lesspheres a,a^(c'est-a-dire I'espaceentierpour lequel f aunevaleur interm^diaire aaetaj,quenoussupposerons remplid'un solide homogene. Suivant lamethode deFourier, enobservant quelesvalours cosmrj.00^7)^.6^^, cosm?;.sin77^.6^^, substitutespour p~^v,sontdessolutionsparticuli^res deliqua- tion(a),pourvu queh^=7n^+n^,jetrouve, pourlasolution du probleme propose, W0>W:.J>^ "'t^^'2.Tir" ou6estlabasedeslogarithmes nep^riens,et 214."Comme exempledeI'usagedecette formule, jeferai Vetantuneconstante. Pour lareduction deI'expression,dam cecas,j'observe que rr 1 1COSmpCOSnq ^e-(m«+rt2)iA* d'ouTond^duit /00 /«00 f"J/jw^<^sm(7;-V)cos(f-f)_^( Idrj'd};'.V \;Jgw =27rcosm7;.cos<.- -ooj—00 et lesigne superieurouinf^rieur ^tantpris,dans laseconds expression,selon que a^estpositif oun^gatif (jeprendsc toujours positif et>aj).Ces reductions faites, I'expressior (6)setrouve r^duite a XIV.] Electric Images. 151 (I)2;=—^/ Idmdn cosmr).coswf.7:372"T~i,2r AJi§J^m.QQ^«•QQ K+Tl'') et (11)t;=^/ Ic?mcZn coswii;cosnJ* suivant lesdeux cas.L'dquation (I)ser^duit a acause delavaleurqu'ontrouvepour Tint^graled^finiequiy estcontenue*. 215." L'expression pour v,dans lesecond cas, setrouve reduite enserieconvergente,siTonsubstitue pour g-^(a-ai) n^g_2A(a-ai) ^g-4/l(a-Oi)^J^^ etpuis,pourchaque terme, savaleur, suivant laformule cit^e dans lecas(I).Ontrouve ainsi v=Vp*[(2a-f)2+,,2+^]J [(y+2a-^)2+,!!+^]i [(2y+2a-f)2+,,2+^]i 1 1 1 [(y-f)2+)j2+f2]i [(2y_f)2+,,2+f23J [(3Y_^)2+,2+^2]i1.1.1 [(|-2ai)2+,,2+^2_iJ [(y+f_2ai)2+,,2+f2]j [(2y+^_2ai)2+r,2+^)i111 ou^[(y+f)2+1j2+|2]i [(2y+^)2+,,2+f2]i [(3y+f)2+,2+f2]i 7=2(a— (Xj). *Lesint^grales definies(c)et(I)sontdescasparticuliers dedeux int^grales multiples dontj'aitrouve lesvaleurs encherchant unedemonstration dela formule(5),tomeXdevotre Journal, page141. J'altrouv6, [above, §182, formula(7)],eneffet, M-l dp^dp^...dKco^ni^p^<iOBm^p^... _{n-l)ir'e-("^i'+»^2'+-)^*u: et /ooToo -00J-00dm^(^m2...C0SWia;iC0sm2rc2...e-(^x'+"»2'+-)^^M d'ouTond^duit imm^diatement lesintegrates cit6es. 152 ElectricLimges. [xiv. Decetteexpression ond^duit facilement ladistribution d'elec- tricit^ surdeuxspheres quisetoucheat. 216. *'Le cas(I)correspond adeuxspheres dont Tune, (a), estendedans deI'autre, (a^).Dans lecas(II),lesolide consid^r^remplit I'espaceentier endehors desdeuxspheres, etlatemperatureestz^roaunedistance infinie. 217."IIyauneinterpretation pourlenouveausystemede coordonn^es(r,6)dansunplan, quiesttres-simple. Eneifet, soient A^A'deuxpoints fixes, etPunpoint quelconque dont ils'agit d'exprimerlaposition. Celapentsefaireaumoyen deTangleAPA\ quej'appelle 6,etdelaraison rdeAPaAP". Quand 6aunevaleur constante, lelieudePestuncerclequi passe parlespoints A,A';etquandraunevaleur constante, lelieudePestuncercle, dont lecentre estdans leprolonge- ment deAA',dun cot^oudeI'autre, suivant quecette valeur estplusgrande ouplus petite que I'unit^, etquialapropri^t^ decouper aangledroit tout cercle decritparlespoints A,A'. "Posons main tenant, pour expliquerlesecondsysteme, *rcos6=^,rsin6= 7]. LelieudeP,quand ^aunevaleur constante, sera telque,siTon mene, deA,ADperpendiculaire aA'P^laraison DP-v-AP sera constante, etTontrouve ainsiquecelieu estuncercle quilouche enA'une droiteperpendiculaireaA'A;etTon trouve semblablement quelelieudeP,quand r)aunevaleur .constante, estuncerclequitouche A'AaupointA'.'' ''Knock, le16septembre 1846. 218."...Depuis quejevous aiecrit ladernierefois, j'ai consider^ leproblemedeladistribution d'^lectricitt^ sur le segmentd'une couchesphdriqueinfiniment mince, faitpar unplan,cecorps^tant compostdematiere conductrice, et j'aitrouv^, enexpression finie, lasolution complete,ensup- posant quelecorps possede unequantitydonnee d'electricit^ etqueladistribution sefaitsous I'influence demasses ^lec- triquesdonnees. J'avais I'intention der^digerdesuite pour vousunpetitMemoire surcesrecherches, maisj'airencontr^ quelquedifficult^ dans Texpositiondelamdthode suivie, et commejesuisapresent tres-occup^ (lescours aGlasgow commencent lel^'novembre, et ilmefaudra beaucoupdo XIV.]Electric Images. 153 prt^paration),ilmefaut differer cette tache*. Jemebornerai pourlemoment aux^noiic^s dequelques-unsdesr«^sultats. 219."SoitSlecorpsconducteur surlequelils'agitdede- terminer ladistribution. Pour premier cas, soitQunpointen dehors deS,surlameme surfacesph^riquedontSfaitpartie, etsupposons que>Sisoitmisencommunication avec lesolpar uu filconducteur infiniment mince(ainsilepotentieldansS seratoujours zero, quels quesoient lescorps^lectrisdsquien soientvoisins).IIs'agitdedeterminer ladistribution d'^lec- tricite surSsous I'influence d'unequantitydonn^e d'^lectricite negative Q,concentr^e aupoint Q.Jed^montre queI'intensite d'eiectricite alameme valeur auxpointsvoisins desdeux cotds delacouche 8yet,endenotantparacette valeur, pourun point quelconque Pde/S,jetrouve _Q^is'-ay oila,5etrsont lesdistances dubord deS,dupointQetdu point P,aunpointCdeSqu'on pent appelersoncentre, etA estladistance entreQetP. IIestremarquable quecette expression necontientpaslerayondelaspheredontSfait partie. Ensupposant quecerayonsoit infini, onaI'expression pourladistribution d'^lectricit^ surundisque circulaire, sous I'influence d'unpointdans sonplan, qui est,enefFet, lameme quecellequeGreen adonneepourcecas. 220."Pour trouver ladistribution dans lecasde8isold et electrise, jeremarque que,silaquantited'^lectricit^ surS esttellequelepotentiel quienr^sulte aunevaleur donnee F, ladistribution surSsera lam^me quecellequiaurait lieu si8 etait situ^ dans I'int^rieur d'une couche^lectrique quiproduit lepotentiel—F,Setant dans I'^tat d'uncorps quin'estpas isoie. Onpent prendre pourcette couche unesphereconcen- triqueavec celle dontSfaitpartie ;ensupposantI'exces du rayon delapremiere spheresur lerayondelasecond einfini- mentpetit, onreduit leproblemealadetermination dela distribution sur S,sous I'influence d'une distribution donnee d'electricite surlaspheredontSfaitpartie,cecorpsSn'^tant *Ithas, infact,been delayedtillDecember 1868andJanuary 1869. Seexv. below. 154} Electric Images, [xiv. pasisol^. Ainsi, parintdgration, jed^duis durdsultat donn^ ci-dessus lesexpressions (oil/estlediam^tre delasphere dontSfaitpartie), pourles intensites surlesdeux cotds, eonvexe etconcave, de8enun point P." NOTEAUSUJET DEL'ARTICLE PRECEDENT; PAR J.LIOUVILLE. 221*. LaLettre deM.Thomson m'asugg^r^ quelquesre- marques quejecrois devoirpresenter ici,parce qu'elles montre- ront, cemesemble, plusclairement encore touteI'importance dutravail dont lejeune g^ometredeGlasgownousadonnd un extraitrapide. Nous r^soudrons dabord leproblemesuivant : ProUeme.—Soient x,y,...,zetf,y,..., fdeuxgroupescon- tenant unnombredgalouinegaldevariables,lespremieres 00,yy...y z,inddpendantes,lesautresf,^,..., ffonctions despre- mieres, ensorteque f=/(^;yv., -2^),v^F{x,y,...,z),..., ^=<f)(x,y,...,z); soitencore p= yjr{j;, y,..., z). Designonsd'ailleurspar f',^',..., f',pcequedeviennent les fonctionsf,-j?,..., f,p,quandonyremplace x,y,...,zpar w, 2/',...,z\Celapose, ondemande dedeterminer lesfonctions y,Ff...y (f>, '\jr,demaniere aavoirg^neralement Pour fixer lesidees, nous nous bornerons aucasdetrois variables x,y,z,etdetrois variables f,17,J";etlaquestionsera deverifierI'^quation *[The original numbering ofM.Liouville's sections hasbeen altered bythe addition of220, formore convenient reference inthepresent volume.] XIV.]Electric Images. 155 Lam^me m^thode r^ussirait pourdeuxgroupes a?,y,...,zet ^fV}"-> ?quelcoDques.IIn'yaurait dechangement quedans quelques details, etseulement silenombre desvariables dtait difff^rent dans lesdeuxgroupes. Ausurplus,nous n'aurons besoinplustardqueducasoUcenombre estlememe depart etdautre, etnesurpasse pas trois, cequinouspermettrad'in- terpr^ter g^om^triquementlesrdsultats denotreanalyse. Donnons hx,y\zdesvaleursparticulieres x^,y^,z^k volenti etrepr^sentons parp^,f^, rj^,^^lesvaleurscorrespon- dantes dep'yf',rj\ ^'.L'dquation (1)nousdonnera (x-x,y+{y-y,f +{z-z,Y Mais, pour plusdesimplicity,nous mettronspartout ?+^o» ^+7?„f+fo,aj+^o^ 2/+2/o»^+^o>aulieudef,97,f,x,y,z,et dememef'+fo'^'+^oj^*^vaulieudef,x\etc.,cequine changerienaux differences f'—f,x'—x,etc.Lavaleur de p^deviendra etr^quation (1)subsistera tellequ'elleest. Enfaisant x'+f+z'==r\ p+y^+^'=p\ 2 '2 onaura /=r^2,/'=-^,Pop Pop etenportantcesvaleurs dansI'dquation (1),ontrouvera ais^- ment -2+-rij-2 (^^+^,^2+"t"^)PP \PPPPPPJ _ Jr1 1 fxXyyzz\ —Po\y+^—^ \^5:;2p+-2^2+— 2Y^j• Maintenant donnons ax\y,zquatre systemesdevaleurs connues avolonte, acbacundesquels repondrontdesvaleurs d^termindes der, ^', 17', f',p\etnous aurons ainsiquatre Equations dupremier degr^ quifourniront lesvaleurs de f1ii p-p-p-p- consider^es commequatre inconnues, enfonction lindaire de 156 Electric Images, [xiv. XyzX ^>7'^»^' Endesignant doneparA,B,G,Ddesconstantes, etparP,Q, RySdespolynomes dupremier degrdenx,y,z,cesvaleurs seront delaforme f_..^ ^_«.« ^-C^^ ^-7)+^7"^"^?' 7~^'^7' p^-^"^?' 7~^'^?- Enfaisant lasomme descarrds des troispremieres,ontrouve unevaleur de—^quidoit etreegaleacelle quedonne la quatrieme Equation.Ainsi lesdeux fonctions r r* doivent etreegales. Mais lapremieredevient une fonction entiere quandonlamultiplie parr^ IIfautdone quela seconde ledevienneaussi, etque,parconsequent, P^+0^+jR* soit^galementdivisibleparr*.Lequotientnepent^videm- ment etrequ'une eonstante, puisquelenum^rateur etled^- nominateur sontdum^medegre.Soitm^eette eonstante, et P^+(3^+^«=mV'=m^{x^+2/'+^0• P,Q,Retant despolynomes dupremier degr^, jefais P=m[ax^hy+cz -\-g)y Q=m(ax+h'y+<iz+g), R=m{a"x+h"y+c"z-{-g"), etj'enconclusparlacomparaisondesdeux membres, d'une part,a^-fa"+a"^=1,ah+a'h'+a'^h"=0, If+h'^^ h"^=1,ac+a'c'+a"c"=0, c=+c^+c'^^=1,hc+ h'c+V'(i'=0, Equationsd'oii r^sultent, comme onsait, les(Equationsinverses a^+6^^c'=1,aa!+W+cd=0, «"'+6"^+c^^= 1,aV^+6^"+cV'=; et,d'autrepart, ag+alg+aV=0,eg^-c'g+c'>"=0, XIV.]Electric Images. 157 Sinous admettions que g,g',g"sont desconstantes r^elles, r^quation g^+g'^+g"^=nous donnerait^^=0,g'=0,g"=0. Mais, dans tous lescas,onarrivera aumeme resultat kI'aide des troispre'c^dentes,enayant ^gard auxEquations decon- dition entre a,h,c,etc.Pourprouver, parexemple, que^=0, ilsuffirad'ajouter entre elles lestroisEquationsdont nous parlous apr^slesavoir multipli^es parlesfacteursrespectifs a,h,c.IInous restedone P=m{ax+hy+cz), Q=m{aw+h'g+c'z), R=m{a"x+h"y+c'z), a,h,c,etc., satisfaisant auxequationsdeconditionci-dessus, les memesqu'onrencontre dans latransformation decoordonndes rectangulairesend'autresrectangulairesaussi. Etcomme les Equations donnentprpr^p r i..(..?)V(..2)V(c,5)-, onenconclut lesformules suivantes ; f= Mais ilfautapresentretablir f— fo>V—Vo^?~?oaulieude ^,rj>?y etx—Xq, 2/— 2/o,z—Zqaulieudex,y,z.Cechange- mentfait,onaura lesformules lesplusgen^rales quipuissent satisfaire aI'^quation (1).Nous avons done lethdoreme suivant : 158 Electric Images. [xiv. Lesformulesg^n^rales quipeuventsatisfaire aI'^quation (1) s'obtiendront enposantd'abord :^=a{x-x,)+h{y-'y,)+c (z-zX Y=a'{a!-x,)+h'{y-y,)+c' (z-z,), z=a"{x-X,)+h"{y-y,)+c"{z- z,), lescoefficients a,6,etc., v^rifiant lesequations decondition a'+o:'+a"'=-l, ab+a'h'-{-a"h" =Q, puisprenant .mxT»^Y ^ mz X'+Y'+Z" XHY^+ Z''X'-1-y''+Z*' etenfin R^ciproquement, onpent ddmontrer que I'^quation (1)est satisfaite decette maniere, ettrouver lavaleur depquicon- vient. D'abord, destroisdemieres formules onconclut facilement lestroisprec^dentesdonnent dem§me /, ^2^/. ^.,r' \^ ,(x'-x)=+ (y-y7+ (z'-z)'*(u--uy^-{v-v)+{w-w) -^^ (^.+;.^^.)(^^.^^Vz-)- enfin, acause desEquationsdecondition entre a,h,etc.,on trouve (x'-xr+(Y'-Y)^+(z'-z)'^=(a;'-^-)'+(y-3/)' +(^'-^)'- IIvient done, enefFet, (r-if+(V-riT+(r-r)'-^'''~ '"^''^ ^'/f"^^''"^^' , lavaleur dep''etant JP=-^^ LI : 1^ m valeurqu'on pourraais^mentexprimerenx,y,z,enobservant queleproduit (x*^+y'^+z^){u^+v^+w^)est^gala (A^+J5'+C)(x'+y'+z^)+2Amx+2BmY+2Cmz+m^ XIV.]Electric Images. 159 etquex,Y,zsontconnus enfonction dex,y,z.Lavaleur qii'ontrouvera ainsipeutsemettre sous laformeV={A'+B'+0')[{x-xj+iy-y,y+{z- z,)"], ^vVv^1^tant desconstantes dont voici lesvaleurs: _ __m{Aa+Ba+Ca") m{Ah+BV-\-CV')Vi-VoA^+B^+C^' m(Ac+Be+00")^1-^0A'+B'+C' Sidonenousregardons plustard x,y,zcomme ^tant lesco- ordonneesrectangulairesd'unpoint quelconque, onvoitquela quantite pseraproportionnellealadistance decepoint (ic,y,z) h,unpointfixe(iCj, 3/^,z^,IIestais^aussi des'assurer que d'- d'- d'- 222. Pour avoirexplicitement ^,tj,feniv,y,z,ilsuffira de remplacer u,v,w,x,Y,zparleurs valeurs. Lapremieresub- stitution fournit ^_^^ ^(x^+T" +z^)+mx ^^°~(J.=^+-5'+0')(x'+Y'+z') +2^mX+2jBmY+20mz+m'^' Ledenominateur estprecisementlavaleur demp^dontonvient dedonner 1'expressionenx,y,z,savoir, m/=(A'+E'+O[(x-xj+{y-y.f+(z- ^.)^. IInerestedoneplus qu'achercher lenum^rateur. Lecaleul deviendra d'ailleurs fortsimplesiTonretranche desdeuxmem- bres laquantity A'+B'+C' caralors lesecond membrepourraser^duire aune fraction ayant pournum^rateur unpolynome dupremier degr^ en X,Y,z,et,parconsequent aussi, enx,y,z.End^signantdone parXuntelpolynome,etposant, pour abr^ger, Xonpourra^crire f—f*="§ > 160 Electric Images. [xiv. T Z etdememe'n—'^—~i->K~K^—%> if,f"etaiit desconstantes, etY,Zdesfonctions lin^aires de Xyy,z,Lespolynomes X,Y,Zs'obtiendraient sanspeine par cequ'onvient dedire;maisonlestrouve sousuneformeplus commode enoperant comme ilsuit. IIestaisedevoirqu'en attribuant unevaleur infinie auneouplusieursdesquantit^s Xj2/,^,ou,siTonveut, enfaisant a?*+2/^+/=X, ona?=r, rj=V,?=T,^qi^-• Sidoneonintroduit cettehypothesedex^-hy^+2^=<x)dans I'equation g^n^rale ilviendra d'ou,eneffa9antlesaccents, Mais, d'unautre cot^, done y2,-17-2 ,y2_ ^JP c'est-a-dire X'+Y' +Z'={x-x,f+{y- y.)'+{z-z^f. Dela,paruncalcul toutsemblable aceluiqu'onaeffectud dans lenumeroprecedent pour I'equation onconclutqu'en reprdsentant par a,/8,7,ol,etc.,desconstantes assujetties auxEquationsdecondition a^+a''+a^=l, a/3+a'yS'+a^'^"=0, 7'+i'+7''=1,^7+ i^'y+/3V= 0, dumemegenre quecelles entre a,6,etc.,ondevraprendre XIV.]Electric Images. 161 Y=^ci'(x^x;)+^'(y-y;) +y'(z-z,\ Z=d\x-X,)+^'{y-y^+ry'\z- z,). Et,r^ciproquement,ilestfacile deverifierqu'en adoptantces valeurs deX,F,Z,lesformules quirdsultent denotreanalyse enfaisant, pour abreger, 2qrBN^^=^'dou^= ~ , entrainerontI'dquation demand^e(1)dont lasolutiong^ndrale estexprimeeainsi d'une maniere nouvelle etplus simple. En eiFet,ontrouve d'abord /rrY(w.yrr^^.^rt^l{^'-^n{y'-'yfHZ'-'Zn puis (Z'-X/+{F-^)«+(^-Zf=(^'-^)^+(2/'-2/)»+(/-^)^ acause desEquationsdecondition entrea,^,etc.Etdela ontire [^ ?;i-wV)-^[^ ^)- (j^2_j_ Y'-{-z^) (X''+Y'^+z^)' c'est-a-direI'equation (1),enprenant ^n n' 223.Onpourraitformer inversement lesvaleurs dex,y,z enf, 77,f;mais ilestclair sans calcul, etapriori, queces valeurs doivents'exprimer pardesformules dumemegenre quecellesquidonnent^,?;,fena;,y,z.Eneffet, 'p^tant une fonction dex, 3/,z,onpentconcevoir cettequantity comme fonction def,77,f.Soitdone 1,1 CT^tant unecertaine fonction def,97,f,etot'lam^me fonction ^6f',7;',f'.L'equation (1)sechangeradans 1'Equationnouvelle d'une forme toute semblable al'equation (1)elle-meme, etqui, T.E. 11 162 Electriclinages. [xiv. parconsequent, donnerax,y,zenf,t^,fdelameme mani^re quer^quation (1)adonnd f,tj,^enx,y,z. 224.Onvoitque,parI'echangedeslettres x,y,zetf,r),f lesunesdans lesautres, unesolutionparticuliere deI'^quation (1),jeveux direune solution dans, laquellelesconstantes auraient desvaleursparticuli^res, endonnera une autre, la plupartdutemps differente, quoiquerentranttoujours,bien entendu, dans letype general indiqu^toutaI'heure. IIest aisd aussi devoirquedeux solutions denudes enfournissent unetroisieme. Supposons, eneffet, qu'en prenant pour f,?;, f,q desfonctions deTJ,V,TT,onait etque,dememe, enprenant pour U,V,W,p,desfonctions de X,y,z,onait ilestclairqu'on pourra exprimeraussiq,^,rj,^enx,y,z,et qu'ilviendra /t'tvA.w ^s^A-iy ^2(^'-^Y+{y-yY+{z-zf (f-?)+KV-V)+(?-g=-fq\py' d'oiiunesolution nouvelle denotreprobleme. Onpent dire,end'autres termes, quediverses transformations quirc^solvent ,ceproblemeetantop^r^es successivement, la transformation unique composde deeetensemble leresout aussi. Etparlamaniere dont nous avons verifid ci-dessus notre solutiong^n^rale,ilestmanifesto quecette solution n'est queler^sultat d'une suite desolutionsparticuli^resainsi ajoutdesentre ellespourainsi dire. 225. IIyaune solutionparticulieredeI'equation (1)que nousdevons dtudiersp^cialement parce qu'elle constitue, apro- prement parler,I'el^ment essentiel denosformulesg^n^rales, etqu'ellenous servira d'ailleurs aenbien montrer lesens g^om^trique.Elleaet^employee parM.Thomson, etconsiste aposer c.__nx _ '^^y y_^^-^^~ x'-\-y'+z^''^~x'-\-y^ +z'''^"Z+TT/' d'oil r^sulte, eneffet, I'equation XIV.]ElectricImages.163 st-a-dire r^quation (1),enprenant ^n Onaalors f+^'+^= ^2^2^^2> et,parconsequent, _n^ nrj _ n^ ^"fT?+r' ^^F+^'+T' ^"rT?+?' valeurs dememecompositionenf,77,fquelesprdc^dentesen a?,2/,z. Onpeut interpreter geometriquementcesformules enre- gardant X,y,z,parexemple, comme descoordonn^es rectangu- laires, et|,rj,fcomme desparametres.Lessurfaces(f),(77),(f), pour lesquelles undecesparametresconserve meme valeur, sont desspheres quisecoupentdeux adeuxortliogonalement, etparI'intersection detroisdesquelles M.Thomson determine lapositiondechaque point (x,y,z)ou(f, 77,f).Sous cepoint devue, ^,77,fsont descoordonneescurvilignes quiserappor- tentalamemefigure quelescoordonneesrectilignes x,y,z, Mais ilestpluscommode, jecrois, d'introduire dans nos re- cherches unedecestransmutations defiguressifamilieres aux geometres,etquionttant cbntribu^ auxprogres delascience dans cesdernierstemps. Latransformation dont ils'agitest bien connue, dureste, etdesplus simples ;c'est celleque M.Thomson lui-meme ajadis employeesous lenomdeprin- cipe desimages^.Considerez x,y,zcomme lescoordonnees d'unpoint quelconque md'unefigure rapportee atrois axes rectangulaires Ox,Oy,Oz, f, 77,fcomme celles d'unpoint /j, d'une autrefigure rapporteeh,trois axesOf, O77,Of,rectangu- laires aussi, etauxquelsnous donnons lamemeorigine 0,et respectivementlesmemos directions, unedecesfigures d^rivant deI'autre, etlepoint fi,enparticulier, correspondant aupoint m,envertu desrelationsparlesquelles f,77,fs'expriment en X,y,z,ouX,y,zenf,77,f.IIestevident quelesdeuxpoints correspondants m, fisontenlignedroite avecI'origine 0,et queleproduit Om.OfJb desrayonsvecteurs Om, O^jlestconstant I*Tome X.deceJournal, page3G4[above, §207]. 11—2 164 Electric Images. [xiv et=n.Une desfiguressededuit done deI'autre enprenan surchacun desrayonsvecteurs menes dupoint aunpoin quelconquedelapremiere figured'autresrayons vecteurs ei raison inverse despremiers;lesextr^mitds decesnouveau: rayonsvecteurs d^terminent lasecondefigure. Nous donne ronsacette transformation lenomdetransformation parrayon vecteursr^ciproques,relativement kI'origine (X_ Si,pourui point m,onaOw=-v/n,onaura aussiOix—Jn, etlespoint'met/I,quisecorrespondentainsi dans lesdeuxfigurescoin cideront. Endisposantdew,onpentfaireensortequ'un poin donndmreste fixedans latransformation;ilsuffit deprendre n—Om^, etalors tons lespointssitues surlasphere dont es lecentre etOm lerayon,resteront fixes aussi, mais tons le autres serontd^plac^s. 226.ATaide decette transformation parrayons vecteurs rM ciproques,onddduira d'unefiguredonnde une infinite d'autrei figures,soitenchangeant I'origined'oiipartentlesrayon vecteurs, soitenprenantdiverses valeurs denavecunemem origine 0,cequinedonne, ausurplus,lieuqu'adesfigure transform ^estoutes semblables entre elles,dumoins tantqu' ngardelememesigne ;carlesfigures quirdpondent adeu: valeurs den^galesetdesignescontraires sontsymfeiquei Onpentd'ailleurs effectuer, I'uneapres I'autre, destransforma tions relatives adesoriginesdiffdrentes. Maisjedisqueno formulesg^ndralesden°222peuvent toujours s'interpreter \\ I'aide d'une seule transformation decetteespece, ensortequ'oj! n'obtiendrait rien devraiment nouveau enajoutantd'autre transformations kcelle-la. Eneffet, dans lecasleplus g^n^ral, nouspouvons encor consid^rer x,y,zetf,77,J*comme lescoordonn^es dedeuxpoint m,fiappartenantadeuxfiguresdifferentes etrapportes adeu: sj^stemesd'axesrectangulairesdes x,y,zetf,97,f.Etvoic comment s'operelatransformation deI'une desfigures dan I'autre. D'abord onpassedex,y,ZyaX,F,^parlesformules Z=a(^- fl^J+^(3/-^0)+7(^- ^o)> z=di'ix-x^+r(2/- 2/0)+i'(^- ^0). IV.]Electriclinages. 165 )r,i\cause des(Equationsdecondition entreor,/?,etc., ce i.issagen'est qu'un changemeutdecoordonneesjrectangulaires 11d'autres coordonneesrectangulaires, quin'altere enrien la )reniierefigurealaquelleilestapplique ;onpentlesupposer iperedavance, etconfondre deslorsX,Y,Zavecx, ?/,z. Delanous ironsauxformules '^~X''i-Y'-hZ''^^X'+Y''-\-Z^'^^X''+Y'+Z'' tnousaurons ainsiunetransformation deX,Y,Zenf— f^, 7~"'7o> ?~ r*>q^^6^o^sregarderons comme descoordonnees ectangulaires prises parrapportauxmemos axes. Cette trans- formation estarayonsvecteursreciproques, comme nous I'avons vu n*"225. Elles'opereenportantsurlesrayonsvecteurs menes deI'origineactuelle deslongueursinversementpropor- tionnelles acesrayons vecteurs; I'anciennefiguresetrouve ainsichang^eencellequiresulte desextremitds detoutes ces longueurs.Passer ensuite de^— ^^,v~ V^^?~ ?''^f>Vj?> n'estqu'un simple deplacementdeI'origine,lesaxes restant parallelesaeux-memes; celaneproduit dans lafiguretrans- formee aucune alteration. Nosformules dun°222resultent done d'une transformation parrayonsvecteursreciproques, combin^e avec deschange- raents ordinaires decoordonnees. De telles transformations ennombrequelconquedonnenttoujoursnaissance auneequa- tiondelaforme(1),etrinterpr^tatiou geometriquedesformules parlesquellesnous avions d'abord lie(n"221) x,y,zetf,i),f semblait endemander deux, relatives adeuxorigines differentes, I'unepourlepassagedex,Y,zau,v,w,I'autrepourlepassage deu,V,wkS,v, ^}i^aisonvoit, parcequiprecede,etgrdce auxformulesplus simples dun°222,qu'uneseule transforma- tion suffitpourconduire aurdsultat leplus general ;iletait importantdeledemontrer. 227. Lesconsiderationsg^om^triquesdont nousvenons de faireusage, pour interpreterlesformulesquiconduisent a I'dquation (1),donnent lieuadesconsequences remarquables dontnous allons direquelques mots. Dans lesdeuxfigures queddterminentrespectivementlescoordonnees x,y,zetles coordonneesf,tj, .f,considerons, d'unepart,deuxpoints quel- 166 Electric Images. [xiv. conques m,m\et,d'aiitrepart,lespoints correspondants /a, fju. SoientDladistance desdeuxpremiers, Acelle desdeux autres, ensorteque A'=(r-fr+(v -'?)'+ (?'-?)'• L'^quation (1),quipourras'ecrire fournit une relation entre ladistance Adedeuxpoints /j,, yi! dans i'une desfiguresetlesquantites D, j^,V-Nous venons dedirequeDestladistance desdeuxpoints m,mcorrespon- dants dans I'autrefigure ;quant^|?etp',cesont,aunfacteur constantpres,lesdistances despoints m,mauncertainpoint fixe. Toute relationmetriqueentre deux ouplusieursdis- tancesAdansTune desfiguresfournira doneimmediatement unerelationanaloguedans I'autrefigure.Mais ilnefautpas croire quelesdiverspoints correspondantsaceux deladroite Asoient surladroite J)\cela arrivepourlespointsextremes parladefinition meme decesdroites, mais n'apas lieu, en general, pourlespointsintermediaires. Engeneral,lasuite despoints correspondantsaceux d'une droite delapremiere figureforme dans lasecondefigure unecirconf^rence decercle, laquellenesereduit aunelignedroite quedansuncaspar- ticulier, celuiousonrayonestinfini. Ayanten^, 97,fI'equationd'une surface oulesequations d'uneligne appartenant ^lapremiere figure,ilsuffit desubsti- tuerk^,7],^leurs valeurspour former enw,y,zI'equationde lasurface oulesEquations delaligne correspondante. On trouve bien facilement, decette maniere, quelesplansetdes spheressetransforment endesspheres quipeuventsereduire adesplans quandlerayondevient infini;que,dememe, des droites etdescirconferences decercle setransforment endes circonferences decercle, etc. Mais, poursuivre lemecanisme decestransformations,ilsuffit deconsid^rer latransformation parrayonsvecteursr^ciproques, quicombinee avecdeschange- ments decoordonn^es donne, comme onI'avu,latransforma- tion laplus g^nerale.Soitdone IV.]Electric Images.1G7 ^~ "> ,„.2 ,^2"~ "712>'^~ ^•^+2/'+^' ~r''yr+^'+r Tensemble desformules relatives alatransformation parrayons vecteursreciproques. Onenconclut immediatement ceque nousvenons d'avancer, concernant lesplansetlesspheres,les droites etlescirconferences decercle, Mais onvoit,deplus, etmeme sans calcul, quelesplans quipassent parlepoint 0, originedesrayons vecteurs, sont lesseulsquirestent desplans dans latransformation; avant etapres,leurpositionestla meme, quoiqueleurs diverspoints,bien entendu,sesoient deplaces poursesubstituer lesunsauxautres, ceuxquidtaient loindeI'origineen^tant apresent devenus voisins, etvice versa. Tout autreplansetransforme enunesphere passant parlepoint (oiilatransformation amene tons lespoints situes aI'infini)etayantsoncentre surlaperpendiculaire au planmenee dupoint ;laperpendiculaireetlediamMre de lasphereontunproduit ^galalaconstante n,etsededuisent ainsi facilement I'une deI'autre. IIestinutiled'ajouter que deuxspheres quicorrespondent adeux plans parallelesse touchent aupoint0.Dememe, deuxspheresainsiposeesse transformeraient endeuxplans paralleles.Mais unesphere quinepasse pasparlepointdoit rester unesphere, puis- qu'elle nepent acquerir aucunpointaTinfini. Les droites passant parlepoint restent desdroites, etconservent leur positioninvariable. Toute autre droite donne lieuaune cir- conference decercle dont leplanestd^termin^parladroite et parlepoint 0,etdont lecentre estsitue surlaperpendiculaire abaissee dupointsurladroite;lediametre estlequotient delaconstante nparcetteperpendiculaire.Lescirconferences provenant dedroitesparallelessont toutestangentes aune parallele meneeparlepoint acesdroites. Onpent voir, enfin, quelatransform(^e d'une circonference estune droiteI 168 Electric Images. [xiv. quandlacirconf^rencepasse parlepoint 0,et,dans toutautre cas,resteunecirconference. Unepropri^te remarquabledecegenredetransformation consiste encequelesdeuxtrianglesformespartroispoints infiniment voisinsquelconquesdelafigure primitiveetles troispoints correspondantsdesatransform^e sont semblables TunaTautre, ensortequesideuxlignessecoupentdans I'une desdeuxfiguressousuncertainangle,leslignes correspon- dantes deI'autrefiguresecouperontsous lememeangle*. La demonstration decettepropri^td reposesurI'dquation (1),a laquellenousavons donn^ laforme PP Supposons,eneffet, quelesdeuxpoints m,m',ou(x,y,z), {x,y\z),soient infiniment voisins, etqueleur distanceDsoit representee pards.Repr^sentons pardacelle deedeuxpoints correspondants /jl,jj!.Comme petpn'aurontpasdedifi"<^rence sensible,ilnousviendra ^dsaa=—5. P Les^Idments da,dsontdone enchaquelieuunrapportcon- stant quidependdepetchange,engeneral, dun lieuaI'autre. Considerons untroisiemepointm"infiniment voisin desdeux premiers,etd^signons pards'etds"sesdistances awetam'; da,da"^tant lesdistancescorrespondantes dans laseconde figure,onaura encore Tfds'da=—^,P ,.ds"da=—Y' P Done da :da' :da"r.dsids: ds". Ainsi,letriangleinfinitesimal mmm" estsemblable autriangle *Delasimilitude destriangles infinimentpetits correspondants,ilresulte encore que lafigure transformee estsemblable alafigure primitive, ouasa sym^trique, dans seselements infinimentpetits. Ens'entenant aupremier cas,quiestproprementcelui denosformules, oiinousprenons naturellement laconstante npositive, onaura, atrois dimensions, unesorte derepresentations descorps, analogue autrac^ descartes geographiques [those according tothe "stereographic projection"], pour lesquelleslerapport desimilitude deselements: correspondantsestvariable aussi d'un lieuaI'autre. XJV.] Electric Images.169 correspondant fi^' [jl". L'anglededsavec dsest,parcons(^- quent,lememe quecelui dedaavecda .Cette demonstration, onlevoit, n'exige pasmeme queI'^quation (1)aitlieupour deuxpointssitu^s aunedistance finie;elledemande seule- ment quecetteequationaittoujourslieupourdeuxpoints infiniment voisins. Ondoitendireautant d'untheoreme que jevais etablir, etquin'estqu'uncorollaire delaproposition prdc^dente. Une surface appartenantaI'une desdeuxfigures^tant donnee, representez-vousleslignes.decourbure decette sur- face, etlesdeux series desurfacesd^veloppables, orthogonales entre elles etalasurface donn^e, quisont formeesparles normales successives. Dans lasecondefigure,lesseries de surfacescoiTespondantesresterontorthogonalesentre elles et^ latransformee delasurface donnee;par suite, envertu du beautheoreme deM.Ch.Dupin,elles traceront encore sur cette transformee deslignesdecourbure. Ceslignes decour- bure r^sulteront ainsi deslignesdecourbure delapremiere surface donnee, etseront imm^diatement connues silesautres lesont. IIsera ais^d'appliquercetheoreme aux surfaces du seconddegre, comme aussi auxsystemes triplesdesurfaces orthogonales queM.Serret aindiquesdansuneNote recente*, etqui,parnotre transformation, endonneront d'autres non moins curieux, etc. Proposons-nous, parexemple,detrouver leslignesdecour- bure delasurfaceenveloppedesspheres quitouchent trois spheres donnees, probleme queM.Ch.Dupinaresolujadis dans laCorrespondancesurVEcole Poly technique,tomeI,page 22.Soient etPlespointsd'intersection decestroisspheres; prenonslepoint pour origine,etoperonsunetransformation parrayonsvecteursreciproques,cequinous foumira une secondefigured'oiinous reviendrons aisement alapremiere- Dans lasecondefigure,lestroisspheresdonnees seront rem- placees partroisplans quisecouperontenunpoint11 correspondant ausecondpointPd'intersection denos trois spheres. Lasurfaceenveloppedesspheres tangentes aces troisplanssera(ensebornant aundesanglessolides etason IPage 241dupresent volume[Liouville's Journal, 18^7]. 170 Electric Images. [xiv. oppose)celled'uncone droit abase circulaireayantsonsommet aupoint 11,etcirconscrit ^unequelconquedesspheres tan- gentes aux troisplans. Leslignes decourbure decette surface coniquesont :1°lesgeneratrices rectilignes quipassent toutes parlepointIT :dans leretour alapremiere figure,cesdroites deviendront descerclespassanttousparlepoint P,dont les tangentesenPferont toutes lememeangleavec latangente aucercle danslequelsetransforme I'axeducone, d'ou resultera unnouveau cone droit, etpassanttoutes aussi avec des cir- constances semblablesparlepoint ;2°descercles, dont les planssont tousparallelesentre euxetperpendiculairesaI'axe ducone, etqui,lorsduretour alapremiere figure,deviendront des cerclescoupant aangledroit ceuxquiresultent de generatrices rectilignes. Leslignesdecourbure delasurface enveloppedesspheres tangentes atroisspheresdonndes sont done descirconferences decercle. Ond^montre avec lameme facility leth^or^me deM.Dupin concemant lacourbe quetrace surchacune des troisspheres donn^es laspherevariablequilestouche. Eneffet,quandles troisspheresdonnees sontremplacees partroisplans,ilest clairquelasuite despoints suivantlesquelslaspherevariable touche unquelconquedesplansestunelignedroitepassant parlepointd'intersection 11.Done, enrevenant aux trois spheres donnees, lacourbe demandee estunecirconference de cerclequipasse parlespointsetP. IIpent arriver, bien entendu, quelespointsetPsoientimaginaires ;mais iln'y aalorsaucun changementessentiel afaire dans cequenous venons dedire, etnosconclusions subsistent. Lacirconstance d'uneorigine imaginaireauraitplus d'inconvenient s'ils'agissaitder^soudre leproblemed'une sphere tangente aquatre autres, enleramenant auprobleme tres-simpledetrouver unesphere tangente aunespheredonn^e etatroisplans donnes;maisonyrem^dierait enaugmentant d'unememequantitylesrayonsdesquatre spheres donnees, ce quinechange paslaposition ducentre delasphere tangente Dememe, ensebornant aconsiderer despointstous situ^f dansunplan passant parI'origine 0,onramenera ladetermi- nation ducercletangentatrois autres acelle d'un cercle qu touche uncercle donn^ etdeux droites donnees. XIV.]Electric Images. 171 Engeneral,lessyst^mesdespheresoudecercles, etsp^ciale- ment despheresoudecercles passant parunpoint donn^, jouissentdeproprietescurieuses dont beaucoupdeviennent intuitives parlatransformation dont nous venons denous occuper. Onpent appliquerenparticuliercetteremarqueaux theoremes queM.Miqueladonnas dans sonMdmoire surles angles curvilignes*.Pour nous borner aucasleplus simple, ilestevidentque,dansuntriangle ABG formepartrois arcs decerclespassanttonsparunmemopoint 0,lasomme des anglesvaut 2droits, puisquenotre transformation rend ce triangle rectilignesans alterer sesangles. 228.Lepassagedesrelations metriquesd'unefigure aI'autre, dans latransformation parrayonsvecteursreciproques, en allant descoordonneesf,77,fauxcoordonneesos,y,z,s'operea. I'aide delaformule ousimplement A=— >, enposant?i=1,cequin'aaucuii inconvenient. Mais en designant par I'origine,dans lasecondefigure seulement, et enemployantlesautres lettres A,B,etc.,pour representor ala foislespointsdelapremiere figureetlespoints correspondants delasecondefigure,cette formule revient adireque,danstoute relation entre desdistances AB,BD, etc.,ilfautremplacerAB chaquedistance tellequeABparjj-a— Tyo*Voila doneune regie pratique tres-commode;cetteregieconvient aussi bienau casduplanquaceluideI'espace. Deuxexemplessufiiront. Quedesdroitespartantd'unpointfixeAcoupent chacune uncercle endeuxpointsBetC,B'etC\etc.,onaura ABXAG=AB'XAG'=constante. Done, dans lafigure transform^e, AB AG AB' AG' OA.OB^ OA.OG^OA .OB'^OA .OG'' etparconsequent,ABAG nn^Tw~constante.I *Tome IX.deceJournal page20[Liouville's Journal, 1844]. 172 Electric Images. [xiv. D'ailleurs lespoints A,B,C,qui^taient enligne droite, se trouvent apresentsurune circonf^^rence decerclepassant par lepoint 0.Nous voyons parlaquelescerclespassant par' deuxpointsfixes A, coupent uncercle donne endeux points B,Gtels,quelerapportdesproduitsdesdistances ABXAG etOBxOGaunevaleur constante pourtons ces cercles. Quelescotes BG,AG,AB d'untriangle rectiligne ABG soient coupes entroispoints A',B',G'parune transversale, on aura AG'XBA'XGB'=BG'xGA'xAB'. Done, dans lafigure transformde, AG' BA' GB' BG' GA' AB V/ /^7~>/^ i'/^/~^ /~\Tif*""/^O/^/^' y^JY"Z^V~/Ak' OA.OG' OB.OA' OG.OB'OB.OG' OG.OA'OA.0B\ cequiredonne AG'xBA'xGB' ^BG'xGA'xAB. Mais cette relations'applique apresent auntriangle curviligneABG form^partrois cerclesquipassenttonsaupointet dont lescotes sontcoupds enA',B,G'parunquatri^mecercle passantaussiaupoint0. IIest,dureste, inutiled'ajouter que AG',BA', etc.,sont lespluscourtes distances despointsAet G',BetA',etc., etnondessegmentsmesures surlescotesdu triangle curviligne. Ong^neraliseraitaisement delameme maniere letheoreme relatif aunpolygene gauche coupe parunplan.Maisenvoila assez surcesujet. 229. Etant denudes deuxspheres quinesecoupent pas,on pent toujours placer I'originesurladroitequijointleurs centres, enunpointrdel tel,qu'apreslatransformation par rayonsvecteursreciproques,cesdeuxspheresseront con- centriques.Prenons ladroite descentrespouraxedesic; designons parhladistance inconnue dupointaucentre de lapremiere sphere,etparh+lsadistance aucentre dela secondesphere;soient k,k'lesrayons.LesEquationsdes deuxspheres seront, avant latransformation, (a^^h-iy +7f-^z'=^k'% etapreslatransformation, quiconsistera aremplacer o),y,zpai XIV.]Electric Images., 173 Xyz x^+fj^z^' ^^qrpq:^' x'-\-y''-\-z'' ellesdeviendront A;'^ Pour quelecentre soit lememe apresent,ilfaut etilsuffit h_h+l ^"®Ji'-k'"{h+lf-k''' d'o^ lh'+(J'+k'-k")h+lk'=0, Equation duseconddegre quidonnerapourhdeux valeurs, enposant G=:{l-k-k') (l-k-{-k')(l+k-¥)(I+k-\-k') ; etilestaisedevoirqueGserapositivesilesdeuxspheres qu'onadonnees d'abord nesecoupent pas. 230. Ceth^oreme pourraetre utile engdom^trie ;mais il aura surtout uneapplication importante dans lesquestionsde physique math^matique. Essayonsicid'indiquer rapidement I'usage,encegenredequestions,delatransformationgenerale quidonneI'^quation (1).LaLettre deM.Thomson nous servira deguide ;nousyajouterons quelques d^veloppements. Lag^n^ralite plusoumoins grandedelasolutionparlaquelle onsatisfait aI'dquation (1)nechange enrien lamarche a suivre, quireste lameme dans tous lescas. Etd'abord deI'^quation A"D onpent conclure, avecM.Thomson, que,siunefonction Ude ^,7],fsatisfait aI'dquation cette.meme fonction, divis^e parpetexprim^eenx,y,z, vdrifieraI'^quationdememe forme d\p-'U ^d\p-'U ^d\p-'U^^ dai^dy''dz^I 174 ElectricImages. [xiv. De1^une liaison entre deuxproblemesdistincts concernant tousdeuxr^quilibredetemperaturedans lescorps homogeneS; mais relatifs adeuxsystemesdontTunr^sulte deI'autreparla transformationquilief,?;,facc,3/,z. Quelepremier systemesoitform^ dedeuxspheres quinese coupent pas,quelatempe'raturesoitdonn^e enchaque point deleurs surfaces, etdemandonsquelleestlaloidestempera- tures permanentes dansI'espace comprisentre elles,siTune estintdrieure aI'autre, oudansI'espaceinfini exterieur atoutes deux, siTune estendehors deI'autre, enajoutantdans ce dernier caslacondition quelatemperaturesoitnulle aTinfini, Onramenera cettequestionaucastres-facile dedeuxspheres concentriques.Cela r^sulte duthdor^me ^tabli ci-dessus efcec montre touteI'importance. Enindiquantcetteapplicationa latheorie delachaleur, M.Thomsonajoute, dureste, avec raisonqu'elles'^tend d'elle-meme ^latheorie deI'electricite. Dans latheorie deI'eiectricite oudumagnetism e,et,en general,dans latheorie deI'attraction, laquantite queG.GrecE etM.Gauss nomment^otentiel,c'est-a-dire laquantity qu'oc obtient enfaisant lasomme deselements attractifs ourepulsift d'une masse divisds parleurs distances aunpoint, joueunrole capital. Onconnait leproblemedeM.Gauss :"Distribuei surunesurface donn^e unemasse attractive ourepulsive,de telle sorte quelepotentielaitenchaque pointdelasurface unevaleur donnde." Onaresolu ceprobleme pourdiff^rentes surfaces, enparticulier pour rellipsoide. Orlasolution relative aune surface quelconquedonne lasolution pourtoutes le.' surfacesquisededuisent decelle-laparunetransformatior pour laquelle r^quation (1)aitlieu.Ayant,eneffet, I'^quatior Xdco'_^ If'A pourlapremiere surface, onaurapourlasecond esurface une Equationdumemegenre, remplagant parleurs nouvellej valeurs Aetdco'.Ona Quant adco',j'observe queleselements lin^airescorrespondants do-etdssont li^sparlaformule 7dsda— —,.P XIV.] Electric Images. 175 Done entre deux^ymentssuperficie onaura (Zci)=— 4-,cZo)'^=Done entre deux^ymentssuperficiels correspondants dw,da, daf 7 f[f\'da'Q suite,jj^^l)='y quir^sout leproblemedeM.Gausspourlasurface trans- formde. Onpentvoir aussiquelesEquations designees par(A), (B), (G)dansmesLettres aM.Blanchet*, etquisontdun sigrand usagedans laplupartdesquestions physico-mathdmatiques concernant Tellipsoide,ontleursanalogues, qu'on end^duit imm^diatement pourlessurfaces transformees derellipsoide-f. Onpentconsid^rer encoreI'^quation df~ dedrf'^d^' etluifaire subir latransformation def,?;,fenx,y,z. rcause deI'dquation quipentsMcrire ontrouve, pardesformules connues, que}aquantite d'U d'Ud'U^ dp"^ drj''"^ d^' est^galea PI' p^dx ^' p^dy' p^dz j \dx dydzJ *VoyezletomeXIdeceJournal. tParmi cessurfaces,ilfaut distinguercellequedonne latransformation par rayons vecteursreciproques, enmettautI'origine aucentrem^me deI'ellipsoide.Onsaitqu'elle estaussi lelieudespieds desperpendiculairesabaissees du centre surlesplans tangents aunautreellipsoide dont lesaxesontpour valeurs lesinverses desvaleurs -desaxesdeI'ellipsoide donne. Unepropriete analogue alieudans leplan, pour lalemniscate parexemple, quipentainsi ^treengendree dedeuxmanicres differentes aumoj'en d'une hyperbole equilatere, circonstance dontM.Chasles atireunheureux parti dans sesrecherches sur lesarcsegoux delalemniscate {Comptes Rendus deVAcademie desSciences, tomeXXI, seance du21juillet 1845). 176 Eleotric Images. [xiv. c'est-^-dire k P[dx'"^df"^dz'J^\dxdx^dydy"^dzdz) ., ,fd\p-'U ^d^.p-'U ^d\p"U\ouenfin kp(^-^+-^+-^ j, enserappelant que cZ^i d'- cZ^i dx'^df^dz''-^'' Parlaonvoitd'abord queI'equation revient hcelle-ci : d\p-'U d\p-'U d^.p''U _^ dx'"^df"^dz'~ ' cequenous savionsdeja.Onvoitensuite queI'equation dfd^"^ drf"^ (Zf'^ setransforme en df~^\dx""^ d'^/'"^cZ/r ou,mieux encore, en df^\dx''^df"^ dz^ )' R^ciproquement,cette derniereEquation, oulecoefficient p varieproportionnellement aladistance dupoint (x,y,z\aun point fixe, seramene aI'equation ^^d^udnr dni df rff"^ df)''"^ d^'' quiestacoefficients constants, resultatquitrouve uneapplica- tion utiledans lath^orie duson. Onpentenfinajouter quelesEquationsaux differences partielles fdU\' /dU\' fdUV ^ et (dU \dxhQ'*{'£H XIV.]ElectricImages. 177 sent (lestransforiiK^es rune deI'autre, cequipourraservir dans lesquestionsdedynamique,otiMM.Hamilton etJacobi ont introduit detellesequationsauxdifferencespartielles. Onmepardonnera, jeI'espere,cesd^veloppements que j'ai crupouvoir donner, alasuite desdeux Lettres siinteressantes deM.Thomson, sans legenerdans sesrecherches. Mon but serarempli, jelerdpete,s'ilspeuventaider abien fairecom- prendrelahauteimportance dutravail decejeune geometre,et siM.Thomson lui-meme veut bienyvoirunepreuve nouvelle deI'amitie quejeluiporteetdeI'estime que j'aipourson talent. T.E. 12 XY.DETERMINATION OFTHEDISTRIBUTION OFELECTEI- CITY ONACIRCULAR SEGMENT OFPLANE OR SPHERICAL CONDUCTING SURFACE, UNDER ANY GIYEN INFLUENCE. [Jan. 1869. Not hithertopublished.} 231. The electricdensityatanypointofthesurface ofan insulatedconducting ellipsoid,electrified and leftundisturbed byexternal influence, is(§11) simply proportionaltothedis- tance ofthetangent plane from thecentre. Ifwetakep=kp astheexpressionofthis law,and callqthewholequantityof electricity communicated, wehave(§14)4i7rkahc =q;sothat theformula forthe electricdensity, p,atanypointPofthe surface interms ofp,thedistance ofthetangent planefrom thecentre, and a,h,cthethree semi-axes,is ^=4^c^(^>= or,interms ofrectangularco-ordinates ofthepoint P, 4.rak (-,+!+ -,) 232. Tofindthe"electrostaticcapacity" (§51, footnote)oi thecharged ellipsoid,letFdenote thepotentialatitssurface. Wehave,by§15(e), Y_ ["^^^du . ^i«^(it'- a'+h'y^/'(u'-a'+0")^^-^^ and therefore thecapacityisthereciprocalofthe definite integralwhichappearsinthisformula. 233.Bytakingc=wefallonthecaseofaninfinitelythin plane ellipticdisc :forwhich wehave r. XV.]DistributionofElectricity,etc. 179 andtherefore 4^7rab(l.(4). Puttingh=ainthis,wehave, foraninfinitelythin circular disc, P='9' (5),4<7ra{a'-ry where adenotes theradius ofthedisc,andptheelectricdensity oneither sideofit,atadistance rfromthecentre. This result was firstgiven byGreen, neartheconclusion ofhispaper"On theLaws oftheEquilibriumofFluidsanalogoustotheElectric Fluid"(Transactions oftheGamhridge Philosophical Societyfin- Nov.12,1832) ;fromwhich Imake thefollowingextract :— 234."Biot {Traite dePhysique,tome ii.p.277)hasrelated "theresults ofsome experiments made byCoulomb onthe "distribution ofthe.electric fluidwhen inequilibrium upona " plateofcopper10inches indiameter, butofwhich thethick- "ness isnotspecified.Ifweconceive thisthickness tobe "verysmall comparedwith thediameter -oftheplate,which "wasundoubtedlythecase,theformulajustfoundoughttobe "applicabletoit,provided w^exceptthosepartsoftheplate *'which areintheimmediatevicinityofitsexterioredge. As "thecomparisonofanyresultsmathematicallydeduced from "thereceivedtheoryofelectricitywith those oftheexperi- "ments ofsoaccurate anobserver asCoulomb mustalwaysbe " interesting, wewillheregiveatable ofthevalues ofthe "densityatdifferentpoints onthesurface oftheplate,calcu- "latedbymeans oftheformula(29), togetherwith thecor- " respondingvalues found fromexperiment:— Distances from 180 DistributionofElectricityonCircular[x.Y. "observed are allsomethingsmaller than thecalculated ones, "which, itisevident, oughttobethe case, since thelatter "have been determined byconsideringthethickness ofthe " plateasinfinitely small, andconsequently theywillbesome- "whatgreaterthanwhen thisthickness isafinitequantity,as "itnecessarilywasinCoulomb'sexperiments." 235. Inthiscase(3)of§232becomes JaU's/{u-a)^Za Hence thecapacityis— .But[§232(3;]thecapacityofa IT globeisnumerically equaltoitsradius;andtherefore the capacityofaninfinitelythin disc islessthan that ofaglobe rrr ofequal radius, intheratio of1to—,or1to1'571. Caven- dishfound theratio 1to1*57,byexperiment* ! 236. Theexpression (5),§233, fortheelectricdensityat an}'- pointPoneither side ofan infinitelythin circular disc ofcon- ductingmaterial electrified and left free fromdisturbing influence, may beputinto aformmore convenient forgeometrical investigation,thus :— LetGbethecentre ofthe disc, so -^^thatCA—a,CP=r,accordingto previousnotation. HenceBP=a+T;PA=a—r; *Myauthority forthis statement isthefollowing entry which Ifind written inpencil onanoldmemorandum-book :— "Plymouth, Mond., July 2,1849. "SirWilliam Snow Harris hasbeen showing meCavendish's unpublished "mss., putinhishands byLord Burlington, andhisworkupon them; a "most valuable mine ofresults. Ifind already thecapacity ofadisc " (circular)wasdetermined experimentally byCavendish as;^-—that ofa ''sphereofsame radius. Nowwehave /•^rdr f'^drcapacityofdisc _. —^^„- ''^ 'TT1"071 2 Itismuch tobedesired that those maniiscriptsofCavendish should be published complete; or,atallevents, that their safe keei^ing and accessi- bility should besecured totheworld. I]Segment ofSpherical Conducting Surface.181 a'-r'=BP.PA^KP.PL\ Lbeanychordthrough P;and(5),with substituted IT forqaccordingto(6),becomes p=^ (7). R37.Consider aplane disc,S\thus electrified, toapotential chweshall, foramoment, denotebyV;and,followingthe suggestionof§210, take itsimage relativelytoaspherical surface ofradiusRdescribed from any pointQascentre. Thisimagewill(§207)beaspherical segment,>Si,electrified (§§210and238)asaninfinitelythinconductingsurface under theinfluence ofaquantity VRofelectricityconcentrated at Q;and(compare §213)thesphericalsurface ofwhichSisa partwillpassthrough Q.Thereader willhave nodifficulty inverifyingthese statements forhimself; but ifhedesires it, hewill findsome further information andexamplesinThom- sonand Tait's NaturalPhilosophy, §§512. ..518. Thus(§515 ofthatwork)ifpbetheelectricdensity oneither sideofthe discatP'j,andpthatoneither sideofitsimageatP,wehave p=w"''^^^' and ifvandvbethepotentialsatanypoint IT',and 11the imageofII',duerespectivelytothediscS'and itsimage, we have(Thomson andTait, §516) "= ^1"''*')• Thisshows that, asthepotential dueto/S"hasaconstant value, V\atallpointsofS',thepotentialdueto/Swill be,atdifferent pointsofS,inverselyastheir distances fromQ ;and ifwetake q=-^RV', anddenotebyVthepotentialdue toelectricity distributed overthetwosides ofS,wehave ^=3^''^' andsoseethatSiselectrified asaninfinitelythinconducting sheet ofthesamefigure would beifconnected with theearth byaninfinitelyfine wire,andinductivelyelectrified onlyby 182 Distribution ofElectricityonCircular [XV. theinfluence ofaquantity—gofelectricityinsulated atQ.. Kow, withourpresentnotation(7)gives r 1 q p= ^ir^JUF.PK'=.^...(11), ifK'L' beanychordthrougliP'ofthe circle boundingthe planedisc >S". 238. LetK,P,andLbetheimagesofK\F\L',sothat KL,theimageofK'L'\ isthearein which >Sfiscutbytheplane through y"'" QandK'L'.Wehave(§207) K'Q=R' R' Hence K'Q:P'Q::PQ:KQ', and therefore thetriangles iT'P'Q,FKQ aresimilar ;andtherefore 'P'QJ{'QKQ.PQ (12).P'K'^KP =KPV: KQ.PQ (Compare §§213, 227.) From this, andthecorresponding expressionfor LP,wehave L'.P\PK'=R'LP .KP ...(13);PQ''LQ.KQ anexpressioninwhich, asthe firstmember hasthesame value foralllines such asL'K'through P',thesecond must have thesame value forallplanes through PQ, cuttingone circle ononesphericalsurface through Pand Q,inKandL.As isconstant,itfollows thatx77~Tv^^^constant*; atheoremPQ -Lt(^-J^Q ofgeometry givenabove(§228)byM.Liouville. Eachmem- *Asaparticular case letQbeeither pole ofthefixed circle. Inthis case LQ=KQ, aconstant. HenceLP .KP isconstant; that is,theproductoi thetwochords from any fixed pointPonaspherical surface tothetwo pointsinwhich any fixed circle onthesurface iscutbyaplane through P andoneofthepoles ofthatcircle,isconstant, however theplane bevaried. This isthesimplest extension tospherical surfaces oftheelementary geo- metrical theorem (Euc.in.35)fortheconstancyoftherectangle under the twoparts ofavarying chord ofafixed circle through afixed point, already used inthetext(§236). XV.]. Segment ofSpherical Conducting Surface. 183 berof(12)maybealtered inform tbus :bisect L'K' inM,and arcLKini\^.Wehave LP.KP =NK'-NP' \(14), LQ,KQ =NQ'-NK'J equationsofwhich the lasttwoarevery easily provedfrom theformula sin(a—(3)sin(a+/3)=sin^a— sin'^yS, bytakingforaand /3theanglessubtended by^NKand\NP atthecentre ofthe circleQKPL ;andagain, bytakingfor aandfitheanglessubtended byJiYQand^NKatthesame point. 239. Using (13)in(11),andtheresult in(8),wefind ^~27r-^Qi^V LP.KP^^' andmodifying by(14), P~27r'QP'\/ NK'-NP'^ ^' Ifwetake forPKQL theplane through PQandCthecentral point (orpole)ofthespherical segment 5,sothatNbecomes C,NKbecomesequaltothechord ofanyarcfromGtothe lip;and(15)becomes which istheresult stated in§219, above. Itisremarkable that thisexpressionisindependentoftheradius ofthespheri- calsurface ofwhich thebowl isapart. Hence, ifwesuppose theradius infinite, wehave thesameexpression (17)forthe electricdensityatanypointPoneither side ofaninfinitely thin circular disc ofradiusa,connected with theearth byan infinitelyfinewire,andinfluenced byaquantity Qofelectricity collected atanypointQintheplaneofthe disc,butoutside itsboundingcircle. Itagreeswith thesolutionpreviously given byGreen forthiscase inhispaperreferred toin§234, above. 240. (Compare §220.) Tofindthedistribution forthecase inwhichSisinsulated, electrified, andremoved from alldis- turbing influence, letVbetheconstantpotential produced throughout Sbythis distribution. Remark that thesame 184 DistributionofElectricityonCircular[xy. distribution ofelectricityonSwould beproduced inductively ifitwere connectedbyaninfinitely fine wire with the earth, and en- closed byany s-urface, EE, rigidly electrified with such aquantity and distribution ofelectricityas(§§ 5, 73,206, 207; alsoThomson and Tait, §499)toproduceauniform potential—Vthroughitsinterior. Now take thisenclosing surface, EE, tobespherical,concentric with that ofwhich >Sfisapart, and ofradiusgreater than that ofthe lastmentioned byan infinitelysmall excess. The electricdensityoftheinducingV distribution willbeuniform alloverEE,andequalto—-— if/bethediameter ofthesurface. TheportionofEEwhich liesinfinitely near totheconvex surface of8willclearly induce onthisconvex surface anequalelectricdensityofcontrary V sign,thatis-,+-— ,..Theremainder ofEE willinduceequal electric densities ontheconcave andconvex sides of8,the amount ofeither ofwhich atanypointPistobeobtainedby integration from(15), (16),or(17),thus:— 241. Letdabeaninfinitesimal element ofE,situated ata pointQanywhere on it.Thequantityofelectricity onthis element is—-tt—^ ;andusingthis for—q in(17),wefind, forthedensityoneither side atP,ofthe electrification induced byit,thefollowing expression:— V da-iCq ^irJFQ's/ a'-CF'' Nowcallingthecentre ofthespherical surface, letCOP bedenoted by 77;COQ by6;thevalue ofeither ofthesewhenPorQisatthelipol thebowl,bya;andtheangle between theplanesofCOP and COQ, hj (j):sothatwehave a'=hf (1-cosol),Cr=^if{l-cosv), CC/=If (1-cos0) and PQ'=hf^ (1-cost; cos^-sin?;sin cos(/>) ; XV.] Segment ofSpherical Conducting Surface. 185 andwemaytake da=ifsineded(i>. Henceif,lastly, pdenote theelectricdensityatP,onthecon- cave sideofthesegment, wehave [^ /2ir ^^ /ddsin-v/fcos a-cos6)I ^Ja Jo1-COST? COS^-SI Stt^/^ {cos 7}-COSa)Ja 70^~<^o^'?°os^-sin97sin0cos0 ButputtingtanJ</>=^,A=l—cosrjcos6,and5=sin7;sin6, wefind ("'^ =4r Jo1-cos7;COS^-sin97sin^COSJqA-Bdt 27r 27r +{A+B)t^ V(-i'-£')~GOS97-cos0' andtherefore V f^'dd sin^V(cosa-cos6) 47ryV(cos 7]—cosa)ja cos7)—cos^ Lastly, putting v^(cosa—cos0)=z,wefind r"c/(9sin^V(cosQc-cos(9) _rV(cosa+i) /cZ^ Ja COS7J—COsd Jo COS7;—COSa+-2;^ =2w(cosa+1)—Vcos 77—cosatan"^ ./—- [. (^Vcos77—cosaj Hence wehave, inconclusion, p=j;| /^£iJL+l__tan-- /'=°'^+^ 1.(18), '^ Ztt/(Ycos7;—cosaVcos77-cosaj^ or,with/andaasabove, andrtodenote thechord CP, p=2^fW-^^-'^--'S~=^}('')= andthesame, withtheaddition of . . .2^/ ••••;^'''' gives (§240)theelectricdensity ontheconvex side;which are exactlytheresults stated above in§220. Twenty-two years agotheseandtheverysimpleformula(17)werecommunicated bymetoM.Liouville withoutproof, andwerepublishedinhis Journal. From thattime tillnowtheyhave notbeenproved, oreven noticed, sofarasIamaware, byanyother writer. 242. Numerical results, calculated from theprecedingfor- mulae(19)and(20), areshown inthefollowingtables :— 186 DistributionofElectricity onCircular[xv. Plane Disc. Curved Disc. xv.]i Segment ofSpherical Conducting Surfaxie. 187 Theconstant coefficient foreach casehasbeen taken soas tomake themean oftheelectric densities ontheconvex and concave sides unityatthemiddlepoint (asinGreen's numbers, §234above, fortheplane disc). The sixpointsforwhich theelectric densities areshown inthetables below are(not thesixpointstowhich Coulomb's observations and Green's numbers quotedin§234 refer, but)themiddlepoint,andthe fivepoints dividingthearcfrom themiddle totheedgeorlip into &ixequal parts. 243.Asecondapplicationoftheprinciplestated in§210, andused in§§237... 239,allows ustoproceedfrom thesolu- tionnowfound forthe electrification ofanuninfluenced bowl todetermine the electrification ofabowl ordiscunder the influence ofelectricityinsulated atapointQ(not,asinthe solution of§239, necessarilyinthesphericalsurface orplane ofthebowl ordisc, but)anywhereintheneighbourhood. Con- sider theimage,>S^,ofanuninfluenced electrified bowl, S', relativelytoasphericalsurface described fromanypointQin itsneighbourhood,ascentre, with radius R.LetD'bethe pointonthesphericalsurface ofS'continued, which isequi- distant from thelip(sothatD'andthemiddlepointofthe conductingsurface S'arethetwopolesofthecircle constitut- ingthelip);D'KP'L' the circle inwhich>S^',andthecon- tinuation ofitsspherical surface, arecutbytheplane through D',Q,andanypointPof /S'atwhich itisdesired tofindthe electricdensity ;andDKPL theimageofD'K' P'L'. Intheannexeddiagrams twocases areillustrated;inone ofwhichSisspherical andconcave towards theinfluencing point,Q ;intheother,Sisplane. Using now forS'allthe notation of§§240, 241,butwith accents added, andtaking advantageof§238, footnote, weseethat 188 DistributionofElectricity onCircular[x^ and f"-a'=D'K'=D'L'=UK' .B'L, Hence(19)becomes ._y{IIJ'K'.DL'^_,/UK'.UL) ^^^.P"2^'W P'KTP'L~^^"^VF'K'.FL']""^^^^' fortheelectricdensityatP'ontheconcave sideof>S".But asin§238,wefind PK'=PK.^^^, P'i'=PX.^^....(22). and D'K'^D'L =DK^-^^=DL^^^..m. Also, ifhdenote theshortest distance fromQtothespherica orplanesurface ofS,and/the diameter ofthissurface(infiniti ofcourse when thesurface isplane,ornegativeifthecon vexity betowardsQ),wehave 7?2 D2 D2JT f'^-k+jni-hif^h)(2*)- Usingthese in(21), puttingV=%,andsubstitutingthe expressionsoobtained forpin(8)of§237,wefind _qh{f-h) \PQ /DK.DL ^SP_Q /DK^DTYS..,^..P~ 27ry.PQ'[DqW PK.PL"^^"^ IDQS/ PK.PL]]^'''^ fortheelectricdensity onthesideof>Sremote fromQ(thatis theconvex orconcave side,when >S^isspherical, accordingas^ iswithin orwithout thecompleted spherical surface).Tb( electricdensity onthesidenextQis[§241(20)]thesame with theaddition ofqh(/— h). 27r/.PQ'^^^* These formulae, (25)and(26), expressthe electricdensityor thetwo sides ofacircularsegmentordisc ofinfinitelythii sphericalorplane conductingsurface connected withtheeartl byaninfinitelyfinewire, and electrified bytheinfluence of ^ quantity—qofelectricityinsulated atapointQanywhereit itsneighbourhood. 244*. Thepositionoftheauxiliary pointD(which appearsii thediagramsastheimageofD',theunoccupied poleofthelif oftheoriginal bowlS')maybefound, without reference toS' byconstruction fromSandQsupposed given ;thus :—Fron (22)of§243wehave KD'.DL ::KQ: QL (27), jLv.] Segment ofSpherical Conducting Surface.189. whereKandLmaybethepointsinwhich thelipofthebowl '8iscutbyanyplane through QD'D, Let, forinstance, this plane passthroughthecentre ofoneofthesphericalsurfaces. [tmust alsopassthroughthecentre oftheother, andbisect 3achbowl;and ifE,F bethepointsinwhich ItcutsthelipofSy(26) appliedtothepresent casegives ED:DF'.:EQ:QF, Hence(Euclid,vi.S) thelinesbisectingthe iangles EDF,EQF cut^'^^--,^_ „.'''' !thebaseEF inthe samepoint ;andDmust beinthecircle which isthelocus ;ofallpointsintheplaneEFQ fulfillingthis condition, being found bythewell-known construction, thus :—Bisect theangle EQGbyQA,meeting EFinA.DrawQBperpendicularto QAyand letitmeetEFproduced,inB.OnBAasdiameter jdescribe acircle, which istherequiredlocus;andDisthe pointinwhich this circle cuts theunoccupied partofthe sphericalorplanesurface ofS. I245.Dbeingfoundbythissimple construction, thesolution oftheproblemiscomplete, without reference toS\thus :—To findtheelectricdensityatanypoint P,draw aplane through QDPy and let itmeet thelipinKandL.Measure DK,DL, PK,PL,PQ,andDQ,andcalculate by(25)and(26). Butwe haveanimportant simplification from thegeometricaltheorem of§238,which shows that DK.DL _Dh.D l PK,PL~ Pk,Pl ^^ ifh,Ibepointsinwhich thelipiscutbyanyplanewhatever through PD. Choose, forinstance, theplane through PD, andCthemiddlepointofS.Then, asD,k,P,G,Ilieallon onecircle, andCisthemiddlepointofthearckPl,wehave (asabove, in§238) Dk.Dl= CD'-Ck'=CD'-a', Pk.Pl= Ck'-CP'=a'-CP'- 190 DistributionofElectricityonCircular[xv where, asbefore, adenotes thechord from themiddlepointt( thelip.Usingthisin(28)and(25)wehave, finally, (29)^_gHf-h)(PQ /CD'-a'tnn-^P^ /CD^-a'-^~ 27ryP^ [DQ\/a'-CF' l^QsJa'-CP forthedensityonthesideremote fromQ;hand/— Abein^ theshorter andlono^er distance. 246. Forthecase inwhich ;S'isaplane disc,or/=oo,thi; becomes __qh_(PQ /Gir-a',JFQ /CD'-a'-]]P~2ir'PQ'\DQ\/a'-CP' lDQ\/ a^-CF'j J^'^"^ andtheaddition(26)toittogivethe electricdensityonth< sidenext toQ, 9^ nr 27rFQ'^ ^ Also, asEFD isastraightlineinthis case, (27)gives QF^QECD=IEF^^~^^(-^2 QE,QFaretobecalculated immediatelyfrom data ofwhateve form, specifyingthepositionofQ;andfromthem andCI found bythisformula,DQistobecalculated. Thusexplicitl; wehaveeveryelementrequiredforcalculatingelectric densitie by(30). 247. Forthecase ofQintheaxis ofthedisc,Disinfinite! CD distant, sothatCD=oo,DQ=oo ,andy^=1.And ifft CFweput r,(30)and(31) give,forthedensityonremote side qh fPQ PQ ] :and forthedensityonnear side, P^2^(^ IfFbeatthecentre ofthe disc,and ifwetakeq=27r^,thes ibecome forremote side,p=,t,(—tan"^-) ^^«"^ i(33) forneai' side, p+j^ XV.] Segment ofSpherical Conducting Surface.191 from which thefollowingnumerical results have been calcu- lated, witha,theradius ofthedisctaken asunity:— Distance of XVI.ATMOSPHEEIC ELECTRICITY. [FromNichol's Cyclopedia,2dEd.(I860).] ; 249. Itmaybepremised,toavoid circumlocution iuthis article, thatevery bodyincommunication with theearth by means ofmatterpossessingelectricconductivity enoughto preventitselectricpotential*fromdiffering sensiblyfrom that oftheearth, willbecalledpartoftheearth. Moist stone, and rock ofallkinds, and allvegetable andanimal bodies, intheir natural conditions, exceptincircumstances ofextraordinary dryness, possess,eithersuperficiallyorthroughouttheir sub- stance, therequisite conductivitytofulfil that condition. On theother hand, various natural minerals and artificial com- pounds, such asglass,—variousvegetable gums,such asIndia- rubber, gutta percha, rosin,— andvarious animalproducts,such assilkandgossamer fibre,—when either inaverydrynatural orinanartificiallydriedatmosphere,resist electrical conduc- tion sostronglythattheymay supportabody,orotherwise form amaterial communication between itandtheearth, and yetallow ittoremaincharged withelectricitytoapotential sensibly differingfrom theearth's, forfractions ofasecond, for minutes, forhours, fordays,oreven foryears,without any fresh excitation orcontinued source ofelectricity. Again, air, whetherdryorsaturated withvapourofwater, andprobably allgasesandvapours,unlessruptured bytoostrong anelectro- motive force, arevery thoroughlydestitute ofconductivity — that istosay,arevery perfectly endowed with theproperty ofresistingthetendencyofelectricitytopassand establish *Two conducting bodies aresaid tobeofthesame electric potential when, ifputinconducting communication with thetwoelectrodes ofanelectrometer, noelectric effect isproduced. When, ontheother hand, theelectrometer shows aneffect, theamount ofthis effect measures thedifference ofpotentials between thetwobodies thus tested. Difference ofpotentialsisalso called electromotive force. vl] Atmospheric Electricity. 193 (qualityofpotentialbetween twobodies nototherwise materi- llyconnected. 250. Hence, when "the surface oftheearth" isspoken )f,thesurfaceseparatingthesolids andliquidsoftheearth remtheairwillbemeant; andwhen themorequalifiedex- )ression ''outer surface oftheearth" isused, inner surfaces )fvesicles, orthesurfaces bounding completelyenclosed spaces )fair,must beunderstood tobeexcluded. Thus, thesurface )famountain peak;thesurface ofacave,uptotheinmost ecesses ofthemost intricatepassages ;thesurface ofatunnel; :hesurface ofthesea,orofalake orriver;allthesurface ofa ^heet ofunbroken water insuchafallasthat ofNiagara ;the surface ofblades ofgrass and flowers, andofsoilbelow;ina \vood, thesurface ofsoil,and oftrunks andleaves oftrees; thesurface ofanyanimalrestingontheearth;theoutside of theroof ofahouse; thewhole inside surface ofaroom with anopenwindow;allbelongtotheouter surface oftheearth. 251.Ontheother hand, themoon, meteoric stones, birds or insectsflying,leaves orfruitfalling,seedwaftedthroughtheair, spray breaking away from acascade orfromwaves ofthesea, theliquid particlesofacloud orafog,presentsurfaces not belongingtothe earth, andbetween which andtheearth's surface differences ofpotential, and lines ofelectric force,may andgenerallydoexist. 252.Thewhole surface oftheearth, asdefined above(§250), isateverymoment electrified inevery part,with theexception ofneutral linesdividing portions which arenegatively (resin- ously) fromportions which arepositively (vitreously)electrified. Thenegativelyelectrifiedportionsareofverymuchgreater extent, atalltimes, than thosepositivelyelectrified;andthere maybetimeswhen thewhole surface isnegatively electrified, because inalllocalities inwhich electrical observations have been hitherto made, withpossiblyoneremarkableexception*, theearth's surface isalwaysfoundnegative, dayandnight, *AtGuajara station, onthePeak ofTeneriffe, "During thewhole period of "observation, bydayandnight, theelectricity wasmoderate inquantity, and "always resinous. Thiswasduring theperiod ofN.E. trade wind, andwithin "its influence, though above itsclouds."—[Professor Piazzi Smyth's Account of theTeneriffe Astronomical Experiment, Philosophical Transactions, 1858, and separate publication ordered bytheLords oftheAdmiralty.] The"electricity" here referred towasthatacquired byaninsulated conductor carrying aburning T.E. 13 194Atmospheric Electricity. [xvi. duringfairweather, andonly occasionally positiveinbroken weather, orduring anactual fallofrain intheimmediate neighbourhood,ifnotexactly ontheplaceofobservation.If, then, atanyonetime there chances tobefairweather over the whole earth, itmaybepresumedthat thewhole outer surface oftheearth isthennegatively electrified, unless, judgingfrom thepossible exception above alluded to,weare still toexpect positiveelectrification insame extremepositions. 253. Asyetnothingisknownregardingtheelectrification ofairitself*, orofclouds orother mattersuspendedintheair, exceptwhat canbeinferred(seebelow, §254)from theelec- trification oftheearth's surface, and itsvariations, withwhich alone, asPeltier hasremarked, theobservations of"atmo- spheric electricity"hithertopublished have dealt (seebelow, §§265, 266).Itisimpossible,inthenature ofthings,to investigatethebodilyelectrification ofanon-conductor byany observation whatever ofelectric action withoutitf,orinany waywhatever, except bysomething equivalenttoadetermina- tion ofthemagnitudeand direction oftheresultant force at every pointofitsmass;]:. Towards thisthorough investigation match intheairatsome distance from theearth. Ifitwerereally negative, theearth's electrification attheplace musthavebeen positive ;butthetestas toquality mayhavebeen deceptive, owing tothehighly insulating condition of both outer andinner surfaces oftheglass shade enclosing thegold leaves, andtothecircumstance ofthetesting piece ofrubbedsealing waxhaving been applied possiblytoonearthegold leaves, instead ofbeside aremote part ofthe insulated rod. Professor Smyth assures the writer, thatheconsiders the electrical experiment asnotsufficiently complete orconfirmed toallow any conclusion tobebuilt on it,and regardsitrather asanindication ofthe importance ofmaking electrical observations with better apparatus,' and more available time forusing it,than the first Teneriffe scientific expedition afforded. *Forknowledge gained since this article waswritten see§§296—301below. tAccording toGreen's remarkable theorems, triply rediscovered byGauss, Chasles, andthewriter ofthis article, alldifferent distributions ofelectricity within asolid, which produce thesame potential atitssurface, producethe same force atevery point withoutit,andtheproblem offinding adistribution ofelectricity within theinterior, toproduce agiven distribution ofpotentialat thesurface, isindeterminate. JLetX,Y,Zbethecomponents oftheresultant force onaunit ofelec- tricity,ifplaced atanypoint x,y,zinamass ofairorother non-conductor; and letpdenote the electrical density ofthesubstance, that istosay,the quantity ofelectricity perunit ofbulk actually possessed bythe airinthe neighbourhood ofthispoint. Then, byawell-knownproposition ofthemathe- matical theory ofattraction, wehave ]^(dXdYdZ\ 4:Tr\dx dydzJ Kvr.] Atmospheric Electricity. 195 3fthedistribution ofelectricitywithin anon-conducting mass, itmayberemarked, thatadetermination ofthenormal com- ponentoftheforce allround aclosed surface isjustsufficient to show theaggregate quantityofelectricity possessed byallthe matter situated within it*.Hence observation inpositionsall round amass ofair isnecessaryfordeterminingthequantity ofelectricity which itcontains; and, therefore, theballoon must beputinrequisitionifknowledgeofthedistribution of electricity throughtheatmosphereistobesoughtfor. 254. Withoutleavingtheearth, however, although wecannot thoroughly investigatethe electrification ofthe air,wecan makeimportantinferences about itfrom observations ofthe electricdensityovertheearth's surface, byaprincipleofjudg- ingwhich maybethusexplained:—Iftheearth weresimply anelectrifiedbody, placedinaperfectly insulating medium of indefinite extent, andnotsensiblyinfluenced byanyother 'electrified matter, orbyreflex influence fromanyconductor or dielectric initsvicinity,itselectricitywould bedistributed *over itssurfaceaccordingtoaperfectlydefinite law,depend- iing solely ontheform ofthe surface, and deducible bya sufficiently powerfulmathematicalanalysis fromsufficiently perfectdata of"geometry" (intheprimitivesense oftheterm), orofwhat, inmore modernlanguage,iscalledgeodesy.If thesurface oftheearth weretruly spherical,thislawwould simply beuniform distribution. Atruly elliptic oblateness of theearth wouldgive,instead ofuniformity,adistribution of electricdensityinsimple proportiontotheperpendicular distance between atangent (thatishorizontal) plane through anypointandtheearth's centre;accordingtowhich theelectric densityattheequator would begreatest, andwould exceed that ateitherpole,where itwould beleast,by3^:adiffer- encewhich, forthepresent, wemay disregard. 255. Thewhole amount ofelectricityover thesurface of anygreat regionofmountainouscountry,orofforest land, *LetNbethenormal component oftheforce atanypoint ofaclosed surface, dsanelement ofthesurface, /thesign ofintegration forthewhole surface, andQthewhole quantity ofelectricity within it.Then, byawell- known theorem ofGreen's, rediscovered asalluded toinapreceding note,wehave ^1..t , 13—2 196 Atmospheric Electricity. [xvi. orofsoilandvegetationofanykind, orofstreets andhouses in atown, orofrough sea,would beveryapproximately thesame asthatonanarea ofunruffled ocean, equaltothe"reduced" area oftheirregularsurface;butthedistribution oftheelec- tricityover hillandvalley,overtheleaves andtrunks oftrees, andthesurfaces ofplants generally, andonthe soilbeneath them, over the roofs, perpendicular walls, andoverhangingoi overshaded surfaces ofbuildings,andthesurfaces ofstreets and enclosed courts between them, andoverthehollows and crests ofwaves inastormy sea,would beextremely irregular, with,, ingeneral, greaterelectricdensityonthemoreprominentand convexportionsofsurfaces, and lessonthemore covered and concave—quite insensible, indeed, inanysuchpositionasthe interior ofacave, orthesoilbelow trees inaforest evenwhere considerable angular openingsofskyarepresented,—orthe roof orfloor ofatunnel, orcovered chamber, evenalthough opentoaconsiderableangleofsky. 256. Ifthus aperfect electro-geodesy gavea"reduced' electric density equaloverthewhole earth,wemightinfer thai the electrification oftheearth isnotinfluenced byanyelec- tricityinthe air.Accordingtowhat hasbeen stated above there mightinthat casebeeither noelectricityinthe air,from theearth's atmospheretotheremotest star,andthelines oi electric forcerisingfrom theearth mighteither beinfinite oi terminate inthesurfaces ofthemoon, meteoric stones, sun planets,and stars; orthere might be,atanydistance con- siderably exceedingtheheightofthehighest mountain, auni- formlyelectrified stratum ofequal quantityandoppositekind totheearth's, balancing throughalltheexteriorspacetheforce duetotheterrestrialelectricity, andlimitingthemanifestations ofelectric force totheatmospherewithin it;ortheremightb( anyoftheinfinitevarietyofdistributions ofelectricityinspace round theearth, bywhich the electricdensityatthe earth'j surface would beuninfluenced. 257. But, inreality,theelectricdensityvariesgreatly,ever inserene weather, over theearth's surface atanyonetime aswemayinfer from(1.)the facts(estabHshedforEurope andprobablytrue inallthetemperatezones ofboth hemi- spheres),that inanyoneplacethe electricdensityofthe cvi.] Atmospheric Electricity. 197 urface observed duringserene weather ismuchgreaterin vinter than insummer, andthat itvariesaccordingtosome- hingofaregular periodicitywith thehours ofthedayand light;and(2.)theconsideration that there isoften serene vcather ofdayandnight,andofsummer andwinter, atone uidthesame time, indifferenttemperate portionsoftheearth. Womay, therefore, consider itasquiteestablished that, even nserene weather, theelectrification oftheearth's surface is avo-elyinfluenced byexternal electrified matter. Although we •aiiiiot(§253)discover theexactlocality and distribution of thisinfluencing electricityfrom itseffects attheearth's surface alone, yetitispossible, from thecharacter ofthedistribution oftheterrestrial electricdensityasinfluenced byit,toassigna superiorlimit toitsheight*.Ifatanyoneinstant theelectric densityreduced tothesealevel were distributedaccordingto asimple "harmonic" law, or,moregenerally, accordingtoa certain definite character ofnon-abruptnessofvariationeasily specifiedinmathematicallanguage f,theexternalinfluencing electricity mightbeatanydistance, howevergreat,forallwe could discover byobservations near theearth's surface. But, little asweknowyetregardingthediurnal law ofelectric variation inserene weather,itis,wemay saywith almost perfect certainty,notsuch ascouldgiveatanyinstant adis- tribution over thewhole earthpossessing anysuchgradual character asthat referred to;and, therefore, wemay,inall probability,from thecharacter ofthediurnal variation itself, saythat itselectricoriginisnotatadistance ofmanyradii from thesurface. Ontheother hand,whenweconsider that intemperate regionsthevelocitywithwhich theearth's surface *Ifatanyinstant theco-efficients oftheseries of"Laplace's functions," expressing theterrestrial electric density reduced tothesealevel, converged ultimately with less rapidity than thegeometricalseries1,— ,— g,.--"^^ might besure that there iselectricityintheairatsome distance from the centre oftheearth, notexceeding mtimes theradius oftheearth's surface. Fortheprinciples onwhich this assertion isfounded, seeashort article, entitled "Note onCertain Points intheTheory ofHeat," Cambridge Mathe- matical Journal, November 1843. +Forinstance,ifinsimple proportion tothecosine oftheangular distance fromanypoint oftheearth's surface, ormore generally,ifexpressible byany finitenumber of"Laplace's functions," orstillmore generally,ifexpressible byaseries of"Laplace's functions," with co-efficients converging ultimately morerapidly thananygeometricalseries.te 198Atmospheric Electricity. [xvii isearned round initsdiurnal course isfrom500to900miles perhour,weseeclearlythatanylawofdiurnal electric varia- tion, established onobservations even sofrequentasonceevery hour, could notpossiblyfixthelocalityoftheorigintowithin 100miles ofthesurface;and aswehave asyetnothingtogo uponinthewayofpublishedobservations morefrequent than three orfourtimes aday,towardsestablishingeither theex- istence orthecharacter ofthediurnal law,wecannot consider it asproved byobservation that theinfluencing electricity which producesitiseven asnear asthe50or100miles limitwhich is commonly (butintheopinionofthewriter ofthis article, most unreasonably) assignedasanendtotheearth'satmosphere. 258. Thegreat suddenness oftheelectric variationsduring broken weather, andtheir closecorrespondencewithbeginnings, changes, andcessations ofrain, hail, orsnow, compelus(bya common sense estimate founded onanunconsciousapplication ofthemathematical lawstated inthefootnotes tothepreced- ing§257)tobelieve that theirorigin agreesinposition with that oftheshowers, and togiveita''local habitation" anda name—Thundercloud. 259.Thewriter ofthis article hasobservedextremely rapid variations ofterrestrial electrificationduring perfectlyserene weather. Thus, inacalmsummernight,withanunvarying cloudlessskyoverhead, andnotthe faintestappearanceof aurorallighttobeseen,hehas,inatemporaryelectric observa- toryintheIsland ofArran, foundlargevariations(asmuch as from acertaindegreetodouble andback)inthecourse of aminute oftime. Theinfluencing electricity bywhich these variations were produced, cannotpossibly (unless ontheex- tremely improbable hypothesisoftheirbeing due tohighly electrified extra-terrestrial matter moving very rapidlywith reference totheearth) have beenveryfarremoved from the earth's surface. Itisnotimpossible,andwehave asyet nothingtomake itdecidedly improbable,thattheywere due tofluctuations upanddown ofaerial strata, perhapsthose of thegreat atmospheric currents, inhigh regionsoftheatmo- sphere. Judging, however, from stillmore recent observations referred tobelow(§262),wemaythink itmoreprobablethat these remarkable variations intheobserved electric force were XVI.] Atmospheric Electricity. 199 duechieflytopositivelyornegativelyelectrified masses moving alongwithin afewmiles ofthelocalityofobservation. 260.Returningtothesubjectofthedistribution ofelec- tricityover theearth's surface atanyinstant, wemayremark, that ifoveranarea ofseveral miles indiameter, ofperfectly level barecountry,orofsea,theelectricaldensityissensibly uniform, wecould not,withoutgoing upinaballoon, and observingthe electric force atpointsinthe airabove, form anyjudgmentwhatever astothedistance from theearth at which theinfluencing electricityissituated. If,ontheother hand,wefind averysensible variation inthe electric density between twopointsofapieceoflevel open country,orat sea,notmanymilesapart, wemayinfer asquitecertain that there isinfluencing electricitynotmanymiles upinthe air,andnotuniformlydistributed inlevel strata. Nothing canbeeasier than tomake this trial—onlytoobserve simul- taneouslywith similar instruments, similarly placed,attwo neighbouring stations, inasuitablelocality—andmost interest- ingandimportantresults aretobederived fromit,assoon as arrangementscanbemade forcontinuingtherequisiteobserva- tionsdayandnight, duringvarious vicissitudes ofweather, especially duringatime ofperfect serenity. 261.Correspondingstatementsapplytoamountainous country,with this modification, thatavery varied, instead of auniform distribution ofelectricdensity, is,insuch alocality, asexplainedabove in§255,thenaturalconsequenceoffreedom from thedisturbinginfluence ofnear electrified masses ofairor cloud. Theproblemofaccurately determining,frompurely geometricdata(§256),thisundisturbed distribution overeven thesmoothest hillside, wouldinfinitelytranscend human mathe- maticalpower, although anapproximatesolution maybereadily givenforanypieceofcountryoverthewhole ofwhich both the inclination andtheratio oftheheightabove thegenerallevel to theradius ofcurvature ofthesurface aresmall. Forarugged mountainouscountry,themostperfect geometric data,andthe most strenuous mathematical efforts, couldscarcelylead us towards anapproximateestimate oftheinequalitiesofelectric density which difl"erent localities mustpresentwithout any disturbance from near electrifiedatmosphere. Hence, ina 200Atmospheric Electricity. [xvi. mountainouscountry—unless wefindelectricity stronginsome locality where from theconfigurationofthesurface, wecorrectly judgeitoughttobeweak ifundisturbed, orweakwhere itought tobestrong,orunless, atleast,wefindsomeverydecided devia- tionfromanysuchamount ofdifference between two stations as,withoutbeingable tomake aprecise calculation, wecan estimate forthedifference duetofigure—wecannot judgeas totheinfluence ofaerial electrification from simultaneous absolute determinations atanyoneinstant alone. Butofoue thingwemaybesure, thatalthoughtheabsolute amounts of theelectrification atanytwo stations not farapartmaydiffer largely, theymust remain inanabsolutelyconstantpropor- tiontooneanother,ifthere isnoelectrified airorcloud near. 262. Hence,ifwefindobservations madesimultaneously by twoelectrometers inneighbouring positions,inamountainous country,tobearalwaysthesame mutualproportion, wemay notbeable todraw anyinference astoelectrified air;butif, onthecontrary, wefind theirproportion varying, wemaybe perfectlycertain thatthere arevaryingelectrified masses ofair orcloud notfar off.Afirstapplicationofthis test isdescribed inthefollowingextract from theProceedingsoftheLiterary andPhilosophical SocietyofManchester forOctober 18,1859 :— "Thefollowingextract ofaletter received from Professor W. "Thomson, F.R.S., Glasgow, Honorary Member oftheSociety, '' etc.,wasreadbyDrJoule :— 'Ihave averysimple"domestic" apparatus bywhich Ican 'observeatmospheric electricityinaneasy way.Itconsists 'merelyofaninsulated canofwater setonatable orwindow 'sillinside, anddischarging byasmallpipethroughafinenozzle 'twoorthree feetfrom thewall. Withonlyabout teninches 'head ofwater andadischargesoslow astogivenotrouble in ' replenishingthecanwith water, theatmosphericefi'ect is 'collected soquicklythatanydifference ofpotentialsbetween 'theinsulated conductor andtheairattheplace where the *stream from thenozzle breaks intodropsisdoneawaywith at 'therate offivepercent,perhalfsecond, oreven faster. Hence *averymoderatedegreeofinsulation issensiblyasgoodas ' perfect,sofarasobservingtheatmosphericeffect isconcerned. *Itiseasy,bymyplanofdrawingtheatmosphereround the ' insulatingstemsbymeans ofpumice-stonemoistened with XVI.] Atmospheric Electricity.201 ' sulphuric acid, toinsure adegreeofinsulation inallweathers, 'bywhich there need notbemore than fivepercent,perhour 'lostbyitfrom theatmospheric apparatusatanytime.Alittle 'attention tokeeptheouterpartoftheconductor clear of * spiderlines isnecessary. The * apparatusIemployedatIn- ' vercloystoodonatable beside 'awindow onthesecond floor, 'which waskept openabout 'aninch toletthedischarg- ' ingtubeprojectoutwithout 'comingincontact with the^^^' '• 'frame. Thenozzle wasonlyabout two feetandahalffrom 'the wall, andnearly onalevel with thewindow sill.The 'dividedringelectrometer stood onthetable besideit,and 'acted inavery satisfactory way (asIhadsupplieditwith a *Leyden phial, consistingofacommon thinwhiteglass shade *which insulated remark- * ably well, instead ofthe 'Germanglass jar—the 'second ofthekindwhich 'Ihad tried, andwhich 'would nothold itscharge *forhalfaday).Ifound 'from13J°to14°oftorsion * requiredtobringtheindex 'tozero,whenurgedaside ' bytheelectromotive force 'oftenzinc-copperwater 'cells. TheLeyden phial 'held sowell,thatthesensi- * bilityoftheelectrometer, 'measured inthatway,did *not fallmore thanfrom '131° to13i°inthreedays. 'Theatmosphericeffect ' ranged from 30°toabove '420°duringthefourdays *which Ihadtotestit;thatFm. 2. 202 Atmospheric Electricity. [xvi. 'istosay,theelectromotive forceperfootofair,measured hori- ' zontallyfromthesideofthehouse, wasfrom 9toabove 126zinc- 'copperwater cells. Theweather wasalmostperfectly settled, 'either calm, orwithslighteastwind,andingeneral aneasterly 'haze inthe air.Theelectrometer twice within halfanhourwent 'above 420°, therebeingatthetime afreshtemporary breeze 'from theeast.What Ihadpreviouslyobservedregardingthe 'effect ofeastwind wasamplyconfirmed.Invariablythe *electrometer showedveryhigh positiveinfineweather, before 'andduringeastwind. Itgenerallyroseverymuchshortly *before aslight puffofwind from thatquarter, andcontinued *hightillthebreeze wouldbegintoabate. Inever once *observed theelectrometergoing upunusually highduringfair 'weather without eastwindfollowing immediately. Oneeven- 'inginAugustIdidnotperceivetheeastwind atall,when 'warned bytheelectrometer toexpectit;butItook the * precautionofbringing myboatuptoasafepartofthebeach, 'andimmediatelyfound bywavescominginthat thewind 'must beblowingashort distance outatsea,althoughitdid 'notgetsofarastheshore. Imade aslight commencement 'oftheelectrogeodesywhich Ipointedoutasdesirable atthe 'British Association, andinthecourse oftwodays, namely, 'October 10thand11th, gotsomeverydecided results. Mac- *farlane, andone ofmyformerlaboratory andAgamemnon *assistants, Russell, camedown toArran forthatpurpose. Mr 'Russell andIwentupGoatfell onthe10th instant, with the ' portableelectrometer(seeFig. 3),andmade observations, while 'MrMacfarlane remained atInvercloy, constantly observing 'andrecordingtheindications ofthehouse electrometer. On 'the11th instant thesameprocess wascontinued, toobserve 'simultaneouslyatthehouse andatoneorother ofseveral *stations onthewayupGoatfell. Ihave notyetreduced all *theobservations;butIseeenoughtoleave nodoubt whatever 'butthat cloudless masses ofairatnogreatdistance from the *earth, certainlynotmore than amile ortwo, influence the 'electrometer largely byelectricity whichthey carry.This I 'conclude because Ifindnoconstancyintherelation between *the simultaneous electrometric indications atthe different 'stations. Between thehouse andthenearest station therela- xvl] Atmospheric Electricity. ^/>.y^t-2^^03 ^ 'tivevariation was least. Between thehouse andastation about 'halfwayupGoatfell, atadistance estimated attwomiles and 'ahalf inaright line,thenumberexpressingtheratio varied 'from about 113to360inthecourse ofabout three hours. On 'two different morningstheratio ofahouse toastation about ' sixty yardsdistant ontheroad beside theseawas97and96 'respectively. Ontheafternoon ofthe11th instant, duringa 'freshtemporarybreeze ofeastwind, blowing upalittlesprayas 'farastheroad station, most ofwhich would fallshort ofthe 'house, theratiowas108infavour ofthehouse electrometer '—bothstandingatthetimeveryhigh—thehouse about 350°. 'Ihave little doubt butthat thiswasowingtothenegative 'electricitycarried bythespray from the sea,which would 'diminishrelativelytheindications oftheroad electrometer'." <^^Theelectrometers referred tointheprecedingextract were ontwo differentplans. Thefirst, or"dividedring electrometer," consists of— (1.)Aringofmetal divided into sectors, ofwhich some—oneormore—areinsulated andcon- nected with theconductor tobeelectrically tested, andthe remainder connected with theearth.(2.)Anindex ofmetal supported byaglass fibre, orawire, stretched intheline of theaxisofthering,andcapableofhavingitsfixed endturned through anglesmeasured byacircle andpointer. (3.)A Leyden phial,with itsinsulatedcoating electrically connected withtheindex.(4.)Acase toprotecttheindex from currents air,andtokeepanartificiallydriedatmosphere round the insulating supports—glazedtoallow theindex tobeseenfrom without, butwith theinner surface oftheglassscreened (electrically) bywire cloth, perforated metal,ortinfoil, todo away withirregularreflections ontheindex. Intheinstru- mentrepresentedinthedrawing (No. 2)above, theringis dividedonlyintotwoparts,which areequal, andseparated by aspaceofairaboutgV^^^^inch. Each ofthese halfringsis supportedontwoglass pillars ;andbymeans ofscrewsacting onafootwhich bears thesepillars,itisadjusted and fixed in itsproper position.Theindex isofthinsheet aluminium, and projectsinonlyonedirection from theglassfibrebearingit. Astiff vertical wire, rigidlyconnected withit,nearlyinthe prolongationofthe fibre, bears acounterpoise considerably 204Atmospheric Electricity. [xvi. below thelevel oftheindex, andheavy enoughtokeepthe index horizontal. Athinplatinumwirehooked tothelower endofthis vertical wire, dipsinsulphuricacid inthebottom oftheLeyden phial. TheLeyden phialischargedeitherposi- tivelyornegatively; and isfound toretain itschargefor _ months, losing, however, gradually,atsome lowrate, lessgenerally than onepercent,per dayofitsamount. Theindex isthus,when theinstrument isinuse,keptinastate of charge correspondingtothepotentialofthe insidecoatingofthephial. When oneofthe halfringsisconnected with theearth, anda chargeofelectricity communicated totheother, theindex moves from ortowards the latter, accordingasthechargecommunicated toitis ofthesame ortheoppositekind tothat ofthe index. This instrument, asanelectroscope, possesses extremesensibility—muchgreater than that ofanyother hitherto constructed; andbytheaidofthetorsion arrangement,it maybemade togiveaccurate metrical results. There aresome difficulties intheuseofit, especiallyasregardsthecomparisonoftheindi- cations obtained with differentdegreesofelec- trification oftheindex, and thereduction oftheresults toabsolute measure, hither- toobviatedonlybyadaily applicationofDelmann's method ofreference toa zinc-copperwaterbattery, which Delmann himselfap- pliesonce for all,toone ofhiselectrometers(unless hisglassfibre breaks, when hemustmake afresh deter- mination ofthesensibility oftheinstrument with its newfibre). Thehighsensi- bilityofthedividedringFig.3.—Portable Atmospheric Electrometer. XVI.] Atmospheric Electricity.205 electrometer renders this testreally very easy,asnotmore than fromtentotwentycells arerequired ;andacomparisonwitha fewgoodcells ofDaniell's maybemadebyitsaid,toascertain theabsolute value andtheconstancyofthewater cells. The difficulty thusmet isaltogetherdoneawaywith inanother kind ofelectrometer, also ''heterostatic," ofwhichonlyonehas yetbeen constructed—theelectrometer oftheportable apparatus shown inthethirddrawing.Inittheindex isattached at right anglestothemiddle ofafineplatinum wire, firmly stretched between theinsidecoatingsoftwoLeyden phials, andconsistssimplyofavery lightbarofaluminium, extend- ingequally onthetwo sides ofthesupportingwire. Itis repelled bytwoshort bars ofmetal, fixed onthetwo sides of thetopofametal tube, which issupported bytheinside coat- ingofthelowerphial, andhasthefinewire initsaxis.A conductor ofsuitableshape, bearing anelectrode, toconnect with thebodytobetested, insulated inside thecase ofthe instrument, intheneighbourhoodoftheindex, andwhen elec- trified inthesameway,orthecontrary way,totheinside coatingsoftheLeyden phials, causes, byitsinfluence, the repulsion between theindex andthefixed barstobediminished orincreased. Theupper Leyden phialismoveable about a fixedaxis,through angles measured byapointer andcircle, andthus theamount oftorsion, inone-half ofthebearing wire, requiredtobring theindex toaconstantposition,inany case, ismeasured. Thesquareroot ofthenumber ofdegrees oftorsion measures thedifference ofpotentials between the conductor tested andtheinnercoatingoftheLeyden phial. Inusing theinstrument, theconductor tested isfirstputin connexion withtheearth, andthetorsionrequiredtobringthe index toitsfixedpositionisread off.This iscalled thezero, orearthreading. The tested conductor isthen electrified, and thetorsionreadingtaken. Intheatmospheric application,this iscalled theairreading. The excess—positiveornegative— ofitssquare root,above that ofthezeroreading,measures the electromotive force between theearth andthepointofair tested. This result, whenpositive shows vitreous, whennega- tiveresinouspotentialintheair;iftheindex isresinous. By theaidofBarlow's table ofsquare roots, theindications ofthe 206Atmospheric Electricity. [xvi. instrument maythusbereduced todefinite measure ofpotential, almost asquicklyastheycanbewritten down. Once forall, thesensibilityoftheinstrument canbedeterminedbycom- parisonwithanabsolute electrometer, oragalvanic battery. Intheportable apparatusaburning match isused—instead of thewater-dropping system,which thewriter findsmore con- venient thananyother forafixedapparatus—toreduce the insulated conductor tothesamepotentialastheairatitsend. 264.Ashasbeenremarked above(§252),itistheelectrifica- tionoftheearth's surface which haseitherdirectlyorvirtually been thesubjectofmeasurement inallobservations onatmo- spheric electricityhitherto made. Themethods which have been followed maybedivided intotwo classes—(1.)Those in which means aretaken toreduce thepotentialofaninsulated conductor tothesame asthat ofthe air,atsomepoint,afew feet oryardsdistant from theearth.(2.)Those inwhich a portionoftheearth(seeabove, §253)isinsulated, removed from itsposition,andtestedbyanelectrometer, inadifferent position,orunder cover. The firstmethod wasveryimperfectly carried outbyBeccaria with hislong'' exploring wire," stretched betweeninsulating supports,orelevatedportionsofbuildings, treetops,orother prominent positionsoftheearth(seeabove, §249) ;also,veryimperfectly bymeans of"Volta's lantern"— anenclosed flame, supportedonthetopofaninsulated conduc- tor.Ontheother hand, itisputinpractice very perfectly, by means ofamatch, orflameburningintheopen air,onthetop ofawell insulated conductor—aplan adopted,after Volta's suggestion, bymany observers;also,evenmoredecidedly, by means ofthewater-dropping system—described inthepreced- ingextract—which hasrecentlyoccurred tothewriter, andhas been foundbyhimboth tobevery satisfactoryinitsaction, andextremely easyandconvenient inpractice. Theprinciple ofeach ofthese methods ofthe first classmaybeexplained bestbyfirstconsideringthemethods ofthesecond class, as follows :— 265. Ifalargesheet ofmetal were laidontheearth in aperfectlyleveldistrict, and ifacircular area ofthesame metal were laidupon it,and, after themanner ofCoulomb's proof plane,were lifted byaninsulated handle, andremoved XVI.]. Atmospheric Electricity. 207 toanelectrometer within doors, ameasure oftheearth's elec- trification, atthetime,would beobtained; or,ifaball,placed onthetopofaconductingrodintheopen air,were lifted from thatposition byaninsulating support,and carried toan electrometer within doors, weshould alsohave, onpreciselythe sameprinciple,ameasure oftheearth's electrification atthetime. Iftheheightoftheballinthissecondplanwereequaltoone- sixteenth ofthecircumference ofthedisc(compare §235)used inthefirstplan,theelectrometric indications would bethesame, providedthediameter oftheball issmall, incomparisonwith theheighttowhich itisraised inthe air,andtheelectrostatic capacityoftheelectrometer issmall enoughnottotakeany considerableproportionoftheelectricityfrom theball inits application. Theidea ofexperimenting bymeans ofadisclaid flatontheearth, ismerely suggestedforthesake ofillustra- tion,andwouldobviouslybemost inconvenient inpractice. Ontheother hand, themethod, byacarrier ball,instead ofa proof plane,ispreciselythemethod bywhich, onasmall scale, Faraday investigated thedistribution ofelectricity induced on theearth's surface(seeabove, §249),byapieceofrubbed shell- lac;andthesame method, appliedonasuitable scale, fortest- ingthenatural electrification oftheearth intheopen air,has given,inthehands ofDelmann ofCreuznach, themost accurate results hithertopublishedinthewayofelectro-meteorological observation*. 266. If,now,weconceive anelevated conductor,firstbelong- ingtotheearth(§249), tobecome insulated, andtobemade tothrowoff,and tocontinuethrowing off,portionsfrom an exposed positionofitsown surface, thispartofitssurface will quickly bereduced toastate ofnoelectrification, andthewhole conductor willbebroughttosuchapotentialaswillallow itto remain inelectricalequilibriuminthe air,with thatportionof itssurface neutral. Inother words, thepotential throughout theinsulated conductor isbroughttobethesame asthat ofthe *Through some misapprehension, MrDelmann himself hasnotperceived thathisownmethod ofobservationreally consists inremoving aportion ofthe earth, andbringingitinsulated withtheelectricity which itpossessed insitu, tobetested within doors, otherwise, hecould nothave objected, ashehas, toPeltier's view. 208Atmospheric Electricity.. [xvi. particular equi-potentialsurface inthe air,whichpasses through thepointofitfromwhich matter breaksaway.Aflame, or theheatedgaspassingfrom aburning match, doesprecisely this :theflameitself, orthehighly-heated gasclose tothe matchbeingaconductor which isconstantly extending out, andgradually becominganon-conductor. Thedropsinto which thejetissuingfrom theinsulated conductor, ontheplan introduced bythewriter, producethesame effects, withmore pointed decision, andwithmore ofdynamical energytoremove therejectedmatter with theelectricity which itcarries from theneighbourhoodofthefixed conductor. 'feS.H' ROYAL INSTITUTION FRIDAY EVENING LECTURE, May 18.1860. C2inH'\'^ 267. Stephen Gray,apensioneroftheCharter-house, after many yearsofenthusiastic andperseveringdevotion toelectric science, closed hisphilosophical labours, about onehundred andthirty years ago,with thefollowingremarkableconjec- ture :—'*That theremaybefound awaytocollect agreater "quantityofthe electrical fire,andconsequentlytoincrease "the force ofthatpower, which, byseveral oftheseexperi- "ments, silicetmagna componere parvis,seems tobeofthe "same nature with that ofthunder andlightning." The inventions ofthe electrical machine andtheLeyden phial immediatelyfulfilled theseexpectationsastocollecting greater quantitiesofelectric fire;andthesurpriseanddelight which theyelicited bytheir mimiclightnings and thunders, andabove allbytheterrible electric shock, hadscarcelysub- sidedwhen Franklin sent hiskitemessengertotheclouds, and demonstrated that theimaginationhadbeen atrueguideto thisgreatscientificdiscovery—theidentityofthenaturalagent inthethunderstorm with themysteriousinfluenceproduced bythesimple operationofrubbingapieceofamber, which, twothousandyears before, hadattracted theattention ofthose 1 XVI.] Atmospheric Electricity. 209 philosophers amongtheancients whodidnotdespisethesmall thingsofnature. 268.Theinvestigationofatmospheric electricity immediately became averypopularbranch ofnatural science;andthedis- coveryofremarkable andmostinteresting phenomena quickly rewarded itscultivators. Thefoundation ofallwenowknow wascompleted byBeccaria, inhisobservations on"the mild electricityofserene weather," nearlyahundredyears ago.It wasnotuntilcomparativelyrecentyearsthat definite quan- titativecomparisonsfromtime totime oftheelectricquality manifested bytheatmosphereinonelocalitywere firstobtained bytheapplicationofPeltier's mode ofobservation with his metricalelectroscope. Themuch more accurate electrometer, andthegreatly improved mode ofobservation, invented by Delmann, havegivenforthe electricintensity,atanyinstant, stillmorepreciseresults;buthave leftsomethingtodesire in pointofsimplicityandconvenience forgeneral use,andhave notafforded anymeans forcontinuous observation, orforthe introduction ofself-recording apparatus. Thespeaker had attemptedtosupply some ofthese wants, andheexplained theconstruction anduseofinstruments, nowexhibited tothe meeting, which hehadplannedforthispurpose. 269.Apparatusfortheobservation ofatmospheric electricity hasessentially twofunctions toperform;toelectrifyabodywith some ofthenaturalelectricity,orwithelectricity produced by itsinfluence; andtomeasure theelectrification thus obtained. 270.Themeasuring apparatus exhibited, consisted ofthree electrometers, which were referred tounder thedesignationsof (I.)Thedividedring reflecting electrometer;(II.)Thecommon house electrometer;and(III.)Theportableelectrometer. (I.)Thedividedring reflectingelectrometer [compare §263, above, and§§444... 456,below]consists of :— (1)Aringofmetal divided intotwoequal parts,ofwhich one isinsulated, andtheother connected with themetal case (5)oftheinstrument. (2)Avery lightneedle ofsheet aluminium hung byafine glass fibre,andcounterpoisedsoastomake itproject onlyto onesideofthisaxisofsuspension. T.E. 14i 210Atmospheric Electricity. [xvi. (8)ALeyden phial, consistingofanopen glass jar,coated outside and inside intheusual manner, with theexception that the tinfoil oftheinnercoatingdoes notextend tothe bottom ofthejar,which isoccupiedinstead byasmallquantity ofsulphuricacid[connectedwith the tinfoilbymeans ofa platinum wire]. (4)Astiffstraightwirerigidlyattached tothealuminium needle, asnearlyasmaybeintheline ofthesuspending fibre, bearingalight platinumwire linked toitslower end, andhanging down soastodipintothe sulphuricacid. (5)Acaseprotectingtheneedle from currents ofair,andfromirregularelectric actions, andmaintaininganartificially driedatmosphereround theglass pillar orpillars supportingtheinsulated half- ringandtheuncoatedportionoftheglass ofthephial. (6)Alightstiffmetallic electrodepro- jectingfromtheinsulated half-ringthrough themiddle ofasmallapertureinthemetal case, totheoutside. (7)Awide metal tube ofsomewhat lessdiameter than the Leyden jar,attached toametalringborne byitsinside coat- ing,andstanding upverticallytoafewinches above thelevel ofthemouth ofthejar. (8)Astiffwireprojecting horizontally from thismetal tube above theedgeoftheLeyden jar,andoutthroughawide hole inthecase oftheinstrument toaconvenientpositionfor applying electricitytochargethejarwith. (9)Avery light glass.mirror, aboutthree-quartersofan inchdiameter, attached byitsback tothewire(4),andthere- forerigidlyconnected with thealuminium needle. (10)Acircularapertureinthecaseshutbyaconvex lens, andalonghorizontal slitshutbyplate glass,with itscentre im- mediatelyabove orbelow that ofthelens,oneofthem above, andtheotherequallybelow thelevel ofthecentre ofthemirror. (11)Alarge apertureinthewidemetal tube(7),onalevel with themirror(9),toallowlightfrom alampoutside the case, entering throughthelens, tofalluponthemirror, andbe XVI.] Atmospheric Electricity. 211 reflected outthroughtheplate-glass window; and three or four finemetal wires stretched across thisaperture toscreen themirror fromirregularelectric influences, withoutsensibly diminishingtheamount oflight falling onandreflected off it. 271.Thedividedring (1)iscutoutofthickstrongsheet metal(generally brass).Itsouter diameter isabout 4inches, itsinner diameter 2^ ;and itisdivided intotwoequal partsby cuttingitalongadiameter with asaw. Thetwohalves are fixedhorizontally;oneofthem onafirmmetalsupport, and theother onglass,soastoretain asnearlyasmay betheir originalrelativeposition,withjustthesaw cut,from-^-^to-^ ofaninch broad, vacant between them. Theyareplaced with theircommon centre asnearlyasmaybeintheaxis ofthe case(5),which iscylindrical,andplaced vertically. TheLeyden jar (3),andthetube(7),carried byitsinsidecoating, have theircommon axis fixed tocoincide asnearlyasmaybewith that ofthecaseanddividedring. Theglassfibrehangs down fromabove inthedirection ofthis axis,andsupports theneedle about aninchabove thelevel ofthedividedring. The stiff wire(4),attached totheneedle, hangs down asnearlyasmay bealongtheaxisofthetube(7). [The following diagrams, placedhere tofacilitatecomparison, representthearrangementof"needle"andquadrants described below in§345, assubstituted inthemodern instrument for thebisectedringandnarrow needle oftheoldelectrometer heredescribed]:— - r 272. Beforeusingtheinstrument, theLeyden phial (3)is charged bymeans ofitsprojectingelectrode(8).When an electrical machine isnotavailable, this isvery easilydonebythe aidofastick ofvulcanite, rubbed byapieceofchamois leather. Thepotentialofthechargethuscommunicated tothephial,is 14—2 212 Atmospheric Electricity. [xvi tobekeptasnearlyconstant asisrequiredfortheaccuracyoi theinvestigationforwhich theinstrument isused. Two or three rubs ofthestick ofvulcanite once aday,ortwice aday^ aresufficient when thephialisofgood glass,wellkept dry. Themost convenient test forthechargeofthephialisa properelectrometer orelectroscope,ofanyconvenient kind kept constantlyincommunication with thechargingelec- trode(8).[Compare §853,below.] The electrometer(II.)istobeordinarilyused forthatpur- poseintheKewapparatus. Failing anysuchgaugeelectro- meter orelectroscope,azinc-copper-water battery often, twenty, ormore small cellsmaybevery convenientlyused(after the manner ofDelmann)totestdirectlythesensibilityofthere- flecting electrometer, which istobebroughttoitsproper degree bychargingitsLeyden phialasmuch asisrequired. 273. Intheuseofthiselectrometer, thetwobodies ofwhich thedifference ofpotentialsistobetested areconnected, one of them, which isgenerallytheearth, with themetal case ofthe. instrument, andtheother with theinsulated halfring. The needlebeing,letussuppose, negatively electrified, willmove towards orfrom theinsulated halfring, accordingasthepoten- tialoftheconductor connected with thishalfringdiffersposi- tivelyornegativelyfrom that oftheother conductor(earth) connected with thecase. Themirror turnsaccordinglyinone direction ortheotherthroughasmallanglefrom itszeroposi- tion,andproducesacorrespondingmotion intheimageofthe lamponthescreen onwhich itisthrown. 274. (II.)Thecommon house electrometer [compare §263, above, and§§374... 377,below].—Thisinstrument consists of:— (1)Athinflint-glass bell,coated outside andinside likea Leyden phial,with theexceptionofthebottom inside, which contains alittlesulphuricacid. (2)Acylindrical metal case, enclosingtheglass jar,cemented toitround itsmouth outside, extending upwardsabout aninch andahalfabove themouth, anddownwards toametal base supportingthewhole instrument, andprotectingtheglass againstthedangerofbreakage. (3)Acover ofplate glass,with ametal rim, closingthetop ofthecylindricalcase oftheinstrument. i fXVI.] Atmospheric Electricity. 213 (4)Atorsion head, after themanner ofCoulomb's balance, -supportedinthecentre oftheglass cover, andbearingaglass tibrewhich hangs downthroughanapertureinitscentre. (5)Alightaluminium needle attached across thelower end ofthefibre(whichissomewhat above thecentre oftheglass bell),andastiffplatinumwire attached toitatright angles, andhangingdown tonearthebottom ofthejar. (6)Avery lightplatinum wire, longenoughtohangwithin one-eighthofaninch orsoofthebottom ofthejar,andtodip inthesulphuricacid. (7)Ametalring,attached totheinnercoatingofthejar, bearing twoplatesinproper positionsforrepellingthetwo ends ofthealuminium needle whensimilarly electrified, and proper stopstolimit theangularmotion oftheneedle towith- inabout 45"from theseplates. (8)Acageoffinebrass wire, stretched onbrass framework, supportedfrom themain caseabove bytwoglass pillars,and partially enclosingthetwoends oftheneedle, andtherepel- ling plates,from allofwhich itisseparated byclearspaces,of nowhere lessthan one-fourth ofaninch ofair. (9)Acharging electrode, attached tothering (7),andpro- jectingover themouth ofthejartotheoutside ofthemetal case(2),throughawideaperture,which iscommonly kept closedbyametalcap,leavingatleast onequarterofaninch ofairround theprojecting endoftheelectrode. (10)Anelectrode attached tothe.cage (8),andprojectingover 214Atmospheric Electricity. [xvi. themouth ofthejartotheoutside ofthemetal case(2),through thecentre ofanaperture, about aquarterofaninchdiameter. 275. This instrument isadaptedtomeasure differences of potential between twoconducting systems, namely ;asone,the aluminium needle(5),therepelling plates (7),andtheinner coatingofthejar ;and, astheother, theinsulatedcage (8).This latter iscommonly connected bymeans ofitsprojectingelectrode (10),with theconductor tobetested. Thetwoconducting systems,ifthroughtheirprojectingelectrodes conDected bya metallic wire,maybeelectrified toanydegree, withoutcausing theslightestsensible motion intheneedle. If,ontheother hand, thetwoelectrodes ofthese twosystemsareconnected withtwoconductors, electrified todifferentpotentials,theneedle moves away from therepelling plates ;andif,byturningthe torsion head,itisbrought back tooneaccurately markedposi- tion, thenumber ofdegreesoftorsionrequiredisproportional tothesquareofthedifference ofpotentialsthus tested. 276.Intheordinaryuseoftheinstrument, theinnercoating oftheLeyden jarischarged negatively, byanexternalapplica- tion ofelectricity throughitsprojectingelectrode(9).The degreeofthechargethus communicated, isdetermined by puttingthecageinconnexion with theearth throughitselec- trode(10),andbringingtheneedlebytorsion toitsmarked position.Thesquareroot ofthenumber ofdegreesoftorsion requiredtoeffect this,measures thepotentialoftheLeyden charge.This result iscalled thereduced earthreading. When theatmosphereinside thejariskept sufficiently dry,—this chargeisretained fromdaytodaywith little loss;notmore, often, thanonepercent, inthetwenty-fourhours. Inusingtheinstrument thechargingelectrode(9)ofthejar isleftuntouched, with theaperture through which itprojects closed over itbythemetalcapreferred toabove. The electrode(10)ofthecage,when anobservation istobemade, isconnected with theconductor tobetested, andtheneedle is brought bytorsion toitsmarkedposition. Thesquarerootof thenumber ofdegreesoftorsion nowrequiredmeasures the difference ofpotentials between theconductor tested andthe interior coatingoftheLeyden jar.The excess, positiveornega- tive, ofthis result above thereduced earthreading, measures XVI.] Atmospheric Electricity.215 theexcess ofthepotential, positiveornegative,oftheconduc- tortested above that oftheearth;orsimplythepotentialof theconductor tested,ifweregardthat oftheearth aszero. 277. (III.) Theportableelectrometer[compare §263,above, and§§863... 373, below]isconstructed onthesame elec- tricalprinciplesasthehouse electrometerjustdescribed. Themode ofsuspensionoftheneedle is,however, essentially different;andavariedplanofconnexion between thedifferent electricalpartshasbeenconsequently adoptedasmore con- venient. Intheportable electrometer, theneedle isfirmly attached atright anglestothemiddle ofafineplatinum wire, tightlystretched intheaxis ofabrass tubewithaperturesin itsmiddle toallow theneedle toprojectonthetwo sides. Oneendoftheplatinumwire isrigidlyconnected with this tube;theother isattached toagraduatedtorsion head. The brass tube carries twometalplatesinsuitablepositionsto repelthetwoends oftheneedle incontrary directions, and metalstopstolimit itsangularmotion within aconvenient range. Theconducting system composedofthese different partsissupportedfrom themetal cover, orroof ofthejar,by threeglassstems. The torsion head iscarried roundbymeans ofastoutglass bar,projecting down from apinioncentered on thelower side ofthiscover, andturned bytheaction ofatan- gentscrewpresentingamilled head, tothehand oftheopera- toroutside. Theconducting systemthus bornebyinsulating supportsisconnected with theoutside conductor tobetested bymeans ofanelectrodepassingoutthroughthecentre ofthe topofthecasebyawideapertureinthecentre ofthepinion.Awirecage, surroundingthecentralpartofthetubeandthe needle andrepelling plates,isrigidlyattached totheinterior coatingoftheLeyden jar.Itcarries twometal sectors, or "bulkheads," insuitablepositionstoattract thetwoends of theneedle, which, however, ispreventedfromtouching them bythelimiting stopsreferred toabove. The effect ofthese attracting plates,astheywillbecalled, istoincreasevery much thesensibilityoftheinstrument. Thesquareroot of thenumber ofdegreesoftorsionrequiredtobringtheneedle loasighted positionnear therepelling plates,measures the 216Atmospheric Electricity. [xvi. difference ofpotentials between thecageandtheconducting system, consistingoftube, torsion-head, repelling plates, and needle. Themetal roofofthejarisattached toastrongmetal case,cemented round theoutside ofthetopofthejar,and enclosingitallround andbelow, toprotectitfrombreakage whenbeingcarried about. There aresufficientaperturesin this case, opened bymeans ofasliding piece,toallow the observer toseetheneedle andgraduatedcircle(torsion-head), whenusingtheinstrument. Ontheoutside oftheroof ofthe jarastoutglass stem isattached, whichsupportsalightstiff metallic conductor, bymeans ofwhich aburningmatch is supported,attheheightoftwoorthree feetabove theobserver. This conductor isconnected bymeans ofafinewirewith the electrometer, inthemanner described above, throughthecentre oftheapertureintheroofAnartificiallydried atmosphere ismaintained around thisglass stem, bymeans ofametal case surrounding it,andcontaining receptaclesofgutta percha,or lead, holding suitably shaped piecesofpumice-stonemoistened withsulphuricacid. The conductor which bears thematch projects upwards throughthecentre ofasufficientlywideaper- ture,andbears asmall umbrella, which bothstopsrainfrom fallinginto thisaperture, anddiminishes thecirculation ofair, owingtowind blowing round theinstrument, fromtaking place, tosogreatadegreeastodoawaywith thedrynessofthein- terior atmosphere requiredtoallow theglass stem toinsulate sufficiently. Theinstrument maybeheldbytheobserver in hishand intheopenairwithout theassistance ofanyfixed stand.Aslingattached totheinstrument andpassingover hisleft shoulder, much facilitatesoperations, andrenders it easytocarrytheapparatustotheplaceofobservation, even if uparuggedhillside,with little riskofaccident. 278. Theburning match intheapparatus which hasjustbeen described, performsthecollectingfunction referred toabove. The collector employedforthestationapparatus, whether the reflectingelectrometer orthecommon house electrometer is used,isaninsulated vessel ofwater, allowed toflow outina finestream throughasmallapertureattheendofapipe pro- jectingtoadistance ofseveral feetfrom thewallofthebuild- inginwhich theobservations aremade. i^.]Atmospheric Electricity. 217 579.Theprincipleofcollecting, whether byfireorbywater, theobservation ofatmospheric electricity,wasexplained by thespeakerthus :—The earth's surfaceis,exceptatinstants, alwaysfound electrified, ingeneral negatively, butsometimes positively. [Quotationfrom Nichol'sCyclopcedia, viz., §265, above, comes here intheoriginal.] After having givensomuch oftheseexplanationsasseemed necessarytoconveyageneralidea oftheprinciplesonwhich theconstruction oftheinstruments ofinvestigation depended, thespeaker proceededtocallattention tothespecial subject proposedforconsideration thisevening. 280.What isterrestrial atmospheric electricity?Isitelec- tricityofearth, orelectricityofair,orelectricityofwateryor otherparticlesintheair ?Anendeavour toanswer theseques- tionswas allthatwasoffered;abstinence fromspeculationasto theoriginofthis electric condition ofouratmosphere, and its physicalrelations with earth, air,andwater, having beenpain- fullylearnedbyrepeatedandvaried failure inevery attempt toseebeyondfacts ofobservation. Inserene weather, the earth's surface isgenerally,inmost localities hitherto examined, foundnegativelyorresinously electrified; andwhen this fact alone isknown,itmightbesupposedthat theglobeismerely electrified asawhole with aresinouscharge,and leftinsulated inspace. 281.But itistoberemarked thattheearth, althoughinsulated initsatmospheric envelope, beinginfactaconductor, touched onlybyaironeofthebestalthoughnotthestrongestofin- sulators, cannot with itsatmospherebesupposedtobeinsulated soastoholdanelectricchargeininterplanetary space.Ithas beensupposed, indeed, thatoutside theearth'srecognised atmo- spherethere existssomethingornothinginspace which con- stitutes aperfectinsulator;butthissupposition seems tohave noother foundation than astrangeidea that electric conduc- tivityisastrengthorapowerofmatter rather than amere non-resistance. Inreality weknow that airhighlyrarefied by theair-pump,orbyotherprocesses,asintheconstruction of the"vacuum tubes," bywhich such admirable phenomenaof electriclight haverecently been seen inthisplace,becomes extremely weak initsresistance tothetransference ofelec- 218Atmospheric Electricity. [xvi. tricity through it,andbeginstoappearrather asaconductor thananinsulator. Onehundred miles orupwardsfrom the earth's surface, the airinspacecannot inallprobability have resisting power enoughtobearanysuch electric forces asthose which wegenerallyfindeven inserene weather inthelower strata. Hence wecannot, with Peltier, regardtheearth asa resinously charged conductor, insulated inspace, andsubject onlytoaccidental influences fromtemporaryelectricdeposits inclouds, orairround it;butwemustsupposethat there is always essentiallyinthehigheraerialregionsadistribution arisingfrom the self-relief oftheouterhighlyrarefied airby disruptive discharge.This electric stratum must constitute very nearlytheelectro-polar complementtoalltheelectricity that exists ontheearth's surface, andinthelower strata ofthe atmosphere;inother words, thetotalquantityofelectricity, reckoned asexcess ofpositiveabovenegative,orofnegative abovepositive,inanylarge portionoftheatmosphere, andon theportionoftheearth's surface belowit,must bevery nearly zero. Thequalityofnon-resistance toelectric force ofthethin interplanetaryairbeing duly considered, wemight regardthe earth, itsatmosphere,andthesurrounding medium asconstitut- ingrespectivelytheinnercoating,thedi-electric(asitwere glass), andtheoutercoatingofagreat Leyden phial, charged negatively;andeven ifwewere toneglecttheconsideration ofpossible depositsofelectricity throughthebodyofthe di- electricitself, weshould arrive atacorrect view oftheelectric indications discoverable atanyonetimeandplaceoftheearth's surface. Infact,anykind of"collector," orplanforcollect- ing electricity from orinvirtue ofthenatural"terrestrial atmospheric electricity," givesaneffectsimply proportionalto the electrification oftheearth's surface thenand there. The methods ofcollecting byfireandwater which thespeaker exhibited, gave definitively,inthelanguageofthemathemati- caltheory,the"electricpotential"ofthe airatthepoint occupied bytheburningendofthematch, orbytheportion ofthestream ofwater where itbreaks intodrops.Ifthe apparatusisused inanopen plane,and care betaken to eliminate alldisturbance due tothepresenceoftheelectro- meter itselfandoftheobserver above theground,theindicated XVI.] Atmospheric Electricity. 219 effect,ifexpressedinabsolute electrostatic measure, anddivided bytheheightofthepointtested above theground,hasonlyto be[accordingtoanoldtheorem ofCoulomb's(seefootnote on §25,above),corrected byLaplace]divided byfourtimes the ratio ofthecircumference ofacircle toitsdiameter, toreduce it toanexpressionofthenumber ofunits, inabsolute electrostatic measure, oftheelectricity perunit ofarea oftheearth's surface atthetimeandplace. Themathematicaltheorydoesaway withevery difficultyinexplainingthevarious andseemingly irreconcilable views which different writers haveexpressed, andexplanationswhich different observers havegivenofthe functions oftheirtesting apparatus.Inthepresentstate of electric science, themost convenient andgenerally intelligible waytostate theresult ofanobservation ofterrestrial atmo- spheric electricity,inabsolute measure, isinterms ofthe number ofelements ofaconstantgalvanic battery, requiredto producethesame difference ofpotentialsasexists between the earth andapointintheairatastatedheight above anopen levelplaneofground.Observations withtheportableelectro- meter hadgiven,inordinaryfairweather, intheisland of Arran, onaflatopenseabeach, readings varyingfrom 200to 400, Daniel's elements, asthedifference ofpotentials between theearthandthematch, ataheightof9feetabove it.Hence, theintensityofelectric forceperpendiculartotheearth's sur- facemust haveamounted tofrom 22to44-Daniel's elements perfoot ofair.Infairweather, with breezes from theeastor north-east, hehadoften found from 6to10times thehigherof these intensities. 282.Even infairweather, theintensityoftheelectric force in theairneartheearth's surface isperpetually fluctuating. The speakerhadoften observedit,especially during calms orvery lightbreezes from the east,varyingfrom40Daniel's elements perfoot tothree orfour times thatamountduringafew minutes;andreturning againasrapidlytothelower amount. Morefrequentlyhehadobserved variations from about 30to about 40,andbackagain, recurringinuncertainperiods of perhapsabout twominutes. Thesegradual variations cannot butbeproduced byelectrified masses ofairorcloud, floating bythelocalityofobservation.Again,itiswellknown that 220Atmospheric Electricity. [xvi. duringstorms ofrain, hail, orsnow, there aregreat andsome- times sudden variations ofelectric force intheairclose tothe earth. These areundoubtedly produced, partlyasthose offair weather, bymotions ofelectrified masses ofairandcloud; partly bythe fall ofvitreouslyorresinouslyelectrified rain, leavingacorresponding deficiencyinthe airorcloud from which itfalls;andpartly bydisruptive discharges (flashesof lightning)between masses ofairorcloud, orbetween either andtheearth. Theconsideration ofthese various phenomena suggestedthefollowing questions,andmodes ofobservation for answering them :— 283. Question1.How iselectricitydistributedthroughthe different strata oftheatmospheretoaheightoffive orsix miles above theearth's surface inordinaryfairweather ?Tobe answered byelectrical observations inballoons atallheights uptothehighest limit, andsimultaneous observations atthe earth's surface. Q.2.Does electrification ofairclose totheearth's surface, orwithin afewhundred feet ofit,sensiblyinfluence the observed electric force ?and ifso,howdoes itvarywith the weather, andwith thetime ofdayoryear?The firstpartof thisquestionhasbeen answeredverydecidedlyintheaffirma- tive, first, forlargemasses ofairwithin afewhundredyards oftheearth's surface, bymeans ofobservations made simul- taneouslyatastation neartheseashore intheisland ofArran, and atoneorother ofseveral stations atdifferent distances, within sixmiles ofit,onthesides andsummit ofGoatfell. After that itwasfound, bysimultaneous observations made at awindow intheNaturalPhilosophy Lecture-Room, andonthe College Tower oftheUniversityofGlasgow,thattheinfluence oftheairwithin 100 feet oftheearth's surface wasalways sensible atboth stations, and often paramountatthelower. Thus, forexample, when, inbroken weather, thesuperficial electrification oftheoutside ofthelecture-room, about 20feet above theground,inaquadrangleofbuildings, wasfound positive,thesuperficialelectrification ofthesides ofthetower, about 70feethigher,wasoften foundnegative,ornearlyzero; andthissometimes evenwhen thepositiveelectrification ofthe sides ofthebuildingatthelower stationequalledinamount r.] Atmospheric Electricity.221 I I^K ordinaryfairweathernegative.This state ofthingscould ^^mllyexist invirtue ofanegativeelectrification ofthecircum- ambient air,inducingapositiveelectrification ontheground [^dsides ofthequadrangle,butnot sufficient tocounter- TSalance theinfluence, onthehigher partsofthetower, ofmore ciistantpositivelyelectrified aerial masses. I^BA longcontinuation ofsuch systemsofsimultaneous obser- vation—notinatownonly,butinvarious situations offlatand ofmountainouscountry, ontheseacoast aswell asfarinland, invarious regionsoftheworld—willberequiredtoobtain the information asked forinthesecondpartofthisquestion. Q.3.Dotheparticlesofrain, hail,andsnow infalling throughtheairpossessabsolutechargesofelectricity?and if so,whetherpositiveornegative,andofwhatamounts indiffer- entconditions astoplaceandweather ?Attemptstoanswer thisquestionhavebeenmade byvarious observers, butasyet without success;as,forinstance, byan" electro-pluviometer," tried atKewmany years ago.Byusingasufficientlywell- insulated vessel tocollect thefalling particles,itisquitecertain thatadecided answer maybeobtained with ease forthecases ofhailandsnow. Inductive effectsproduced bydrops splash- ingawayfrom thecollecting vessel,ifexposedtotheelectric force oftheairinanopen position,orinductive effects ofthe oppositekindproduced bydrops splashing awayfrom surround- ingwalls orscreens andfallingintothecollecting vessel,ifnot inanexposed position, make itlesseasytoascertain theelec- tricalqualityofrain;but,bytaking means toobviate the disturbingeffects ofthese influences, thespeaker hopedto arrive atdefinite results. 284. Itwould havebeenmoresatisfactorytohavebeen able toconclude adiscourse onatmospheric electricityotherwise than inquestions, butnoother form ofconclusion would havebeen ^Btallconsistent with thepresentstate ofknowledge. ^B285. Thediscourse wasillustrated bytheuseofthemirror ^ectrometer reflectingabeam oflightfrom the electriclamp, ^^Kid throwingitonawhite screen, where itsmotions were measured byadivided scale. Theprincipleofthewater- ^droppingcollector wasillustrated byallowingajetofwater to ^^k)w byafinenozzle intothemiddle ofthelecture-room, from I 222Atmospheric Electricity. [xvt. anuninsulated metal vessel ofwater andcompressed air,and collectingthedropsinaninsulated vessel onthe floor. This vessel wasconnected withthetestincj electrode ofthereflectinor electrometer;and itwasthenfound toexperienceacontinually increasing negative electrification, when fixedpositivelyelec- trified bodies were intheneighbourhoodofthenozzle. Ifthe sameexperiment weremade inordinaryfairweather inthe open air,instead ofunder theroofandwithin thewalls ofthe lecture-room, thesame result would beobserved, without thepresenceofany artificiallyelectrifiedbody. The vessel fromwhich thewater wasdischargedwasnext insulated;and other circumstancesremaining unvaried, itwasshown that this vessel becamerapidlyelectrified toacertaindegreeof positive potential, andthefalling dropsceased tocommunicate anymoreelectricitytothevessel inwhichtheyweregathered. 286.Theinfluence ofelectrified masses ofairwasillustrated bycarryingabout theportable electrometer, with itsmatch burn- ing,todifferentpartsofthelecture-room, while insulated spirit-lampsconnected with thepositive andnegativecon- ductor ofanelectrical machine, burned onthetwo sides. The speakerobserved theindications ontheportableelectrometer; butthepotentialsthusmeasured were seenbytheaudience marked onthescalebythespotoflight ;thereflectingelectro- meter being keptconnected with theportableelectrometer in allitspositions, bymeans ofalongfine wire. Itwasfound that,when theburning match wasononeside ofacertain surfacedividingtheairofthelecture-room, thepotentialindi- cated waspositive, andontheother sidenegative. 287. Thewater-droppingcollector constructed forthe self- registering apparatustobeused atKew,hadbeenpreviously setupontheroofoftheRoyal Institution, andaninsulated wire(Beccaria's"Deferent Wire")leddown tothereflecting electrometer onthelecture-room table. The electric force in theairabove theroofwasthus tested several times duringthe meeting ;and itwasatfirstfound tobe,asithadbeen during several days preceding, somewhat feeblepositive (corresponding toafeeblenegativeelectrification ofthe earth's surface, or ratherhousetops,intheneighbourhood). Thiswasanot unfrequentelectrical condition ofdays,such asthese hadbeen XVI.] Atmospheric Electricity.223 ofdull rain, with occasional intervals ofheavier rainand of cessation. The naturalelectricity wasagainobserved by means ofthereflectingelectrometerduringseveral minutes near theendofthediscourse;andwasfound, instead ofthe weakpositivewhich hadbeenpreviously observed, tobe strong positiveofthree orfourtimes theamount. Uponthis thespeaker quoted*ananswer which Prior Cecahadgivento aquestionBeccaria hadputtohim"concerningthestate of electricitywhen theweather clearsup.""' If,when therain asceased (thePrior said tome)astrong excessive-]-elec- tricity obtains, itisasignthattheweather willcontinue fair "'for severaldays;iftheelectricityisbut small, itisasign "'thatsuchweather willnot last somuch asthatwholeday, "*andthat itwillsoonbecloudy again,oreven willagain "'rain.'" Theclimate ofthiscountryisverydifferent from that ofPiedmont, where Beccaria and hisfriend made their observations, buttheir rule astothe"electricityofclearing weather" hasbeen foundfrequentlyconfirmed bythespeaker. Hetherefore considered that, althoughitwas stillrainingat thecommencement ofthemeeting,the electrical indications theyhad seen gavefairpromisejfortheremainder ofthis evening,ifnotforalonger period. There canbenodoubt but that electric indications, whensufficiently studied, will be foundimportantadditions toourmeans forprognosticatingthe weather;andthespeaker hopedsoon toseetheatmospheric electrometergenerally adoptedasauseful and convenient weather-glass. 288. Thespeakercould notconclude withoutguardinghim- selfagainst anyimputationofhaving assumed theexistence of two electric fluids orsubstances, because hehadfrequently spokenofthevitreous andresinous electricities. Dufay's very important discoveryoftwomodes orqualitiesofelectrification, ledhisfollowers tooreadilytoadmit hissuppositionoftwo distinct electric fluids. Franklin, ^pinus,and Cavendish, *From Beccaria's first letter'•OnTerrestrial Atmospheric Electricity during Serene Weather."—Garzegna dlMondavi, May 16,1775. +i.e.,vitreous, orpositive. XAttheconclusion ofthemeetingitwasfound thattherainhadactually ceased. Theweather continued fairduring theremainder ofthenight, and three orfour ofthefinest days oftheseason followed. 224Atmospheric Electricity. [xvi. with ahypothesisofone electric fluid, openedthewayfora juster appeciationoftheunityofnature inelectric phenomena. Beccaria, with his"electricatmospheres," somewhat vaguely struggledtoseedeeperintotheworkingofelectricforce, but hisviews found littleacceptance, andscarcely suggestedin- quiryoreven meditation. Theeighteenth century made a school ofscience foritself, inwhich, forthenotunnatural dogmaoftheearlier schoolmen,"matter cannot actwhere itis not,"wassubstituted themost fantastic ofparadoxes,contact does not exist. Boscovich'stheory wastheconsummation of theeighteenth centuryschool ofphysicalscience. Thisstrange ideatookdeep root,andfrom itgrewupabarren tree,exhaust- ingthe soilandovershadowingthewhole field ofmolecular investigation,onwhich somuchunavailinglabour wasspent bythegreatmathematicians oftheearly partofournineteenth century.IfBoscovich'stheory nolonger cumbers theground, itisbecause onetruephilosopher required morelightfortrac- inglines ofelectric force. 289.MrFaraday's investigationofelectrostatic induction influences nowevery departmentofphysical speculation, and constitutes anerainscience. Ifwecannolonger regard electric andmagneticfluidsattractingorrepellingatadistance asrealities, wemaynow alsocontemplateasathingofthe pastthat belief inatoms andinvacuum, against which Leib- nitz soearnestlycontended inhismemorablecorrespondence withDrSamuel Clarke. 290.Wenowlookonspaceasfull.Weknow thatlightis propagatedlikesoundthrough pressureandmotion. Weknow that there isnosubstance ofcaloric—thatinscrutably minute motions cause theexpansionwhich thethermometer marks, andstimulate oursensation ofheat—that fire isnotlaidupin coalmore than inthisLeyden phial,orthisweight:there is potentialfireineach. Ifelectric forcedependsonaresidual surface action, aresultant ofaninner tensionexperienced by theinsulating medium, wecanconceive thatelectricityitself istobeunderstood asnotanaccident, butanessence ofmatter. Whateverelectricity is,itseemsquitecertain thatelectricity inmotion ISheat; and that acertainalignmentofaxes of revolution inthismotion ismagnetism. Faraday's magneto- I] Atmospheric Electricity. 225 optic experiment makes thisnotahypothesis,butademon- strated conclusion*. Thus arifle-bullet keepsitspointfore- most; Foucault'sgyroscopefinds theearth's axis ofpalpable rotation;and themagneticneedle shows thatmore subtle rotatory movement inmatter oftheearth, which wecall ter- restrial magnetism:allbyoneandthesamedynamicalaction. 291. Itisoften asked, arewetofallbackonfactsandpheno- mena, andgiveupallidea ofpenetratingthatmysterywhich hangsround theultimate nature ofmatter ?This isaquestion thatmust beanswered bythemetaphysician, and itdoesnotbe- longtothedomain ofNaturalPhilosophy. But itdoesseem that themarvellous train ofdiscovery, unparalleledinthehistory ofexperimental science, which thelastyearsoftheworld has seen toemanate fromexperimentswithin these walls, must lead toastageofknowledge,inwhich laws ofinorganicnature willbeunderstood inthissense—thatonewillbeknown as essentiallyconnected with all,and inwhichunityofplan through aninexhaustiblyvaried execution, willberecognised asauniversallymanifested result ofcreative wisdom. 292.[Postscript,withdiagram, communicated tothePhiloso- phical Magazinein1861;butnow firstpublished.] MrBalfour Stewart, Director oftheKewMeteorological Observatory, has,since thecommencement ofthepresent year (1861), broughtintoregular andsatisfactory operationthe self- recording atmosphericelectrometer withwater-droppingcollec- tor,described intheprecedingabstract :aspecimenofthe results isexhibited intheaccompanying photographiccurves. I'I'Mlvr MM II IMMI IMI II II II II II IIII IIM I S301S3 4S6~ *See"Dynamical Illustrations oftheMagnetic andtheHelipoidal Kotatory Effects ofTransparent Bodies onPolarized Light." ByProf.W.Thomson.— Proceedings oftheRoyal Society, June 12,1856. T.E. 15 226Atmospheric Electricity. [xvi, 293.Thediagramexhibits thevariations oftheelectric force oftheatmosphere,asphotographically recorded bythedivided ringelectrometer attheKewObservatoryfortwo succes- sivedays, commencing onthe28th ofApril 1861. The preparedsensitive paper wasmade tomoveverticallyata uniform ratebymeans ofclock-work, while aspotoflight (the imageofaportionofagas-flamereflected from themirror of thedividedring electrometer) movedhorizontallyacross it accordingtothecontinually varyingelectric force oftheatmo- sphere,andmarked thecurvephotographically. Thedatum line,showingthepositionthespotoflight would have ifthe electric forcewere zero,isproduced byanimagefrom thesame source oflightreflected from afixed mirror attached tothe case oftheelectrometer. Thenumbers indicate hours reckoned fromnoon aszero,upto23.Thesamepaper is,forthesake ofeconomy, generallyused tobeartherecord fortwodays. Thus thedistance ofthespotoflightfrom thedatumline, ononeside orother, indicates, andthephoto-chemicalaction records, foreach instant oftime the electricpotential, positive ornegative,oftheatmosphereatthepoint where thestream of water dischargedfrom theinsulated vessel breaks intodrops. ONELECTRICAL "FREQUENCY." [From Report ofBritish Association, Aberdeen Meeting, 1859.] 294. Beccaria found thataconductor insulated intheopen airbecomes charged sometimes withgreater andsometimes with lessrapidity, andhegavethename of" frequency"toex- presstheatmospheric quality onwhich therapidityofcharg- ingdepends.Itmight seem natural toattribute thisquality toelectrification ofthe airitself round theconductor, orto electrifiedparticlesintheairimpinging upon it;buttheauthor gavereasons forbelievingthattheobserved effects areentirely duetoparticles flying away from thesurface oftheconductor, inconsequenceoftheimpactofiion-electrified particles against it.Hehadshown inaprevious communication thatwhen no electricityofseparation (or,asitismoregenerally called, "fractionalelectricity,"or"contactelectricity ")iscalled into I XVI.] Atmospheric Electricity. 227 play,thetendencyofparticles continually flyingofffrom a conductor istodestroyallelectrification atthepartofitssur- facefromwhich theybreak away. Hence aconductor insulated intheopen air,andexposedtomist orrain,with wind, will tend rapidlytothesame electricpotentialasthat oftheair, beside thatpartofitssurface from which there isthemost frequent dropping,orflying away,ofaqueous particles. The rapid chargingindicated bytheelectrometer under cover, after puttingitforaninstant inconnexion with theearth, isthere- fore, inreality,duetoarapid dischargingoftheexposed parts oftheconductor. Theauthor hadbeen ledtothese views by remarkingtheextremerapiditywithwhich anelectrometer, connected byafinewirewith aconductor insulated above the roof ofhistemporaryelectricobservatoryinthe island of Arran, became charged, reachingitsfullindication inafew seconds, andsometimes inafraction ofasecond, afterbeing touched bythehand, duringagaleofwind and rain. The conductor, averticalcylinderabout 10incheslongand4inches diameter, with itsupper end flatandcornerslightly rounded off,stoodonly8feetabove the roof, or,inall,20feetabove theground,andwasnearlysurrounded bybuildings risingto ahigherlevel. Even with somoderate anexposureasthis, sparks werefrequently produced between aninsulated andan uninsulatedpieceofmetal, which mayhavebeenabout^^^thof aninchapart,within theelectrometer, andmore than once a continuous line offirewasobserved intheinstrumentduring nearlyaminute atatime, while rainwasfallingintorrents outside. ONTHENECESSITY FORINCESSANT RECORDING, ANDFOR (SIMULTANEOUSOBSERVATIONS INDIFFERENT LOCALI- fcriES, TOINVESTIGATE ATMOSPHERIC ELECTRICITY. B [From Report ofBritish Association, Aberdeen Meeting, 1859.] 95. Thenecessityforincessantly recordingtheelectric con- ition oftheatmospherewasillustrated byreference toobser- vationsrecently made bytheauthor intheisland ofArran, by which itappearedthatevenunder acloudless sky,without any 15—2 228Atmospheric Electricity. [xvi. sensible wind, thenegativeelectrification ofthesurface ofthe earth, alwaysfoundduringserene weather,isconstantly vary- ingindegree. Hehadfound itimpossible,atanytime, to leave theelectrometer withoutlosingremarkable features of thephenomenon. Beccaria, Professor ofNatural Philosophy intheUniversityofTurin acentury ago,used toretire to Garzegna when hisvacation commenced, and tomake inces- sant observations onatmospheric electricity, nightandday, sleepingintheroom with hiselectrometer inalofty position, fromwhich hecould watch theskyallround, limited bythe Alpine range onone side,andthegreat plainofPiedmont on theother. Unlessrelaysofobservers canbegottofollow his example,and totakeadvantageofthemore accurate instru- mentssupplied byadvanced electric science, aself-recording apparatusmust beappliedtoprovidethedat^requiredfor obtaining knowledgeinthismostinterestingfield ofnature. The authorpointedout certainsimpleandeasily-executed modifications ofworkingelectrometers(exhibitedtothemeet- ing),torender themself-recording. Healsoexplainedanew collecting apparatusforatmospheric electricity, consistingof aninsulated vessel ofwater, dischargingitscontents ina finestream from apointedtube. This stream carries away electricityaslongasanyexists onitssurface, where itbreaks intodrops. Theimmediateobjectofthisarrangementisto maintain thewhole insulated conductor, includingtheportion oftheelectrometer connected with itandtheconnecting wire, inthecondition ofnoabsolutecharge ;that istosay,with as muchpositive electricity onone side ofaneutral line asof negativeontheother. Hence thepositionofthedischarging nozzle must besuch, that thepointwhere thestream breaks intodropsisinwhat would betheneutral lineofthecon- ductor, ifJirst perfectly dischargedunder temporary cover, and then exposedinitspermanent open position,inwhich itwill becomeinductivelyelectrified bytheaerial electromotive force. Iftheinsulation ismaintained inperfection,thedroppingwill notbecalled onforanyelectrical effect, andsudden orslow atmospheric changeswill allinstantaneouslyandperfectlyin- duce theircorrespondingvariations intheconductor, andgive their appropriateindications totheelectrometer. The neces- I] AtmospheriG Electricity. 229 saryimperfectionoftheactual insulation, which tends tobring theneutral linedownwards orinwards, orthecontraryeffects ofaerial convection, which, when theinsulation isgood, gene- rally preponderate,andwhich insome conditions oftheatmo- sphere, especially during heavy wind andrtiin, areoftenvery large,arecorrected bythetendencyofthedroppingtomain- taintheneutral lineintheonedefiniteposition. Theobjects tobeattained bysimultaneous observations indifferent localities alluded towere—(1)tofixtheconstant foranyobservatory, bywhich itsobservations arereduced toabsolute measure of electromotive forceperfoot ofair; (2)toinvestigatethedis- tribution ofelectricityinthe airitself (whether onvisible clouds orinclearair)byaspeciesofelectricaltrigonometry,of which thegeneral principles wereslightlyindicated. Apor- table electrometer, adaptedforballoon andmountain observa- tions, withaburning match, regulated byaspringsoastogive acone offireintheopen air,inadefinitepositionwith refer- ence totheinstrument, was exhibited. Itiseasily carried, with orwithout theaidofashoulder-strap,andcanbeused bytheobserverstanding up,andsimply holdingtheentire apparatusinhishands, without astand orrest ofanykind. Itsindicationsdistinguish positivefromnegative,andarere- ducible toabsolute measure onthespot. Theauthor gavethe result ofadetermination which hehadmade, with the assist- ance ofMrJoule, ontheLinks, apieceoflevel ground near thesea,beside thecityofAberdeen, about 8A.M.onthepre- ceding day(September 14),under acloudlesssky,andwitha light north-west windblowing,with theinsulatingstand ofthe collecting partoftheapparatusburied intheground, andthe electrometer removed toadistance of5or6yards, andcon- nected byafine wire with thecollectingconductor. The heightofthematch was3feetabove theground, andthe observer attheelectrometerlayonthegroundtorender the electrical influence ofhisownbody onthematch insensible. Theresult showed adifference ofpotentials between theearth (negative) andthe air(positive) atthematchequaltothat of 115elements ofDaniell'sbattery, and, therefore, atthattime andplace, theaerial electromotive forceperfootamounted to that ofthirty-eightDaniell'scells, or12cellspercentimetre. 230Atmospheric Electricity. [xvi. OBSERVATIONS ONATMOSPHERIC ELECTRICITY. [From theProceedings oftheLiterary andPhilosophical Society ofManchester, March, 1862.] 296. 1findthatatmospheric electricityisgenerally negative within doors, andalmostalwayssensible tomydividedring reflectingelectrometer. Iuseaspirit-lamp, onaninsulated stand afew feetfrom walls, floor, orceilingofmylecture room, andconnect itbyafinewire with theinsulated half ringoftheelectrometer. Adecidednegativeeffect isgenerally found, w^hich shows apotentialtobeproducedinthecon- ductors connected with theflame, negative relativelytothe earth byadifference amountingtoseveral times thedifference ofpotentials (orelectromotive /orce) between twowires ofone metal connected with thetwoplatesofasingle element of Daniell's. Ihave tested that thesjpirit-lamp givesnoidio- electric effect amountingtosomuch asthe effect ofasingle cell. The electric effect observed istherefore notdue to tliermal orchemical action intheflame. Itcannot bedueto contact electrifications ofmetallic orother bodies inconductive communication with the walls, floor, orceiling, because the potentialsofsuchmustalwaysfallshort ofthedifference of potentials produced byasinglecell. Ihave taken care to distinguishtheobserved natural effect fromanythingthat can beproduced byelectricaloperationsforlecture orlaboratory purposes. Thus Iobservegenerallyinthemorningbeforeany electricaloperations have beenperformed,andfindordinarily results quitesimilar tothose observed ontheMonday mornings when the electrical machine hasnotbeen turned since the previous Friday. The effect, when there hasbeennoartificial disturbance, hasalwaysbeenfound negative, excepttwoorthree times, since themiddle ofNovember; buttrustworthyobser- vations have notbeenmade onmore than aquarterofthe number ofdays. 297.Afewturns oftheelectrical machine, wdthaspirit-lamp onitsprime conductor, oraslightly charged Leyden phial,with itsinsidecoating positive putinconnexion with aninsulated spirit-lamp,isenoughtoreverse thecommonnegativeindica- tion. Anothervery striking wayinwhich thismaybedone istoputanegatively charged Leyden phialbelow aninsulateii XVI.] Atmospheric Electricity. 231 riarne(acommongas-burner,forinstance). The flame, becom- ingpositivelyelectrified byinduction, keeps throwing off,by thedynamic powerofitsburning, portionsofitsowngaseous matter, anddoes notallow them tobeelectricallyattracted down totheLeydenphial,butforces them torise. These, on cooling, become, likecommon air,excellent non-conductors*, and, mixingwiththeairoftheroom, giveapreponderanceofpositive influence tothetestinginsulated flame(thatistosay,render the airpotential positiveattheplace occupied bythisflame). 298. Halfanhour, oroftenmuch more, elapsesaftersuchan operation,before thenaturalnegativelyelectrified airbecomes againparamountinitsinfluence onthetestingflame. 299. That eitherpositiveornegative electricity maybe carried, eventhroughnarrowpassages, byair,Ihave tested by turning anelectric machine, with aspirit-lamp onitsprime conductor, forashort time inaroomseparatedfrom thelecture room byanoblique passageabout twoyards longandthen stoppingthemachine andextinguishingthelamp;soasto send alimitedquantityofpositive electricityintothe airof thatroom. When thelecture-room window waskept open,and thedoorleadingtotheadjoiningroom shut, thetesting spirit- lamp showed thenaturalnegative. When thewindow was closed, andasmall chink(aninch orlesswide) openedofthe door, theindicationquickly becamepositive.Ifthedoorwas then shut,andthewindowagain opened,thenatural effect was slowlyrecovered. Acurrent ofair,tofeed thelecture-room fire,wasfoundentering byeither door orwindow when the other was shut. This alternatepositiveandnegativeelectric ventilation mayberepeated manytimes withoutrenewingthe positive electricityoftheadjoining room byturningthe machine afresh. *Ifindthatsteam from akettleboiling briskly onacommon fire isan excellent insulator. Iallow ittoblow foraquarter ofanhour ormore against aninsulated electrified conductor, without discovering that ithas anyeffect ontheretention ofthecharge. Theelectricity ofthesteam itself, insuch circumstances, asistobeexpected from Faraday's investigation,is notconsiderable. Common airloses nearlyallitsresisting power atsome temperature between that ofboiling water and red-hot iron, andconducts continuously (not,asIbelieve isgenerally supposed tobethecase, bydis- ruption) asglass does atsome temperature below theboiling point, with "so great ease astodischarge anycommon insulated conductor almost completely inafewseconds. 232Atmospheric Electricity. [xvr. 300. Theoutofdoors airpotential,astestedbyaportable electrometer inanopen place,orevenbyawaterdropping nozzle outside, twoorthree feetfrom tbewalls ofthelecture room, wasgenerally onthese occasionspositive, andtheearth's surfaceitself, therefore, ofcourse, negative ;—thecommon fair weather condition, which Iamforced toconclude isdue toa paramount influence ofpositive electricityinhigher regionsof theair,notwithstandingthenegative electricityofthe airin thelower stratum neartheearth's surface. Onthetwoorthree occasions when thein-dooratmospheric electricity wasfound positive, and, therefore, thesurface ofthefloor, walls, and ceil- ing negative,thepotentialoutside wascertainly positive, andtheearth's surface outofdoorsnegative,asusual infair weather. 300'. Extract from letter addressed toGeneral Sabine :— "During myrecent visit toCreuznach Ibecameacquainted withMrDellman ofthatplace, whomakesmeteorological, chiefly electrical, observations forthePrussian Government, and Ihadopportunitiesofwitnessinghismethod ofelectrical observation. Itconsists inusingacopperballabout 6inches diameter, tocarry awayanelectrical effect from aposition about twoyards above theroof ofhishouse, depending simply ontheatmospheric 'potential'atthepointtowhich thecentre oftheball issent; and itisexactlythemethod ofthe'carrier ball'bywhichFaraday investigatedtheatmospheric potential intheneighbourhoodofarubbed stick ofshell-lac, andother electrified bodies{Experimental Researches, Series xi.1837). Thewholeprocess onlydiffers fromFaraday'sinnotemploying thecarrier balldirectly,astherepellerinaCoulomb-electro- meter, butpattingitintocommunication withtheconductor of aseparateelectrometer ofpeculiarconstruction. Thecollecting partoftheapparatusissosimple andeasily managedthatan amateur could, forafewshillings,setoneuponhisownhouse, ifatallsuitable asregardsroofandwindows;and,ifprovided withasuitable electrometer, couldmake observations inatmo- spheric electricitywith asmuch ease asthermometric orbaro- metric observations. Theelectrometer usedbyMrDellman is ofhisown constructi<m(describedinPoggendorfFs Aiinalen, 1853, Vol.Lxxxix., alsoVol.Lxxxv.), and itappearstobevery XVI.] Atmospheric Electricity. 233 satisfactoryiuitsoperation.Itis,Ibelieve, essentially more accurate andsensitive than Peltier's, and ithasagreat advan- tageinaffordingavery easyandexact method forreducingits indications toabsolute measure. Iwasmuch struck with the simplicity and excellence ofMrDellman's wholesystemof observation onatmospheric electricity;and ithasoccurred to methattheKewCommittee might bedisposedtoadopt it,if determined tocarryout electrical observations. When Itold MrDellman that Iintended tomake asuggestiontothis effect, heatonce offered tohave anelectrometer,ifdesired, made under hisown care. Iwish also tosuggesttwoother modes ofobserving atmospheric electricity which have occurred tome, aspossessingeach ofthem some advantagesoveranyofthe systemshitherto followed. Inoneofthese Iproposetohave anuninsulatedcylindricaliron funnel, about 7inches diameter, fixed toaheightoftwoorthreeyardsabove thehighest part ofthebuilding, andalight moveable continuation(likethe telescopefunnel ofasteamer)ofayardandahalf ortwoyards more, which canbeletdown orpushed upatpleasure.Insu- latedbysupportsatthetopofthefixedpartofthefunnel, I would haveametal stemcarryingaballlikeDellman's, stand- ingtosuchaheightthat itcanbecovered byahiugedlidon thetopofthemoveablejointofthefunnel, when thelatter is pushed up ;andafinewire fixed tothelower endoftheinsu- lated stem, andhanging down, intheaxis ofthefunnel tothe electrometer. When theapparatusisnotinuse,themoveable jointwould bekeptatthehighest,with itsliddown, andthe balluninsulated. Tomake anobservation, theballwould be insulated, the lidturned uprapidly, andthemoveablejoint carryingitletdown, anoperation which could beeffected ina fewseconds byasuitable mechanism. Theelectrometer would immediatelyindicate aninductive electrification simply propor- tional totheatmospheric potentialattheposition occupied by thecentre ofthe ball,andwould continue toindicate ateach instaut theactualatmospheric potential,however variable, as longasnosensible electrification ordiselectrification hastaken place through imperfectinsulation orconvection byparticlesof (lust orcurrents ofair(probablyforaquarterorahalf ofan hour,when care istaken tokeeptheinsulation ingood order). 234Atviospheric Electricity. [xvi. Thismightbethebestform ofapparatusformakingobserva- tions inthepresenceofthunder-clouds. But Ithink thebest possible planinmostrespects,ifitturns outtobepracticable, ofwhich Icanhave little doubt, willbetouse,instead of theordinaryfixed insulated conductor with apoint,afixed conductor ofsimilar form, buthollow, andcontainingwithin itself anapparatusformaking hydrogen, andblowingsmall soap-bubblesofthatgasfrom afinetubeterminatingasnearly asmaybeinapoint,ataheightofafewyardsinthe air. With thisarrangementtheinsulation wouldonlyneed tobe goodenoughtomake the loss ofacharge byconductionvery slow incomparison with convective lossbythebubbles;sothat itwould beeasytosecureagainst anysensible error from defective insulation. If100 or200bubbles, each-^inch in diameter, areblown from thetopoftheconductorperminute, the electricalpotentialinitsinterior willvery rapidlyfollow variations oftheatmospheric potential, andwould beatany instant thesame asthemean fortheatmosphere during some periodofafewminutespreceding. The action ofasimple pointis(as,Isuppose,isgenerally admitted) essentiallyunsa- tisfactory,andasnearlyaspossible nugatoryinitsresults. I amnotaware how flame hasbeen found tosucceed, but I should think notwell inthecircumstances ofatmospheric observations, inwhich itisessentiallyclosed inalantern; and Icannot seeonanytheoretical ground how itsaction inthese circumstances canbeperfect,likethat ofthesoap-bubbles.I intend tomake atrial ofthepracticabilityofblowingthe bubbles;and ifitproves satisfactory,there cannot beadoubt oftheavailabilityofthesystemforatmosphericobservations." [Addition,Feb.1857.]—Theauthor hasnowmade various trials onthelast-mentionedpartofhisproposal,andhehas notsucceeded infinding anypracticable self-regulating appa- ratus forblowingbubbles anddetaching them onebyonefrom thetube. Hehasseen reason todoubt whether itwillbe possibletogetbubbles sosmall asthoseproposed above, torise ata]l;buthehasnotbeen ledtobelieve that, ifitisthought worth while totry,itwillbefoundimpracticabletoconstruct aself-acting apparatuswhich willregularlyblowanddischarge separately, bubbles ofconsiderably larger diameter, and soto I XVI.] Atmospheric Electricity. 235 secure theadvantages mentioned, although with aproportion- ately larger consumptionofthegas. Ontheother hand, hefinds that,bytheaidofanextremely sensitive electrometer which hehasrecently constructed, he willbeable, inallprobability withgreateaseand atvery small cost, tobringintopractice the firstofhistwoplans,con- structed onaconsiderablysmaller scale asregards heightthan proposedinthepreceding statement. ONSOMEREMARKABLE EFFECTS OFLIGHTNING OBSERVED INAFARM-HOUSE NEAR MONIEMAIL, CUPAR-FIFE. (From Proceedings ofthePhilosopldcal Society ofGlasgow.) 301. Thefollowingisanextract from aletter, addressed last ;iutumn tomebyMrLeitch, minister ofMoniemailparish:— "Moniemail Manse, Cupar-Fife, 26thAugust, 1849. "... Wewere visited onthe11th inst. with aviolent thunder-storm, which didconsiderable damagetoafarm-house inmyimmediateneighbourhood.Icalledshortlyafter- wards andbrought awaythewires and thepaperwhich I enclose. . . . "Ihavesomedifficultyinaccountingfortheappearanceof the wires. You will observe thattheyhave beenpartially fused, andwhen Igotthem firsttheyadheredcloselytoone another. You will findthat the flatsidesexactlyfit.They were both attached toone crank, andranparalleltoone another. Thequestion is,howweretheyattracted sopower- fullyastobecompressed together? . . . "You willobserve thatthepaperisdiscoloured. This has been done, notbyscorching,butbyhaving some substance deposited onit.There waspainted wood also discoloured, on which thestratum wasmuch thicker. Itcouldeasilybe rubbed off,when yousawthepaint quitefresh beneath. . . . ''Thefarmer showed meaprobangwhich hungonanail. 236 Atmospheric Electricity. [xvi. Thehandleonlywas left.The rest, consistmgofatwisted cane, hadentirely disappeared. Byminute examination I found asmallfragment,which wasnotburnt, butbroken off." [The copperwires andthestainedpaper,enclosed withMr Leitch's letter, were laidbefore theSociety.] Theremarkable effects oflightning,described byMrLeitch, are allextremely interesting. Those with reference tothe copper wires arequiteoutofthecommon class ofelectrical phenomena; nothingofthekindhaving,sofarasIamaware, been observedpreviously,either asresultingfrom natural dis- charges,orinexperiments onelectricity.Itisnotimprobable that theyaredue totheelectro-magneticattraction which must have subsisted between thetwowiresduringthe dis- charge,itbeingawell-known factthatadjacent wires, with currents ofelectricityinsimilar directionsalong them, attract oneanother. Itmay certainlybedoubted whether the in- appreciablyshort timeoccupied bythe electricaldischarge could havebeen sufficient toallow thewires, afterhavingbeen drawn into contact, tobepressedwith sufficient force tomake them adheretogether,andtoproducetheremarkableimpres- sions whichtheystill retain. Ontheother hand, theelectro- magneticforcemust have beenvery considerable, since the currents inthewires werestrong enough nearlytomeltthem, and since they appeartohave been softened, ifnotpartially fused; theflatteningandremarkableimpressions might readily havebeenproduced byeven aslightforcesubsistingafter the wirescame incontact. Thecircumstances with reference totheprobang,described byMrLeitch, afford aremarkable illustration ofthe well- known fact, thatanelectricaldischarge, when effectedthrough thesubstance ofanon-conducting (thatistosay,apowei^fully resisting) solid, shattersit,withoutproducing anyconsiderable elevation ofitstemperature;notleaving marks ofcombustion, ifitbeofanordinarycombustible material such aswood. DrRobert Thomson, atmyrequest, kindlyundertook to examine thepaper removed from thewall ofthefarm-house, andenclosed with hisletter tomebyMrLeitch;soas, if possible, bytheapplicationofchemical tests, todiscover the stainingsubstancedepositedonitssurface. MrLeitch, inhis XVI.] Atmospheric Electricity. 237 letter, hadsuggestedthat itwould beworth while totry whether this case isanexampleofthedepositionofsulphur, which Fusinieri believed hehaddiscovered insimilar circum- stances.Accordinglytests forsulphur wereapplied, butwith entirely negativeresults. Stains presentingasimilarappear- ancehadbeensometimes observed onpaperintheneighbour- hood ofcopper-wires throughwhichpowerful dischargesin experiments with thehydro-electricmachine hadbeenpassed ; andfrom this itwassuggestedthat thestainingsubstance might havecome from thebell-wires. Tests forcopper were accordingly applied,andthe results were mostsatisfactory. Thefront ofthepaperwasscrapedindifferentplaces,soasto remove some ofthepigmentinpowder; andthepowders from thestained, andfrom thenotstainedparts, wererepeatedly examined. Thepresenceofcopperintheformer wasreadily made manifest bytheordinarytests :inthe latter, notraces of coppercould bediscovered. Theback ofthepaper presented agreen tint,havingbeen tornfrom awallwhich hasprobably beenpaintedwith Sclieele'sgreen ;andmatterscraped away fromanypart ^ofthebackwasfound tocontaincopper. Since, however, thestains infront weremanifestly superficial,the discolouration being entirely removed byscraping, and since there wasnoappearancewhatever ofstainingattheback of thepaper,norofanyeffect oftheelectricaldischarge,itwas impossibletoattribute thestains tocopper produced from the Scheele's greenonthewallbelow thepaper. DrThomson, therefore, considered themostprobable explanationtobe, thatthestains ofoxide ofcopper must havecome from the bell-wire. Toascertain how farthisexplanation could be supported bythecircumstances ofthe case, Iwrote toMr Leitch asking him forfurtherparticulars, especiallywith re- ference tothispoint, andIreceived thefollowing answer :— "MONIEMAIL, CuPAR-FlFE, SOthNov. 1849. ". . . .Ireceived yourletterto-day, andimmediately called atHall-hill, intheparishofCollessie, thefarm-house which hadbeen struck bythelightning.... 288 Atmospheric Electricity. [xvi. "IfindthatDrThomson'ssuggestionisfullyborne outby the facts. Iatfirstthoughtthat thebell-wire didnotrun alongtheline ofdiscolouration, butInow findthatsuchwas thecase. . . . [Fromadrawingandexplanation whichMrLeitchgives,it appearsthat theware runsvertically alongacorner ofthe room, from the floor, toabout ayardfrom theceiling, where itbranches into two, connected with twocranks near one another, andclose totheceiling.] "The efflorescence [thestainspreviouslyadvertedto]was oneach side ofthisperpendicularwire. Insomeplacesit extended more thanafootfrom thewire. Thedeposit seemed tovaryinthicknessaccordingtothesurface onwhich itwas deposited. There wasnone ontheplasteronthe roof. It wasthinnest uponthewall-paper,andthickest uponthewood facingofthedoor*. This lastexhibited various colours. On thethickestpartitappeared quiteblack;where therewasonly aslight film, itwasgreenoryellow.. . . "Imaymention thatthethunder-storm wasthatofthe11th ofAugustlast. Itpassedovermost ofScotjand, andhas rarely beensurpassedforterrificgrandeuratleastbeyondthe tropics.Itcommenced about nine o'clock p.m.,and inthe course ofanhour itseemed todieaway altogether. Thepeals becamevery faint, andtheintervals between theflashes and thereports very great,when allatonce aterrificcrashing peal was heard, which didthedamage. Thestorm ceased with thispeal. "Theelectricity must have been conductedalongthelead ontheridgeofthehouse, andhavedivergedintothree streams; onedownthroughtheroof,andthetwoothersalongtheroofto thechimneys. Oneoftheseappearstohave struck alargestone outfrom thechimney, and tohave been conducted down the chimneytothekitchen, where itlefttraces uponthe floor. It hadbeenwashed over before Isawit,but stillthetraces were visible ontheArbroathflags. The stains were ofalighter *These remarkable facts areprobably connected with theconducting powers ofthedifferent surfaces. Theplaster ontheroof isnotsogood aconductor asthewall-paper, with itspigments ;andthepainted wood isprobably abetter conductor than either.—W.T. XVII.]Soundproduced bytheDischarge ofaCondenser. 239 tintthan thestone, andthegeneral appearance wasasifapail ofsomelight-colouredfluidhadbeen dashed over thefloor, so astoproducevarious distinct streams. Allalongthecourse of thedischarge, andparticularlyintheneighbourhoodofthebell- wires, there were small holes inthewallabout aninchdeep, likethemarks thatmightbemade byafingerinsoftplaster. "Most ofthewindows were shattered, and allthefragments ofglasswereontheoutside. Isupposethismust beaccounted forbytheexpansionoftheairwithin thehouse. "Thewindow-blind ofthestaircase, which wasdown atthe time,was riddled, asifwith small shot. Thediameter ofthe spacesoriddled wasabout afoot.Onminute examination I found that theholes were notsuch ascouldreadilybemade byapointed instrument orapellet. Theyw^ereangular,the clothbeingtornalongboth thewarpandthewoof. "Thehouse wasshattered fromtoptobottom. Two ofthe serving-maidsreceived apositive shock, butsoon recovered. Astrongsmell ofwhat wassupposedtobesulphur wasper- ceivedthroughoutthehouse, butparticularlyinthebed-room inwhich theeffects Idescribed before tookplace." XVII. SOUND PRODUCED BYTHEDISCHARGE OFA CONDENSER. [LETTER TOPROFESSOR TAIT.] KiLMiCHAEL, Brodick, IsleofArran, Oct. 10,1863. 302.Yesterday evening, when engagedinmeasuringthe electrostaticcapacitiesofsomespecimensofinsulated wire designedforsubmarinetelegraph cables,Ihadoccasion fre- quentlytodischarge, throughagalvanometer coil,acondenser consistingoftwoparallel platesofmetal, separated byaspace ofairabout '007 inch across, andchargedtoadifference ofpotentials equaltothat ofabout 800 Daniell's elements. Iremarked ataninstant ofdischargeasharp sound, with a very slight prolonged resonance, which seemed tocome from 240 Discharge ofaCondenser.[xvii. theinterior ofthecasecontainingthecondenser, andwhich struckmeasresemblingasound Ihadrepeatedly heard before when thecondenser hadbeenovercharged andaspark passed across itsair-space. But Iascertained that thissound was distinctlyaudible when there wasnosparkwithin thecon- denser, andthewholedischargetookplace fairly throughthe 2000yardsoffine wire, constitutingthegalvanometercoil. I arrangedthecircuit sothat theplace where thecoiitact was made toproducethedischarge wassofarfrommyearthat the initiating sparkwasinaudible;but still Ihearddistinctlythe same sound asbefore from within thecondenser. 303. Usinginstead ofthegalvanometercoileither ashort wire ormyownbody (asintakingashock fromaLeyden phial), Istillheard thesound within thecondenser. Theshock was imperceptible except byaveryfaintprick onthefingerinthe placeofthespark,and(thedirect sound ofthespark being barely,ifatall,sensible)there was stillaveryaudible sound, alwaysofthesame character, within thecondenser, which I heard atthesame instant as1feltthesparkonmy finger. MrMacfarlane could hear itdistinctly standingatadistance ofseveral yards.Wewatched forlightwithin thecondenser, butcould seenone. Ihave since ascertained thatsuddenly chargingthecondenser outofoneofthespecimensofcable chargedforthepurpose producesthesame sound within the condenser; alsothat itisproduced bysuddenly reversingthe chargeofthecondenser. 304. Thus itisdistinctly provedthataplateofairemits a sound onbeing suddenly subjectedtoelectric force, oronexpe- riencingasudden changeofelectric forcethroughit.Thisseems amost natural result when viewed inconnexion with thenew theory putforward byFaradayinhisseriesregardingthepart played byairorother dielectric inmanifestations ofelectric force. Italsotends toconfirm thehypothesisIsuggestedto account fortheremarkable observation maderegarding light- ning,w^hen youtoldmeofitabout ayear ago,andother similar observations which Ibelieve have beenreported, prov- ingasound tobeheard attheinstant ofaflash oflightning inlocalities atconsiderable distances fromanypartoftheline ofdischarge, andwhichbysome have beensupposedtode- I XVII.]Measurement oftheElectrostatic Force. 241 monstrate anerror inthecommontheoryofsound. Imay addthatMrMacfarlane tellsmehebelieves hehasheard, at theinstant ofaflash oflightning,asound asofaheavy body strikingtheearth, andimaginedatfirst thatsomethingclose tohimhadbeen struck, butheard theordinarythunder ata msible time later. . III.MEASUREMENT OFTHEELECTROSTATIC FORCE PRODUCED BYADANIELL'S BATTERY. roeeedings Royal Society, Feb. 23andApril 12,1860, orPhil.Mag. 1860, second half-year.] 805. Inapaper*'0nTransient Electric Currents," published inthePhilosophical MagazineforJune 1853, [Mathematical andPhysical Papers,Art.LXII.]Idescribed amethod for measuringdifferences ofelectricpotentialinabsolute electro- static units, which seemed tomethebestadaptedforobtaining accurate results. The "absolute electrometer" which Iex- hibited tothe British Association ontheoccasion ofits meetingatGlasgowin1855, wasconstructed forthepurpose ofputtingthismethod intopractice, and, asIthenexplained, wasadaptedtoreduce theindications ofanelectroscopic*orof atorsion electrometer toabsolute measure. 806. Thewant ofsufficientlyconstant andaccurate instru- ments ofthelatter class haslongdelayed mycarryingoutof theplansthen setforth. Efforts which Ihavemade toproduce electrometers tofulfil certain conditions ofsensibility,con- venience, andconstancy,forvariousobjects, especiallythe electrostatic measurement ofgalvanic forces, andofthe differ- ences ofpotential requiredtoproduce sparksinair,under definite conditions, andtheobservation ofnaturalatmospheric electricity, have enabled menow tomake abeginningofabso- lutedeterminations, which Ihopetobeable tocarryoutsoon inamuch more accurate manner. Inthemeantime, Ishall giveaslight descriptionofthechief instruments andprocesses *Ihave used theexpression "electroscopic electrometer," todesignate an electrometer ofwhich theindications aremerely read ofifineach instance byasingle observation, without thenecessity ofapplying anyexperimental process ofweighing, orofbalancing bytorsion, orofotherwise modifying the conditions exhibited. T.E. 16 242 MeasurementoftheElectrostatic Force[xviii. followed, and state theapproximateresultsalready obtained, asthesemaybemade thefoundation ofvariousim^jortant estimates inseveral departmentsofelectrical science. 307. The absolute electrometer alluded toabove (compare §358, below),consists ofaplanemetallic disc, insulated ina horizontalposition,with asomew^hat smallerplanemetallic disc hung centrallyoverit,fromoneendofthebeam ofabalance. Ametal caseprotectsthesuspendeddiscfrom currents ofair, andfromirregularelectric influences, allowingalightvertical rod,rigidlyconnected with thedisc atitslower end,and sus- pendedfrom thebalance above, tomove upanddownfreely, throughanaperture justv/ideenoughnottotouch it.Inthe side ofthecasethere isanotheraperture, through whichpro- jectsanelectroderigidlyconnected withthelower insulated disc. Theupperdisc iskeptinmetallic communication with thecase. 308. Inusingthisinstrument toreduce theindications ofan electroscopicortorsion electrometer toabsolute electrostatic measure, theinsulatedpartoftheelectrometer iskeptin. metallic communication with theinsulated disc, while the casesenclosingthetwoinstruments arealsokeptinmetallic communication with oneanother. Acharge,eitherpositiveor negative,iscommunicated totheinsulatedpartofthedouble apparatus.The indication ofthetested electrometer isread off,andatthesame time theforcerequiredtokeepthemove- able disc atastated distance from thefixed discbelowit,is weighed bythebalance. Thispartoftheoperation is,asI anticipated, somewhat troublesome, inconsequenceofthein- stabilityoftheequilibrium, butwith alittle care itmaybe managedVvdth considerableaccuracy. Theplanwhich Ihave hitherto followed, hasbeen tolimit theplayofthearm ofthe balance toaverysmallarc,bymeans offirmstops suitably placed,thusallowingarangeofmotion totheupperdisc throughbutasmallpartofitswhole distance from thelower. Acertainweightisputintotheoppositescale ofthebalance, andtheindications ofthesecond electrometer areobserved when the electric force isjustsufficient todraw down the upperdiscfromrestinginitsupper position,andagainwhen insufficient tokeepitdown with thebeampressedon its lowerstop. Thisoperationisrepeatedatdifferent distances, [I.].produced hyaDanielVsBattery. 243I^pithusnoconsiderable errordependingonawant ofparallel- ismbetween thediscs could remain undetected. Itmaybe remarked that theupperdisc iscarefully balanced bymeans ofsmallweightsattached toit,soastomake ithangasnearly aspossible paralleltothelower disc. Thestemcarryingitis graduatedtohundredths ofaninch(-254ofamillimetre) ; andbywatchingitthroughatelescopeatashort distance, it iseasytoobserve^ofamillimetre ofitsvertical motion. 309. Ihaverecently appliedthismethod toreduce toab- solute electrostatic measure theindications ofanelectrometer forming partofaportable apparatusfortheobservation of atmospheric electricity.Inthisinstrument(compare §263) avery lightbarofaluminium attached atright anglestothe middle ofafineplatinum wire, which isfirmlystretched be- tween theinsidecoatingsoftwoLeyden phials,oneoccupying aninvertedposition above theother, experiencesandindicates theelectrical forcewhich isthesubjectofmeasurement, and which consists ofrepulsionsincontrarydirections onitstwo ends, produced bytwo short bars ofmetal fixed onthetwo sides ofthetopofametal tube, supported bytheinside coat- ingofthelowerphial. 310.Theamount oftheelectrical force(orrather, asitshould becalled incorrect mechanicallanguage, couple)ismeasured by theangle through which theupper Leyden phial must be turaed round anaxis coincident with theline ofthewire, so astobringtheindex toamarkedposition. Anindependently insulated metalcase,bearing anelectrodeprojecting outwards, towhich thebodytobetested isapplied,surrounds theindex andrepelling bars,butleaves freeaperturesabove andbelow, forthewire topassthroughitwithout touching it;andby otheraperturesinitssides andtop,itallows themotions of theindex tobeobserved, andtheLeyden phialstobecharged ordischargedatpleasure, bymeans ofanelectrodeappliedto oneofthefixed barsdescribed above. When bymeans ofsuch anelectrode theinsidecoatingsoftheLeyden phialsarekept connected with theearth, this electrometer becomes aplain repulsion electrometer, onthesameprincipleasPeltier's, with theexception that theindex, supported byaplatinumwire instead ofonapivot,isdirectedbyelasticityoftorsion instead 16—2 244 Measurement oftheElectrostatic Force[xviii. ofbymagnetism ;andthe electrical effect tobemeasured is produced byapplyingthe electrified bodytoaconductor con- nected with afixed metal caseround theindex andrepelling bars, instead ofwith these conductors themselves. 311. This electrometer, beingofsuitablesensibilityfordirect comparisonwith theabsolute electrometeraccordingtothe processdescribed above, isnotsufficientlysensitive tomeasure directlytheelectrostatic effect ofanygalvanic batteryoffewer thantwohundred cells withmuchaccuracy. Nothavingat thetimearrangementsforworkingwith amultiple batteryof reliable character, Iused asecond torsion electrometer ofa higher degreeofsensibilityasamedium forcomparison,and determined thevalue ofitsindications bydirect reference toa Daniell'sbatteryoffrom sixtotwelve elements ingoodwork- ingorder. This electrometer, inwhich alight aluminium index, suspended bymeans ofafineglass fibre, kept constantly electrified bymeans ofalight platinumwirehanging down from itanddippingintosomesulphuricacid inthebottom of acharged Leyden jar,exhibits theeffects ofelectric forcedue toadifference ofpotentials between twohalves ofametallic ring separatelyinsulated initsneighbourhood,willbesuffici- entlydescribed inanother communication totheRoyal Society. Slight descriptionsoftrialinstruments ofthiskindhavealready beenpublishedintheTransactionsofthePontifical Academy ofBorne*, and inthesecond edition ofNichol'sCyclopcedia (article Electricity, Atmospheric), 1860(§§249, 266,above). 312. Ihopesoon tohave another electrometer onthesame general principle,butmodified from those hitherto made, so astobemore convenient foraccurate measurement interms of constant units. Inthemeantime, Ifind that,byexercising sufficient care, Icanobtain goodmeasurements bymeans of thedividedringelectrometer oftheform described inNichol's Cyclopcedia (§263,above). 313. Intheordinaryuseoftheportable electrometer, acon- siderablechargeiscommunicated totheconnected inside coat- ingsoftheLeyden phials, andthealuminium index isbrought toanaccurately markedposition bytorsion, while theinsulated *Accademia Pontificia deiNuovi Lyncei, February 1857. XVIII.] produced byaDanielVsBaUery. 245 metal casesurroundingitiskeptconnected with theearth. Thesquareroot ofthereadingofthetorsion-head thus ob- tained measures thepotential,towhich theinsidecoatingsof thephialshave been electrified.If,now, themetal case referred toisdisconnected from theearth andputincon- nexion with aconductor whosepotentialistobetested, the squarerootofthealteredreadingofthetorsion-headrequired tobringtheindex toitsmarkedpositioninthenewcircum- stances measuressimilarlythe difference between this last potential andthat oftheinsidecoatingsofthephials. Hence theexcess ofthelattersquarerootabove theformerexpresses indegree and inquality (positiveornegative)therequired potential.Thisplanhasnotonlythemerit ofindicatingthe qualityoftheelectricitytobetested, which isofgreat import- ance inatmospheric observation, but italso affords amuch higher degreeofsensibilitythan theinstrument haswhen used asaplain repulsionelectrometer;and,onaccount ofthis last- mentionedadvantage,itwasadoptedinthecomparisonswith thedividedringelectrometer. Ontheother hand, theportable electrometer wasused initsleast sensitive state, that istosay, with itsLeyden phialsconnected with the earth, when the comparisonswith theabsolute electrometer weremade. 314. Thegeneralresult oftheweighingshitherto made, is thatwhen thediscs oftheabsolute electrometer were atadis- tance of•.5080 ofacentimetre, thenumber ofdegreesoftorsion intheportableelectrometer was'20924 times thenumber of grammes' weight requiredtobalance theattractive force;and thenumber ofdegreesoftorsion was"4983 times thenumber ofgrammes' weightfound inother series ofexperimentsin which thedistance between thediscswas'762 ofacentimetre. Accordingtothelawofinverse squaresofthedistances to which theattraction between twoparalleldiscs issubject when aconstant difference ofpotentialsismaintained between them*, theforce atadistance of'254 ofacentimetre would havebeen T^V5> accordingtothe first ofthepreceding results, or,accord- ingtothesecond, ys^q^^^*^^number ofdegreesoftorsion. Themean ofthese is-^i-^,or'0777;andwemayconsider this *See§11ofElements ofMathematical Theory ofElectricity appended to thecommunication followingthisinthe"Proceedings." 246 Measurement oftheElectrostatic Force[xviii. number asrepresenting approximatelythevalue ingrammes' weightat'254 ofacentimetre distance between thediscs ofthe absolute electrometer, correspondingtoonedegreeoftorsion oftheportableelectrometer. Bycomparingtheindications of theportableelectrometer with those ofthedividedringelectro- meter, andbyevaluatingthose ofthe latter interms ofthe electromotive force ofaDaniell'sbattery chargedintheusual manner, Ifindthat284times thesquareroot ofthenumber ofdegreesoftorsion intheportableelectrometer isapproxi- matelythenumber ofcells ofaDaniell'sbattery which would produceanelectromotive force(or,which isthesamething,a difference ofpotentials) equaltothat indicated. Hence the attraction between thediscs oftheportable electrometer, ifat '254 ofacentimetre distance, andmaintained atadifference of potentials amountingtothatproduced by284cells, is'0777 ofa gramme. The effect of1000 cellswould therefore betogivea force of'965 ofagramme,since theforce ofattraction ispropor- tional tothesquareofthedifference ofpotentials between the discs. The diameter oftheopposedcircular areas between which the attraction observed tookplace, was14"88centi- metres. Itsareawastherefore 1740 square centimetres, and therefore theamount ofattractionpersquare decimetre, accord- ingtotheprecedingestimate for"254 ofacentimetre distance and1000 cells' difference ofpotential,is'554 ofagramme. Hence, withanelectromotive force ordifference ofpotentials produced by1000 cells ofDanielFsbattery,theforce ofattrac- tionwould be3o7grammes weight persquaredecimetre between discsseparatedtoadistance of1millimetre. [The force ingrammes weightisequalto"000.357 xn^,ifthearea ofeach oftheopposedsurfaces isequaltoasquare whose side is11times thedistance between them, providednbealarge number.] 315. This result differsverymuch fromanestimate Ihave madeaccordingtoWeber's comparisonofelectrostatic with elec- tro-magneticunits andmytheoretical estimate of2,500,000 Britishelectro-magneticunits fortheelectromotive force ofa singleelement ofDaniell's. Ontheother hand, itagreesto aremarkable degreeofaccuracywith direct observations made forme,during myabsence inGermany, byMrMacfarlane, in XVIII.] produced hyaDanielVsBattery. 247 themonths ofJune andJuly 1856, ontheforce ofattraction produced bythe directapplicationofaminiature Daniell's battery,ofdifferent numbers ofelements, from 93to451, appliedtothesame absolute electrometer with itsdiscs at •2006 ofacentimetre asunder. These observationsgave forcesvarying, onthewhole, very closely accordingtothe squareofthenumber ofcells used;andthemean result re- ducedaccordingtothislawto1000 cellswas1*516 grammes. Keducingthis tothedistance of1millimetre, anddividing by174, thearea insquare decimetres, wefind 3*51grammes persquaredecimetre atadistance of1millimetre. 316. Althoughtheexperiments leadingtothisresult were executed withgreatcarebyMrMacfarlane, Idelayed publish- ingitbecause ofthegreat discrepanceitpresentedfrom the estimate which Ideduced from Weber's measurement, pub- lished whilemypreparationswere inprogress.Icannot doubt itsgeneralcorrectness now,when itissodecidedlycon- firmed bytheelectrometric experiments1havejust described, which have been executedchiefly byMrJohn Smith and MrJohnFerguson, workinginmylaboratorywithmuch abilitysince themouth ofNovember. Iam stillunable to explainthediscrepance,but itmay possiblybeowingtosome miscalculation Ihavemade inmydeductions from Weber's result. Glasgow College, Jan.18,1860. [Addition, April 1870.—Fromexperimentsofthepresent date, performed byMrWilliam Leitch andMrDugald M'Kichan, with thenewAbsolute Electrometer(§364,below), itisdeduced thatwith thedifference ofpotentials produced by1000 Daniell's cells inseries, theforce ofattraction would be57grammes persquaredecimetre between discsseparated toadistance of1millimetre, instead of3*57grammesasfound in§314. Thisnewmeasurement, with Maxwell's correction ofWeber's number, which diminishes itbyabout 8percent. [Report ofBritish Associationfor1869, page438 :—Committee onElectricalStandards), seems toreduce toasnearlyasmay benothing, thediscrepancefromm}^thermo-dynamicestimate ofDecember 1851[Philosophical Magazine)referred toin§318, 248 MeasurementoftheElectrostatic Force[XVIII. below.Calculatingfrom itby§339,wefind374 forthedif- ference ofpotentials,orelectromotive force inc.g.s.absolute electrostatic measure, produced by1000 elements ofDaniell's.] Postscript, April 12,1860. 317. Ihave since found that Ihadinadvertentlymisinter- preted Weber's statement intheratio of2to1.Ihadalways, asitappearstomemost natural todo,regardedthetransference ofnegative electricityinone direction, and ofpositiveelec- tricityintheother direction, asidenticalagencies,towhich, in ourignoranceastotherealnature ofelectricity, wemayapply indiscriminatelytheoneexpressionortheother, oracombina- tion ofthetwo. Hence Ihavealways regardedacurrent of unitstrengthasacurrent inwhich thepositiveorvitreous electricityflows inonedirection attherate ofaunit ofelec- tricity perunit oftime; orthenegativeorresinouselectricity intheother direction atthesame rate;or(accordingtothe infinitely improbable hypothesisoftwo electricfluids)the vitreouselectricityflows inonedirection atanyrate lessthan aunitpersecond, andtheresinous intheoppositedirection at arateequaltotheremainder oftheunitpersecond. Ihave only recently remarked thatWeber'sexpressionsarenotonly adaptedtothehypothesisoftwo electric fluids, butthatthey alsoreckon asacurrent ofunitstrength, what Ishould have called acurrent ofstrength 2,namely,aflow ofvitreous electricityinonedirection attherate ofaunit ofvitreous electricity perunit oftime, and oftheresinous electricityin theother directionsimultaneously,attherate ofaunit of resinouselectricity perunit oftime. 318. Weber's result astotherelation between electrostatic andelectro-magnetic units, whencorrectly interpreted,Inow findwould beinperfectaccordance withmyown results given above,iftheelectromotive force ofasingleelement ofthe Daniell'sbatteryusedwere 2,140,000 Britishelectro-magnetic units instead of2,500,000, asaccordingtomythermo-dynamic estimate. This isasgood anagreementascould beex- pected when the difficulties oftheinvestigations,andthe uncertainty which still exists astothetruemeasure ofthe XVIII.] produced hyaDanielVsBattery. 249 electromotive force oftheDaniell's element areconsidered. Itmust indeed beremarked that theelectromotive force of Daniell's batteryvaries bytwoorthree ormorepercent,with variations ofthesolutions used;that itvaries alsovery sensibly withtemperature;andthat itseems also tobedependent,to some extent, oncircumstances not hitherto elucidated. A oroughexamination oftheelectromotive force ofDaniell's andother forms ofgalvanic battery,isanobjectofhighim- portance, which, itistobehoped,willsoonbeattained. Until thishasbeen done, atleast forDaniell'sbattery,theresults of thepreceding papermayberegardedashavingabout asmuch accuracyasisdesirable. 319. Imay state, therefore, inconclusion, thattheaverage electromotive forcepercelloftheDaniell's batteries which I have used, producesadifference ofpotentials amountingto •00296[correctedto-00374, April 1870,]in[c.g.s.]absolute electrostatic measure. Thisstatement isperfectly equivalentto thefollowinginmore familiar terms :— Onethousand cells ofDaniell'sbattery,with itstwopoles connected bywires withtwoparallel platesofmetal 1millimetre apart,andeach asquaredecimetre inarea, produces anelec- trical attractionequaltotheweightof3'57[correctedto5*7] grammes. XIX.—MEASUREMENT OFTHEELECTROMOTIVE FORCE REQUIRED TOPRODUCE ASPARK INAIRBETWEEN PARALLEL METAL PLATES ATDIFFERENT DISTANCES. [Proceedings Eoyal Society,Feb.23andApril 12,1860, orPliil. Mag., 1860, second half-year. ] 320.The electrometers used inthisinvestigation were the absolute electrometer andtheportableelectrometer described in mylastcommunication totheRoyal Society, andtheopera- tions were executed bythesamegentlemen, MrSmith and MrFerguson. The conductors between which thesparks passedwere twounvarnishedplatesofacondenser; ofwhich onewasmoved byamicrometer screw, givingamotion of ^-gofaninch(aboutonemillimetre) perturn,andhavingits head divided into40equal partsofcircumference. The readingsonthescrew-head could bereadilytaken totenth partsofadivision, that istosay,toabout^J^ofamillimetre onthedistance tobemeasured. Thepointfromwhich the sparkwouldpassinsuccessive trials being somewhat vari- able, and often near theedgesofthe discs, athin flat pieceofmetal, madevery slightlyconvex on itsupper surface likeanextremelyflatwatch-glass, was laidonthe lowerplate.Itwasthen found thatthespark always passed between thecrown ofthisconvexpieceofmetal andtheflat upper plate. Thecurvature oftheformer was sosmall, that thephysicalcircumstances ofitsown electrification near its crown, theoppositeelectrification oftheopposedflatsurface inthepartsnear thecrown oftheconvex, andthe electric pressureonortension intheairbetween them could not,it wassupposed,differsensibly from those between twoplane conductingsurfaces atthesame distance andmaintained at thesame difference ofpotentials. XIX.] Measurement ofElectromotive Force. 251 321.Thereadingofthescrew-head correspondingtothe positionofthemoveable discwhen touchingthemetal below, was alwaysdeterminedelectrically bymakingasuccession ofsparks pass,andapproachingthemoveable discgradually bythescrew until allappearanceofsparksceased. Contact wasthuspro- duced without anyforce ofpressurebetween thetwobodies capableofsensibly distortingtheirsupports. With these arrangementsseveral series ofexperiments were made, inwhich thedifferences ofpotentials producing sparks across different thicknesses ofairweremeasured firstbythe absolute electrometer, andafterwards bytheportabletorsion electrometer. ThefollowingTables exhibit theresults hither- toobtained :— 322.Table I.—December 13,1859. Measurements hyabsolute electrometer ofmaximum electrostaticforces*across astra- tumofairofdifferentthicknesses. Area ofeachplateofabsolute electrometer =174square centimetres. Distance betweenplatesofabsolute electrometer^ -508ofacentimetre. Lenpith of 252 MeasurementofElectromotive Force[xix. lengthofair isrequiredtoproduceasparkatshort distances than atlong.When itisconsidered thattheabsolute electri- fication ofeach oftheopposedsurfaces*depends simply on theelectromotive forceperunitlengthofthespacebetween them, or,which isthesamething,theresultant electrostatic force intheairoccupyingthatspace,itisdifficult even tocon- jecture anexplanation. Withoutattemptingtoexplain it,we areforced torecognisethe factthat athinstratum ofair is strongerthan athick oneagainstthesamedisi-uptivetension intheair,accordingtoFaraday'sview ofitscondition astrans- mittingelectric force, oragainstthesameliftingelectricpres- surefrom itsbounding surfaces, accordingtotheviews ofthe eighteenth century school, asrepresented byPoisson. The same conclusion isestablished byaseries ofexperimentswith thepreviously-described portabletorsion electrometer substi- tuted fortheabsolute electrometer, leadingtoresults shown inthefollowing Table :— 324. Table II.—January 17,1860. Measurementshyportable torsion electrometerofelectromotiveforces producing sparks across astratum,ofairofdifferentthicknesses. XIX.] requiredtoproduceaSpark.253 325. The series ofexperimentshere tabulated stopsatthe distance 18thousandths ofaninch,because itwasfound thatthe force intheelectrometercorrespondingtolonger sparksthan that,wastoostrongtobemeasured withcertainty bytheport- able electrometer, whether from theelasticityoftheplatinum wire, orfrom therigidityofitsconnexion with thealuminium index beingr liable tofailwhen more than 85°or90°oftorsion wereapplied. Sofarasitgoes,itagrees remarkablywellwith theotherexperimentsexhibited inTableI.,asisshown bythe following comparative Table, inwhich, alongwith results of actual observation extracted from Table II.,areplacedresults deduced fromTable I.byinterpolationforthesamelengthsof spark:— Table III.—Experiments ofDecember 13,1859, and January 17,1860, compared. Col. 1. 254 MeasurementofElectromotive Force[xix. Table IV.—January 21,1860. Measurementshyportable torsion electrometer ofelectromotiveforces producing sparks across astratumofairofdifferentthicknesses. I XIX.] requiredtoproduceaSpark.25i especiallyasregardsthedifferences between those shown in Table II.andthoseshown inTables IV.andV.,which, agree- ingonthewholecloselywithoneanother, fallconsiderably short oftheformer. 826.Table VI.—Summary ofresults reduced toabsolute measure. 1Col. 1. 256 MeasurementofElectromotive Force[xix. Appendix(§§327-338). 327. Inorder that thedifferentexpressions," potential," **electromotive force,""electric force," or"electrostatic force," "pressureofelectricityfrom ametallic surface balanced byair," used inthepreceding statement, maybeperfectly understood, I addthefollowing explanations anddefinitionsbelongingtothe ordinaryelements ofthemathematical theoryofelectricity:— 328. Measurementofquantities ofelectricity.—Theunitquan- tityofelectricityissuchaquantity, that,ifcollected inapoint, itwillrepelanequal quantitycollected inapointataunit distance with aforce equaltounity. 329. [Inabsolute measurements theunit distance isone centimetre; andtheunit force isthat force which, acting ona grammeofmatter duringasecond oftime, generatesavelocity ofonecentimetre persecond. Theweightofagrammeat Glasgowis981'4 ofthese units offorce. Theweightofa grammeinanypartoftheearth's surface maybeestimated with about asmuchaccuracyasitcanbewithout aspecial experimenttodetermine itfortheparticular locality, bythe following expression:— Inlatitude X,average weightofagramme =978-024 X(1+-00.5133 xsin'\)absolute kineticunits.] 330. Electricdensity.—ThistermwasintroducedbyCoulomb todesignatethequantityofelectricity perunit ofarea inany partofthesurface ofaconductor. Heshowed how tomeasure it,thoughnotinabsolute measure, byhisproof plane. 331. Resultant electricforceatanypointinaninsulating fluid [compare §65,above].—Theresultant force atanypointinair orotherinsulatingfluid intheneighbourhoodofanelectrified body,istheforce which aunit ofelectricity concentrated at thatpoint wouldexperienceifitexercised noinfluence onthe electric distributions initsneighbourhood. 332. Relation between electricdensityonthesurface ofacon- ductor^ and electricforceatpointsintheairclose toit.—Accord- ingtoapropositionofCoulomb's, requiring, however, correction, and firstcorrectly given byLaplace,theresultant force atany pointintheairclose tothesurface ofaconductor isperpendi- XTX.TX.] requiredtoproduceaSpark. 257 cular tothesurface andequalto47ryc),ifpdenotes theelectric densityofthesurface intheneighbourhood (§87,Cor.). 333. Electric pressure fromthesurface ofaconductor balanced byair.—Athin metallic shell orliquid film, asforinstance a soap-bubble,ifelectrified, experiencesarealmechanical force inadirectionperpendiculartothesurface outwards, equalin amount perunit ofarea to27r/3^ pdenoting,asbefore, the electricdensityatthepartofthesurface considered(§88). This forcemaybecalled either arepulsion (asaccordingto theviews oftheeighteenth century school)oranattraction effected bytension ofairbetween thesurface oftheconductor andtheconducting boundaryoftheairinwhich itisinsu- lated, asitwouldprobablybeconsidered tobebyFaraday ; butwhatever maybetheexplanationofthemodusoperandi by which itisproduced,itisarealmechanical force, andmaybe reckoned asinCol.5ofthepreceding Table, ingrammes weight persquarecentimetre. Inthecase ofthesoap-bubble,for instance, itseffect willbetocause aslight enlargementofthe bubble onelectrification with either vitreous orresinous elec- tricity,andacorresponding collapseonbeing perfectlydis- charged.Ineverycasewemay regarditasconstitutinga deduction from theamount ofair-pressure which thebody experiences when unelectrified. Theamount ofthisdeduction beingdifferent indifferentparts accordingtothesquareofthe electricdensity,itsresultant action onthewhole bodydisturbs itsequilibrium, and constitutes infacttheresultant ofthe electric forceexperienced bythebody. 334. Collectedformulce ofrelation between electricdensityon thesurface ofaconductor, electric diminution ofair-pressure upon it,and resultantforceintheairclose tothesurface.—Let, as before, pdenote the first ofthese three elements,letpdenote thesecond reckoned inunits offorceperunit ofarea,and let Rdenote thethird. Thenwehave R= 47r/), 335. Electricpotential [differenceofpotentials being what, afterGermanusage,isstillsometimes called ''electromotive force."{Addition, April 1870.)]—Theamount ofworkrequired T.E. 17 258 Measurement ofElectromotive Force [xix. tomove aunit ofelectricity againstelectricrepulsion fromany- onepositiontoanyotherposition,isequaltotheexcess ofthe electricpotentialofthesecondposition above the electric potentialofthe firstposition. Cor. 1.The electricpotentialatallpointsclose tothesurface ofanelectrified metallic body hasonevalue, since anelectri- fiedpoint, possessingsosmall aquantityofelectricityasnot sensiblytoinfluence theelectrification ofthemetallic surface, would,ifheld near thesurface inany locality, experiencea force perpendiculartothesurface initsneighbourhood. Cor. 2.The electricpotential throughouttheinterior ofa hollow metallicbody,electrified inanywaybyexternal influ- ence, or,ifinsulated, electrified either byinfluence orbycom- munication ofelectricitytoit,isconstant, since there isno electric force intheinterior insuch circumstances. [Itiseasily shown bymathematical investigation,that the electric forceexperienced byanelectricpoint containing an infinitelysmallquantityofelectricity, whenplaced anywhere inTtEe neighbourhoodofahollow electrified metallic shell, graduallydiminishes tonothingiftheelectricpointbemoved graduallyfrom the exteriorthroughasmallapertureinthe shell intotheinterior. Hence theonevalue ofthepotential close tothesurface outside, mentioned iii_Cor. 1^isegaaLlO theconstant valuethroughouttheinterior mentioned inCor.2.] 336.Interpretation ofmeasurementhyelectrometer. -^^werj kind ofelectrometer consists ofacageorcasecontainingamove- ableandafixed conductor, ofwhich oneatleast isinsulated and putinmetallic communication, bywhat Ishall calltheprin- cipalelectrodepassing through anapertureinthecaseorcage, with theconductor whoseelectricityistobetested. Inevery properlyconstructed electrometer, the electric forceexperi- enced bythemoveablepartinagiven positioncannot be electricallyinfluencedexcept bychangingthe difference of potentialsbetween theprincipalelectrode andtheuninsulated conductor orconducting systemintheelectrometer. Even thebest ofordinary electrometers hitherto constructed donot fulfil this condition, astheinner surface oftheglassofwhich thewhole orpartoftheenclosingcase isgenerally made, is liable tobecome electrified, andinevitablydoesbecome so ] requiredtoproduceaSpark. 259 when anyvery highelectrification isdesignedlyoracciden- tally introduced, even foraveryshort time;theconsequence ofwhich isthatthemoving bodywillgenerallynotreturn to itszeroposition when theprincipalelectrode isperfectlydis- insulated. Faraday longagoshowed how toobviate thisradi- caldefect bycoatingtheinterior oftheglass casewithafine network oftinfoil;and itseemsstrangethateven atthepre- sentdayelectrometers forscientific research, as,forinstance, fortheinvestigationofatmospheric electricity,should becon- structed with sobadandobvious adefect uncuredbysosimple andperfectaremedy. When itisdesired toleave theinterior oftheelectrometer asmuchlightaspossible, and toallow it tobeclearlyseen from anyexternalposition with aslittle embarrassment aspossible,acagemade likeabird'scage, with anextremelyfinewireonametal frame, inside theglassshade used toprotecttheinstrument from currents ofair, etc.,may besubstituted withadvantageforthe tinfoil networkliningof theglass.Itappears, therefore, thataproperly constructed electrometer isaninstrument formeasuring, bymeans ofthe motions ofamoveable conductor, thedifference ofpotentials oftwoconducting systemsinsulated from oneanother, ofone ofwhich thecase orcageoftheapparatus formspart.Itmay beremarked inpassing,that itissometimes convenient in special researches toinsulate thecase orcageoftheapparatus, andallow ittoacquireapotential differingfrom that ofthe earth, andthat then, asalways,thesubjectofmeasurement is thedifference ofpotentialsbetween theprincipalelectrode and thecase orcage,while intheordinaryuseoftheinstrument thepotentialofthe latter isthesame asthat oftheearth. Hence wemayregardtheelectrometer merelyasaninstrument formeasuringdifferences ofpotential between twoconducting systems mutuallyinsulated;andtheobjecttobeaimed atin perfecting anykind ofelectrometer (moreorlesssensitive asit may be,accordingtothesubjectsofinvestigationforwhich it istobeused), is,thataccurate evaluations inabsolute measure, ofdiffer^ences ofpotential^ mayheimmediatelyderivable fromits indications. 837.Relation between electrostatic forceandvariationofelectric potential.—§335,otherwise stated, isequivalenttothis :—The 17—2 2G0 Measurement ofElectromotive Force [xix. average componentelectrostatic force inthestraightline of airbetween twopointsintheneighbourhoodofanelectrified bodyisequaltotheir difference ofpotentialsdivided bytheir distance. Inother words, the rate ofvariation ofelectric potential perunit oflengthinanydirection isequaltothe componentoftheelectrostatic force inthat direction. Since theaverageelectrostatic force inthelinejoiningtwopointsat which thevalues ofthepotentialareequalisnothing,the direction oftheresultant electrostatic force atanypointmust beperpendiculartotheequipotentialsurfacepassing through thatpoint;orthelines offorce(whicharegenerally curves) cuttheseries ofequipotentialsurfaces atright angles. The rate ofvariation ofpotential perunit oflength alongalineof force isthereforeequaltotheelectrostatic force atanypoint. 338. Stratumofairbetween twoparallelornearly parallel planeorcurved metallicsurfaces maintained atdifferent poten- tials.—Letadenote thedistance between themetallic surfaces oneach sideofthestratum ofairatanypart,andVthediffer- ence ofpotentials.Itiseasily shown thattheresultant elec- trostatic force issensiblyconstant throughthewhole distance, from theonesurface totheother; andbeinginadirection V sensibly perpendiculartoeach,itmust(§337)beequalto—.a Hence(§332)theelectricdensity oneach oftheopposedsur- V faces isequalto7— .This isGreen'stheoryoftheLeyden phial. 339. Absolute Electrometer. —Asaparticularcaseof§338, letthediscs beplane andparallel:and letthedistance be- tween them besmall incomparisonwith their diameters, or with thedistance ofanypartofeither from anyconductor differingfrom itinpotential. The electricdensitywillbe uniform over thewhole ofeach oftheopposedsurfaces and V . . . equalto7— ,being positiveononeandnegativeontheother; andinallotherpartsofthesurface ofeach theelectrification willbecomparatively insensible. Hence theforce ofattraction between themperunit ofarea(§§333and334)willbe^— 5;oTra XIX.] requiredtoproduceaSpark.261 ifAdenote thearea ofeither oftheopposed surfaces, the whole force ofattraction between them istherefore A^—^. Hence,iftheobserved forcebeequaltotheweightof-m;grammes atGlassjow, wehave mV^xw^^A- —i,87ra , , ^ ^^ /981-4 XSttXw; andtherefore k=aa/ -|. Addition, dated Apkil 12,1860. 340. Experimentsonpreciselythesameplanasthose of Table I.December 13,havebeenrepeated bythesametwoex- perimenters,with different distances from75to1*5ofacenti- metre between theplatesoftheabsolute electrometer, and results havebeen obtainedconfirmingthegeneralcharacter of thoseshown intheprecedingTables. The absolute evaluations derived from these later series must bemore accurate than those deduced above from the singleseries ofDecember 13,when thedistance between the platesintheabsolute electrometer wasonly'5ofacentimetre. Itherefore, bypermission,addthefollowing Table ofabsolute determinations :— 262 MeasurementofElectromotive Force.[xix. electrostatic forceprecedingaspark,atthegreater than atthe smaller distances. Itseems mostprobable that atstillgreater distances the electrostatic force willbefound tobesensibly constant, asitwascertainly expectedtobeatalldistances. Thelimitingvalue towhich theresults shown inthe last Table seem topoint must besomethingnotmuch lessthan 130. Thiscorrespondstoapressureof68grammes weight per squaredecimetre. Wemaytherefore conclude that theordi- nary atmospheric pressureof103,200 grammes persquaredeci- metre, iselectricallyrelievedbythesubtraction ofnotmore than 68,ontwovery slightly convex metallic surfaces, before theairbetween them iscracked andaspark passes, provided thedistance between them isnot lessthanJofacentimetre. Bytakingintoaccount theresult ofmypreceding communica- tion totheRoyal Society, wemayalsoconclude thataDaniell's batteryof5510 elements canproduceaspark between two slightlyconvex metallic surfaces at^ofacentimetre asunder inordinary atmosphericair. XX.ELECTEOMETEES ANDELECTEOSTATIC MEASUEEMENTS. 340'from British Association ReportofGlasgow 1855Meeting, §§341—389 from BeportofDundee 1867Meeting, being part ofReport ofCommittee on Standards ofElectricalResistance.]II^B840'. Inthiscommunication three instruments were de- ^cribed and exhibited totheSection :the first astandard electrometer, designedtomeasure, byaprocessofweighing the mutual attraction oftwoconducting discs, thedifference of electricalpotentialbetween twobodies withwhichtheyare connected, aninstrument which willbeuseful fordetermining theelectromotive force ofagalvanic batteryinelectrostatic measure, and forgraduating electroscopicinstruments soas toconvert their scale indications into absolute measure;the second anelectroscopic electrometer, which maybeused for indicatingelectricalpotentialsinabsolute measure, inordinary experiments, and, probablywithgreat advantage,inobser- vations ofatmospheric electricity ;andthethird, forwhich a scientific friend hassuggestedthename ofElectroplatymeter, aninstrument which maybeappliedeither tomeasure the capacitiesofconductingsurfaces forholding chargesofelec- tricity,ortodetermine theelectric inductivecapacitiesofinsu- latingmedia. 341.Anelectrometer isaninstrument formeasuringdiffer- ences ofelectricpotentialbetween twoconductorsthrough effects ofelectrostatic force, and isdistinguishedfrom thegal- vanometer, which, ofwhateverspecies, measures differences of electricpotentials through electromagneticeffects ofelectric currentsproduced bythem. When anelectrometermerely indicates theexistence ofelectricpotential,withoutmeasuring itsamount,itiscommonlycalled anelectroscope ;butthe name electrometer isproperly applied whengreaterorless degreesofdifference areindicated onanyscale ofreckoning, ^-—1 264OnElectrometers and Electrostatic Measurements, [xx. ifapproximately constant, evenduringasingleseries ofexperi- ments. The firststeptowards accurateelectrometryinevery case istodeduce from thescale-readings, numbers which shall beinsimple proportiontothedifference ofpotentialstobe determined. Thenextand laststepistoassignthe corre- spondingvalues inabsolute electrostatic measure. Thus,when foranyelectrometer the firststephasbeen taken,itremains onlytodetermine thesingleconstant coefficient bywhich the numbers, deduced from itsindications assimply proportional todifferences ofpotential, must bemultipliedtogivediffer- ences ofpotentialinabsolute electrostatic measure. This co- efficient willbecalled, forbrevity,theabsolute coefficient of theinstrument inquestion. 342. Thus, forexample,thegold-leafelectrometer indicates differences ofpotential between thegoldleaves andthe solid wallsenclosingtheair-spaceinwhichthey move. Ifthis solidbeofother thansufficiently perfect conducting material, ofwoodandglass,orofmetal andglass,forinstance, asinthe instrumentordinarily made, itisquite imperfect andindefinite initsindications, and isnotworthyofbeing even called an electroscope,asitmayexhibit adivergence when thedifference ofpotentialswhich theoperatordesires todiscover isabsolutely zero. Itisinterestingtoremark(§336) thatFaradayfirst remedied this defectbycoatingtheinterior oftheglasscase withtinfoil, cutawaytoleaveapertures proper and sufficient toallow indications tobeseen,butnotenoughtocause these indications todiffersensibly fromwhattheywould beifthe conducting envelope werecompletelyclosed aroundit;and thatnot tillalongtime after didanyother naturalist, mathe- matician, orinstrument-maker seem tohave noticed thedefect, oreven tohaveunconsciouslyremedied it. 343. Electrometers maybeclassified ingenera andspecies accordingtotheshape andkinematic relations oftheirparts ; butasinplantsandanimals aperfect continuityofinterme- diatespecieshasbeenimagined between therudimentary plantandthemostperfect animal, soinelectrometers we may actuallyconstructspecies havingintermediatequalities continuous between themostwidelydifferentgenera. But, notwithstanding,some such classification asthefollowingis I ]OnElectrometers and Electrostatic Measurements. 26 i convenient with reference totheseveral instruments commonly inuseandnow tobedescribed :— I.Eepulsionelectrometers. Pair ofdivergingstraws asusedbyBeccaria, Volta, and others, lastcentury. Pair ofdiverging goldleaves(Bennet). Peltier's electrometer. Delmann's electrometer. Old station-electrometer, described inlecture tothe Royal Institution, May1860 [§§274-275, above]; alsoinNichol'sCyclopaedia,article" Electricity, Atmo- spheric" (edition, 1860) [§263, above], and inDr Everett's paperof1867,"OnAtmospheric Electricity" (Philosophical Transactions). Symmetricalelectrometers. Bohnenberger'selectrometer. Divided-ringelectrometers. III. Attracted discelectrometers. »Absolute electrometer. Long-rangeelectrometer. Portable electrometer. Spring-standardelectrometer. 344. Class I.issufficientlyillustrated bytheexamples referred to;and itisnotnecessarytoexplain anyofthese instrumentsminutelyatpresent,asthey are, forthepresent atallevents, superseded bythedivided-ring electrometer and electrometers ofthethird class. There areatpresent onlytwoknownspeciesofthesecond class;but itisintended toinclude allelectrometers inwhich asymmetricalfield ofelectric force isconstituted bytwo symmetricalfixed conductors atdifferent electricpotentials, andinwhich theindication oftheforce isproduced bymeans ofanelectrified bodymoveablesymmetricallyineither direction from amiddlepositioninthis field. This definition isobviously fulfilled byBohnenberger's well-known instrument*. *Asingle gold leafhanging between theupper ends oftwoequal andsimilar dry^piles standing vertically onahorizontal plate ofmetal, onewith its positive andtheother with itsnegative poleup. 2G6 071Electrometers and Electrostatio Measurements,[xx. 845.Myfirstpublished descriptionofadivided-ringelectro- meter istobefound intheMemoirsoftheRoman Academy of Sciences* forFebruary 1857; butsince thattime Ihavemade great improvementsintheinstrument—first,byapplying a lightmirror toindicate deflections ofthemoving body; next, bysubstitutingfor tvt^ohalfringsfourquadrants, andconse- quentlyforanelectrified body projectingononesideonlyof the axis,anelectrified body projecting symmetrically onthe two sides, andmoveable round anaxis;andlastly, byvarious mechanical improvements, andbytheaddition ofasimple gaugetotestthe electrification ofthemoveablebody,andof areplenishertoraise this electrification toanydesireddegree. 346. Intheaccompanying drawings,Plate I.fig.1repre- sents thefront elevation oftheinstrument, ofwhich thechief bulk consists ofajarofwhiteglass (flint) supported onthree legsbyabrassmounting, cemented round theoutside ofits mouth, which isclosed byaplateofstout sheet-brass, with alantern-shapedcoverstandingover awideapertureinits centre. Forbrevity,inwhat follows these threepartswillbe called thejar,themain cover, andthelantern. Fig.5representsthequadrantsasseenfrom above;they areshown inelevation ataand 6,fig. 1,andinsection atcand d,fig.2.Theyconsist offourquartersofaflatcircular box ofbrass, with circularaperturesinthecentres ofitstopand bottom. Theirpositionintheinstrument isshown infigs. 1,2,and 6.Each ofthefourquadrantsissupported ona glassstempassing downwardsthroughaslotinthemain cover ofthejar,from abrass mountingontheoutside ofit,and admits ofbeing drawn outwards foraspaceofabout 1centi- metre(Iofaninch) from thepositions theyoccupy when the instrument isinuse,which areapproximatelythose shown in thedrawings.Three ofthem aresecured intheirproper posi- tionsbynuts{e,e,e)ontheoutside ofthechief flatlidofthe jarshown infig.4.Theupper endofthestem, canyingthe fourth, isattached toabrasspiece (/,fig.6)restingonthree shortlegsontheupperside ofthemain cover, twoofthese legsbeing guided byastraight V-grooveat{g)togivethem *Accademia Pontificia deiNiiovi Lincei. I.]OnElectrometers and Electrostatio Measurements. 267 freedom tomove inastraightlineinwards oroutwards, andto prevent anyother motion. This brasspieceispressedout- wards anddownwards byaproperly arranged spring (/?,),and iskeptfromslidingoutbyamicrometer-screw(^)turningin afixed nut. Thissimplekinematic arrangement gives great steadiness tothefourth quadrant when thescrew isturned inwards oroutwards, andthen leftinanyposition ;and atthe same timeproducesbut little frictionagainsttheslidingin either direction. Theopposite quadrantsareconnected intwo pairsbywires, asshown infig.5;andtwostout vertical wires {I,m),called thechief electrodes, passing throughholes inthe roof ofthelantern, arefirmly supported bylong perforated vulcanite columnspassing throughthose holes, andserve to connect thepairsofquadrantswith theexternal conductors whose difference ofpotentialsistobetested.Springs (w,6)at thelower ends ofthese columns, shown infigs.1and 2,main- tain metallic contact between thechief electrodes andthe uppersides oftwocontiguous quadrants (aandh)when the lantern issetdown initsproper position, butallow thelantern toberemoved, carryingthechief electrodes withit,andtobe replacedatpleasure withoutdisturbingthequadrants. The lantern alsocarries aninsulatedcharging-rod (p),ortemporary electrode, forchargingtheinnercoatingofthejar(§351)toa smalldegree,tobeincreased bythereplenisher (§852), or,it may be,formaking special experimentsinwhich thepotential oftheinteriorcoatingofthejaristobemeasured byaseparate electrometer, orkeptatanystated amount ofdifference from that oftheoutercoating. When notinusethistemporary electrode issecured inapositioninwhich itisdisconnected from theinnercoating. 347. Themain coversupportsaglasscolumn{q,fig.2) projecting vertically upwards throughitscentralaperture, totheupper end ofwhich isattached abrasspiece (r),which bears above itafixedattractingdisc(s),tobedescribed later (§353) ;andprojecting down from itafixedplate bearing thesilk-fibresuspensionofthemirrorit),needle(?i), etc.,seen infigs.1and 2,and fixed guardtubes{v,lu),tobedescribed presently. Tothemain cover also isattached thecircular level(fig. 6),which isadjustedtoindicate thepositionofthe 268 OjiElectrometers and Electrostatic Measurements, [xx. instrument inwhich thequadrantsarelevel, andtheguard- tubesjustmentioned vertical. Itslower surface which rests onthecover isslightly rounded, likeaconvex lens, soasto admit ofaslightfurtheradjustment (seeendof§348,Addition) byvaryingtherelativepressureofthethree screws bywhich it isfastened down tothecover. 348.Themoveable conductor oftheinstrument consists ofa stiffplatinumwire(x),about 8centimetres(3|inches) long, withtheneedlerigidlyattached inaplane perpendiculartoit, andconnected withsulphuricacid inthebottom ofthejarby afineplatinumwirehanging down from itslower endandkept stretched byaplatinum weight under thelevel oftheliquid. Theupperend ofthe stiffplatinumwire issupported bya singlesilk-fibre sothat ithangs downvertically. Themirror isattached toitjustbelow itsupperend. Thus themirror, theneedle, andthe stiffplatinum stem constitute arigidbody having very perfect freedom tomove round avertical axis (the line ofthebearing fibre), andyetpractically preventedfrom anyother motion intheregularuseoftheinstrument bythe \veightofitsownmassandthat oftheloosepieceofplatinum hangingfrom itbelow thesurface oftheliquidinthejar.A verysmall magnetisattached totheneedle, which, bystrong magnetsfixed outside thejar,isdirected tooneposition,about which itoscillates after itisturnedthrough anyangleround thevertical axis,andthen lefttoitself. Theexternal magnets are soplacedthatwhen there ismagnetic equilibriumthe needle isinthesymmetrical position shown infigs.5and6 with reference tothequadrants*. [Addition, April 1870.—The success oftheexperimentsre- ferred tointhefootnote hasledtotheadoptionofthebifilar suspensioninalltheQuadrant Electrometers nowmade. Itis representedinthemargin. The stiffplatinumwirewhich carries themirror andneedle hasacrosspieceatitsupper end, to which areattached thelower ends ofthetwosuspendingsilk fibres;theother endsbeingwound uponthetwopins c,d,which maybeturned intheir sockets byasquare-pointed key,to *RecentlyIhavemade experiments onabifilar suspension withaview to superseding themagnetic adjustment, which promisewell. I ]OnElectrometers and Electrostatic Measurements. 269 equalizethetensions ofthe fibres, andmake theneedlehang midway between theupperandunder surfaces ofthequadrants. Thepins c,d,arepivotedinblocks carried bysprings e,f,to allowthem tobeshiftedhorizontally when adjustingthepositionofthe pointsofsuspension. Thescrews a,b, which traverse these blocks, have their points bearing againstthefixedplate behind, sothatwhen aorhisturned inthedirection ofthehands ofa watch, theneighbouring pointofsus- pensionisbrought forward, andcon- versely. The needle maythusbe made toturnthroughanangle,till itliesinthesymmetrical position representedinfig. 5,PlateI.,when allelectrical disturbance hasbeen guarded against byconnectingthe quadrantswith theinside andout- side ofthejar.The conicalpinh passesbetween thetwosprings and screws into theplate behind; by screwingitinwards thepointsof suspensionaremade torecede from each otherlaterally, and thesensibilityoftheneedle toadeflecting coupleisdiminished, andconversely. Themethod employedtotestthesymmetryofthesuspen- sion issuggested bytheconsideration that ifthetension be equallydistributed between thetwo fibres, thesensibilityof theneedle tothesamedeflecting couplewillbelessthan if thewhole orthegreater partoftheweight weresupported byone fibre;also, thesensibility beingaminimum, asmall deviation from theconditions which make itsowillproduce theleastchangeofsensibility, bytheknownpropertyofa maximum orminimum. Totestwhether these conditions are attained, raise firstonesideoftheinstrument alittle(oneturn ofthefoot-screw onthat side isusually sufficient), andthen produce anequaldeviation intheoppositedirection from the position markedbytheattached level(§347) ;and ineach 270OnElectrometers and Electrostatic Measurements, [xx. positionoftheinstrument observe thedeflection oftheimage onthescaleproduced bysome constant difference ofpotentials, asthatbetween thetwopolesofaDaniell's cell. This deflection oughttobeverynearly equalinthethreepositions, butexactly equalinthetwodisturbedpositions, andsomewhatgreaterin these than inthemiddle orlevelposition. When theinstru- ment isfaroutofadjustment,thedeviation willbegreaterin oneofthedisturbedpositionsand lessintheother than inthe middleposition. When itisbutslightlyoutofadjustment, thedeflections inthedisturbedpositions maybothsomewhat exceed that inthemiddleposition,buttodifferentdegrees. Anapproximationtosymmetrythus faratleast should be obtained bymerely turningthepins (c,d)intheir sockets as already directed, throughtheminutest anglessensible tothe operator,withoutalteringtheadjustmentofthespirit-level on thecover. When thathasbeen done, thelevelonthecover oughttobeadjusted (§347)bysuccessive trials toindicate thepositionoftheinstrument such thatwhenequallydis- turbed from itinopposite directions, thedeflections obtained areequallyinexcess ofthedeflection obtained intheindicated position.] 349. Theneedle(u)isofthin sheet aluminium cuttothe shapeseen infigs.5and6;theverythinnest sheet thatgives therequisitestiffness beingchosen. Itsarea is4^squarecenti- metres, andweight '07ofagramme.Ifthefourquadrants areinaperfectly symmetrical positionroundit,and iftheyare keptatoneelectricpotential byametallic arcconnectingthe chief electrodes outside, theneedle maybestronglyelectrified without beingdisturbed from itspositionofmagnetic equili- brium; but ifitiselectrified, and iftheexternal electrodes bedisconnected, andanydifference ofpotentialsestablished between them, theneedle willclearly experienceacouple turningitround itsvertical axis, itstwoends beingdriven from thepositive quadrants towards thenegative,ifitisitself positivelyelectrified. Itiskept positiverather than negative intheordinaryuseoftheinstrument, because Ifind that when aconductor withsharp edgesorpointsissurrounded byanotherpresenting everywhere asmooth surface, amuch greaterdifference ofpotentialscanbeestablished between I ]OnElectrometers and Electrostatic Measurements. 271 them, withoutproducing disruptive discharge,ifthepointsand edgesarepositivethan iftheyarenegative. 350. Themirror{t)serves toindicate, byreflectingarayof h'ghtfrom alamp,small angularmotions oftheneedle round thevertical axis. Itisavery light, concave, silveredglass mirror, beingofonly8millimetres(^ofaninch) diameter, and 22milligrammes (Jofagrain) weight.Ihad formany years experienced great difficultyingettingsuitable mirrors formy form ofmirrorgalvanometer;buttheyarenowsuppliedin verygreat perfection byMrBecker, ofMessrs Elliott Brothers, London. [Addition, May1870.—Ihave notsucceeded in getting more oftheselightgroundconcave mirrorsgiving good images,after afewsupplied byMrBecker atthetimewhen thereport was written. Thelightest groundmirrors that MrBecker canguaranteetogivegood images, weigh -^^of agramme (-^^ofagrain).These answer wellenoughfor theelectrometers, because thealuminium needleweighing -^ ofagramme (1^^ grain),andbeingofmuchgreaterlinear dimensions, itsmoment ofinertia isnotlargelyincreased by theaddition ofamirror ofthatweight ;andtheyarepreferred forthispurposetotheexquisite lightmirrorssupplied by MrWhite, asbeing stronger and less liable towarpinbeing mounted. But forgalvanometers,andespecially telegraph- signal galvanometers,itisimportantthat themirrors bethe very lightest possible. Theonlymirrors suitable forthis purpose which Icannow obtain aresupplied byMrWhite. They give very perfect images,andweigh -glyofagramme (foofagrain) without themagnets, andgijofagramme with themagnetsattached. MrWhiteproduces themby cuttingoutandsilveringalargenumber ofcircles ofthe thinnestmicroscope glass, attachingthemagnets (fouronthe back ofeachmirror), andfinally testingfortheimage. Outof fifty tried, about tenorfifteen aregenerallyfoundsatisfactory.Amirror may giveagood imagebefore themagnetsare attached, andbecome warpedoutofshapeandgiveabad imageafter themagnets have been cemented toit.]The focus forparallel raysisabout 50centimetres(20inches) from themirror, andthus theraysofthelamp placedata distance of1metre(or40inches)arebroughttoafocus at 1 272OnElectrometers and Electrostatic Measurements,[xx. thesame distance. Thelampisusually placedclose behind the vertical screen alittle below orabove thenormal line ofthemirror, andtheimageisthrown onagraduatedscale extending horizontally above orbelow theapertureinthescreen through which thelampsends itslight. When themirror isatitsmagneticzeroposition,thelampissoplacedthat its image is,asnearlyasmay be,inaverticalplane with itself, andnotmore thananinchabove orbelow itslevel, sothat there isaslittleobliquityaspossibleinthe reflection, andthe linetraversed bytheimageonthescreenduringthedeflection is,asnearlyasmay be,straight. The distance ofthelamp andscreen from themirror isadjustedsoastogiveasperfect animageaspossibleofafinewirewhich isstretchedvertically intheplaneofthescreen across theaperture through which thelampshines onthemirror;andwithMrBecker's mirrors, aswithMrWhite's selectedgalvanometer mirrors, Ifind iteasytoread thehorizontal motions ofthedarkimageto anaccuracyofthetenth ofamillimetre. Intheordinary useoftheinstrument awhitepaper screen, printed from a copper-plate,divided tofortieths ofaninch, isemployed, and thereadingsarecommonly taken toabout aquarterofascale- division; butwith alittlepractice theymay,when somuch accuracyisdesired, beread with considerableaccuracytothe tenth ofascale-division.Formerlyaslitinfront ofthelamp wasused, butthewiregivingadark lineinthemiddle ofthe imageoftheflame isavery great improvement,first intro- duced byDr.Everett(inconsequenceofasuggestion made byProfessor P.G.Tait) inhisexperimentsontheelasticityof solidsmade intheNaturalPhilosophy LaboratoryofGlasgow University*. 851. Thechargeoftheneedle remainssensiblyconstant from hour tohour, andeven from daytoday,invirtue of thearrangement bywhich itiskeptincommunication with sulphuricacid inthebottom ofthejar,theoutside ofthe *ADrummond light placed about 70centimetres from themirror gives animage, onascreen about 3metres distant, brilliant enough forlecture- illustrations, andwith sufficient definition toallow accurate readings ofthe positions onascalemarked bytheimage ofafine vertical wire infront of thelight. I ]OnElectrometers and Electrostatic Measurements. 273 jarbeingcoated with tinfoil andconnected with theearth, so that itisinrealityaLeyden jar.Thewhole outside ofthe jar,evenwhere notcoated with tinfoil,isintheordinaryuse oftheinstrument, especiallyinourmoist climate, kept virtually atonepotential throughconductionalongitssurface. This potentialisgenerally, byconnectingwires ormetalpieces, kept thesame asthat ofthebrasslegsandframework oftheinstru- ment. Topreventdisturbance incase ofstronglyelectrified bodies being broughtintotheneighbourhoodoftheinstrument, awire iseither wrappedround thejarfromtoptobottom, ora cageornetwork ofwire, oranyconvenient metal case,isplaced round it;butthisoughttobeeasilyremoved oropenedatany time topermittheinterior tobeseen.When theinstrument islefttoitself fromdaytodayinordinary use,theneedle, connected with theinnercoatingofthejarasjust described, loses, ofcourse, unlessreplenished, somethingofitscharge; butnotingeneralmore thanJpercent, perday,when thejar isofflint-glass made inGlasgow. Ontryingsimilarjarsof green glassIfound thattheylosttheir charge morerapidly perhour than thewhiteglass jarspermonth. Ihave occa- sionally,butvery rarely,found whiteglass jarstobeasdefec- tive asthose green ones,and itispossiblethat thedefect I found inthegreen jarsmayhavebeenanaccident tothejars tested, andnotanessentialpropertyofthatkind ofglass. 852. Ihaverecently made theveryuseful addition ofa replenishertorestoreelectricitytothejarfrom time totime whenrequired.Itconsists of(1)aturningvertical shaft of vulcanitebearing twometalpiecescalled carriers(h, b,figs. 17and18) ;(2)twosprings {d,d,figs. 16and18),con- nectedbyametallicarc,makingcontact withthecarriers once everyhalfturn ofthe shaft, andtherefore called connectors; and(3)twoinductors(a,a)withreceiving springs (c,c)attached tothem, which make contact with thecarriers onceeveryhalf turn, shortlybefore theconnectingcontacts aremade. The inductors(a,a,figs.16and18)arepiecesofsheet metal bent into circularcylindrical shapesofabout 120° each;theyare placed soastodeviate inthemanner shown inthedrawing frompartsofacylindricalsurface coaxial with theturning- shaft, leaving gapsofabout 60°oneach side. Thediameter of T.E. 18 I 274OnElectrometers and Electrostatic Measurements, [xx. thiscylindricalsurface isabout 15millimeters (about fofan inch). The carriers{h,h,figs.17and18)arealso ofsheet metal bent tocylindrical surfaces, butnotexactlycircular cylinders; and are soplacedonthebearingvulcanite shaft thateach isrubbed bythecontactspringsover averyshort space, about 1millimeterbeyonditsforemostedge,when turned intheproperdirection forreplenishing. Thereceiving springs (c, c,figs.17and18)make their contacts with each carrier immediatelyafter ithasgotfairly under cover, asitwere, of theinductor. Each carrier subtends anangleofabout 60°at theaxis oftheturning-shaft. Theconnectingcontacts are completed justbefore the carriers commenceemergingfrom being under cover oftheinductors. The carriers maybesaid tobeunder cover oftheinductors when theyarewithin the angleof120^subtended bytheinductors oneach side ofthe axis. One oftheinductors isinmetallic communication with theoutsidecoatingofthejar,theother withtheinside.Figs. 16,17,and18illustrate sufficientlytheshapeofcarriers and thesuccession ofthecontacts. Thearrow-head indicates the direction toturn forreplenishing. When itisdesired to diminish thecharge,thereplenisheristurned backwards. A smallcharge having beengiventothejarfromanindependent source, thereplenisher when turned forwards increases thedif- ference ofpotentialsbetween thetwoinductors andtherefore between thetwocoatingsofthejarconnected withthem bya constantpercentage perhalf turn, unless itisraised tosohigh adegreeastobreak down theair-insulation bydisruptivedis- charge. The electric action isexplained simplythus:—The carriers, when connected bytheconnecting springs,receiveop- posite charges byinduction, ofwhichthey deposit large propor- tions thenext timethey touchreceiving springs. Thus, for example,ifthejarbecharged positively,thecarrieremerging from theinductor connected with theinner coatingcarries a negative charge round tothereceiving springconnected with the outsidecoating,while theother carrier, emergingfromtheinduc- torconnected withtheoutsidecoating,carries apositive charge round tothereceiving spring connected with theinsidecoating. Ifthecarriers arenotsufficientlywellunder cover ofthein- ductorsduringboth thereceivingcontacts andtheconnecting XX.] OnElectrometers and Electrostatic Measurements. 275 contacts torender thecharges which they acquire byinduction duringtheconnectingcontactsgreaterthan thatwhichthey carryaway withthem from thereceiving contacts, therotation, even intheproperdirection forreplenishing,doesnotincrease, but,onthecontrary,diminishes thechargeofthejar.The deviations oftheinductors from thecircularcylinder,referred toabove, have been adoptedtogive greater security against this failure. Asteelpivotfixed tothetopofthevulcanite shaft, andpassing throughthemain cover, carries asmall milled head(?/, fig.1)above, ontheoutside, which isspun rapidlyround ineither direction bythefinger,andthus in lessthan aminute asmallchargeinthejarmaybedoubled. Thediminution ofthecharge, when theinstrument isleftto itself fortwenty-four hours,issometimes imperceptible;but when anyloss isdiscovered tohavetakenplace,even iftothe extent of10per cent., afewmoments' useofthereplenislier suffices torestoreit,andtoadjustitwithminute accuracyto therequired degree byaidofthegaugetobedescribed pre- sently. Theprincipleofthe" repJenisher" isidentical with that ofthe"doubler" ofBennet. Intheessentials ofitscon- struction itisthesame asYarley's improvedform ofNichol- son's" revolvingdoubler." 353. Thegaugeconsists ofanelectrometer ofClass III. Themoveable attracted disc isasquare portionofapieceof verythin sheet aluminium oftheshape shown atainfig.4. Itissupportedonastretched platinumwirepassing through twoholes inthesheet, andoveraverysmallprojecting ridge ofbent sheet aluminiumplacedinthemanner shown inthe magnified drawing, fig.3.Theends ofthiswire arepassed throughholes incurvedsprings, shown infig. 4,andarebent round them soastogiveasecure attachment without solder, andwithout touchingthestraightstretchedpartofthewire. Theends oftheplatinumwire(y5, y8)areattached bycement tothesprings, merelytoprevent them frombecoming loose, carebeingtaken that thecement does notpreventmetallic contact between somepartoftheplatinumwireandone orboth ofthebrasssprings.Ihaveconstantlyfound fine platinum wirerendered brittle byordinarysolderappliedtoit. Theuseofthesespringsistokeeptheplatinumwire stretched 18—2 276OnElectrometers and Electrostatic Measurements,[xx. "withanapproximatelyconstant tension fromyeartoyear,and atvarious temperatures.Their fixed ends areattached to roundpins,which areheldwith their axes inalinewith the fibrebyfriction, inbearings forming partsoftwpadjustable brasspieces (7,7)indicated infig.4;thesepiecesareadjusted once foralltostretch thewirewith sufficient force, andtokeep thesquareattracted disc initsproper position. Theround pins bearingthestretching springsareturnedthrough very smallangles bypressingontheprojecting springswith the finger. Theyaresetsoastogiveaproper amount oftorsion tendingtotilttheattracted disc(2)upwards,andthelongend ofthealuminium lever(S),ofwhich itforms apart,downwards. Thedownward motion ofthelongend islimited byaproperly placed stop.Anotherstop (e)above limits theupward motion, which takesplaceunder theinfluence ofelectrification inthe useoftheinstrument. Averyfineopaqueblack hair(thatofa small black-and-tan terrier Ihavefoundmuchsuperiortoany hithertotried)isstretched across theforkedportionofthe sheetaluminium inwhich thelongarmofthelever terminates. Looked athorizontally from theoutside oftheinstrument itis seen, asshown infig.7,PlateI.,againstawhitebackground, marked withtwoveryfineblack circles. Thesesight- plates intheinstruments, asnowmade byMrWhite, areofthesame material astheordinary enamel watch-dials, with blackfigure? onawhiteground. Thewhitespace between thetwo circle.' should beaverylittle lessthan thebreadth ofthehair. The sight-plateissettobeasnear thehair asitcanbewithoui impedingitsmotion inanypartofitsrange;itisslightl} convex forwards, and issoplacedthatthehair isnearer to ii when inthemiddle between theblack circles thanwhen ir anyotherpartofitsrange.Itisthusmadevery easy,ever withoutoptical aid,toavoid anyconsiderable error ofparallax inestimatingthepositionofthehairrelativelytothetw( black circles. Byasimple plano-convexlens(<^,fig. 2),witl theconvex sideturned inwards, itiseasy,intheordinaryus< oftheinstrument, todistinguishamotion upordown ofth< hairamountingto-^-^ofaninch. With alittle care Ihav«, ascertained, DrJouleassisting, thatamotion ofnomore thai^— t—ofaninch from one definite central position can b'50.000 ^ XX.]OnElectrometers and Electrostatic Measurements. 277 securelytested without theaidofothermagnifying powerthan that given bythesimplelens. The lensduringuse isina fixedposition relativelytotheframeworkbearingtheneedle, but itmaybedrawn outorpushedintosuitthefocus ofeach observer. Togivegreat magnification,itoughttobedrawn out sofarthat thehairandsight-plate behind maybebut little nearer tothelensthan itsprincipal focus, andtheobserver's eyeoughttobeataveryconsiderable distance from theinstru- ment, nolessthan20centimetres(8inches) togetgoodmag- nification;andashort-sighted personshould usehisordinary concaveeye-lensclose tohiseye.Thereason forturningthe convexityofthesmall plano-convexlensinwardsis,thatwith suchalens soplaced,iftheeyeoftheobserver istoohighortoo low,thehairseems tohimcurvedupwardsordownwards, and heisthusguidedtokeephiseyeonalevelsufficientlyconstant todoawaywith allsensible effects ofparallaxonthepositionof thehairrelativelytotheblack circles. Theframeworkcarry- ingthestretched platinum wireandmoveable attracted disc is above thebrass roof ofthelantern, inwhich asquare aperture iscuttoallow thesquare portion constitutingtheshortarmof thealuminium balance tobeattracted downwardsbythefixed attractingdisc(§347),tobepresentlydescribed. Asideview oftheattracting plate,thebrass roofofthelantern, thealu- minium balance, thesight-plate,thehair,andtheplano-convex lens isgiveninsection(fig. 2) ;alsoaglass upperroof topro- tectthegaugeandtheinterior oftheinstrument below from dustanddisturbancebycurrents ofair,towhich, without this upper roof, itwould beexposed, throughthesmall vacantspace around themoveable aluminiumsquare. The fixedattracting disc isborne byavertical screwscrewingintotheupperbrass mounting (z,fig.2)(§347),connected with theinnercoatingof theLeyden jarthroughtheguard tubes, etc.,and issecured in anyposition bythe"jam nut,"shown inthedrawingatz, fig.2.This disc(s)iscircular, andabout 38millimetres(IJ inch)indiameter, and isplaced horizontallywith itscentre under thecentre ofthesquare apertureintheroof ofthe lantern. Itsdistance from thelower surface oftheroofandof themoveable attracted discmaybefrom 2Jto5millimetres (from -^jjto^ofaninch), and istobeadjusted, alongwiththe 278OnElectrometers and Electrostatic Measurements, [xx. amount oftorsion intheplatinumwirebearingthealuminium balance-arm, soastogivetheproper sensibilitytothegauge. Thesensibilityisincreased bydiminishingthedistance from theattractingtotheattractedplate, andincreasingtheamount oftorsion. Or,again,thedegreeofthepotentialindicatedby itwhen thehair isinthesighted positionisincreased byin- creasingthedistance between theplates,orbyincreasingthe amount oftorsion. Ifthe electrification oftheneedle istoo great,itsproper positionofequilibrium becomes unstable;or before thisthere issometimes aliabilitytodischarge byaspark across some oftheair-spaces. Theinstrument worksextremely wellwiththeneedlecharged but little lessthan togiveriseto oneorboth ofthese faults, andIadjustthegauge accordingly. Sd4!. Thestrengthofthefixed steeldirecting magnetsisto beadjustedtogivethedesired amount ofdeflection withany stated difference ofpotentialsmaintained between thetwo chief electrodes, when thejarischargedtothedegreewhich bringsthehair ofthegaugetoitssighted position.Inthe instrumentsalready made, thedeflection* byasinglecellof Daniell's amounts toabout 100scale-divisions(ofJ^-ofaninch eachandatadistance of40inches),ifthemagneticdirective force issuch astogiveaperiodofvibrationequaltoabout 1*5 seconds, when thejarisdischargedandthefourquadrants areconnected with oneanother andwith theinnercoatingof thejar.Lowerdegreesofsensibility maybeattained better by increasingthemagnetic directingforcethanbydiminishingthe chargeofthejar. Thus, forinstance, when itistobeused formeasuringandphotographically recordingthepotentialof atmospheric electricityatthepointwhere thestream ofthe water-dropping collector"!* breaks intodrops,themagnetic directingforcemaybemade from10to100timesgreaterthan thatjustdescribed. When this istobedone itmaybecon- venient toattach asomewhat morepowerful magneticneedle than thatwhich hasbeenmade inthemost recent instruments where ahighdegreeofsensibilityhasbeenprovidedfor.But it *That istosay, thenumber ofscale-divisions overwhich theluminous image moves when thechief electrodes aredisconnected from oneanother and putinmetallic connexion with thetwoplates, ofaDaniell's battery. tSeeRoyalInstitution Lecture, May 18,1860{§§278,279,above), orNichol's Cyclopadia,article "Electricity, Atmospheric" (Edition 1860) (§262, above). I]OnElectrometers and Electrostatic Measurements. 279 istoberemarked that ingeneralthedirecting-forceoftheex- ternal steelmagnetscannot betoostrong,asthestrongeritis theless isthedisturbance produced bymagneticbodies moving intheneighbourhoodoftheinstrument*. Inlaboratory work, where numerousmagnetic experimentsarebeing performedin theimmediateneighbourhood, andintelegraphfactories where there isconstant disturbance bylarge moving masses ofiron, the artificialmagneticfield oftheelectrometeroughttobe made very strong. Toallow this,andyetleave sufficient sensibilitytotheinstrument, thesuspended magnetic needle hasbeenmade smaller and smaller, until itisnowreduced to twosmallpiecesofsteel sidebyside, 6millimetres(Jofan inch) long.Forameteorological observatoryallthat isneces- sary is,that thedirecting magneticforcemaybesogreatthat thegreatestdisturbanceexperiencedinmagnetic storms shall notsensiblydeflect theluminousimage. 355. Thesensibilityofthegaugeshould besoadjustedthat avariation inthechargeofthejar,producing aneasily per- ceivedchangeinthepositionofthe hair, shallproduce no sensible changeinthedeflection oftheluminous image pro- duced bythegreatestdifference ofpotentials between the quadrants,which istobemeasured intheuseoftheinstru- ment. Ibelieve theinstrumentsalready made,whenadjusted tofulfil these conditions, maybetrusted tomeasure the dif- ference ofpotentials produced byasinglecellofDaniell's to anaccuracyofaquarter percent. Itmust beremembered that theconstancyofvalue oftheunit ofeach instrument dependsnotonlyontheconstancyofthepotentialindicated bythegauge,butalsoontheconstancyofthemagneticforce inthefieldtraversed bythesuspended magnet,andonthecon- stancyofthemagnetic moment ofthelatter. Aseach ofthese maybeexpectedtodecreasegraduallyfromyeartoyear (al- though very slowlyafter the firstfewhours orweeks), rigorous methods must beadoptedtotakesuch variations intoaccount,if theinstrument istobetrusted asgiving accurately comparable indications atalltimes. Theonlymethod hitherto provided *Allembarrassment from thissource willbedoneaway with ifthebifilar planbeadopted (see §348,Addition). 280OnElectrometers and Electrostatic Measurements,[xx. forthismostimportant objectconsists intheobservation of thedeflectionproduced byameasured motion ofoneofthe quadrants bythemicrometer screw(i)when thefourquadrants areputinmetallic communication with oneanother through theprincipalelectrodes;thejarbeing broughttooneconstant potential byaidofthegauge,andtherefore theforce producing thedeflectionbeingconstant. Theamount ofthedeflection will showwhether ornottheforce ofthemagneticfieldhaschanged, and willrender iteasyatanytime toadjustthestrengthofthe magnets,ifnecessary,tosecure thisconstancy. But toattain thisobject bythese means, thethree quadrantsnotmoved by themicrometer screw must beclamped bytheirfixing-screws sothattheymaybealwaysinthesameposition. 356.Theabsoluteconstancyofthegauge cannot bealtogether reliedupon.Itcertainly changestoasensibledegreewithtem- perature ;andindifterent instruments, toverydifferentdegrees, andeven indifferent directions, aswillbeseen(§877)incon- nexion with thedescriptionoftheportableelectrometer tobe givenlater. But thistemperaturevariation doesnotamount in ordinarycasesprobablytoasmuch asonepercent.;and itis probablethat after ayearortwoanycontinued secular variation oftheplatinumtorsionspringwillbequiteinsensible. Itisto beremarked, however, that secularexperiments ontheelasticity ofmetals arewanting, andoughtatleast tobecommenced in ourgeneration. Inthemeantime itwillbedesirable, bothon account ofthetemperaturevariation andofthepossiblesecular variation inthecoupleoftorsion, tocheck thegauge byaccu- ratemeasurements ofthetime ofoscillation oftheneedle with itsappurtenances. Themoment ofinertia ofthisrigid body, exceptinsofarasitmaybeinfluenced byoxidation ofthe metal, ofwhich Ihave asyetdiscovered nosigns,maybe regardedasconstant, andtherefore theamount ofthedirect- ingcouple due tothemagnets maybedetermined withgreat accuracy byfindingtheperiodofanoscillation when thefour quadrantsareputinconnexionthroughthechargingrodwith themetalmounting bearingtheguard plates,etc. Ihave not asyetputintopractice anyoftheobvious methods, founded onthegeneral principleofcoincidences used inpendulum observations, fordeterminingtheperiodoftheoscillation;but I.]OnElectrometers and Electrostatic Measurements. 281 thoughnotmore than twentyorthirty completeoscillations canbecounted, itseems certain thatwith alittle trouble the Iiriodofoneofthemmaybeeasilydetermined toanaccuracy aboutj^^percent. 357. [Addition, May 1870.—Themost direct andobvious ethod ofusingtheQuadrant Electrometer istoconnect the TOchief electrodes, with thetwobodies whose difference of ^potentialsistobemeasured, andoneofthem with thecaseof ^^feieinstrument. With theinstruments made atthepresent ^aate, adifference ofpotentials equaltothat oftheopposite poles ofasingleDaniell's cellgives,when measured inthismanner, adeflection oftheimageoverabout 60scale-divisions, more orlessaccordingtothedistance atwhich thepointsofsus- pensionofthe silkfibres havebeenadjusted (§848,Addition). The difference ofpotentials due tosixcells inseries would thus deflect theimagetotheextremityofthescale, andbethe greatestdifference ofpotentialsthatcould bemeasured bythe electrometer, ifthese were theonlyconnexions available for measurements. Asecond andmuch lowergradeofsensibility isobtained bysimply raising,soastodisconnect from the quadrant beneathit,theelectrode connected with the Thisbeing done,itrequiresa batteryofabout 10or15cells toproducethe deflectionpre- viously produced byasinglecell. Several stilllowergradesofsen- sibilityhave beenprovidedfor intheinstrumentsrecently made, bytheaddition ofaninduction- plate,insulateddirectlyoverone ofthequadrants behind the mirror. Thesketch inthemar- ginrepresentsavertical section throughtheinduction-plate (e), insulating glass stem{%)by which itissupported,itselec- trode(a),thequadrant (c),and mainglass stem{q).The line ABmthehorizontalplanbe-case. 282OnElectrometers and Electrostatic Measurements, [xx. low isthelineofsection, passing throughthecentres ofthe electrode andinsulating stem oftheinduction-plate,andthat ofthemainglass stem, -which areinonestraightline. The plan representsthatpartofthemain cover asseenfrom above, when thelantern andupper works areremoved. Theplate (b) whichsupportsthemain stem(q)hasbeenenlargedtobear alsotheinsulating support (i)oftheinduction-plate. The outline oftheinduction-platefallswithin that ofthequadrant beneathyt by'IGofacentimetre (^ofaninch)allround. It isdistant '48ofacentimetre (^ofaninch) from theupper surface ofthequadrant. Thedimensions inthefigurearehalf full size. With anelectrometer fitted with theinduction-plate,the usual connexions forthe first ordirect method ofmeasure- ment arethesame asabove mentioned. Theelectrode ofthe induction-plate maybeconnected with that ofthequadrant beneathit,orwith the case, oritmaybeinsulated, without sensibly affectingthe indications ofthe instrument. For thesecondgradeofsensibilitytheinduction-plateiscon- nected with the case, andthe difference ofpotentialsto bemeasured isestablished between itandthedistantpair ofquadrants,thenearerpairbeinginsulated byraisingtheir electrode. Tofreethelatter from theinducedchargewhich theycommonlyreceive bytheactofraisingtheir electrode, adisinsulator isprovided, consistingofalightarm orspring w^hichmaybeturned soastomake contact with thequadrant bymeans ofasmall milled headprojectingabove thecover. Foracertain lowergradethearrangementisthesame, except thatthedistantpairofquadrants,instead oftheinduction-plate, isconnected with thecover, andthedifference ofpotentialsto bemeasured isestablished between thecover andtheinduction- plate. With thisarrangementthedeflections measure about fivetimes the difference ofpotentials producingthesame deflections bythesecondgrade. Theconnexions maybefurther varied soastoproduceother degreesofsensibility givingindicationsperfectly trustworthy and available forcomparativemeasurements. The different methods offormingtheconnexions, with orwithout anin- ductor, areindicated inthefollowing table, whereRmeans the I tfX.]OnElectrometers and Electrostatic Measurements. 283 lectrode ofthepairofquadrants markedRW inthefigure, Lthat ofthepairLL\and/that oftheinduction-plate; Gis econductor ledfrom oneofthebodiesexperimented upon, theconductor ledfrom theother andconnected totheouter etallic case oftheinstrument, which maybeinsulated from etable ifnecessary byplacingasmall block orcake ofclean raffin under each ofthethree feetonwhich theinstrument nds;{R)or(X)means that theelectrode ofRR' orLL' is beraised soastobedisconnected from itspairofquadrants, usinthegradeofdiminished powerorsensibility standing first inthetable ontheright,theelectrode Lisraised, one conductor isconnected withR;/andtheother with thecase oftheinstrument. Thegrade standinglastinthetable, in whichLandRareboth raised,istheleast sensitive ofall.In each ofthese methods thecorrectness oftheindications has been verified bymeasurements takensimultaneously withthe Standard Electrometer(§379), themeasured difference of potentials beingthat oftheearth and ofaLeyden jarfitted with areplenisher, bymeans ofwhich itspotential wasvaried oastomake thedeflected imagestand atallpoints between eextremityofthescaleandthezeroposition. Theworking ofthereplenisher being suspendedatintervals toallow an accurate readingtobetaken ofthepositionoftheimage and theindication oftheStandard Electrometer, thesubsistence of acorrectproportion between thedeflection andthemeasure- ment obtained from theStandard Electrometer w^asverified at allpointsoftherange. WITHOUT INDUCTOK. FullPower, LC R0\"^ \_L0] Diminished Power.poi DimiWITH INDUCTOE. PullPower. rzoi VRC {ro\^^Uo Grades ofDixninished Power. {L)\.io\ RIG[f3 jhedPc XIG RO]L[LICl 701 L0\ 284OnElectrometers and Electrostatic Measurements,[xx. Scale USt BightThefacilityafforded bythenum- berofthesearrangementsforvary- ingthesensibilityoftheinstru- ment even toamoderate orslight degree withoutalteringtheadjust- ment ofthe fibres, willbefound useful insome kinds ofobserva- tions. Forinstance, ifitbede- sired toobserve thefluctuations ofavarying potential,adegreeof sensibility which throws thede- flectedimage nearlytotheex- tremityofthescale willcause the fluctuations tobetwice assensible andaccuratelyread asifthede- flection wereonlyhalfasmuch, as theywillbearthesameproportion tothewhole deflection inthetwo cases. Itisintended infuture tomake theinduction-platesmaller and more distant fromthequadrant,in order todiminish theinductive effect andpermitofthemeasure- ment offrom100to5000 cellsbytheleast sensitive method. Insome electrometers alsothe firsttwogradesofsensibility may beconsidered sufficient, andtheinduction-plate dispensed with.] Absolute Electrometer. 358. Theabsolute electrometer(fig. 11,PlateII.)andthe other instruments ofClass III.arefounded onamethod of experimentingintroduced bySirWilliam Snow Harris, and described inhis firstpaper ''On theElementaryLaws of Electricity*," thirty-four years ago.Inthese experiments aconductor, hungfrom onearm ofabalance andkeptin metallic communication with theearth, isattracted byafixed insulated conductor, which iselectrified, and, forthesake of keepingitselectricpotential constant,isconnected with the Philosophical Transactions, 1834. IX.]OnElectrometers and Electrostatic Measurements. 285 ^.inner coatingofaLeyden battery. The first result which eannounced is,that,when other circumstances remain the me,theattraction varies with thesquareofthequantity ofelectricitywithwhich theinsulatedbodyischargedand isindependentoftheunopposed parts."Itisreadilyseen " that, inthecase ofMrHarris'sexperiments,itwillbe "soslightontheunopposed portionsthat itcould notbe perceivedwithoutexperimentsofaveryrefined nature, such asmightbemade bytheproof planeofCoulomb, whichis, infact,with aslight modification, theinstrumentemployed byMrFaradayintheinvestigation. Now tothedegreeof approximationtowhich the electrification oftheunopposed partsmaybeneglected,thelawsobserved byMrHarris when theopposedsurfaces areplanemaybereadily deduced from themathematicaltheory. Thus letvbethepotentialinthe interior ofA,thecharged body,aquantity which willdepend solelyonthestate oftheinteriorcoatingofthebatterywith which, inMrHarris'sexperiments, Aisconnected, and will therefore besensiblyconstant fordifferentpositionsofA relative totheuninsulatedopposed body B.Letabethe distance between theplane opposedfaces ofAandB^and letSbethearea oftheopposed partsofthesefaces, which willingeneral bethearea ofthesmaller,iftheybeunequal. "When thedistance aissosmall thatwemay entirely neglect "theintensityonalltheunopposed partsofthebodies,itis " readilyshown, from themathematicaltheory,that(since the "difference ofthepotentialsatthesurfaces ofAandBisv) "theintensityoftheelectricity produced byinduction atany " pointoftheportionofthesurface ofBwhich isopposedto "Ais-— ,theintensityatanypoint which isnotsosituated " beinginsensible. Hence theattraction onanysmall element " ft),oftheportion 8ofthesurface ofB,willbeinadirection " perpendiculartotheplaneandequalto27r ij— jo)*.Hence "thewhole attraction onBis ^ird' SeeMathematical Journal, vol. iii.p.275(VII. above, §§146, 147). 286OnElectrometers and Electrostatic Measurements,[xx, "This formulaexpressesallthelaws stated byMrHarris *'asresults ofhisexperimentsinthecasewhen theopposed "surfaces areplane*." 359. Aftermanytrials tomake anabsolute electrometer founded ontherepulsion between two electrifiedspherical conductors forwhich Ihadgivenaconvenient mathematical formula in§-4ofthepaper justquoted (§^0, above),itoccurred tometotakeadvantageofthefactnoticed byHarris, buteasily seen asanimmediateconsequenceofGreen's mathematical theory,thatthemutual attraction between twoconductors used asinhisexperimentsisbut little influencedbytheform ofthe unopposed parts;andin1853, inapaper"OnTransient Electric Currentsf,"Idescribed amethod formeasuringdifferences of electricpotentialinabsolute electrostatic measure founded on that idea. The"absolute electrometer," which Iexhibited to theBritish Association atitsGlasgow Meetingin1855, wascon- structed forthepurposeofputtingthese methods intopractice. Thisinstrument consists ofaplane metal disc insulated ina fixed horizontalpositionwith asomewhat smaller fixed metal dischung centrallyoverit,from oneend ofthebeam ofa balance. Intwopapers |entitled "Measurement ofElectro- static Forceproduced byaBattery," and"Measurement ofthe Electromotive ForcerequiredtoproduceaSparkinAirbetween Parallel Metal Plates atDifferent Distances," publishedinthe Proceedings oftheRoyal SocietyforFebruary 1860, Idescribed applicationsofthiselectrometer, inwhich, forthe firsttime I believe, absolute electrostatic measurements were made. The calculations ofdifferences ofpotentialinabsolute measure were made accordingtotheformulaquoted above(§358)frommy oldpaperon"TheElementary Laws ofStaticalElectricity." 860. Thisformula isrigorous onlyifthedistance between thediscs isinfinitelysmall incomparison with their diameters; andtherefore, inmyearliestattempttomake absolute electro- static measurements, Iusedverysmall distances. Ifound *"OntheElementary Laws ofStaticalElectricity," Cambridge andDublin Mathematical Journal, 1846;andPhilosophical Magazine, July, 1854(II.above, §27). tPhilosophical Magazine, June, 1853. JXVni. andXIX. above, §§310—340. I X.]OnElectrometers and Electrostatic Measurements, 287 great difficultyinsecuringthatthedistance should benearly- enough equal between differentpartsoftheplates, and in measuringitsabsolute amount with sufficientaccuracy; and found besides serious inconveniences inrespectofsensibility and electricrange:later Imade agreat improvementinthe instrument bymaking onlyasmall central area ofoneofthe discs moveable. Thus the electricpartoftheinstrument becomes twolarge parallel plateswith acircularaperturein oneofthem, nearlyfilledupbyalightcircular discsupported properlytoadmit ofitselectrical attraction towards theother being accurately measured inabsolute units offorce. The disc andtheperforated plate surroundingitwillbecalled, for brevity,thediscandtheguard-plate. The faces ofthese two nexttheotherplatemust beasnearlyaspossibleinoneplane when thedisc ispreciselyinthepositionformeasuringthe electric forceupon it,which, forbrevity,willbecalled its sighted position. Thespace between thediscandtheinner edgeofitsguard-ring must beaverysmallpartofthediameter oftheaperture, andmust beverysmall incomparisonwith the distance between theplates;butthediameter ofthediscmaybe greater than, equal to,orlessthanthedistance between theplates. 861. Mathematicaltheoryshows thattheelectric attraction experienced bythedisc isthesame asthatexperienced bya certainpartofoneoftwo infiniteplanesatthesame distance, with thesame difference ofelectricpotentials,thisareabeing veryapproximatelythemean between thearea oftheaperture andthearea ofthe disc,andthat theapproximationisvery good, even should thedistance between theplatesbeasmuch asafourth orfifth,andthediameter ofthedisc asmuch as three-fourths ofthediameter ofthesmaller ofthetwoplates. This conclusion willbereadilyassented towhenweconsider that* theresultant electric force atanypointintheairbetween thetwoplatesisequal numericallytotherateofconduction of heatperunitarea across thecorresponding spaceinthefollow- ingthermalanalogue.Letasolid ofuniform thermal conduc- tivity replacealltheairbetween andaround theplates; andin *"OntheUniform Conduction ofHeat through Solid Bodies, and itscon- nexion with theMathematical TheoryofElectricity," Cambridge Mathematical Journal, Feb. 1842; andPhilosophical Magazine, July, 1854(I.above, §§1—6). 288OnElectrometers and Electrostatic Measurements,[xx. placeoftheplatesletthere behollowspacesinthis solid. Let these hollowspacesbekeptattwouniformtemperatures, differing byanumber ofdegrees equal numericallytothe difference ofpotentialsintheelectricsystem,thespacecorre- spondingtothediscandguard-ring beingatonetemperature, andthatcorrespondingtotheopposite plateattheother tem- perature;and letthethermalconductivityofthe solid be unity.Ifweattempttodraw theisothermal surfaces between thehollowcorrespondingtothecontinuousplateontheone side,andthatcorrespondingtothe discandguard-ringon theother, weseeimmediatelythattheymust beverynearly plane, fromverynearthedisc allthewayacross tothecorre- spondingcentralportionoftheopposite plate,butthat there willbeaconvexitytowards theannularspace between thedisc andguard-ring. 362. Thusweseethattheresultant electric force will, toa V verycloseapproximation, beequaltoy:forallpointsofthe airbetween theplatesatdistances from theouter bounding edges exceeding twoorthree times thedistance between the plates, and atdistances from theinterstice between theguard- ringand disc not lessthan thebreadth ofthis interstice. Hence,ifpdenote the electricdensityofanypointofthe plateordisc farenoughfrom theedges, wehave V Buttheoutward forceexperienced bythesurface ofthe electrified conductorperunit ofarea atanypointis27^/^^and therefore ifFdenote the forceexperienced byanyareaA ofthefixedplate, nopartofwhich comes near itsedge,we have which willclearly beequaltotheattractionexperienced by themoveable disc, ifAbethemean areadefined above. This givesV=D .—J-,theformulabywhich difference ofpoten- tials inabsolute electrostatic measure iscalculated from the m XX.] OnElectrometers and Electrostatic Measurements. 28^ result ofameasurement ofthe force F,which,itmust be remembered, istobeexpressedinkinetic units. Thus ifW bethemass ingrammestowhich theweightisequal, wehave F=gW, where gistheforce ofgravityincentimetres persecondper second. Thedifficulty which, infirstapplyingthismethod about twelveyears ago,Ifound inmeasuring accuratelythedistance Dbetween theplatesand inavoidingerror from their not being rigorously parallel,Inowelude bymeasuring only differ- ences ofdistance, anddeducingthedesired results from the difference ofthecorrespondingdifferences ofpotentials. Thus letVbethedifference ofpotentialsbetween theplatesre- quiredtogivethesame forceF)when thedifference ofpoten- tials isVinstead ofF,wehave 63.Theplanofproceedingwhich Inowuse isasfollows : "Eachplate (fig. 11,PlateII.)isinsulated; oneofthem, the continuous one, forinstance, iskeptatapotential differing from theearthbyafixedamount tested byaidofaseparate idiostatic* electrometer*f ;theotherplate (theguard -ring and moveable discinmetallic communication with oneanother)is alternately connected with theearth andwith thebodywhose potentialistobemeasured. Thelowerplateismoved upor downbyamicrometer screw until themoveable discbalances inadefiniteposition,indicated bythehair(with background ofwhite with blackdots)seen throughalens, asshown in fig.11.Before and aftercommencingeach series ofelectrical experiments,aknown weightisplacedonthedisc,andasmall wire rider onthelever from which thedischangsisadjusted tobringthehair toitssighted position when there isnoelectric force. This lastcondition issecured byputtingthetwoplates *See§385,below. t[ALeyden jarwithanidiostatic gauge andreplenisherfitted tothecover bywhich itisclosed hasbeenfound very suitable forthispurpose. Thegauge canbeadjusted toahigher degreeofsensibility than isattainable inan electrometer forgeneral purposes, astheStandard orthePortable Electrometer, andthemicrometer movements andgraduationsofthese electrometers arenot required.— May, 1870.]i T.E. 19 290OnElectrometers and Electrostatic Measurements, [xx. inmetallic communication with oneanother. Fortheelectric experimentstheweightisremoved, sothatwhen thehair is inthesighted positiontheelectric attraction onthemoveable disc isequaltotheforce ofgravity ontheweight. The electric connexions suitable inusingthisinstrument fordetermining inabsolute electrostatic measure thedifference ofpotentials maintained byagalvanic battery between itstwoelectrodes are indicated infig.11.Nodetails astothecase forpreventing disturbance bycurrents ofair,and formaintainingadryatmo- sphere, byaidofpumice impregnatedwithstrong sulphuric acid, areshown, becausetheyarebynomeans convenient in theinstrument atpresentinuse,which hasundergonesomany transformations thatscarcely anypartoftheoriginalstructure remains. Ihopesoon toconstruct acompactinstrument con- venient forgeneraluse.Theamount offorcewhich isconstant ineach series ofexperiments maybevaried fromoneseries to another bychangingthepositionofthesmall wire rideronthe lever. The electric systemhere described isheterostatic(§385 below),there beinganindependentelectrification besides that whose difference ofpotentialistobemeasured. NewAbsolute Electrometer. [§364... 367added May, 1870.] 364. Plate III. isasketch inperspectiveofthis instru- ment, one-third ofthe full size.AsintheAbsolute Electro- meterjust described, the electricsystemisheterostatic; with this addition, that thepotentialoftheauxiliary chargeis tested andmaintained, notbyaseparateelectrometer and electric machine, butbyanidiostaticarrangement forming partoftheinstrument itself. This consists ofaLeyden jar, formingthecaseoftheinstrument;agauge;andareplenisher. TheLeyden jarisawhite(flint) glass cylinder,coated inside andoutside with tinfoil tonearlytheheightofthe circular plate {A) ;apertures beinglefttoadmit therequisite lightto theinterior andallow theindications ofthevertical scale(r) anddivided circle{t)toberead.Abrassmountingiscemented round theupperrimofthejar,towhich isscrewed thecover XX.] OnElectrometers and Electrostatic Measurements. 291 ofstout sheet-brass (0),which closes thejaratthetop.By another brass mountingcemented round itslower rim,thejar isfastened down tothecast-ironsole-plate (D)which closes itslower end. Thesole-plateissupportedonthreelegssimilar tothose shown infig.13,Plate II.Thecover {C)supports thereplenisher (E),andthealuminium balance-lever ofthe idiostatic oraufre, which areidentical inconstruction with those described in§§352,353,butonalargerscale. The airinside iskept drybyaidofpumicesoaked withstrong sulphuric acid, contained inglassvesselsplacedinthebottom ofthe jar. Themoveable discorbalance(c)hangsinacircularaperture intheplate {A),which rests onthree fixedsupports {z,z,.) cemented totheinterior surface ofthejar,andinmetallic con- nexion with theinsidecoating ;themanner ofsupportisthat ofthehole, slot,andplane,described in§380, (2),below. Thisperforated plateorguard-plate supports onabrasspillar theattracting plate (F)oftheidiostaticgauge,which thus tests thepotentialoftheguard-plate, balance, andinside coat- ing.Thispotentialiskeptconstantduring anyseries ofex- periments byusingthereplenisher accordingtotheindications ofthegauge,which ismadeextremelysensitivebyaproper adjustmentofthedistance from theattracting plate {F)to thebalance-lever and ofthetorsion bywhich the electrical attraction isbalanced(seeendof§353). Thereplenisherhas metallic contact with theguard-plate throughthespring (e). Thejarischarged byaninsulatedcharging-rodletdown for theoccasionthroughahole inthecover. 365. The balance(c)isalightaluminium disc,about 46 millimetres indiameter, strengthened byanelevated rimand radial ribsonitsupper surface, buthavingitslower surface plane andsmooth. Itnearlyfillstheapertureintheguard- plate,sufficient clearance beingleft('75ofamillimetre all round)toallow ittomove upanddown without risk offric- tion. Itissupported bythree delicate steelsprings, each of which consists oftwoparts ;theupper endoftheupper part isattached tothelowerextremityofaverticalinsulating stem(i)directly above thecentre ofthe disc,where thecor- responding endofthelowerpartisfixed. Theopposite ends, 19—2 292 0)1Electrometers and Electrostatic Measurements, [xx. whichproject considerably beyondthecircumference ofthe disc, arerivetedtogether. One ofthesesprings (s)isshown inthefigure.Theirgeneralformmaybecomparedtothat of coach-springs. Thepointofattachment oftheir upper parts ismovedvertically byakinematic arrangement preciselythe same asthatemployedinthePortable Electrometer(§369). Theinsulating stem(i)isattached toabrass tube(a),which slidesupanddown inVguides bytheaction ofamicrometer screw. This micrometer screw isworked bymeans ofthe milled head (m)projectingabove thecover {C) ;theguidesfor thetube(a)andindex{x)which moves upanddown with the tube, aresimilar tothoserepresented morefullyinfig.10, Plate II.,and ar-erigidlyattached toastrongbrassplate (6) lyingacross themouth ofthejarbelow thecover, andresting upontheflangeofthebrassmounting,towhich itisfastened byscrews. Theplate (6)issoadjustedthat thebalance may hangconcentric with theperforationintheguard-plate. The tube{a)issimilar inconstruction tothatrepresentedinfig.8, Plate II.,and described in§369, below. Themicrometer screw carries ahorizontal circular disc{d)graduated by100 equal angulardivisions. Anapertureisleft inthecover through which itsindications canberead offbyreference to afixedmark onthesloping edgeoftheaperture. This, together with thescale(/),each division ofwhichcorrespondstoone fullturn ofthemicrometer screw, measures thevertical distance throughwhich thetube(a)andthepointsofattachment ofthe springsaremoved. Metallic communication between thebalance andtheguard- plateismaintained byalight spiralwire attached tothepillar {g)and totheupper supportofthesprings, which isabrass piece cemented totheinsulating stem. Anarm,notseen in thefigure, projectsfrom theguard-plateover thedisc sothat itsextremityisbetween thecentre ofthediscandtheupper end,bent horizontally,ofanuprightfixed tothedisc; thus servingasastoptoconfine themotion ofthediscbetween certain limits,Averyfineopaqueblack hair(§353)isstretched between twosmalluprights (oneofwhich isseen inthefigure) standinginthecentre ofthe disc.Anachromatic convex lens{h),fixed ontheguard-plate,standsopposite,andpro- I XX.]0)1Electrometers and Electrostatic Measurements. 293 duces animageofthehair intheconjugate focus, which is justovertheouteredgeoftheguard-plate. Thetwoopposed screw-points {k)areadjustedtotouch each side oftheimage thusthrown bythe lens,which, ontheprincipleoftheastro- nomicaltelescope,isobserved through aneye-lens (I),attached outside ofthejartotheupperbrassmounting. Bythis arrangementtheerror ofparallaxinobservingthepositionof thehairrelativelytothetwopointsisavoided; theposition oftheeyemaybevaried inanydirection withoutcausing any changeintheapparentrelativepositionofthehair(image) and points.Inadjustingthese differentparts,itisarranged that when theimageofthehair isexactly between thetwopoints, orinwhat iscalled thesighted position,theunder surfaces of thebalance andguard-plate maybeasnearlyaspossibleinone horizontalplane. Thebalance andspringsareprotected,intheuseofthe instrument, fromdisturbingelectrical forces, byabrass cover intwohalves{y,y),oneofwhich isrepresented displacedin thefigure,toshow theinteriorarrangements. Thetwohalves, whenplaced together,form acircular box,withanaperturein front inwhich thelens[h)stands, andanotheraperture behind toadmitlightfrom theskyorfrom alamp placedoutside of thejarinthelineofthehair, lens,andpoints. 366. The electricalpartoftheinstrument iscompleted by thecontinuousattracting plate {B),under andparalleltothe guard-plateandspring-balance. This isastiff circular brass plate withpartscutouttoallow ittomovefreely past the fixedsupports {z,z,,)oftheguard-plate. Anelectrode{n) projecting throughahole inthesole-plate from aninsulating stem(p)iskeptinmetallic communication byaspiral wire withanarmprojecting from thecentre ofthecontinuousplate. Theplate {B)issupported byabrasspillar (g),fromwhich itis insulatedbyashortglassstem. Itismovedvertically bythe micrometer screw{xo)(step ^-^ofaninch) ;and thismotion is measured byavertical scale(r)andhorizontalgraduatedcircle (i)attached tothescrew. Thescrewprojects below thesole- plate, and isworked bythemilled head(it),thenut(v)being fixed inthecentre ofthesole-plate. Thepillar [rj)moves in 294OnElectrometers and Electrostatic Measurements,[xx. Vorring guides,and restsupontheupper end ofthescrew inthemannerrepresentedinfig.14,Plate II. 867. Before thisinstrument isavailable forabsolute electro- static measurements, the forcerequiredtomove thebalance through anyfixed vertical distance(thepointofsuspension being unmoved) must beknown. This isascertainedbyweighings conducted inthefollowing manner :—Thecover(C)isremoved, and allelectrical forceuponthebalance isguarded against by puttingtheelectrode{n)inmetallic communication with the guard-plate.The balance isthenbrought, byturningthe micrometer circle{d),tothesighted position ;andthereading onthescale(/)andgraduatedcircle{d)isnoted.Aknown weightisthen distributed symmetricallyover thedisc(^of agrammehasbeen usedhitherto), whichdisplacesitbelow the sighted position.Itisnow raised tothesighted position by turningthe disc{d),andthealtered micrometerreadingis noted. Thedifference between thetworeadingsmeasures the distance through which thegiven weight displacesthebalance inoppositiontothetension ofthesprings; andconversely, when thebalance hasbeendisplaced throughthesame distance byelectrical attraction between itand thecontinuousplate belowit,thisknownweightisthemeasure oftheforce exerted uponit.Ithasbeen thusfound byrepeated weighings,thata weightofY^Qofagramme displacesthebalancethrough a distancecorrespondingtotwo fullturns ofthemicrometer screw andafraction ofonedivision ofthecircle,intheinstru- mentbelongingtotheLaboratoryoftheGlasgow University. This distancehaving been ascertained with allpossible care andatdifferenttemperatures,inview ofthepossibleeffect of temperature ontheelasticityofthesprings,theplanofpro- ceedingtoabsolute electrostatic measurements isasfollows, the Aveights being removed andcovers{y,y,C)replaced. Allelectrical influencehaving beenremovedbyawire led from theelectrode{n)throughthehole inthecover(C)tothe guard-plate,thebalance isbroughttothesighted position. Startingfrom thispoint,itisraisedbythemicrometer screw through anydistance which hasbeen ascertained tocorrespond toaknownweight, e.g.thedistancejustmentioned. This cor- ".1.]OnElectrometers and Electrostatic Measurements. 295 responds exactlytotheremoval oftheweight (§363)inthe useoftheAbsolute Electrometeralreadydescribed. Thejar isthencharged,andthepotentialiskeptconstantduringthe experiments byusingthereplenisher accordingtotheindica- tions ofthegauge, which, asalready said, hasbeenmade extremelysensitive forthepurpose. Theattracting plate [B) isconnected byitselectrode[n)alternatelywith theoutside coatingofthejar(which maybeeither connected with the earth orinsulated) andwith thebodythedifference ofwhose potentialfrom that oftheoutsidecoatingistobemeasured. Ineach casethebalance isbroughttothesighted position by movingtheplate {B)upordownbythemicrometer screw(w), andthereadingonthevertical scale(r)andgraduatedcircle[t] isnoted. The difference ofthetworeadings givesthediffer- ence ofthetwo distances between balance andattracting plate,fromwhich thedifference ofpotentialsisdeduced by theformula attheendof§362. Inmeasuringthedifference ofpotentials between thepolesofavoltaicbattery,itisfound veryconvenient toconnect thepoles, throughaSteinheil(or doubleBavarian) key,either with theoutercoatingofthejar (orearth),theother with theinsulated electrode(n).The reading beingtaken andthekeyreversed, thedifference of readings,itisevident, measures adifference ofpotentials double that ofthepolesofthebattery. Two observers are convenient, onetowatch thegauge andusethereplenisher accordingly,theother totakethereadings. IPoETABLE Electrometer. 368. Intheordinaryuseoftheportableelectrometer(figs. 8,9,and10,PlateII.),the electricsystemisheterostatic and quitesimilar tothat oftheabsolute electrometer, when used in themanner described above in§363. Butthebalance isnot adaptedforabsolute measure oftheamount offorce ofattrac- tionexperienced bythemoveable disc;onthecontrary,itis preciselythesame asthatdescribed forthegaugeofthequad- rant electrometer in§353above, onlyturnedupside down.I 296OnElectrometers and Electrostatic Measurements,[xx. Thus, intheportable instrument, thesquaredisc(/)forming partofthelever ofthinsheetaluminium isattractedupwards byasolid circular disc ofsheet-brass(g),thick enoughfor stiffness. Every partofthealuminium leverexceptthis square portionisprotected from electric attraction byafixed brassplate {hh)with asquare hole init,asnearlyasmaybe stopped bythesquare partofthesheetaluminium destined to experiencetheelectric attraction, allotherpartsofthealumi- nium balance-lever being below thisguard- plate. Thealumi- nium lever (ik),asshown infigs.8and 10,isshapedsothat when thehair{I)attheend ofitslongarm isinitssighted position,theuppersurfaces ofthefixedguard-plate Qi)and moveable aluminiumsquare (/)areasnearlyasmaybeinone plane. Themode ofsuspensionispreciselythesame asthat described(§353) forthegaugeofthequadrant electrometer. Intheportable instrument, careful attention isgiven bythe maker tobalance thealuminium leverbyaddingtoitsmall masses ofshellac orother convenient substance, sothat its centre ofgravity maybeintheline ofitsplatinum-wire axis, or,moreproperly speaking,insuch apositionthattheinstru- ment shallgive,whenelectrified, thesame ''earth-readings" when held inany positions,eitherupright,orinclined, orin- verted(§375below). Thus thecondition ofequilibriumof thebalance, when thehair isinitssighted position,isthatthe moment ofelectric attraction round theaxis ofsuspensionshall beequaltothemoment ofthecoupleoftorsion, the latter beingasconstant asthepropertiesofthematter concerned (platinum wire, brassstretching-springs, etc.)willallow. 369. Theguard-plate carrying, bytheplatinum-wiresus- pension,thealuminium balance,isattached tothebottom of asmallglassLeyden jar(mm),and isinpermanentmetallic communication with itsinsidecoatingoftinfoil. The outside tinfoil coatingofthisjarisinpermanentmetallic communica- tionwith theoutside brassprotectingcase. Theupper open mouth ofthiscase isclosedbyalidorroof,which bears onits inner sideafirmframeprojecting downwards. Thisframe has twoYnotches, inwhich astout brass tube(o)slides, keptin theYsbyaproperly placed spring {p)[(May, 1870)better twosprings,onepressing directlytowards eachY],givingit I.]OnElectrometers and Electrostatic Measurements. 297 freedom toslideupanddown inonedefinite line*. Firmlyfixed intheupper endofthistube isanut(a,fig.8),which ismade tomoveupanddown byamicrometer screw. Thelower end oftheshaft ofthisscrew hasattached toitaconvexpieceof polishedsteel(6,fig. 8),which ispressed uponahorizontal agate plate rigidlyattached totheframework above mentioned byastiff brasspiece projectingintotheinterior ofthebrass tubethroughaslotlongenoughtoallow therequisite range ofmotion. Thisarrangementwillbereadily understood from theaccompanying drawings.Ifchasbeendesigned upon obvious geometrical principles, which have been hithertoneglected,so farasIknow, inallmicrometer screw mechanisms, whether forastronomical instruments orotherpurposes. The screw- shaft isturned byamilled head, fixed toitatthetopoutside of theroof oftheinstrument;andtheangles throughwhich itis turned arereadonacircle divided intoonehundredequal parts ofthecircumference(orZ^'Qeach)byreference toafixedmark ontheroof oftheinstrument. The hole intheroofthrough which thescrew-shaftpassesiswideenoughtoallow theshaft toturnwithouttouching it,andthelower edgeofthegradu- ated circleturningwith thescrew iseverywhere verynear the uppersideofthe roof,butmust nottouch itatanypoint.A second nut(c,fig.8)above the effective nut fitseasily, but somewhataccurately,inthehollow brass tube, and isprevented fromturninground inthetubebyaproper projectionand slot. Thus thescrew isrenderedsufficiently steady,with reference tothesliding tube;that istosay,itsaxis ispreventedfrom anybutexcessivelysmall deviations from the axis ofthe slidingtubeandfixedguides ;andwhen thenut iskeptfrom being turned round itsproper axis, itformsalongwith the slidingtubevirtuallyarigid body.Acarefully arranged *Inconsequence ofsuggestions byMrJenkin, itisprobable thatthespring may bedoneaway with, andtheVsreplaced byrings approximately fitting round thetube, butleavingitquite free tofalldown byitsown weight. In consequence ofthesymmetrical positionoftheconvex endofthescrew over thecentre oftheattracted disc, slight lateral motions ofthetubeproduce no sensible effect ontheelectric attraction. \_May, 1870.—Various trials bothon theportable andstationary instruments have butvery partiallyfulfilled this anticipation ;andhave confirmed thepractical value oftheVs. Thecon- structional advantages oftherings andgeometrical merits oftheVsareeasily combined.! 298OnElectrometers and Electrostatic Measurements,[xx. spiral spring pressesthetwonuts asunder, and socauses the upperside ofthethread ofthescrew-shaftalwaystopress againsttheunder side ofthethread oftheeffective nut,thus doing awaywithwhat istechnicallycalled inmechanics"lost time." Inturningthemicrometer screw, theoperator presses itsheadgently downwards with hisfinger,tosecure that its lower endbearsfirmly upontheagate plate.Itwould bethe reverse ofanimprovementtointroduce aspringattached to theroof oftheinstrument outside topressthescrew-head downwards, inasmuch ashowever smooth thetopofthescrew- shaft mightbemade, andhowever smooth thespring pressing itdown, there would stillbeavery injuriousfriction impeding thepropersettlement oftheslidingtube into itsVs.Astiff fork{q)stretchingover thegraduatedcircle isfirmlyattached totheroof outside, topreventthescrew frombeinglifted upbymore than averysmallspace; about-^-^ofaninch atmost. Inusingtheinstrument, theobserver should oc- casionally pullupthescrew -head andpressitdownagain, andgiveitsmall horizontal motions, tomake sure thatwhen itisbeingused itispressedinproperlytoitsVsanddown upontheagate-plate. Alongarm {d,figs.8and10) (ortwo arms oneabove theother), firmlyattached tothesliding-tube, carries anindex which moves upanddown with it.Two fixed guiding-cheeks oneach side ofthisindexpreventthetube frombeingcarried round toofarineither direction when the screw isturned :oneofthese cheeks isgraduatedsothateach division isequalinlengthtothestepofthemicrometer screw; thisenables theoperatortoascertain thenumber oftimes he hasturned thescrew. These twocheeks must never simul- taneously pressuponthesliding-pointer ;onthecontrary, they must leave itaslightamount oflateral freedom tomove. If thisdoes notamount to'36ofadegree,theamount of"lost time"produced byitwillnotexceed-^^ofadivision ofthe micrometer circle, and willnotproduce anysensible error in theuseoftheinstrument. Aglassrodcemented tothelower endofthetubeprolongsitsaxisdownwards, and bears the continuousattracting-plateoftheelectrometer atitslower end. Theobject aimed atinthemechanismjustdescribed isto preventthenutandotherparts rigidlyconnected with itfrom I OnElectrometers and Electrostatic Measurements. 299 anyother motion thanparalleltoonedefinite line,andtoleave itfreedom tomove inthis line,unimpeded byanyother friction than thatwhich isindispensableinthearrangementforkeeping theslidingtube initsVs. 370. Iftheinner tinfoilcoveringoftheLeyden jarwere completed uptotheguard-plate bearingthealuminium bal- ance-lever, thelongarmofthisleverbeingintheinterior ofa hollow conductor wouldexperiencenoelectric influence, andno forcefrom theelectrification oftheLeyden jar,orfromseparate electrification oftheupper attracting plate, or,morestrictly speaking,the electricdensityandconsequentelectric force on thelongarm ofthelever would beabsolutelyinsensible to themost refined testwecouldapply,because ofthesmallness ofthegapbetween themoveable aluminiumsquare andthe boundaryofthesquare apertureintheguard -plate. But to seethehaironthelongendofthelever, andtheAvhite back- groundwith black dotsbehindit,anotinconsiderableportion oftheglassunder theguard-platemust becleared oftinfoil outside and inside. Thus the electricpotentialoftheinner coatingoftheLeyden jarwill notbecontinuedquiteuni- formlyovertheinner surface ofthebaredportionoftheglass, andadisturbanceaffecting chieflythemost sensitivepartof thelever willbeintroduced. Todiminish this asmuch as possible withoutinconveniently impeding vision, adouble screen ofthin wirefencing,inmetallic communication with theinner tinfoilcoating and theguard -plate,isintroduced between theendoftheleverandtheglass through which itis observed. 371.Avery light spiral spring (?•)connects theupperattract- ingplate with abrasspiece supported uponafixed vertical glasscolumnprojecting downwards from theroof oftheinstru- ment. This brasspiecebears astout wire(s),called themain electrode, projecting vertically upwards alongthe axis ofa brass tubeopenateach end, fixed inanapertureintheroof soastoproject above andbelow, asshown infig.9. 872. Thetopofthemain electrode bears abrasssliding piece {t),which, when raised alittle, serves forumbrella and wind-guard withoutdisturbingtheinsulation; andwhenpressed down closes theaperture andputsthe electrode inmetallic 300OnElectrometers and Electrostatic Measurements, [xx. connexion with theroof oftheinstrunient. When theinstru- ment istobeused foratmospheric electricity (unlessatafixed station), asteel wire, about 20centimetreslong,isplacedin theholeonthetopoftheslidingbrasspiece justmentioned, and isthusheld intheverticalposition. Aburning match is attached toitsupper end,which hasthe effect ofbringingthe potentialofthechief electrode andupper attracting plate, etc., alltothepotentialofthe airatthepoint where thematch burns*. Theinstrument iseither held intheobserver's hand, oritisplaced uponafixedsupport, and caretaken that its outer brass case isinconnexion with theearth. When the difference ofpotentialsbetween twoconductors istobetested, oneofthese isconnected withthebrass case oftheinstrument, andtheother with thechief electrode, theumbrellabeing kept up.Ifboth ofthese conductors must bekeptinsulated from theearth, thebrass case oftheelectrometer must beputonan insulating stand, andthemicrometer screw turned byaninsu- latinghandle. 373.Aleadcup {ee, fig.8),supported bymetalpillarsfrom theroofandcarrying piecesofpumice-stone,held intheir place byIndia-rubber bauds, completestheinstrument. The inner surface oftheglassmust beclean, andparticlesofdust, minute shreds orfibres, etc.,removed ascarefullyaspossible, especiallyfrom thelower surface oftheupper attracting-plate, andtheuppersurface oftheguard-plateandaluminiumsquare facingitfrom below. Thepumiceisprepared -bymoistening itwith afewdropsofstrong pure sulphuricacid. Ordinary sulphuricacid ofcommerce should beboiled withsulphateof ammonia tofree itfrom volatile acidvapours,andtostrengthen itsufiiciently byremovingwater iftheacidbenotofthe strongest.There should notbesomuch acidappliedtothe pumiceastomake ithave theappearanceofbeing moist, but theremust beenoughtomaintain asufiiciently dryatmosphere within theinstrument forvery perfectinsulation oftheLeyden jar,which Ifinddoes notingenerallosemore ofitscharge *SeeNichol's Cyclopcedia, article "Electricity, Atmospheric," 2ndedition, 1860(§266,above); or"Eoyal Institution Lecture onAtmospheric Electricity," May, 1860{§§277, 278, above). 1 ]OnElectrometers and Electrostatic Measurements. 301 than fivepercent,perweek, when thepumiceisproperlyim- pregnatedwith acid. Thus there isnotendencyoftheliquidto dropoutofthepumice; andthepumice being properlysecured bytheIndia-rubber bands, theinstrument maybethrown about withany force, short ofthatwhich mightbreak theglass jaror either oftheglass stems, withoutdoing anydamage ;butto insure this hardiness thesheet aluminium ofwhich thebal- ance ismade must beverythin. After several weeks' usethe pumice may begintolook moist, andevenslighttraces of moisture maybeseenontheoutside oftheleadcup,inconse- quenceofwatery vapourattracted bythesulphuricacidfrom theatmosphere;butthepumiceshould thenbetaken outand dried. Atallevents thismust bedone ingood time, before enoughofliquidhascollected togiveanytendencytodrop. Inallclimates inwhich Ihave hitherto tested theinstrument, Ihave found thepumiceeffective forinsulation and safe inkeepingalltheliquidtoitself fortwomonths. But MrBeckerhaving reportedtomethatmany instruments havebeen returned tohim inaruinous condition fromdrops ofsulphuricacidhaving become scatteredthroughtheirmetal work, Inow cause tobeengraved conspicuouslyontheouter caseoftheinstrument "pumice dangerous, ifnotdried ONCE AMONTH;"alsoaframecarryingacard, onwhich thedates of dryingareinscribed, tobeplacedinaconvenientpositionon theroofoftheinstrument. 374. Topreparetheinstrument foruse,theinnercoatingof theLeyden jarmust becharged throughacharging rod,insu- lated inavulcanite orglass tube,and letdown fortheoccasion through ahole intheroof oftheinstrument, byaidofasmall electrophorus, whichgenerally accompaniestheinstrument, or byanelectrical machine. Igenerally prefertogiveanegative chargetotheinnercoating,asIhave notfound anyphysical reason, such asthatmentioned in§349above, topreferaposi- tivechargetoanegative charge; andthenegative charge gives increasedreadingsofthemicrometer, intheordinaryuseofthe instrument, tocorrespondtopositive chargesoftheprincipal electrode, aswillbepresently explained.Before commencing tocharge thejar,theupper attracting-plateshould bemoved tonearlythehighest positionofitsrange bythemicrometer 302OnElectrometers and Electrostatic Measurements,[xx. screw, otlierwise toostrongaforce ofelectric attraction maybe putuponthealuminiumsquare;and besides, thejarwill dis- chargeitself between theupper plate andtheextreme edgeof thealuminiumsquare, when itispulled verymuch above the level oftheguard-plate bytheelectric attraction. Ihave not foundanyinjuryorchangeofelectric value ofthescale-divi- sions toarise fromanysuchrough usage ;but still, toguard against such apossibility,Iproposetoaddtotheguard-plate checks toprevent thecorners ofthealuminium fromrising much, ifatall,above itslevel, and toconduct thedischarge andprotect thealuminium andplatinumfrom the shock, incase oftheupper plate being broughttoonear thelower. When theinstrument isbeing charged,orwhen itisoutofuse atanytime, theumbrella shouldalwaysbekeptdown;but it must beraised toinsulate theprincipal electrode, ofcourse, beforeproceedingtoapplythis toabodywhose difference of potentialfrom abody connected with thecase oftheinstru- ment istobemeasured. 375. Inusingtheinstrument theumbrella mustveryfre- quentlybelowered, ormetallic communication established in anyother convenient waybetween thechief electrode andthe outer brass case, themicrometer screw turned until thehair takes itssighted position,andthereading taken, thehundreds beingreadontheinterior verticalscale, andtheunits(orsingle divisions ofthecircle) onthegraduatedcircle above. The number thus found iscalled theearth-reading.Itmeasures thedistance fromanarbitraryzeropositiontothepositionin which theupper attracting-plate must beplacedtogivethe amount ofelectric force onthealuminiumsquare which bal- ances thelever initssighted position. Aconstant added to theearth-reading,orsubtracted fromit,gives (§341)anumber simply proportionaltothedifference ofpotentials between the upperandlowerplate ;that istosay,between thetwocoat- ingsoftheLeyden jar.The vertical scale andmicrometer circle arenumbered, sothat increased distances between the plates giveincreased readings ;andthezeroreadingshould correspondasnearlyasmaybetozero distance between them; althoughintheinstruments hitherto made nopains havebeen taken tosecure this condition, evensomewhatapproximately. 1.]0)1Electrometers and Electrostatic Measurements. 303 Ifitisdesired toknow theconstant, anelectricalexperiment must bemade todetermine it,which isdone with ease;but this isnotnecessaryfortheordinaryuseoftheinstrument, which isasfollows :— 376. First, anearth-readingistaken, then theupperelec- trode isinsulated byraisingtheumbrella, orotherwise break- ingconnexion between theprincipalelectrode andtheouter metal case oftheinstrument. Theprincipalelectrode andthe outer casearethenconnected with thetwobodies whose differ- ence ofpotentialistobedetermined, andthemicrometer screw isturned until thehair isbroughttoitssighted position. The readingofhundreds onthevertical scaleandunits onthecircle isthen taken.Lastly,theprincipalelectrode isagaincon- nected with the.case oftheinstrument andanother earth-read- ingistaken. Ifthesecondearth-readingdiffers from thefirst, theobserver must estimate themostprobable earth-readingfor themoment when thehairwasinitssighted position, with the upper plate andthemetal case inconnexion with thetwo bodies whose difference ofpotentialistobemeasured. The estimatedearth-readingistobesubtracted from thereading taken inconnexion with thebodies tobetested. This differ- encemeasures(§362)therequireddifference ofpotentialsbe- tween them inunits oftheinstrument. Thevalue oftheunit oftheinstrument oughttobeknown inabsolute electrostatic measure; andthedifference ofreading found inanyexperi- ment istobemultiplied bythis,which iscalled(§341)the absolute coefficient oftheinstrument, togivetherequired difference ofpotentialsinabsolute measure. Itsohappens that, intheportableelectrometers ofthekindnow described which havebeen hitherto constructed, theabsolute coefficient is somewhere about "01,sothatoneturn ofthescrew, orone hundred divisions ofthe circle, correspondstosomewhere about one electrostatic unit, with agrammefortheunit ofmass, a centimetre fortheunit ofdistance, andasecond fortheunit oftime;butthedifferent instruments differ fromoneanother byasmuch astenortwenty percent, intheir absolute coeffi- cients. Inallofthese Ihave found between three andfour Daniell's cells tocorrespondtotheunit division;that isto say,between three hundred and fourhundred cells toafull 304OnElectrometers and Electrostatic Measurements,[xx. turn ofthescrew. Withgreat care, theobserver maymeasure small differences ofpotentials bythisinstrument tothetenth partofadivision(ortoabout halfaDaniell'scell). With a verymoderate amount ofpractice and care,anerror ofasmuch ashalfadivision maybeavoided ineachreading. 377. Butthere areimperfectionsintheinstrument itself which make itdifficult orimpossibletosecureveryminute accuracy, especiallyinmeasurementsthroughwideranges. (1)Inthe firstplace,Iamnotsure that theend ofthe needlecarryingthehair isprotected sufficiently bythewire fences(§370)from electric disturbance toprovide against any errorfrom thissource, whichpossiblyintroduces seriousirregu- larities. (2)Inthesecondplace,thecapacityofthejarinthesmall portable instrument isnotsufficient tosecure thatthepotential ofitsinnercoatingshall not differsensiblywith thedifferent distances towhich theupper plateisbrought,tobalance the aluminium lever with thehairinitssighted position. Buton thispointitistoberemarked thattheelectricdensityonthe uppersurface oftheguard-plateisinitscentralparts always thesamewhen thehair isinitssighted position;and itis thereforeonlythecomparativelysmall difference ofthequantity ofelectricityonthis surface, towards therim,correspondingto different distances oftheattractedplate,thatcauses difference ofpotentialintheinnercoatingofthejar.But iftheupper attracting -plate bekeptforseveral minutes atany distance, differing byafewturns ofthescrew, from thatwhichbrings thehair toitssighted position,theelectricity creeps alongthe inner unconnected surface oftheglasssoastodiminish the chargeoftheinner metalliccoating,orincrease it,according asthedistance istoogreatortoosmall. Ifthen quicklythe screw beturned andtheearth-reading taken, itisfound smaller orgreater,asthecasemay be,thanpreviously ;butafter afew minutes more itreturns toitspreviousvaluevery approxi- mately.Error from thissource maybepracticallyavoided by takingcarenever toallow thehair toremain formore thana fewminutes farfrom itssighted position;never sofar, for instance, asabove thecentre oftheupper,orbelow thecentre ofthelowerspot. I]OnElectrometers and Electrostatic Measurements. 305 '(3)Athird source oferror arises from changeoftempera- tureinfluencingtheindications. Inmost oftheinstruments hitherto made Ihavefound that thewarmth ofthehandpro- duces inafewminutes averynotable augmentationofthe earth-reading (asitwere anincreasedchargeinthejar) ;but inthelastinstrument which Ihave tested(White,No.18)I findthereverse effect, theearth-reading becomingsmaller as theinstrument iswarmed, orlarger when itiscooled. Ihave ascertained that thesechangesarenotdue tochangesinthe electriccapacitiesoftheLeyden jars ;and Ihave found that thechange,ifany,ofspecificinductivecapacityofglass by changeoftemperatureisexcessively small, incomparisonto whatwould berequiredtoaccount forthetemperatureerrors ofthese instruments, whichprobablymust bedue tothermo- elasticpropertiesoftheplatinum wire, orofthestretching- springs,orofthealuminium balance-lever, ortoacombination ofthe effectsdepending onsuchproperties;but Ihave en- deavoured invain, forseveralyears, andmademany experi- ments, todiscover theprecisecause. Itsurelywillbefound, andmeans invented forremedyingtheerror,nowwhen Ihave aninstrument inwhich theerror isintheoppositedirection to that ofmost oftheother instruments. Itisofcourse much greaterinsome instruments than inothers: insome itisso great that theearth-readingisvaried byasmuch astwenty divisions bythewarmth ofthehand inthecourse offiveor tenminutes aftercommencingtousetheinstrument,ifithas beenpreviouslyforsome time inacoldplace.Itsinfluence maybeeliminated, notquite rigorously, butnearly enoughso formostpractical purposes, byfrequently taking earth-readings (§375)andproceeding accordingtothedirections of§376. (4)Afourth fault intheportableelectrometer is,thatthe diameter oftheguard-plate andupper attracting disc,which ought tobeinfinite, arenotsufiiciently great,inproportionto thegreatest distance between them, torender thescalequite uniform initselectric valuethroughout. Acareful observer will,however, remedythegreater partoftheerrorduetothis defect, bymeasuring experimentallytherelative(orabsolute) values ofthe scale-division indifferentpartsoftherange. Therewill,however, remain uncorrected someirregularity, due T.E. 20 306OnElectrometer's and Electrostatic Measurements,[xx. toinfluence ofthedistribution ofelectricityovertheuncoated inner surface, intheinstruments ashitherto made, inallof which theinner surface ofthejariscoated with tinfoilonly below theguard-plate,sothattheuppersurface oftheguard- platemaybeseenclearly,inorder that theobserver may alwaysseethat allisinorder about thealuminiumsquare and aperture roundit;andparticularlythat there arenoinjurious shreds orminute fibres. Buttheirregularinfluence ofthe electrification oftheuncoatedglass,iffound sensible, willbe rendered insensible bycontinuingthe tinfoilcoatinganinch above theuppersurface oftheguard-plate. 378. Allfaults, exceptthetemperature error, dependonthe smallness oftheinstrument; and iftheobserver chooses to regardasportableaninstrument ofthirtycentimetres(ora foot) diameter, with allother dimensions, and alldetails of construction, thesame asthose oftheinstrument described above, hemayhave aportableelectrometerpracticallyfree from three ofthefour faults described. Itisscarcelyto beexpectedthat asmall instrument(12Jcentimetreshigh, and8Jcentimetres indiameter) whichmaybecarried about in thepocket canbefreefrom such errors. Buttheyareso farremedied astobeprobablynotperceptible,inthelarge stationary instrument which Inowproceedtodescribe. Standard Electrometer. 879. This instrument(figs. 12,13,and 14,PlateII.)differs- from theportable electrometeronlyindimensions, andincer- tainmechanical details, which arearrangedtogive greater accuracy bytaking advantageoffreedom from theexigencies ofasmallportable instrument. Itisatpresentcalled thei standard electrometer, inanticipationofeither remedying,oil oflearningtoperfectlyallow for,thetemperature error, and oil finding bysecularexperiments ontheelasticityofmetals, that* theirproperties used intheinstrument aresatisfactoryasre- gardsthepermanence fromyeartoyear,andfromcenturytc century,ofthe electric value ofitsreading.Itisaninstru- mentcapableofbeing applied with greatease toveryaccurate I ]OnElectroineters and Electrostatic Measurements. 307 measurements ofdifferences ofpotential,interms ofitsown unit. Thevalue oftheunit foreachsuch standard instrument ought,ofcourse, tobedetermined with thegreatest possible accuracyinabsolute measure;anduntil confidence canbefelt astoitssecularconstancy,determinations shouldfrequently bemadebyaidoftheabsolute electrometer. 380. TheLeyden jarofthestandard electrometer consists ofalargethinwhite-glass shade coated inside andoutside to within 6centimetres ofitslip,andplacedovertheinstrument asanordinary glass shade, toprotect against dust, currents of ail',andchangeofatmosphere.Itmayberemoved atpleasure fromthecast-iron soleoftheinstrument, andthen theinterior works areseen, consistingof— (1)Acontinuous disc ofbrasssupportedonaglass stem, inprolongationofastout brass rodortubesliding vertically inVs,inwhich itiskeptbyaspring [better bytwosprings (§369)], andrestingwith itslower flatendontheupperend ofamicrometer screw shaft, shown infig.13,where thescrew, graduated circle, andstout brass rodareasseen intheinstru- ment;themanner inwhich thelower endoftherodortube isconstructed tokeeptheround upperendofthescrew-shaft inpositionisshown insection infig.14. (2)Restingonthreeglass columns, aguard-plate with a square apertureinitscentre, andcarryingonitsupperside thestretching springs andthinplatinumwiresuspensionofan aluminium balance-lever, shapedlikethose ofthegauge (§353) andtheportable (§368) already described, butsomewhat larger. Thetopsofthethreeglass columns arerounded; around holeandashort slot inlinewith thishole arecutin theguard-plate, andreceive therounded ends oftwoofthe columns, which aresomewhatlongerthan thethird. The flat smooth lower surface oftheguard-platerestssimply onthe topofthethirdglass column. Thediameter oftheround hole andthebreadth ofthe slotintheguard-plate maybeabout -j^ofthediameter ofcurvature oftheupper hemispherical rounded ends oftheglass columns, sothatthebearing portions oftherounded ends intheround holeand intheslotrespec- tivelymaybeinclined somewhere about 45"totheplaneofthe 20—2 808OnElectrometers and Electrostatic Measurements,[xx. plate. This well-known buttoooften neglected geometrical arrangement gives perfectsteadiness tothesupported plate, withoutputting anytransverse strain uponthesupporting glass columns, such aswasalmost inevitable, andcaused the breakageofmany glass stems, before themental inertiaopposing deviations from theordinary instrument-maker's plan (ofscrew- ingtheguard-platetobrass mountings cemented tothetopsof theglass columns) wasovercome. Ithasalsotheadvantage ofallowingtheguard-platetobelifted offandreplacedina moment. (8)Principalelectrodeprojectingdownwardsthroughahole inthesoleoftheinstrument, andrigidly supportedfrom above byabrass mounting cemented tothetopofathick vertical glass column, connected byalight spiral springwith thelower attracting platemoved upanddown bythemicrometer screw. Theaperture round theprincipalelectrode maybeordinarily stopped byaperforated column ofwellparaffinedvulcanite projecting some distance above andbelow theaperture,which Ifind toinsulateextremely well, even inthesmoky, dusty, and acidulated atmosphereofGlasgow. When anextremely perfectinsulation oftheprincipalelectrode and connected attracting plateisrequired,thevulcanitestopper surrounding itmaybewithdrawn from theaperture,sothat theonlycom- munication between theelectrode andthecase oftheinstru- mentmaybealongthetwoglass columns intheartificially dried interioratmosphereofthecase;butfrom daytoday when theinstrument isout ofuse,theapertureround the principalelectrode should bekept carefully stopped,ifnotby avulcanite insulator byaperforated cork; (althoughIfind but little lossofinsulation, eitheralongtheinnerglasssurface oftheLeyden jaroralongthethreeglass columns, when thif precautionisneglected). (4)Temporary charging-rodenclosed inandsupported byi verticalinsulating column ofparaffined vulcanite, oraglasj tube well varnished outside andthickly paraffinedinside Thisinsulating columnbearingthecharging-rodisturne( round tillahorizontalspring projectingfrom itsdipper em touches theinnercoatingofthejar,when this istobechargec fromanindependent source, orwhen, foranyotherexperimehta XX.] OnElectrometers and Electrostatic Measurements. 309 reason, itistobeputinconnexion with aconductor outside thecase oftheinstrument. (5)Asmall replenisherofthekind described forthequad- rant electrometer (§352),butwithmuch widerair-spacesto prevent discharge bysparks. (6)Alarge glassorlead dish tohold aslargemasses of pumiceasmay be,which aretobekept sufficiently impreg- nated with strong sulphuricacid. 381.Aconsiderable portionofthejarabove theguard-plate isleftuncoated toallow theobserver toseeeasilythehairand white backgroundwith black dots; also several other smaller partsoftheglassabove theguard -plateare leftuncoated to admitlighttoallow asmall circular levelontheuppersideof theguard-platetobeseen. Thelongarm ofthealuminium balance-lever isverythoroughly guarded bydoublecages and fences ofwire(§370),sothat itcanexperience nosensible influence from electric disturbingforceswhen thecovering jar isputinpositionand electric connexion isestablished between itsinner coatingandtheguard-plate byprojectingflexible wires orslipsofmetal. 382. Thealuminium square plateissomewhatlarger, and theplatinum bearingwiresomewhatlongerinthisinstrument than intheportable electrometer, torender itsensible tosmaller differences ofpotential.Thestepofthescrew isthesame as intheportable (J^ofaninch), andonedivision(yj^jofthe circumference ofthescrew-head) correspondstoadifference of potentials which, roughly speaking,isequaltoabout that ofa singlecellofDaniell's. The effectiverangeoftheinstrument isaboutsixtyturns ofthescrew, andtherefore about 6000 cells ofDaniell's. That oftheportableelectrometer isabout 15 turns ofthescrew(equivalenttoabout 5000cells). Neither ofthese instruments hassufficient rangetomeasure thepoten- tial towhich Leyden jarsarechargedinordinaryelectric experiments,orthose reached bytheprimeconductor ofa powerfulelectric machine. Thestationaryinstrument with its longscrew and itslarge plates now described, wouldgofar towards meetingthiswant ifitsaluminium leverandplatinum suspensionweremade onthesame scale asthose oftheport- able electrometer: but foraninstrument never wanted to 310OnElectrometers and Electrostatic Measurements,[xx. directlymeasure differences ofpotentials oflessthantwo or three thousandcells, theheterostatic(§385) principleisin generalnotuseful, andtherefore Ihave constructed thefollow- ingverysimpleidiostatic(§385)instrument, which isadapted tomeasure with considerableaccuracydiiferences ofpotential from4000 cellsupwards,toabout 80,000 cells. Long-range Electrometer. 383. Inthis(fig. 15,PlateVI.)thecontinuousattracting- plateisabove, andtheguard-plate withaluminium balance below, asintheportableelectrometer;but,asinthestandard stationary electrometer, theupper plateisfixedandthelower plateismoved upanddown byamicrometer-screw. The mechanism ofthescrew and slide has allthesimplicity and consequent accuracyofthat ofthestandard electrometer. In theonlylong-rangeinstrumentyetconstructed thestepofthe screw isthesame asthat oftheothers(-^g-ofaninch). In future instruments itwould bewelleither tohave alonger step ortohave asimple mechanism(which canbeeasily added)to giveaquickmotion;asintheuseofthepresent instrument, theturningofthescrewrequiredforgreat changesofthe potentialmeasured isverytedious. Theguard-plate projects bymore thananinch allround beyondtherimoftheupper attracting-plate;partlytoobviate thenecessityofgivingita thick rim,which would berequiredtoprevent brushes and sparksfromoriginatinginit,ifithadonlythesame diameter asthecontinuousplate above, andpartlytoguard theobserver fromreceivingasparkorshock inmeasuringthepotentialoi anelectric machine orofaLeyden battery, andtopreventhis hairfrombeingattracted totheupper plate. Thus theguard- plateisallowed tobenothicker than suffices forstiffness, and this allows theobserver toseethehair attheend ofthe aluminium balance-lever without theleverbeing made ofs dynamically disadvantageous shape,aswould benecessaryi: theguard -platewere thick, orhadathick rimadded toit Noglasscase isrequiredforthisinstrument. The smallnesf oftheneedle andthegreatnessoftheelectric forceactingor 'I ^^a]OnElectrometers and Electrostatic Measurements. 311 aresuch that Ifind inpractice nodisturbance toanyincon- venientdegree byordinarycurrents ofair;althoughitand all these attracted discinstruments show theinfluence ofsudden changeofbarometricpressure,such asthatproduced byopen- ingorshuttingadoor. Ifnotkeptunder aglass shadewhen outofuse,thelower surface oftheupperattracting-plate, and thelower surface oftheguard-plate andattracted aluminium square,should becarefullydustedbyadrycoolhand. Gene- rally speaking,none ofthevital electricorgansofanelectro- meter should betouched byacloth, asthis isalmost sure to leave shreds fatal totheirhealthyaction. [(Addition, 1870)Iintend tocover thewhole instrument with aglass shade, wellvarnished over alarge space round an apertureinitstop,intowhich aninsulated electrode forthe upper platewillbecemented: because with theinstrument openasitisatpresent great difficultyhasbeenexperiencedin measuring hightension onaccount ofdustandshreds which impairtheinsulation.] 384. The effectiverangeofthisinstrument isabout 200 turns ofthescrew. Rathergreaterforce oftorsion isgiven than intheportable electrometer, andarather smaller attracted disc maybeused, sothatupwardsoffour cellsmaybetheelectric value ofone division. Theinstrument initspresentstate measuresnearlybutnotquitethehighest potentialIcan ordinarily produceintheconductor ofagoodWinter's electric machine, which sometimesgives sparks andbrushes afootlong. 385.The classification ofelectrometersgiven above isfounded ontheshapeandkinematic relations oftheir chieforganic parts ;but itwillberemarked thatanotherprincipleofclassi- fication ispresented bythedifferent electricsystems used in them, whichmaybedivided intotwoclasses :— I.Idiostatic, that inwhich thewhole electric forcedepends ontheelectrification which isitself thesubjectofthetest. II. Heterostatic, inwhich, besides the electrification tobe tested, another electrification maintained independentlyofitis takenadvantageof Thus, forexample,thelong-rangeelectrometer(§§383,384) issimply idiostatic, and isnotadaptedforheterostatic use;but hofthemmaybeusedidiostatically. Theabsolute electro- I 312OnElectrometers and Electrostatic Measurements, [xx meter wasatfirstsimplyidiostatic(§§358-362); morerecently ithasbeen used heterostatically, and isabout toacquire (§363) special organs adaptedforheterostatic use;asyet,however, no speciesoftheabsolute electrometer promising permanencehas come intoexistence. [See§§364-367 describingaheterostatic absolute electrometer ofaspecieswhich(Jan. 1871) promises tobepermanent.] 386. Itisinstructive totrace theoriginofvarious hetero- static speciesofelectrometers bynatural selection. Abody hanging,orotherwisesymmetrically balanced, inthemiddle ofasymmetricalfield offorce, butfree tomove inonedirec- tion ortheother inalinetangeutialtoaline offorce, moves iaonedirection ortheopposite when electrifiedpositivelyor negatively. Bohnenberger's arrangementofthiskind hasa convenient andapproximatelyconstant field offorce; and his instrument waschosen inpreferencetoothers whichmayhave been equally sensitive, butwere lessconvenient andconstant, and itbecame apermanent species. 387. Bennet'sgold-leaf electroscope,constructed with care tosecure good insulation, electrifiedsufficientlytoproducea moderate divergence,hasbeen often used totest,byaidofthis electrification, thequalityoftheelectrification ofanelectrified body broughtintotheneighbourhoodofitsupper projecting electrode, causing,ifitselectricityisofthesamesignasthat ofthegold leaves, increase ofdivergence ;ifoftheopposite sign,diminution. Byconnectingtheupperelectrode with the inner coatingofaLeyden jarwith internalartificiallydried atmosphere,thechargeofthegoldleavesmaybemade tolast with little lossfromdaytoday;andbyinsulating Faraday's metal cage (§342)round thegold leaves, andalternatelycon- nectingitwith theearth andwithaconductor whose difierence ofpotentialsfrom theearth istobetested, anincrease ora diminution ofdivergenceisobservedaccordingasthis differ- ence isnegativeorpositive,thegoldleavesbeing positive. Hence (throughPeltier's andDelmann's forms) theheterostatic stationaryandportable repulsion electrometers, described (§§274-277, 263 above)intheKoyalInstitution Lecture on"Atmospheric Electricity," and inNichol'sC?/clopcedia, article''Electricity, Atmospheric," alreadyreferred to,ofwhich I .]OnElectrometers and Electrostatic Measurements. 313 onespeciesstillsurvives inKing's College, Nova Scotia, andin theNatural PhilosophyClassroom ofEdinburgh University. Thesame form oftheheterostatic principle appliedtoSnow Harris's attracted disc electrometer gavetheportableand standard electrometers described above. 388.Amodification ofBohnenberger's electroscope,inwhich thetwoknobs onthetwosides ofthehanging goldleafbecame transformed into halves ofa circularcylinder,with itsaxis horizontal andthegoldleaf hung onawire insulated in aposition coincidingwith its axis;-producingaspeciesde- signedfortelegraphic pur- poses,butwhich didnot ac- quire permanence bynatural selection, and isonlyknown toexist inone fossilspecimen. Inthisinstrument thewire bearingthegoldleafwasconnected with acharged Leyden jar,andthesemi-cylinders with thebodies whose difference of potentialwas tobetested. Butvarious modifications ofthe divided-cylinderordivided-ringclass with theaxis vertical andplaneofmotion horizontal have donesomepractical work, andonespecies,thenewquadrantelectrometer(§346), pro- mises tobecomepermanent. 389. The heterostaticprincipleinoneform orother is essential todistinguish betweenpositiveandnegative. As remarked above(§387), theoriginal typeofthisuseofitisto befound intheoldsystemoftestingthequalityofthecharge taken bythedivergingstraws orgoldleaves oftheelectroscopes used fortheobservation ofatmospheric electricity ;which was donebybringingapieceofrubbedsealing-waxintotheneigh- bourhood, andobserving whether thiscaused increase ordiminu- tionofthedivergence. Adoubt which still exists astothesign (§252)oftheatmospheric electricityobserved byProfessor Piazzi SmythonthePeak ofTeueriffe, isowingtotheimper- fection ofthiswayofapplyingtheprinciple.Itis,indeed. 314OnElectrometers and Electrostatic Measurements,[xx. electricity that therubbedsealing-wax acquires. And, again (§342),itisnotcertain thattheglasscaseenclosingthegold leaves, especiallyifveryclean andsurrounded byaverydry natural atmosphere,screens themsufficientlyfrom direct in- fluence ofthepieceofsealing-waxtomake sure that the divergence duetovitreouselectricitycould notbeincreased bythepresenceoftheresinouslyelectrifiedsealing-waxifheld nearer thegoldleaves than theupper projectingstem. 390.The heterostaticprinciple hasaverygreat advantage asregards sensibilityoveranysimpleidiostaticarrangement, inasmuch as,forinfinitelysmall differences ofpotentialtobe measured, theforce isasthesquaresofthedifferences inany idiostatic arrangement,but issimply proportionaltothediffer- ences ineveryheterostatic arrangement. XXI.ATMOSPHEEIC ELECTRICITY* !WAPPARATUS FOROBSERVING ATMOSPHERIC ELECTRICITYf. roceedings Literary andPhilosophical Society ofManchester, March8,1859.] 391.BrJoule readanextract fromaletter hehadsometime JOreceived from Professor W.Thomson.—"Ihavehadanap- paratusforAtmospheric Electricity putupontheroof ofmy lecture-room, andgotagoodtrial ofityesterday, whichproved mostsatisfactory.Itconsists ofahollow conductorsupported byaglassrodattached toitsown roof,withaninternal atmo- sphere keptdrybysulphuricacid :thelower endoftheglass rodisattached tothetopofaniron bar,bywhich thehollow inductor isheldabout twofeetabove theinclined roof ofthe milding. Acan,openatthetop,slides upanddown onthe ronbarwhichpasses throughahole inthecentre ofitsbottom, id,being supported byatube withpulleys,etc.below, can isilyberaised orlowered atpleasure. Awireattached tothe isulated conductorpasses throughawide hole inthebottom of lecan,and isheldbyasuitable insulatedsupportinside the milding,sothat itmaybeledawaytoanelectrometer below. ?omake anobservation, thewire isconnected with theearth, rhilethecan isup,andenvelopestheconductor—itsposition rhentheinstrument isnotinuse.The earth connexion is lenbroken, andthecan isdrawn down abouteighteeninches, [mmediatelytheelectrometer shows alargeeffect(fromfiveto fteen degrees onmydividedring electrometer, inthestate it jhanced tobein,requiring more thanonehundreddegreesof )rsion tobringitback tozero, inthefewobservations Imade), henthesurface oftheearth is(asusualwhen theskyiscloud- *Thetwo articles constitutmg thischapter were accidentally omitted from 3hapter XVI. tItwaswith theinsulated conductor ofanapparatusofthiskind after- mrds setupintheisland ofArran that theobservations described in§294 reremade. 316 Atmospheric Electricity. [xxi. less) negative,theelectrometer showspositive electricity. But when anegativecloud(natural,orofsmoke) passes over, the indication isnegative.The insulation issogoodthat the changes maybeobserved foraquarterofanhour ormore, and when thecan isputuptheelectrometer comessensiblytozero again, showing scarcely anysensible change when theearth connection ismade, before makinganew start." DrJoule stated thathehadrecentlywitnessedexperiments with Professor Thomson's newAtmospheric Electrometer, the merit ofwhich consisted initsextreme sensitiveness, and thefacilitywith which accurate observations could bemade with it. NOTES ONATMOSPHERIC ELECTRICITY*. [FromthePhilosophical Magazine, Fourth Series, Nov. 1869.] 392.Twowater-droppingcollectors foratmosphericelec- tricitywereprepared,andplaced,one atawindow ofthe Natural Philosophy Lecture-room, andtheother atawindow oftheCollegeTower oftheUniversityofGlasgow. Adivided rincr-electrometer wasused atthelast-mentioned station;an electrometer adaptedforabsolute measurement, nearlyinthe formnow constructed asanordinary house electrometer, was used inthelecture-room. Four students oftheNatural Philo- sophy Class, Messrs Lorimer, Lyon, M'Kerrow, andWilson, afterhaving perseveredinpreliminary experiments andarrange- ments from themonth ofNovember, devoted themselves with much ardour andconstancy during February, March, and Apriltothework ofobservation.During periodsofobserva- tion, atvarious times ofday, early and late,measurements were completedandrecordedevery quarter minute orevery half minute,—thecontinual variations ofthephaenomenon rendering solitary observations almostnugatory. During several hours eachday, simultaneous observation wascarried ononthisplanatthetwo stations. Acomparisonofthe *Read before theBritish Association, June, 1860. fxi.] Atmospheric Electncity. 317 esiilts manifested oftengreat discordance, andnever complete agreement.Itwasthus ascertained that electrification ofthe r,ifnotofsolidparticlesinthe air(which have noclaim exclusive consideration inthisrespect),between thetwo stations andround them, atdistances fromthem notvery greatincomparisonwith their mutual distance, waslargely operativeintheobserved phsenomena.Itwasgenerallyfound hatafter theindications hadbeennegativeforsome time at thstations, thetransition topositivetookplaceearlier by several minutes atthetower station(upper)than atthelecture- room (lower). Sometimes duringseveral minutes, preceded and followedbypositive indications, there werenegativein- dications atthelower, while there wereonly positiveatthe upper.Inthese cases thecircumambient airmust have con- tainednegative (orresinous) electricity. Ahorizontal stratum ofairseveral hundred feetthick overhead, containingasmuch positive electricity percubic foot asthere must have been of negative percubic foot oftheairabout theC.ollege buildings those occasions, wouldproduceelectrical manifestations at ;heearth's surface similar incharacter andamount tothose ordinarilyobservedduringfairweather. I393.Beccaria hasremarked ontherareoccurrence ofnegative atmosphericindicationsduringfairweather, ofwhich hecan onlyrecord sixduringaperiodoffifteenyearsofvery persever- ingobservation byhimself andthePrior Ceca. Onsome,if notall,ofthose occasions there wasasquallyand variable wind, changingaboutrapidly between N.E, andN.W. On severaldaysofunbroken fairweather inAprilandMayofthe present yeartheatmosphericindication wasnegative during shortperiods, andoneach occasion there wasasudden change ofwind, generallyfrom N.E. toN.W., W,,orS.W. Forinstance, ontheSrdofMay,after awarm, sunny, andverydryday,with agentleN.E. breeze, andslight easterlyhaze inthe air,Ifound, about 8.30 P.M., theexpected positive atmosphericindication. After dark(nearly anhourlater)itwassocalm that Iwasable tocarry anunprotectedcandle intotheopenairandmake an observation withmyportableelectrometer. TomysurpriseI found asomewhatstrong negative indication, which Iobserved forseveral minutes. Althoughthere wasnosensible wind in 318Atmospheric Electricity. [xxi. thelocalitywhere Istood*, Iperceived bytheline ofsmoke from ahighchimneyatsome distance thatthere wasadecided breeze fromW.orS.W.Alittle later agentle S.W.wind set inallround, andwith the aidofalantern Ifoundstrong positive indications, which continued aslongasIobserved. Duringallthistime theskywascloudy,ornearlyso.That reversed electric indications should often beobserved about the time ofachangeofwindmaybeexplained,withaconsiderable degreeofprobability,thus :-— 394. Thelower airuptosomeheight above theearth must ingeneralbemore orless electrified with thesame kind of electricityasthat oftheearth's surface;and, since thisreaches ahigh degreeofintensity onevery tree-topandpointed vegetable fibre, itmust therefore causealways more orlessof thephaenomenonwhich becomesconspicuousasthe" lightof Castor andPollux"known totheancients, orthe"fireofSt. Elmo"described bymodern sailors intheMediterranean, and which consists ofaflowofelectricity,ofthekindpossessed by theearth, intothe air.Hence infairweather thelower air must benegative, althoughtheatmospheric potential,even close totheearth's surface, isstillgenerally positive. But if aconsiderable area ofthislower stratum iscarried upwards intoacolumn overanylocality bywindblowinginwards from different directions,itseffectmayforatimepredominate,and giverisetoanegative potentialintheair,andapositiveelec- trification oftheearth's surface. 395. Ifthisexplanationiscorrect, awhirlwind(suchasis oftenexperiencedonasmall scale inhotweather) must diminish, andmay reverse, theordinary positiveindication. 396. Since thebeginningofthepresent month Ihavehadtwo orthree opportunitiesofobservingelectrical indications with myportableelectrometer during daythunder-storms. Icom- menced theobservation oneach occasion afterhavingheard thunder, and Iperceived frequent impulsesontheneedle which caused ittovibrate, indicating sudden changesofelec- tricpotentialattheplace where Istood. Icould connect the largeroftheseimpulses with thunder heard some timelater, *About fivemiles south ofGlasgow. II.] Atmosphenc Electricity. 319 withabout thesamedegreeofcertaintyasthebrighterflashes oflightning duringathunder-storm bynightareusuallyre- cognisedasdistinctly connected with distinctpealsofthunder. Bycountingtime Iestimated thedistance ofthedischargenot nearer onanyoccasion than about four orfivemiles. There were besides manysmallerimpulses ;andmostfrequentlyI observed several ofthese between oneofthelargerandthe thunder withwhich Iconnected it.Thefrequencyofthese smaller disturbances, which sometimes kepttheneedle ina constant state offlickering,oftenprevented mefromidentify- ingthethunder inconnexion withanyparticularoneofthe impulsesIhadobserved.Theydemonstrated countless dis- charges,smaller ormore distant than those thatgiveriseto audible thunder. Onnone ofthese occasions have Iseenany lightning. Theabsolutepotentialatthepositionoftheburn- ingmatch wassometimespositiveandsometimesnegative ; andthesudden change demonstrated bytheimpulses onthe needle were, sofarasIcouldjudge,asoften augmentationsof positiveordiminutions ofnegative,asdiminutions ofpositive oraugmentationsofnegative.This afternoon, forinstance (Thursday, June28),Iheard severalpealsofthunder, andI found theusualabrupt changesindicatedbytheelectrometer. Forseveral minutes theabsolutepotentialwassmallpositive, withtwoorthreeabrupt changestosomewhatstrong positive, falling back toweakpositive,andgathering againtoadis- charge.Thiswasprecisely what thesame instrument would have shownanywherewithin afewyardsofanelectrical machine turnedslowlysoastocause aslow succession of sparks from itsprime conductor toaconductor connected with theearth. 397. Ihaverepeatedlyobserved theelectricpotentialinthe neighbourhood ofalocomotive engineatwork onarailway, sometimesbyholdingtheportableelectrometer outatawindow ofoneofthecarriagesofatrain, sometimes byusingitwhile standing ontheengine itself, andsometimes whilestanding on theground beside the line. Ihave thus obtained consistent results, totheeffect thatthesteam from thefunnel wasalways negative, andthesteam from thesafety-valve always positive. Ihave observedextremely strongeffects ofeach class fromI 320Atmospheric Electricity. [xxi. carriages even farremoved from theengine.Ihave found strong negativeindications intheairafteranengine had dis- appeared round acurve, and itscloud ofsteam haddissolved outofsight. 398. Inalmost allpartsofalarge manufactorj^, withsteam- pipes passing through them forvariousheating purposes,I have found decided indications ofpositive electricity.In most ofthese localities there wassomeslight escapeofhigh- pressure steam, whichappearedtobetheoriginofthepositive indications. 399.'These phaenomena seem inaccordance withFaraday's observations ontheelectricityofsteam, which showedhigh- pressure steamescapingintotheairtobeingeneral positive, butnegative when itcarriedglobulesofoilalongwith it. VI-ryiy^ynS^,£K<*^ XXII.—NEWPROOF OFCONTACT ELECTRICITY. proceedings Literanj andPhilosophical Society ofManchester^ Jan. 21,1862.] Thefollowingextract ofaletter from Professor W.Thomson, D.,etc., tothePresident, wasread :— '400."About twoyears agoIWTote toyouthatametal bar, insulated soastobemoveable about anaxisperpendicularto theplaneofametalringmade uphalfofcopperandhalf of zinc, thetwohalvesbeingsolderedtogether,turns from the zinctowards thecopper whenvitreously electrified, andfrom thecoppertowards thezincwhenresinouslyelectrified. [See diagramof§270(4).] "Ifthecopperhalfandthezinc half oftheringareinsu- latedfromoneanother, and iftheyareconnected bymeans of wires withtwopiecesofonemetal maintained atanystated difference ofpotential byproper apparatusfordividingthe electro-motive force ofthetwoplatesofaDaniell's element into 100parts, from60to70ofthosepartsarerequiredtoreduce thezinchalfringandthecopperhalfringtosuch astate that themoveable barremains atrestwhether itiselectrified vitreouslyorresinously. "Ifthecopperhalfringisoxidized byheat, theamount of electro-motive forcethenrequiredtoneutralize thetwohalves ismuch increased. If,afteroxidizingthecopperonedayby heat, Ileave theapparatustillthenextday,the effect is generally diminished, though somethingofitstill remains. Afteragain heatingthecopper bylayingitforsome time ona red-hot ironheater andallowingittocool, Ifound the effect almostexactly 100parts.Ihavenodoubt thatbymakingthe coatofoxideverycomplete andthickenough, andbycleaning thezincperfectly,Ishallbeable togetconsiderablyabove the electro-motive force ofasingleDaniell's element. Iremembered perfectly whatyoutoldmealongtimeagoaboutheatingthe T.E. 21i 322 NewProof ofContactElectricity. [xxii. coppersofabattery andgettingastrong effect, forsome time equaltothat oftheDaniell'scell,when Itried the effect of oxidizingthecopper platebyheat. "Ibelieve there arealso electrical effects ofheatitself; so that ifonehalf ofaringofonemetal ishotandtheother iscold, theneedle willshow adifferenceaccordingasitis charged positivelyornegatively. "Fornearly twoyearsIhave feltquitesure that theproper explanationofvoltaic action inthecommon voltaicarrange- ment isverynear Volta's, which fellinto discredit because Volta orhisfollowersneglectedtheprincipleofconservation offorce. Inowthink itquitecertain thattwometalsdipped inoneelectrolytic liquidwill(when polarizationisdoneaway with)reduce twodrypiecesofthesame metals, when connected each toeachbymetallic arcs, tothesamepotential. "There cannot beadoubt that thewholethingissimply chemical action atadistance. Zinc andcopperconnected by ametallic arcattract oneanother fromanydistance. Sodo platinum plates coated withoxygenandhydrogen respectively. Icannow telltheamount oftheforce, andcalculate howgreat aproportionofchemicalaffinityisusedupelectrolytically, before twosuch discscome withinjoVo*^^^^^^"^^ ^^^^^ another, oranyless distance down toalimit within which molecular heterogeneousness becomes sensible. This, ofcourse, willgiveadefinite limit forthe sizes ofatoms, orrather, asI donotbelieve inatoms, forthedimensions ofmolecular structures."[Inanarticle onthe"Size ofAtoms" published in"Nature" forMarch 31,1870,ithasbeen shown, bythe principleofreckoninghereproposed,that"platesofcopper "andzinc ofathree-hundred-millionth ofacentimetre thick, " placedclosetogether alternately, form anearapproximation "toachemical combination,ifindeed such thinplatescould "bemade withoutsplitting atoms."] :ill.—ELECTROPHORIC APPARATUS, ANDILLUSTRATIONS OFVOLTAIC THEORY. ONASELF-ACTING APPARATUS FORMULTIPLYING ANDMAIN- ITAININGELECTRIC CHARGES, WITH APPLICATIONS TOILLUS- TRATE THEVOLTAIC THEORY. [From theProceedings oftheRoyal SocietyforJune 20,1867.] 401. Inexplainingthewater-droppingcollector foratmo- spheric electricity,inalecture intheRoyalInstitution in1860 (§285,above),Ipointedouthow,bydisinsulatingthewaterjar andcollectingthedropsinaninsulated vessel, aself-actingelec- triccondenser isobtained.If,owingtoelectrified bodies inthe neighbourhood,thepotentialintheairround theplacewhere the stream breaks intodropsispositive,thedropsfallaway nega- tivelyelectrified;orvice versa, ifthepotentialisnegative,the dropsfallaway positivelyelectrified. Thestream ofwater descendingdoes notinanywaydetract from thechargesof theelectrified bodies towhich itselectric action isdue, pro- videdalwaysthese bodies arekept properlyinsulated;butby thedynamical energyoffluid-motion, andworkperformed by gravity uponthedescending drops, electricity maybeunceas- ingly produced onthesameprincipleasbytheelectrophorus. But, asintheelectrophorus therewasnoprovision except good insulation formaintainingthechargeoftheelectrified body orbodies from which theinductionoriginates,thiswant is supplied bythefollowing reciprocal arrangement,inwhich the bodycharged bythedropsofwater ismade theinductor for another stream, thedropsfromwhich intheir turnkeepup thechargeoftheinductor ofthe first. 402.Tostems connected with theinsidecoatingsoftwo Leydenphials areconnected metalpieces, which, toavoid cir- cumlocution, Ishall callinductors and receivers. Each stem bears aninductor andareceiver, theinductor ofthefirstjarbeing 21—2I 324Electrophone Apparatus, [xxin. verticallyover thereceiver ofthesecondjar,and vice versa. Each inductor consists ofavertical metalcylinder (fig. 1)open ateach end. Each receiver consists ofavertical metalcylinder Fig. 1.^P®^^*^^^^ ^^^>butpartially stoppedinitsmiddle byasmall funnel(fig. 1),with itsnarrow mouth pointing downwards, andsituated alittle above the middle ofthecylinder. Two fine vertical streams yofuninsulated water arearrangedtobreak into !drops,oneasnear asmaybetothecentre ofeach inductor. Thedropsfallalongtheremainder ofthe axisoftheinductor, andthence downwards, alongthe upper partoftheaxisofthereceiver oftheotherjar, untiltheymeet thefunnel. Thewater re-forms into dropsatthefinemouth ofthefunnel, which fallalong thelowerpartoftheaxis ofthereceiver andare carried offbyaproperdrain below theapparatus. Suppose nowasmallpositive chargeofelectricity be giventothe firstjar.Itsinductor electrifiesnega- i tivelyeachdropofwaterbreaking awayinitscentre from thecontinuous uninsulated water above;all bInductor.'thcscdrops giveuptheirelectricitytothesecondjar, whentheymeet thefunnel initsreceiver. Thedrops falling awayfrom thelower finemouth ofthefunnelcarryaway excessivelylittleelectricity, howeverhighlythejarmaybe charged ;because theplacewheretheybreak away is,asitwere, intheinterior ofaconductor, andtherefore hasnearlyzero elec- trification. Thenegativeelectrification thusproducedinthe secondjaracts,throughitsinductor, onthereceiver ofthe first jar,toaugmentthepositiveelectrification ofthe firstjar,and causes thenegativeelectrification ofthesecondjartogoon morerapidly, andsoon.Thedynamicalvalue oftheelectrifi- cations thusproducedisdrawn from theenergyofthedescend- ingwater, and isveryapproximately equaltotheintegralwork donebygravity againstelectric forceonthedrops,intheirpath from thepoint wheretheybreak awayfrom theuninsulated water above, tocontact with thefunnel ofthereceiver below. Inthe firstpartofthiscourse eachdropwillbeassisted down- wardsbyelectricrepulsionfrom theinductivelyelectrified water andtubeabove it;butbelow acertain pointofitscourse xxiir.] and Illustrations ofVoltaic Theory, 325 theresultant electric forceuponitwillbeupwards, and, ac- cordingtotheordinary wayofviewingthecompositionof electric forces, mayberegardedasbeingatfirstchiefly upward repulsionofthereceiver diminished bydownwardrepulsion from thewater andtube, andlatterlythesum ofupwardre- pulsionofthereceiver andupwardattraction oftheinductor. Thepotential methodgivestheintegral amount, beingtheex- cessofworkdoneagainstelectric force, above workperformed hy electric forceoneachdropinitswholepath.Itisofcourse equal tomY,ifmdenote thequantityofelectricitycarried byeach drop,asitbreaks from thecontinuous water above, andVthe potentialoftheinner coatingofthejarbearingthereceiver, thepotentialoftheuninsulated water being taken aszero. The practicallimit tothecharges acquirediswhen oneofthem issostrongastocausesparkstopassacross some ofthe separating air-spaces,ortothrow thedropsofwater outof theirpropercourse andcausethem tofalloutside thereceiver through which theyoughttopass.Itiscurious, after com- Fm. 2. mencing withnoelectricity exceptafeeblechargeinoneofthe jars,onlydiscoverable byadelicate electrometer, toseeinthe 326Electrophoric Apparatus, [xxiii. course ofafewminutes asomewhatrapidsuccession ofsparks passinsomepartoftheapparatus,ortoseethedropsofwater scattered about overthelipsofoneorboth thereceivers. 403. TheLeyden jarsrepresentedinthesketch(fig. 2)are open-mouthed jarsofordinaryflintglass, which, whenvery dry, Igenerallyfind toinsulateelectricitywithwonderfulperfection. Theinsidecoatingsconsist ofstrong liquid sulphuric acid,and heavyleadtripods with vertical stemsprojecting upwardsabove thelevel ofthe acid, which, byarmsprojecting horizontally above thelipofthejar,bear theinductors and receivers, as shown infig.2.Lids ofgutta perchaorsheet metal close the mouth ofeachjar,exceptasmallair-spaceoffromJto^of aninchround theprojectingstems. Ifatube(fig. 3)beadded Fig. 3. SASulphuric Acid. tothe lidtopreventcurrents ofairfromcirculatinginto the interior ofthejar,theinsulation maybesogoodthattheloss maybenomore thanonepercent, ofthewholechargeinthree orfourdays. Twosuchjarsmaybekeptpermanently charged fromyeartoyearbyveryslowwater-dropping arrangements, adropfrom each nozzle onceeverytwoorthree minutesbeing quitesufficient. 404.Themathematicaltheoryoftheaction, appended below*, isparticularly simple,butneverthelesscuriously interesting. *Let c,c'bethecapacitiesofthetwojars, Z,Vtheir rates oflossperunit IIII.] and IllustrationsofVoltaicTheory. 327 405.Thereciprocalelectrostatic arrangement nowdescribed presentsaninteresting analogytotheself-sustainingelectro- magnetic system recently broughtbefore theE-oyal Society by MrC.W.Siemens andProfessor Wheatstone, andmathemati- cally investigated byProfessor Clerk Maxwell. Indeed itwas from thefundamentalprincipleofthiselectromagnetic system thatthereciprocal partoftheelectrostatic arrangementoccurred tomerecently. Theparticularform ofself-acting electrophorus condenser now described, Ifirst constructed many years ago. Imaytake thisopportunityofdescribing anapplicationof ittoillustrate averyimportantfundamentalpartofelectric theory,Ihopesoon tocommunicate totheKoyal Societya descriptionofsome otherexperiments which Imade seven years agoonthesamesubject,andwhich Ihopenow tobeable toprosecutefurther. 406. Using onlyasingleinductor andasingle receiver, as shown infig.1,lettheinductor beputinmetallic communication withametal vessel orcistern whence thewater flows;and let thereceiver beputincommunication with adelicate electro- scopeorelectrometer. Iftheliningofthecistern andtheinner metallic surface oftheinductor bedifferent metals, anelectric effect isgenerallyfound toaccumulate inthereceiver and electrometer. Thus, forinstance, iftheinner surface ofthe potential ofcharge, perunit oftime, andD,D'thevalues ofthewater- droppers influenced bythem. Let+uand-v'betheir potentials attime t; Vand v'being ofone sign intheordinary useoftheapparatus described inthetext. Theaction isexpressed bythefollowing equations:— c-^^=I>v-lv; c-r-=Dt7-?V. dt dt Ifc,D, I,c',D',Vwere allconstant, thesolution ofthese equations would be, forthecase ofcommencing with the first jarcharged topotential 1,andthe second zero, _{c'p+V)e?t—(cV 4-V)e(yt ,_ePt_€<rt c(p-<r) c'{p-a) with thecorresponding symmetrical expression forthecase inwhich the second jarischarged, and the first atzero, inthebeginning; theroots of thequadratic {cx+l){c'x+V)-'DI>'= being denoted bypand a.When IV>DD', both roots arenegative ;and theelectrification comes tozero intime, whatever maybetheinitial charges. ButwhenW<DD',oneroot ispositive andtheother negative, andultimately thecharges augment inproportion toeptifpbethepositiveroot. 328Electrophoric Apparatus, [xxiii. Fig. 4.inductor bedrypolished zinc,andthevessel ofwater above be copper,thereceiveracquiresacontinually increasing charge ofnegative electricity. There islittle ornoeffect, eitherposi- tive ornegative,iftheinductorpresentasurface ofpolished coppertothedrops wheretheybreak from thecontinuous water above :but ifthecoppersurface beoxidized bytheheat ofalamp, until, instead ofabrightmetallic surface ofcopper,itpresents aslate-coloured surface ofoxide ofcoppertothedrops,these becomepositively electrified, asisproved bya continually increasing positive chargeexhi- bitedbythe electrometer. When theinner surface oftheinductor isofbrightmetallic colour, either zinc orcopper, there seems tobe little difference inthe effect whether itbewet with water orquite dry;also Ihave notfound aconsiderable differenceproduced bylining theinner surface oftheinductor with moist or drypaper. Copper filings fallingfrom acopper funnel andbreaking away from contact inthe middle ofazinc inductor, inmetallic commu- nication wdth acopper funnel, asshown infig.4, producearapidly increasing negative chargein asmall insulated cancatching them below. Thequadrant divided-ringelectrometer* in- dicating, bytheimageofalamponascale, angularmotions of asm.all concave mirror(iofagraininweight)such asIuse ingalvanometers,isveryconvenient forexhibitingthese results. Itssensibilityissuch that itgivesadeflection of100 scale- divisions (^ofaninch each) oneither side ofzero, asthe effect ofasinglecellofDaniell's;thefocusing, bysmall con- cave mirrors suppliedtomebyMrBecker, beingsogoodthat adeflection caneasilyberead withaccuracytoaquarterofa scale-division. ByadoptingPeltier's method ofasmallmag- netic needle attached totheelectric moveablebody (or"needle"), andbyusingfixed steelmagnetsoutside theinstrument togive directingforce(insteadoftheglass-fibre suspensionoftheaCopper Filings. 6Inductor— Zinc. cReceiver. *See Nichol's Eiwyclopcedia, 1860, article"Electricity, Atmospheric;" orProceedings oftheRoyal Institution, May 1860, Lecture onAtmospheric Electricity [§§249... 293, above]. :ili.]and IllustrationsofVoltaicTheory. 829 ivided-ringelectrometers described inthearticles referredto), andbygivingameasurable motion bymeans ofamicrometer screw tooneofthequadrants,Ihave afewweeksagosucceeded inmakingthisinstrument intoanindependent electrometer, instead ofamereelectroscope,oranelectrometer invirtue ofa separate gauge electrometer, asintheKewrecording atmo- spheric electrometer, described intheRoyalInstitution lecture. 407.Revertingtothearrangementdescribed above ofacopper selofwaterdischargingwater indrops from anozzlethrough aninductor ofzinc inmetallic connection with thecopper,let thereceiver beconnected withasecond inductor, thisinductor insulated;and letasecond nozzle, fromanuninsulated stream ofwater, discharge drops throughittoasecond receiver. Let thissecond receiver beconnected withathird inductor used to electrifyathird stream ofwater tobecaughtinathirdreceiver, andsoon.Wethushave anascendingscale ofelectrophorus actionanalogoustothebeautiful mechanical electricmultiplier ofMr.C.F.Yarley,with which, bypurelyelectrostatic induc- tion,heobtained arapidsuccession ofsparks fromanordinary singlevoltaic element. This result iseasilyobtainedbythe self-acting arrangement now described, with theimportant modification inthevoltaic elementaccordingtowhich nochemi- calaction iscalled intoplay,andwork donebygravityissub- stituted forworkdonebythecombination ofchemical elements. ONAUNIFORM ELECTRIC CURRENT ACCUMULATOR^ [From thePhilosophical Magazine, January 1868.] 408.Conceive aclosed circuit, CTABC,accordingtothe following description: —Oneportion^ofit,TA,tangentialtoa circular disc ofconductingmaterial andsomewhatlongerthan theradius;thecontinuation, AB,atright anglestothis in theplaneofthewheel, ofalength equaltotheradius;and thecompletionofthecircuit byafork,BC,extendingtoan axlebearingthewheel. Ifallofthewheel were cutaway exceptaportion, CjT,from theaxle tothepointofcontact at thecircumference, thecircuit would form asimple rectangle, GTAByexceptthe bifurcation ofthe sideBC.Leta 830 OnaUniformElectric[xxiii. fixedmagnetbeplacedsoastogivelines offorceperpen- dicular tothewheel, inthepartsofitbetween Gthecentre andTthepointofthecircumference touchedbythe fixed conductor; and letpower beapplied >^^^^nm^]^^^\tocause thewheel torotate inthe //' \\direction towards A.Accordingto // \\Faraday's well-knowndiscovery,a /'1current isinduced inthe circuit in // such adirection that themutual ''^^ electromagneticaction between itand thefixedmagnetresists themotion of thewheel. Now themutual elec- tromagneticforce between theportions ABandGTof the circuit isrepulsive, accordingtothewell-known elemen- tarylaw ofAmpere, and therefore resists theactual motion ofthewheel; hence, ifthemagnetberemoved, there will stillbeelectromagneticinductiontendingtomaintain the current. Letussupposethevelocityofthewheel tohavebeen atfirstnogreater than thatpracticallyattained inordinary ex- perimentswith Barlow'selectromagneticdisc. Asthemagnet isgradually withdrawn letthevelocitybegradually increased soastokeepthestrengthofthecurrent constant, and,when themagnetisquite away,tomaintain thecurrentsolely by electromagneticinduction between thefixedandmoveablepor- tions ofthe circuit. If,when themagnetisaway,thewheel beforced torotate faster than thelimiting velocityofourpre- vioussupposition,thecurrent willbeaugmented accordingto thelawofcompound interest, andwouldgoonthusincreasing without limit were itnotthattheresistance ofthecircuit would becomegreaterinvirtue oftheelevation oftemperature pro- duced bythecurrent. Thevelocityofrotation whichgivesby induction anelectromotive forceexactly equaltothatrequired tomaintain thecurrent, isclearly independentofthestrength ofthecurrent. Themathematical determination ofitbecomes complicated bythenecessityoftakingintoaccount thediffusion ofthecurrent through portionsofthediscnotinastraightline between GandT;but itisverysimple andeasyifweprevent this diffusion bycuttingthewheel intoaninfinite number of infinitelythinspokes,agreatnumber ofwhich aretobesimul- IXIII.]Current Accumulator. 331 neouslyincontact with the fixed conductor atT,The Hnearvelocityofthecircumference ofthewheel inthelimiting ebears tothevelocitywhich measures, inabsolute measure, eresistance ofthecircuit, aratio(determinable bythesolu- tion ofthemathematical problem)whichdepends onthepro- ^fcortionsoftherectangle GTAB, and isindependentofits ^Hbsolutedimensions. ^H.409.Lastly, supposethewheel tobekept rotatingatany ^Hpnstant velocity, whether above orbelow thevelocitydeter- mined bytheprecedingconsiderations;andsupposethecurrent tobetemporarilyexcited inanyway (forinstance, bybringing amagnetintotheneighbourhood andthenwithdrawing it); thestrengthofthiscurrent wdlldiminish towards zero orwill increase towardsinfinity, accordingasthevelocityisbelow or above the criticalvelocity. Thediminution oraugmentation would follow thecompoundinterest law iftheresistance inthe circuit remained constant. The conclusionpresentsuswith thiswonderful result :that ifwecommence withabsolutely no j^B|lectriccurrent andgivethewheel any velocityofrotation ^^^xceedingthe criticalvelocity,the electricequilibriumisun- stable :aninfinitesimal current ineither direction wouldaug- ment until, byheatingthe circuit, the electric resistance be- comes increased tosuchanextent that theelectromotive force ofinductionjustsuffices tokeepthecurrent constant. 410. Itwillbedifficult, perhaps impossible,torealize this result inpractice,because ofthegreat velocity required, andthe difficultyofmaintaining goodfrictional contact atthecircum- ference, without enormous friction, andconsequentlyfrictional generationofheat. 411. Theelectromagnetic augmentation andmaintenance of acurrent discoveredbySiemens, andputinpractice byhim, withtheaidofsoft iron,andproved byMaxwell tobetheoreti- cally possiblewithout soft iron, suggestedthesubjectofthis communication totheauthor, and ledhim toendeavour to arrive atasimilar result withonlyasingle circuit, andnomaking andbreakingofcontacts;and itisonlythese characteristics that constitute thepeculiarityofthearrangement which he now describes. 332 OnVolta-ConvectionhyFlame.[xxiii. Fig. 1.ONVOLTA-CONVECTION BYFLAME. [From thePhilosophical Magazine, January 1868.] 412. InNichol'sCyclopcedia,article" Electricity, Atmo- spheric" (2d edition), and intheProceedings oftheRoyal Institution May 1860 (Lecture onAtmospheric Electricity), [§§249... 293,above]theauthor hadpointedoutthat the effect oftheflame ofaninsulatedlampistoreduce the lamp and otherconductingmaterial connected with itto thesamepotentialasthat ofthe airintheneighbourhood oftheflame, andthattheeffect ofafinejetofwater froman insulated vessel istobring thevessel andotherconducting material connected with ittothesamepotentialasthat ofthe airatthepoint where thejet breaks intodrops.Inarecent communication totheRoyal Society"OnaSelf-acting Ap- paratusforMultiplying and MaintainingElectricCharges, withapplicationsto illustrate theVoltaicTheory," [§§401... 407, above,]anexperimentwas described inwhich awater- dropping apparatuswasem- ployedtoprovethe difference ofpotentialinthe air,inthe neighbourhoodofbrightmetallic I isisi— IL.-isi^ surfaces ofzine and copper \j' Imetallicallyconnected with oibe ^^ I another, which istobeexpected from Volta'sdiscoveryofcontact- electricity.Inthepresent com- munication asimilarexperiment isdescribed, inwhich theflame of aspirit-lampisused instead ofajetofwaterbreakingintodrops. 418.Aspirit-lampisplaced onaninsulated stand connected Avith averydelicate electrometer.Copperand zinccylinders, inmetallic connection withthemetal case oftheelectrometer, arealternatelyheldverticallyinsuch apositionthat the I III] OnVolta-ConvectionbyFlame. 3:'^3 flame burnsnearlyinthecentre ofthecylinder,which isopen atboth ends. Iftheelectrometerreading,with thecopper cylinder surroundingtheflame, iscalled zero, thereading observed with thezinccylinder surroundingtheflame indicates positiveelectrification oftheinsulated standbearingthelamp. 414. Itistoberemarked thatthedifferential method here followed eliminates theambiguityinvolved inwhat ismeant by thepotentialofaconducting system composed partlyofflame, partlyofalcohol, andpartlyofmetal. Inamerelyillustrative experiment,which theauthor hasalready made, theamount of difference made bysubstitutingthezinccylinderforthecopper cylinder round theflame wasrather more than halfthe differ- ence ofpotentialmaintained byasinglecellofDaniell's. Thus, when thesensibilityofthequadrant divided-ringelectrometer (§406)wassuch thatasinglecellofDaniell'sgaveadeflection of79scale-divisions, thedifference ofthereading when thezinc cylinder wassubstituted forthecopper cylinder round thein- sulated lampwas39scale-divisions. From otherexperiments oncontact-electricity made sevenyears agobytheauthor, and agreeing with results which havebeenpublished byHankel, it appearsthatthedifference ofpotentialsintheairintheneigh- bourhood ofbrightmetallic surfaces ofzincandcopperin metallic connexion with oneanother isaboutthree-quartersof that ofasinglecellofDaniell's. Itisquitecertain thatthe differenceproducedinthemetal connected with theinsulated lampwould beexactly equaltothetrue contact difference of themetals,iftheinterior surfaces ofthemetalcylinderswere perfectlymetallic(freefrom oxidation oranyothertarnishing, such asbysulphur, iodine, oranyotherbody) ;providedthe distance oftheinner surface ofthecylinderfrom theflame were everywheresufficient topreventconduction byheated airbe- tween them, andprovidedthelengthofthecylinder were infinite(or,practically, anything more than three orfourtimes itsdiameter). 415. Theauthorhopesbeforelongtobeable topublisha completeaccount ofhisoldexperimentsoncontact-electricity, ofwhich aslightnoticeappearedintheProceedingsofthe Literary andPhilosophical SocietyofManchester[§400, ve].I S34 ElectricReplenisher. [xxiii. ONELECTKIC MACHINES FOUNDED ONINDUCTION AND CONVECTION. [From thePhilosophical Magazine, January 1868.] 416.Tofacilitate theapplicationofaninstrument, which I haverecently patented,forrecordingthesignalsoftheAtlantic Cable, asmall electric machinerunning easily enoughtobe driven bythewheelwork ofanordinary Morse instrument was desired; and Ihave therefore designedacombination ofthe electrophorus principlewith thesystemofreciprocalinduction explainedin[§§401...407]arecent communication tothe Royal Society {Proceedings, June 1867), which maybebriefly described asfollows :— 417.Awheel ofvulcanite, withalargenumber ofpiecesof metal(called carriers, forbrevity)attached toitsrim,iskeptro- tating rapidly round afixed axis. The carriers arevery lightly touched atoppositeends ofadiameter bytwofixedtangent springs. One ofthesesprings (the earth-spring)isconnected with theearth, andtheother(thereceiver-spring)withanin- sulatedpieceofmetal called thereceiver, which isanalogous tothe"prime conductor" ofanordinaryelectric machine. Thepointofcontact oftheearth-springwith thecarriers is exposedtotheinfluence ofanelectrifiedbody (generally anin- sulatedpieceofmetal) called theinductor. When this is negatively electrified, each carrier comes awayfrom contact with theearth-spring, carrying positive electricity, which itgives up, throughthereceiver-spring,tothereceiver. The receiver and inductor areeach hollowed outtoaproper shape, andarepro- perly placedtosurround, each asnearlyasmay be,thepointof contact ofih.Qcorresponding spring. 418. The inductor, forthegood workingofthemachine, should bekeptelectrified toaconstantpotential. This is effected byanadjunctcalled thereplenisher,which maybe appliedtothemain wheel, butwhich, foralarge instrument, oughttobeworked byamuch smaller carrier-wheel, attached either tothesame ortoanotherturning-shaft. 419. Thereplenisherconsistschieflyoftwoproperly shaped piecesofmetal called inductors, which arefixed intheneighbour- hood ofacarrier-wheel, such asthatdescribed above, andfour xxiil]ElectricReplenisher.335 fixedsprings touchingthecarriers attheends oftwodiameters. Two ofthesesprings (called receiver-springs)areconnected respectivelywith theinductors; andtheother two(called con- necting springs)areinsulated andconnected withoneanother (oneoftheinductors isgenerallyconnected withtheearth, and theother insulated). Theyaresosituated thattheyaretouched bythecarriers onemergingfrom theinductors, andshortlyafter Fig. 1. Section.. Elevation. thecontacts with thereceiver-springs.Ifanydifference of potential between theinductors isgiventobegin with, the action ofthecarriers, asiseasily seen, increases itaccordingto thecompound-interest lawaslongastheinsulation isperfect. Practically,inafewseconds afterthemachine isstartedrunning, brightflashes andsparks begintoflyabout invariouspartsof theapparatus,evenalthough theinductors andconnectors have beenkeptfordaysascarefully dischargedaspossible. Forty elements ofadrypile (zinc, copper, paper), appliedwith one poletooneoftheinductors, andtheother foramoment tothe connecting springsandtheother inductor, maybeused tode- termine, ortosuddenly reverse, thecharacter(vitreousor I 336 ElectricReplenisher. [xxiii. resinous)oftheelectrification oftheinsulated inductor. The onlyinstrumentyetmade isaverysmall one(withcarrier- wheel 2inches indiameter), constructed fortheAtlantic Fig. 2. Telegraph application ;but itsaction hasbeen sostartlingly successful thatgoodeffectmaybeexpectedfromlargermachines onthesameplan. 420.When thisinstrument isused toreplenishthechargeof theinductor intheconstant electric machine, described above, oneofitsowninductors isconnected withtheearth, andtheother with theinductor tobereplenished. When accurateconstancy XXIII.] Applications oftheElectricReplenisher. 337 isdesired, agange-electroscopeisappliedtobreak andmake contact between theconnector-springsofthereplenisher when thepotentialtobemaintained rises above orfallsbelow a certain limit. 421. Several usefulapplicationsofthereplenisherforscien- tificobservation wereshown bytheauthor attherecent meeting oftheBritish Association (Dundee),—among others, tokeepup thechargeintheLeydenjarforthedivided-ringmirror-elec- trometer, especially when thisinstrument isused forrecording atmospheric electricity. Asmallreplenisher, attached tothe instrument within thejar,isworked byalittle milled headon theoutside, afewturns ofwhich will suffice toreplenish the lossoftwenty-fourhours. Postscript, Nov. 23,1867. 422.Ashasbeen stated, thismachine wasplanned originally forrecordingthesignalsoftheAtlantic Cable. The small ''replenisher" representedinthediagramshasproved perfectly suitable forthispurpose. The firstexperiments onthemethod forrecording signalswhich Irecently patented weremademore thanayearagobyaidofanordinary plate- glassmachine worked byhand. Thisdaythesmall" replenisher"hasbeen connected with thewheelworkdrawingtheMorsepaper onwhichsignals arerecorded, and, withonlytheordinary driving-weightas moving power,hasproved quitesuccessful. 423.The scientificapplications indicated when thecommuni- cation wasmade totheBritish Association havebeen tested with- inthelastfewweeks, andespecially to-day,with theassistance ofProfessor Tait. Thesmallreplenisherisnowmade aspart ofeachquadrant electrometer. Itispermanently placedinthe interior oftheglassLeyden jar ;andafewturnsbythefinger appliedtoamilled head ontheoutside ofthe lidarefound sufficient toreplenishthe lossoftwenty-fourhours.Asmall instrument hasalsobeenmade andtested forputtinginprac- ticetheplanofequalizing potentials, describedverballyinthe communication totheBritish Association, which consisted ina mechanicalarrangementtoproduceeffects ofthesame char- acter asthose ofthewater-dropping system,described several T.E. 22i 338 Potential- Equalizer. [xxiii. years agoattheRoyalInstitution*. Theinstrument isrepre- sented intheannexed sketch(fig. 3).ATandAT aretwo springs touching acircular row ofsmall brasspegs-|- insulated from oneanother inavulcanite disc. Thesespringsareinsu- lated, oneorboth,andareconnected with thetwoelectrodes of Fig. 3. theelectrometer —oroneofthem withtheinsulatedpartofthe electrometer, andtheother with themetalenclosingthecase when there isonlyoneinsulated electrode. Oneapplicationis totestthe" pyro-electricity"ofcrystals ;thusacrystaloftour- maline, PiV,bymeans ofametal armholdingitsmiddle, issup- ported symmetricallywith reference tothedisc inaposition paralleltothelineTT',andjoiningthelines ofcontact ofthe springs. When warmed(asisconveniently donebyametal plateataconsiderable distance fromit),itgivesbyordinary tests, asiswellknown, indications ofpositiveelectrification to- *Lecture onAtmospheric Electricity, Proceedings oftheRoyal Institution, May1860. Seealso^'icho^s Cyclopedia,article "Electricity, Atmospheric" [§§249... 293]. t[Inow findasmaller number oflarger discs tobepreferable, asconsider- able disturbances areproduced bythenumerous breakings ofcontact unless thetwospringsareinprecisely thesame condition astoquality and clean- ness ofmetal surface. Thin stiffplatinum pins attached tothediscs, and veryfineplatinum springs touching them asthey pass,willprobably give good and steadyresults ifthesprings arekept very clean. The smallest quantityoftheparaffin (with which, asusual inelectric instruments, the vulcanite iscoated),ifgetting oneither spring, would probably produce im- mense disturbance.— December 23,1867.] XXIII.] Applications ofPotential-Equalizer. 339 wards theoneendP,andofnegativeelectrification towards the other endN.Thewheel inthearrangement nowdescribed is keptturningatarapidrate;andtheeffect ofthecarrier isto produceinthesprings TA,T'A' thesamepotentials, approxi- mately,asthose which would exist intheairatthepoints T,T' ifthewheel andspringswere removed. Thesprings being connected with theelectrodes ofthedivided-ring quadrant electrometer, thespotoflightisdeflected totheright,letus say.After continuingtheapplicationofheat forsome time thehotplateisremoved, andalittle later thespotoflight goes tozeroandpassestothe left,remainingthere foralong time, andindicatingadifference ofpotentialsbetween thesprings,in thedirection A'TpositiveandATnegative. Theelectrometer beingofsuchsensibilityastogiveadeflection ofabout 100 scale-divisions totherightorleftwhen tested byasingle gal- vanic cell,andhavingarangeof300 scale-divisions oneach side, itisnecessarytoplacethetourmaline atadistance of several inches from thedisc tokeeptheamount ofthedeflec- tionwithin thelimits ofthescale. 424. Anotherapplicationofthisinstrument isforthe experimental investigationofthe voltaictheory, according tothegeneral principledescribed[§406]inthecommuni- cation totheKoyal Society alreadyreferred to.* Inittwo inductors areplacedasrepresentedinfig.4.The inner Fig. 4. surface ofeach ofthese isofsmooth brass;andoneofthem islinedwholly,orpartially,with sheet zinc, copper, silver, orother metal tobetested. Thus, toexperiment upon thecontact difference ofpotentials between zincandcopper, I*Proceedings oftheRoyal Society, IMay 18G7. 22—2 3-iOApplications ofPotential-Equalizer. [xxiii. one ofthe inductors iswhollylined with sheet zinc or with sheetcopper, andthetwoinductors areplacedinme- tallic communication with oneanother. Thespringsareeacli inmetallic communication with theelectrodes ofthequadrant mirror electrometer, andthewheel iskept turning. Thespot oflightisobserved totakepositions ditfering, accordingasthe liningiszinc orcopper, by72Jpercent, ofthedifferencepro- ducedbydisconnectingthetwoinductors fromoneanother and connecting them with thetwoplatesofasingleDaniell's cell, when either thezincorthecopper liningisleftinoneofthem. These differences areveryapproximatelyinsimple proportion tothedifferences ofpotentials between thepairsoftheopposite quadrants oftheelectrometer inthedifferent cases. The dif- ference between theeffects ofzincandofcopperinthisarrange- ment isofcourse inthedirectioncorrespondingtothepositive electrification ofthequadrants connected with thespringwhose pointofcontact isexposedtothezinc-linedinducingsurface. Itmust beremembered, however, asistobeexpectedfrom Hankel's observations, thatthedifference measured willbemuch affected byaslight degreeoftarnishing byoxidation, orother- wise, oftheinner surface ofeither inductor. When the coppersurface isbroughttoaslate-colour byoxidation under theinfluence ofheat, thecontact difference between itand polishedzincamounts sometimes, asIfound inexperiments made sevenyears ago [§400, above],to125, that ofasingle cellofDaniell's beingcalled 100. 425.Ausefulapplicationofthe little instrumentrepresented infig.4isfortestinginsulation ofinsulated conductors ofsmall capacity,asforinstance, shortlengths (2or3feet) ofsubmarine cable,when theelectrometer used issuch that itsdirectappli- cation totheconductor tobetested wouldproduceasensible disturbance initscharge, whetherthroughthecapacityoftheelec- trometer beingtoogreat,orfrom inductive effects duetomotion ofthemoveablepart,orparts, especiallyiftheelectrometer is "heterostatic" [§385].Inthisapplicationoneoftheinduc- tors iskeptinconnection with ametalplateinthewater sur- roundingthespecimenofcable tobetested;andtheother is connected with thespecimen,orissuccessively connected with thedifferent specimensunder examination. Thespringsare I XXIII.] OntheReciprocal Electrophorus.341 connected with thetwoelectrodes oftheelectrometer asusual. Thesmall constantcapacityoftheinsulated inductor, andthe practically perfectinsulation which may^vith easebesecured forthesingle glassorvulcanite stembearing it,aresuch that theapplicationofthetesting apparatustothebodytobetested produceseither nosensiblechange,orasmallchangewhich canbeeasilyallowed for. Itwillbeseenthatthesmall metal pegscarried awaybytheturning-wdieelfrom thepointofthe insulatedspring,inthearrangementlastdescribed, correspond preciselytothedropsofwaterbreaking awayfrom thenozzle inthewater-droppingcollector foratmospheric electricity. 426.Aformbearingthesame relation tothatrepresented inthedrawingsthataglass-cylinderelectric machine bears toa plate-glass machine oftheordinarykind willbemoreeasily made, andwillprobablybefoundpreferable, when thedimensions arenotsogreatastorender itcumbrous. Init,itisproposed tomake thecarrier-wheelnearlyafter thepatternofamouse- mill, with discs ofvulcanite instead ofwood for itsends. The inductor and receiver oftherotatory electrophorusor thetwoinductor-receivers ofthereplenisher, may,when this patternisadopted,bemeretangent planes;but itwillprobably befound better tobendthemsomewhat toacurvedcylindrical shapenotdiffering verymuch from tangent planes. When, however, great intensityisdesired, thebestpatternwillpro- bablybehadbysubstitutingforthecarrier-wheel anendless rope ladder, asitwere, with cross bars ofmetal andlongitudinal cords ofsilkorother flexibleinsulatingmaterial. This, byan actionanalogoustothat ofthechain-pump,willbemade to move withgreat rapidity, carrying electricity from aproperly placed inductor toaproperly shaped andproperly placedre- ceiver atadistance from theinductor which maybeasmuch tasseveral feet. ONTHERECIPROCAL ELECTROPHORUS, [From thePhilosophical Magazine, April 1868.] 427.Having beeninformedbyMr.FleemingJenkin thathe hadheard from Mr.Clerk Maxwell thattheinstrument which I described under thename" Replenisher,"inthePhilosophical 342 OntheBeciprocal Electrophorus. [xxiii. MagazineforJanuary 1868, wasfounded onpreciselythesame principleasaninstrument "forgenerating electricity"which hadbeenpatented someyears agobyMr. C.F.Yarley,I wassurprised ;forIremembered hisinductive machine which hadbeen somuch admired attheExhibition of1862, and whichcertainlydidnotcontain thepeculiar principleofthe "Replenisher."But Itooktheearliestopportunityoflooking intoMr.Varley's patent (1860), andfound, aswas tobeex- pected,thatMr.Maxwell wasperfectly right. Inthatpatent Mr.Varleydescribes aninstrumentagreeinginalmostevery detail with thegeneral descriptionofthe"Replenisher" which 1gaveinthe article ofthePhilosophical Magazine already referred to.Theonlyessential difference isthatnocontacts aremade inMr.Varley's instrument, but, instead, thecarriers pass,each atfourpointsofitscircularpath,within such short distances offourmetallicpiecesthatwhen asufficientintensity ofchargehasbeen reached, sparks passacross theair-intervals. Hence togiveacommencement ofaction toMr.Varley'sinstru- ment, oneoftheinductors must becharged fromanindepen- dent source toaconsiderablepotential (thatofseveral thousand cells forinstance),tomake surethatsparkswillpassbetween thecarriers andthemetalpiece (correspondingtooneofmy connecting springs) which itpassesunder theinfluence ofthat inductor. Inmy" Eepienisher,"however welldischargedit maybetobegin with, electrification enoughisreached after a fewseconds(onthecompoundinterestprinciple,withanin- finitesimalcapitaltobegin with)toproduce sparks andflashes invariouspartsoftheinstrument. InMr.Varley's instrument, whatcorrespondstomyconnector isdescribed asbeingcon- nected with theground;andtheeflect istoproduce positive andnegativeelectrification ofthetwoinductors. Inthis re- spectitagreeswith theself-acting apparatusformultiplying andmaintainingelectriccharges,described inacommunication totheRoyal SocietylastMay.* From thisarrangementI passedtothe'"' Replenisher"byusingawheel with carriers as asubstitute forthewater-droppers,andarrangingthat the connectors might beinsulated andoneoftheinductors con- *Proceedings oftheRoyal Society, 1867 ;or,Phil. Mag., November 1867. XXIII.] OntheBeciprocal Electrophorus. 343 nected with theearth, which, ofcourse, maybedone inMr. Varley's instrument, andwhich renders itidentical with mine, with theexceptionofthedifference ofspring-contactsinstead ofsparks.This difference isessential forsome oftheapplica- tions ofthe'' Replenisher," which Idescribed, andhavefound very useful, especiallythesmall internalreplenish er,forreple- nishing, when needed, thechargesoftheLeyden jarofmy heterostatic electrometers. But thereciprocal-electrophorus principle, which seemed tomeanoveltyinthecommunication totheRoyal SocietyandinthePhilosophical Magazinearticle oflastJanuaryreferred to,had,asInow find,been invented and published byMr.Varley long before, inhispatentof1860, when itwas, Ibelieve, reallynew toscience. 428. Postscript. —Glasgow College, March 20,1868.—In lookingfurther intoMr.Varley's patent,Ifindthathedescribes anarrangementformaking spring-contactsinstead ofthenarrow air-spacesforsparks,—andthatheuses thespring-contactsto enable him tocommence with averysmall difference ofpoten- tials,andtomagnify onthecompoundinterestprinciple. He even states thathecancommence with such adifference of potentials ascanbeproduced byasinglethermo-electric element, andbytheuseofhisinductive instrument canmultiplythisin ameasuredproportionuntilhereaches adifference ofpotentials measurable byanordinaryelectrometer. Thus itappearsthat his anticipationofallthat Ihavedone inmy"Replenisher"iseven morecompletethan Isupposed whenwritingthepreceding. 429. Second Postscript(1870).—Onhaving hadmyatten- tion called toNicholson's"Revolving Doubler," Ifindinitthe samecompoundinterestprincipleofelectrophoricaction. It seems certain that thediscoveryisNicholson's, andabout one hundredyearsold. Holtz's now celebrated electric machine, which isclosely analogousinprincipletoVarley'sof1860, is,I believe, adescendant ofNicholson's. Itsgreat power depends ontheabolition byHoltz ofmetallic carriers, andofmetallic make-and-break contacts. Itsinductiveprincipleisidentical with that ofVarley'searlier andmyown later invention. It differs fromVarley's andmine inleavingtheinductors tothem- selves, andusingthecurrent inthe"connecting"arc(§419), which, whensparksaretobeproduced,isbroken. XXIV.— AMATHEMATICAL THEORY OFMAGNETISM. [Abstract from theProceedings oftheBoijal Society, June 1849.] 430.Thetheoryofmagnetism was first mathematically treated inacompleteformbyPoisson. Brief sketches ofhis theory, withsomesimplifications, have been given byGreen andMurphyintheirworks onElectricity andMagnetism.In allthesewritingsahypothesisoftwomagneticfluids hasbeen adopted, andstrictlyadhered tothroughout. Nophysical evidence canbeadduced insupportofsuch ahypothesis; butonthecontrary,recent discoveries, especiallyinelectro- magnetism,render itextremely improbable. Hence itisof importance that allreasoningwith reference tomagnetism should beconducted without assumingtheexistence ofthose hypotheticalfluids. 431.Thewriter ofthepresent paperendeavours toshow thata complete mathematicaltheoryofmagnetism maybeestablished uponthe solefoundation offactsgenerally known, andCou- lomb'sspecial experimentalresearches. Thepositive partsof thistheory agreewith those ofPoisson's mathematicaltheory, andconsequentlytheelementarymathematical formulae coin- cidewith those which havebeenpreviously given byPoisson. Thepaperatpresentlaidbefore theRoyal Societyisre- stricted totheelements ofthemathematicaltheory, exclusively ofthosepartsinwhich thephenomenaofmagneticinduction areconsidered. Theauthorhopestohave thehonour oflayingbefore the Societyacontinuation, containing someoriginalmathematical investigationsonmagnetic distributions, andatheoryofinduc- tion, inferromagneticordiaraagneticsubstances. i XXIV.] AMathematical Theory ofMagnetism. 345 [Transactions oftheRoyal SocietyforJune 1849,andJune 1850.] Introduction. 432.The existence ofmagnetismisrecognised bycertain phenomenaofforce which areattributed toitastheir cause. Other physicaleffects arefound tobeproduced bythesame agency;asintheoperationofmagnetismwith reference to polarized light, recentlydiscovered byMrFaraday ;butwe must stillregard magneticforce asthecharacteristic ofmag- netism, and,howeverinterestingsuch other phenomena may beinthemselves, however essential aknowledgeofthemmay beforenablingustoarrive atany satisfactoryideasregarding thephysical nature ofmagnetism,and itsconnexion with the general propertiesofmatter, wemust stillconsider theinvesti- gationofthelaws, accordingtowhich thedevelopmentand theaction ofmagneticforce areregulated,tobetheprimary objectofaMathematical Theoryinthisbranch ofNatural Philosophy. 433.Magnetic bodies, whenputnearoneanother, ingeneral exertverysensible mutual forces;butabody which isnot magneticcanexperience noforce invirtue ofthemagnetismof bodies initsneighbourhood.Itmayindeed beobserved that abody, if,willexert aforceuponanother bodyA;andagain, onathirdbody B',although whenAandBarebothremoved toaconsiderable distance fromM,nomutual action canbe discovered between themselves;butinallsuch casesAandB are,when intheneighbourhoodofM,temporarily magnetic ; andwhen both areunder theinfluence ofMatthesame time, theyarefound toactupononeanother with amutual force. Allthese phenomenaareinvestigatedinthemathematical theoryofmagnetism, which, therefore, comprehends two dis- tinct kinds ofmagneticaction—themutual forces exercised between bodiespossessing magnetism,andthemagnetization induced inother bodiesthroughthe influence ofmagnets. The First Part ofthispaperisconfined tothemoredescriptive andpositivedetails ofthesubject,with reference totheformer class ofphenomena.After asufficient foundation hasbeen laidinit,bythemathematicalexpositionofthedistribution of magnetisminbodies, andbythedetermination andexpression 346 AMathematicalTheory ofMagnetism. [xxiv. ofthegenerallaws ofmagnetic force, aSecond Part willbe devoted tothetheoryofmagnetization byinfluence, ormagnetic induction. FIRST PART.— ONMAGNETS, ANDTHEMUTUAL FORCES BETWEEN MAGNETS. Chapter I.—Preliminary Definitions andExplanations. 434.Amagnetisasubstance whichintrinsically possesses magnetic properties. Apieceofloadstone, apieceofmagnetized steel, agalvanic circuit, areexamplesofthevarieties ofnatural and artificialmagnetsat present known;butapieceofsoftiron, orapieceofbismuth tem- porarily magnetized byinduction, cannot, inunqualified terms, be called amagnet.Agalvaniccircuit isfrequently,forthesake ofdistinction, called an"electro-magnet;" but,accordingtotheprecedingdefinition ofa magnet,thesimple term, withoutqualification, maybeappliedto suchanarrangement. Ontheother hand, apieceofapparatuscon- sistingofagalvanic coil,with asoftironcore, althoughoften called simply "an electro-magnet,"isinrealityacomplex arrangement involving anelectro-magnet (whichisintrinsically magneticaslongas the electric current issustained)andabody transiently magnetized byinduction. 435. Inthefollowing analysisofmagnets,themagnetismof every magneticsubstance considered willberegardedasab- solutely permanentunder allcircumstances. This condition is notrigorouslyfulfilled either formagnetizedsteel orforload- stone, asthemagnetismofanysuchsubstance isalwaysliable tomodification byinduction, andmaytherefore beaffected either bybringinganother magnetinto itsneighbourhood,or bybreakingthemass itself andseparatingthefragments. When, however, weconsider themagnetismofanyfragment taken from asteel orloadstonemagnet,thehypothesiswillbe that itretains without anyalteration themagneticstate which itactuallyhadinitspositioninthebody. Thegeneral theory ofthedistribution ofmagnetism founded upon conceptionsof thiskind, willbeindependentofthetruth orfalseness ofany suchhypothesiswhich maybemade forthesake ofconveni- XXIV.] AMathematical Theory ofMagnetism. 347 ence instudyingthesubject; butofcourse anyactualexperi- ments inillustration oftheanalysisorsynthesisofamagnet would beafifected byawant ofrigidityinthemagnetismof thematteroperatedon.Forsuch illustrations electro-magnets [withoutiron orother magnetic substance]areextremely appropriate,asinthem, except duringthemotion bywhich anyalteration intheir form orarrangementiseffected, no appreciableinductive action can exist. 436. Inselectingfrom theknown phenomenaofmagnetism those elementaryfactswhich aretoserve forthefoundation of thetheory,allcomplexactions depending ontheirregularities ofthebodies made useofshould beexcluded. Thus ifwe were toattemptanexperimental investigationoftheaction between twoamorphous fragmentsofloadstone, orbetween twopiecesofsteelmagnetized byordinary processes, we shouldprobablyfailtorecognisethesimplelawsonwhich the actionsresulting from suchcomplicatedcircumstances depend ; andwemust look forasimplercase ofmagneticaction before wecanmake ananalysis whichmaylead totheestablishment ofthefundamentalprinciplesofthetheory. Muchcomplica- tion willbeavoided ifwetakeacase inwhich theirregularities ofone, atleast, ofthebodies donotaffect thephenomena tobe considered. Now, theearth, aswas firstshownbyGilbert, is amagnet ;and itsdimensions aresogreatthat there isno sensible variation initsaction ondifferentpartsofany ordinary magnet uponwhich wecanexperiment, andconse- quently,inthecircumstances, nocomplicacy depending onthe actual distribution ofterrestrialmagnetism. Wemay therefore, withadvantage, commence byexaminingtheaction which the •earthproduces uponamagnetofanykind atitssurface. 437.Atavery early periodinthehistoryofmagneticdis- coverytheremarkablepropertyof" pointingnorth andsouth" wasobserved tobepossessed byfragmentsofloadstone and magnetizedsteel needles. Toform aclearconceptionofthis phenomenon, wemust consider thetotal actionproduced by theearth uponamagnetofanykind, andendeavour todis- tinguish between the effects ofgravitation which theearth exerts uponthebodyinvirtue ofitsweight,andthose which result from themagnetic agency. 348 AMathematical Theory ofMagnetism. [xxiv. 438. Inthe firstplace,itistoberemarked thatthemag- netic agencyoftheearthgivesrise tonoresultant force of sensible magnitude, upon anymagnetwith reference towhich wecanperform experiments [thatistosay,small enoughto beasubjectforlaboratory experiments],asisproved bythe followingobserved facts :— (1.)Amagnet placedinanymanner, andallowed tomove with perfectfreedom inanyhorizontal direction(bybeing floated, for example,onthesurface ofaliquid), experiences noaction which tends tosetitscentre ofgravityinmotion, andthere istherefore no [directly observable]horizontal forceuponthebody. (^)Themagnetismofabodymaybealtered inanyway,without affectingitsweightasindicated byabalance. Hence there canbe no[directly observable]vertical forceuponitdependingonitsmag- netism. 439. Itfollows thatanymagneticaction which theearth canexert uponamagnet [ofdimensions suitable forlaboratory experiments] must be[sensibly]acouple. Toascertain the manner inwhich this action takesplace,letusconceive a magnettobesupported byitscentre ofgravity* and leftper- fectlyfree toturnround thispoint,sothat, without anycon- straintbeingexerted which could balance themagnetic action, thebodymaybeincircumstances thesame asifitwere with- outweight. Themagneticaction oftheearthuponthemagnet givesrisetothefollowing phenomena:— (1.)Themagnetdoesnotremain inequilibriuminevery positionin which itmaybebroughttorest, asitwould dodid itexperiencano action butthat ofgravitation. (2.)Ifthemagnetbeplacedinapositionofequilibriumthere isa certain axis(which,forthepresent, wemayconceive tobefound by trial),such that ifthemagnet beturned roundit,through anyangle, andbebroughttorest,itwillremain inequilibrium. *Theordinary process forfinding experimentally thecentre ofgravityof abodyfailswhen there isanymagnetic action tointerfere with theeffects ofgravitaton.Itis,however, forourpresent purpose,sufficient toknow that thecentre ofgravity exists;that is,that there isapoint such that the vertical line oftheresultant action ofgravity passes through it,inwhatever position thebody beheld. Ifitwere ofanyconsequence, aprocess some- what complicated bythemagnetic action, foractually determining, byex- periment, thecentre ofgravity ofamagnet might beindicated, and thus theexperimental treatment ofthesubject inthe textwould becompleted. xxiY.] AMathematical Theory ofMagnetism. 349 (3.)Ifthemagnetbeturnedthrough 180", about anaxisperpen- dicular tothis,itwillagainbeinapositionofequilibrium. (4.)Anymotion ofthemagnet whatever, which isnotofeither of thekindsjust described, norcompoundedofthetwo,willbringit intoapositioninwhich itwillnotbeinequilibrium. (5.)Thedirecting couple experienced bythemagnetinanyposi- tiondepends solelyontheangleofinclination oftheaxisdescribed in (1.)tothelinealongwhich itlieswhen themagnetisinequilibrium; being independentofthepositionoftheplaneofthisangle, andof thedifferentpositionsintowhich themagnetisbrought byturningit round that axis. 440.From these observations wedraw theconclusion thata magnet always experiencesadirecting couplefrom theearth unless acertain axisbelongingtoitisplacedinadeterminate position.This lineofthemagnetiscalled itsmagnetic axis.* 441. The direction towards which themagneticaxisofthe magnettends invirtue oftheearth's action, iscalled"theline ofdip,"or"the direction ofthetotal terrestrialmagnetic force," atthelocalityoftheobservation. 442.Nofurtherexplanation regarding phenomena which dependonterrestrial magnetismisrequiredinthepresent chapter ;but,asthefactshave been stated inpart,itmaybe righttocompletethestatement, asfarasregardstheaction experienced byamagnetofanykindwhen held indifferent positionsinagiven locality, bymentioningthefollowing conclusions, deduced inaveryobvious manner from the generallaws ofmagneticaction stated below, and verifiedfully byexperiment:— Ifamagnetbeheld with itsmagneticaxis inclined atany angletothelineofdip,itwillexperienceacouple,themoment ofwhich isproportionaltothesineoftheangleofinclination, actinginaplane containingthemagneticaxisandthelineof dip.Thepositionofequilibriumtowards which thiscouple tends tobringthemagneticaxis isstable, and ifthedirec- tionofthemagneticaxisbereversed, themagnet maybeleft balanced, but itwillbeinunstableequilibrium. *Any lineinthebody parallel tothismight, with asgood reason, becalled amagnetic axis, butwhenweconceive themagnet tobesupported byitscentre ofgravity, themagnetic axis isnaturally taken asalinethrough this point. [Seeaddition to§444.] 350 AMathematical Theory ofMagnetism. [xxiv. 443. The directive tendencyobserved inmagneticbodies beingfound todependontheirgeographical position, and to berelated, insomedegree,totheterrestrialpoles,received the name ofpolarity, probablyonaccount ofafalsehypothesisof forces exercised bythepole-star*orbytheearth'spolesupon certainpointsoftheloadstone orneedle, thence called the *' polesofthemagnet." Theterms" polarity"and" poles"are still retained, buttheuseofthem, which hasvery generally been made, isnearlyasvagueastheideas fromwhichthey hadtheirorigin. Thus,when themagnetisanelongated mass, itsends arecalledpolesifitsmagneticaxisbeinthedirection ofitslength;nodefinitepoints,such asthose inwhich the surface ofthebodyiscutbythemagnetic axis, being pre- ciselyindicated bytheterm asitisgenerallyused.If,how- ever, thebodybesymmetricalabout itsmagnetic axis,and symmetrically magnetized,whetherelongatedinthat direction ornot,thepoles mightbedefinitelytheendsofthemagnetic axis(orthepointsinwhich thesurface iscutby it),unless themagnetbeannular andnotcutbyitsmagneticaxis(aring electro-magnet,forinstance),inwhich casetheordinarycon- ceptionofpolesfails. Notwithstandingthisvagueness, how- ever, thetermspoles andpolarityareextremely convenient, and,with thefollowing explanations, theywHllfrequentlybe made useofinthispaper:— 444. Let beanypointinamagnet,and letKOS bea straightlineparalleltothelinedefined above asthemagnetic axisthroughthecentre ofgravity.Ifthepoint 0,however ithasbeen chosen, becalled thecentre ofthemagnet,theline NS,terminated either atthesurface, oneach side, orinany arbitrary manner, iscalled themagnetic axis,andtheends N,S,ofthemagneticaxisarecalled thepolesofthemagnet. f *InthepoemofGuiot deProvence(quotedinWhewell's HUtory ofthe Inductive Sciences, vol. ii.p.46),aneedle isdescribed asbeing magnetized and placedinoronastraw(floating onwater itistobepresumed)— "Puis setome lapointe toute Contre I'estoile sans doute." tAdefinition ofpoles atvariance with this isadoptedinsome special cases, especiallyinthat oftheearth considered asagreat magnet, butthemanner in which theterm willbeused inthispaperwillbesuch astoproduce noconfusion onthisaccount.i XXIV.] AMathematical Theory ofMagnetism.351 [Addition^1871.—Later, §494, apropercentral axis, tobe called themagnetic axis,andapointinitwhichmaybecalled themagnetic centre, willbedefined accordingtopurely mag- netic conditions.] 445. Thatpole(marked N)whichpoints,onthewhole, from thenorth, and, innorthern latitudes, upwards,iscalled thenorthpole,andtheother{S),whichpointsfrom thesouth, iscalled thesouthpole. 446.The sides ofthebodytowards itsnorth poleandsouth polearesaid topossess"northernpolarity"and"southern polarity" respectively,anexpression obviouslyfounded onthe ideathatthesurface ofamagnet mayingeneralbecontem- platedasalocus ofpoles. 447. Ifamagnetic bodybebroken upintoanynumber of fragments,each morsel isfound tobeacomplete magnet, presentinginitself allthephenomenaofpolesandpolarity. Thispropertyisgenerally contemplated when, inmodern writingsonphysical subjects, polarityismentioned asa property belongingtoasolidbody ;andacorrespondingidea isinvolved inthetermwhen itisappliedwith reference tothe electric state which MrFaradaydiscovered tobeinduced in non-conductors ofelectricity ("dielectrics") whensubjectedto theinfluence ofelectrified bodies.* However different arethe physicalcircumstances ofmagneticand electricpolarity,it appearsthatthepositivelaws ofthephenomenaarethesame,-|- andtherefore themathematical theories areidentical. Either subject mightbetaken asanexampleofavery important branch ofphysical mathematics, which mightbecalled"A Mathematical TheoryofPolar Forces." 448. Although wehave seen thatanymagnet,ingeneral, experiencesfrom theearth anactionsubjecttocertainvery simple laws, yettheactual distribution ofthemagnetism which itpossesses may beextremely irregular. Wemay certainlyconceive that ifthemagnetizedsubstance bea regular crystalofmagneticiron ore,themagnetismisdistri- *Faraday's Experimental Researches inElectricity,Eleventh Series, tSeeapaper"On theElementary Laws ofStatical Electricity," published intheCambridge andDublin Mathematical Journal(vol. i.)inDecember 1845. 852 AMathematicalTheory ofMagnetism. [xxiv. butedthroughitaccordingtosomesimple law;butbytaking anamorphous andheterogeneous fragmentoforepresenting magnetic properties, bymagnetizinginanywayanirregular mass ofsteel,byconnecting anynumber ofmorsels ofmagnetic matter soastomake upacomplex magnet,orbybendinga galvanic wire intoanyform,wemayobtainmagnetsinwhich themagnetic propertyisdistributed inanyarbitrary manner, howeverirregular. Excludingforthepresentthelast-men- tioned case, letusendeavour toform aconceptionofthe distribution ofmagnetisminactually magnetized matter, such assteel orloadstone, andtolaydown theprinciples according towhich itmayinanyinstance bemathematically expressed. 449. Ingeneral wemayconsider amagnetascomposedof matter which ismagnetized throughout, since, ingeneral,itis found thatanyfragmentcutoutofamagnetic mass isitself a magnet possessing properties entirelysimilar tothose which havebeen described aspossessed byanymagnetwhatever. It may be,however, that asmallportioncutoutofacertain positioninamagnet, may present nomagnetic phenomena; and ifwecutequal andsimilarportions from differentposi- tions,wemayfindthem topossess magnetic properties differing toanyextent both inintensityand inthedirections oftheir magneticaxes. 450. Ifwefind thatequaland similarportions,cutin parallel directions, from anydifferentpositionsinagiven magnetic mass, possess equal and similarmagnetic properties, themass issaidtobeuniformly magnetized. 451. Ingeneral, however, theintensityofmagnetization must besupposedtovaryfrom oneparttoanother, andthe magneticaxes ofthedifferentpartstobenotparalleltoone another. Hence, tolaydowndeterminatelyaspecificationof thedistribution ofmagnetism throughamagnetofanykind, wemustbeable toexpresstheintensityandthedirection of magnetizationateachpoint.Beforeattemptingtodefine a standard forthenumericalexpressionofintensityofmagneti- zation, itwillbeconvenient toexamine theelementarylaws upon which thephenomenaofmagneticforcedepend,since it isbythese effects that the.nature andenergyofthemagnetism towhich they areduemust beestimated. XXIV.] AMathematical Theory ofMagnetism. 353 Chapter II.—OntheLawsofMagnetic Force, andonthe DistributionofMagnetisminMagnetized Matter. 452.Theobjectoftheelementary magnetic researches of Coulomb wasthedetermination ofthemutual action between twoinfinitely thin, uniformly andlongitudinally magnetized bars. Themagnetswhich heusedwere instrictness neither uniformlynorlongitudinally magnetized,such astate being unattainable byanyactualprocessofmagnetization; but, as the bars wereverythincj/lindricalsteel wires, andwere symmetrically magnetized,theresultant actions weresensibly thesame asiftheywere inreality infinitely thin,andlongi- tudinally magnetized ;andfromexperiments which hemade, itappearsthattheintensityofthemagnetization must have beenverynearlyconstant from themiddle ofeach ofthebars towithin ashort distance from either end,where agradual decrease ofintensityissensible*. 453. These circumstanceshavingbeen attendedto,Coulomb wasable todeduce from hisexperiments thetruelaws ofthe phenomena,andarrived atthefollowingconclusions :— (1)Iftwo thinuniformly andlongitudinally magnetized barsbeheld nearoneanother, anaction isexerted between them which consists offour distinct forces, alongthefour linesjoiningtheir extremities. (2)The forces between likeends ofthetwobars arere- pulsive-f-, (3)The forces between unlike ends areattractive. (4)Ifthebarsbeheld sothatthefourdistances between their extremities, twoand two, areequal,the four forces ;between them willbeequal. (5)Iftherelativepositionsofthebars bealtered, each force willvary inverselyasthesquareofthemutual distance of thepolesbetween which itacts. *Seenoteon§469,below. tHence weseethepropriety oftheterms north andsouth applied tothe opposite polarities ofamagnet, asexplained above. Thus wedesignate the polarity, ortheimaginary magnetic matter ofthenorthern andsouthern magnetic hemispheres oftheearth, asnorthern andsouthern respectively;and since thepoles ofordinary magnets which arerepelled bytheearth's northern orsouthern polarity must besimilar, these also arecalled northern orsouthern, asthecasemay be. T.E. 23 354 AMathematicalTheory ofMagnetism. [xxiv. 454. Toestablish astandard forestimatingthestrengthofa magnet,letusconceive twoinfinitelythinbars tobeplacedso that either endofonemaybeatunit ofdistance fromanend oftheother. Then,ifthebarsbeequally magnetized,each uniformly andlongitudinally,tosuch adegreethattheforce between those ends shall beunity,thestrengthofeach bar- magnetisunity*. 455. Ifanynumber, m,ofsuch unit bars, ofequal length, beputwith like endstogether,soastoconstitute asingle complex bar,thestrengthofthemagnetsoformed isdenoted bym. Ifthere beanynumber ofthinbar-magnetsofequal length, andeach ofthem ofsuch astrengththatqofthem, with like endstogether, would constitute aunit-bar;and ifpofthose barsbeputwith likeendstogether,thestrengthofthecomplex magnet soformed willbe- . 456. Ifasingle infinitelythinbarbemagnetizedtosucha degree that inthesamepositionsitwouldproducethesame effects asacomplexbarofanystrength m(anintegeror fraction),thestrengthofthismagnetisdenoted bym. 457. Iftwocomplex bar-magnets,ofthekind described above, beputnear oneanother, each barofone will acton each baroftheother with thesame forces asifalltheother barswere removed. Hence,ifthedistance between thetwo poles beunity, and ifthestrengthsofthebarsberespectivelymandmf(whether these numbers beintegralorfractional), the force between thosepoleswillbemm'. If,now, the relativepositionofthemagnetsbealtered, sothatthedistance between twopolesmaybe/,the force between them will, accordingtoCoulomb's law,be mm 7^ *TheEoyal Society, initsInstructions formaking observations onTerres trialMagnetism, adopts onefootastheunit oflength; andthat force which ifacting onagrain ofmatter, would inonesecond oftimegenerate one foo' persecond ofvelocity, astheunit offorce;which isconsequently vet} nearly_l-_ oftheweight, inanypart ofGreat Britain orIreland, ofon< grain. [Note, 1871.—The British Association's Committee on Electri( Measurement have recently adopted thecentimetre asunit oflength,ant thegramme asunit ofmass, instead ofthefootandgrain.] XXIV.] AMathematical Theorij ofMagnetism. 355 Accordingtothedefinitiongivenabove ofthestrengthofa simple bar-magnet,itfollows thatthesameexpression gives theforcebetween twopolesofanythinuniformly andlongi- tudinally magnetized bars, ofstrengths mandm'. 458. Themagnetic moment ofaninfinitely thin, uniformly andlongitudinally magnetized bar,istheproductofitslength into itsstrength. 459. Ifanynumber ofequally strong, uniformly andlongi- tudinally magnetized rectangularbars ofequal infinitelysmall sections, beputtogetherwith like ends towards thesame parts,acomplex uniformly magnetizedsolid ofanyformmay beproduced.Themagnetic moment ofsuchamagnetisequal tothesum ofthemagnetic moments ofthebars ofwhich itis composed. 460. Themagnetic moment ofanycontinuous solid, uni- formly magnetizedinparallel lines, isequaltothesum ofthe magneticmoments ofallthethinuniformly andlongitudinally 'magnetizedbars intowhich itmaybedivided. Itfollows thatthemagnetic moment ofanypartofauni- formly magnetizedmass isproportionaltoitsvolume. 461. Theintensity ofmagnetizationofauniformly magnet- ized solid isthemagnetic moment ofaunit ofitsvolume. Itfollows thatthemagnetic moment ofauniformly mag- 'Qetized solid, ofanyform and dimensions, isequaltothe 'productofitsvolume intotheintensityofitsmagnetization. 462. Ifabody bemagnetizedinanyarbitrary regularor •irregular manner, aportion maybetaken inany position,so small inallitsdimensions thatthedistribution ofmagnetism throughitwillbesensiblyuniform. Thequotientobtained by dividingthemagnetic moment ofsuchaportion,inanyposi- tionP,byitsvolume, istheintensity ofmagnetizationofthe substance atthepointP;andalinethrough Pparalleltoits lines ofmagnetization,isthedirection ofmagnetization,atP. Ohapter III.—0)1theImaginary MagneticMatter bymeansof which thePolarity ofaMagnetized Bodymayherepresented. 463. Itwillveryoften beconvenient toreferthephenomena )fmagneticforce toattractions orrepulsions mutuallyexerted 23—2 356 AMathematical Theory ofMagnetism. [xxiv. betweenportionsofanimaginary magnetic matter, whicli, as weshall see,maybeconceived torepresentthepolarityofa magnetofanykind. Thisimaginarysubstancepossesses none oftheprimary qualitiesofordinary matter, and itwould be wrongtocall iteither asolid, orthe"magneticfluid" or "fluids"; but,without making anyhypothesis whatever, we maycall it"magnetic matter," ontheunderstandingthat it possesses onlythepropertyofattractingorrepelling magnets, orotherportionsof"matter" ofitsown kind, accordingto certain determinate laws,whichmaybestated asfollows :— (1)There aretwokinds ofimaginary magnetic matter, northern andsouthern, torepresent respectivelythenorthern andsouthern magnetic polaritiesoftheearth, orthesimilar polaritiesofanymagnetwhatever. (2)Likeportionsofmagnetic matterrepel,andunlikepor- tions attract, mutually. (3)Anytwosmallportionsofmagnetic matter exert a mutual forcewhich variesinverselyasthesquareofthedis- tance between them. (4)Two units ofmagnetic matter, ataunit ofdistance from oneanother, exert aunit offorce, mutually. 464. Ifquantitiesofmagneticmatter bemeasured numeri- callyinsuch units, and ifthepositiveornegative signbe prefixedtodenote thespeciesofmatter, whether northern (which, byconvention, wemaycallpositive)orsouthern, all theprecedinglaws areexpressedinthefollowing proposi- tion :— Ifquantities mand m',ofmagnetic matter heconcentrated respectivelyatpoints atadistance, f,fromoneanother, theywill repelwithaforce algebraically equalto mm' 465. Itappearsfrom theexplanations givenabove thatthe circumstances ofauniformly magnetizedneedle mayberepre- sented ifweimagine equal quantitiesofnorthern andsouthern magneticmatter tobeconcentrated at itstwopoles, the numerical measure oftheseequal quantities beingthesame as that ofthe" strength"ofthemagnet. Themutual action between two needles would thus be XXIV.] AMathematical Theory ofMagnetism. 357 reduced toforces ofattraction andrepulsionbetween theportions ofmagneticmatter bywhich theirpolesarerepresented. 466.Anymagneticmasswhatevermay,aswehave seen,be regardedascomposedofinfinitelysmallbar-magnets putto- getherinsuch awayastoproducethedistribution ofmag- netism which itactually possesses ;andhence, bysubstituting- imaginary magnetic matter forthepolesofthese magnets, weobtain adistribution ofequal quantitiesofnorthern and southern magneticmatterthroughthemagnetized substance, bywhich itsactual magneticcondition mayberepresented. Thedistribution ofthismatter becomes verymuchsimplified, from thecircumstance thatwehave ingeneralunlikepolesof theelementary magnetsincontact, bywhich theopposite kinds ofmagneticmatter arepartially (orinaclass ofcases wholly'^) destroyed throughthe interior ofthebody. The determination oftheresultingdistribution ofmagnetic matter, whichrepresentsinthesimplest possible manner thepolarity ofanygiven magnet,isofmuch interest, andeven importance, inthetheoryofmagnetism,andwema}'^therefore make this anobjectofinvestigation,before goingfurther. 467. Let itberequiredtofindthedistribution ofimaginary magneticmatter torepresentthepolarityofanynumber of uniformly magnetized needles, S^N^, 8^N^,...S^N^,ofstrengths fi^, fi^,...fi^respectively, when theyareplaced together,endto end(notnecessarilyinthesamestraight line). IfAdenote theposition occupied byS^when thebars are intheirplaces;if]S\andS^areplacedincontact atK^^;N^ andS^,atK^;and soonuntilwehave the lastmagnet, with itsend>S^^,incontact withN^_^,atZr^_j,and itsother end,N^, free, atapointB;weshallhave toimagine fi^units ofsouthernmagneticmatter tobeplacedatA; fi^units ofnorthern, andfj,^units ofsouthern matter atK^^ ; fi^units ofnorthern, andfi^ofsouthern matter SitK^; /jb^_^units ofnorthern, andfi^ofsouthern matter atK^_^; andlastly, jjb^units ofnorthern matter atB. *Inallcaseswhen thedistribution is"solenoidal."Seebelow, Chap. v. §499;communicated totheRoyal Society, June 20,1850. 358 AMathematical Theory ofMagnetism. [xxiv. Hence thefinal distribution ofmagneticmatter isasfollows :— — fjb^SitA /^i-/^2 K f^n-^-H'n ^n-1 andfjb^B. 468. Thecomplex magnet AKJ^^...K^_^Bconsists ofa number ofparts,each ofwhich isuniformly andlongitudinally magnetized, and itwillactinthesamewayasasimplebarof thesamelength, similarly magnetized ;andhence themagnetic matter whichrepresentsabar-magnet AB oithiskind iscon- centrated inaseries ofpoints,attheends ofthewhole bar,and atalltheplaceswhere there isavariation inthestrength*of itsmagnetization. 469. Ifthelengthofeachpartthrough which thestrength ofthemagnetismisconstant, bediminished without limit, and iftheentire number ofthepartsbeincreasedindefinitely,a straightorcurvedinfinitelythinbarmaybeconceived tobe produced, which shallpossessadistribution oflongitudinal magnetism varying continuously from oneendtothe othei accordingtoanyarbitrarylaw. Ifthestrengthofthemagnet- ismatanypointPofthisbarbedenotedby [jl,and if[/x]anc (ft)denote thevalues of/aatthepointsAandB,the investi- gationof§467,with theelementary principlesandnotation o thedifferential calculus, leads atonce tothedetermination o theultimate distribution ofmagneticmatter bywhich such i bar-magnet mayberepresented. Thus ifAPbedenoted bys fjbwillbeafunction of5,which maybesupposedtobeknown and itsdifferential coefficient willexpressthecontinuous dis tribution ofmagnetic matter whichreplacesthegroupo materialpointsatK^,K^, etc.; sothattheentire distributioi ofpolarityinthebarand atitsends willbeasfollows:—ii *This expressionisequivalent totheprodiict oftheintensity ofmagnetisx tion into thesection ofthebar;andbyretainingitweareenabled toinclude cases inwhich thebar isnotofuniform section. i XXIV.] AMathematical Theory ofMagnetism. 359 any infinitelysmalllength, cr,ofthebar,aquantityofmatter equalto dfjb and, besides, terminal accumulations, ofquantities — [//-]atA, and(jbu)atB. Itfollows thatif,through anypartofthelengthofabar, thestrengthofthemagnetismisconstant, there willbeno magnetic matter tobedistributed throughthisportionofthe magnet; but ifthestrengthofthemagnetism varies, then, accordingasitdiminishes orincreases from thenorth tothe southpoleofanysmallportion,there willbeadistribution of northern orsouthernmagneticmatter torepresentthepolarity which results from thisvariation. Correspondinginferences maybemadeconversely,with re- ference tothedistribution ofmagnetism, when thedistribution oftheimaginary magneticmatter isknown. ThusCoulomb found that hislongthincylindrical bar-magnetsacted upon oneanother asifeachhadasymmetricaldistribution ofthe twokinds ofmagnetic matter, northern within alimitedspace from oneend,andsouthern within alimitedspacefrom the other, theintermediatespace (constituting generallythegreater partofthebar)being unoccupied ;fromwhich weinfer that novariation inthemagnetism wassensible throughthemiddle partofthebar,butthat,throughalimitedspace oneachside, theintensityofthemagnetization must have decreasedgradu- allytowards theends*. *This circumstance wasalluded toabove, in§452. Interesting views on thesubject ofthedistribution ofmagnetism inbar-magnets areobtained by taking arbitrary examples toillustrate theinvestigation ofthe text. Thus wemay either consider auniform barvariably magnetized, orathinbarof varying thickness, cutfrom auniformly magnetized substance;and, accord- ingtothearbitrary data assumed, various remarkable results may beob- tained. Weshall seeafterwards thatanysuch data, howeverarbitrary, may beactually produced inelectro-magnets, andwehave therefore themeans ofillustrating thesubject experimentally, inascomplete amanner ascan beconceived, although from thepractical non-rigidityofthemagnetism of magnetized substances, ordinary steel orloadstone magnets would notafford suchsatisfactory illustrations ofarbitrarycases asmight bedesired. The distribution oflongitudinal magnetism insteel needles actually magnetized indifferent ways, andespecially'•magnetized tosaturation," hasbeen the I 860 AMathematical Theory ofMagnetism. [xxiv. 470. The distribution ofmagnetic matter whichrepresents thepolarityofauniformly magnetized bodyofanyform,may beimmediately determined ifweimagineitdivided into in- finitelythin bars, inthedirections ofitslines ofmagnetization; foreach ofthese bars willbeuniformly andlongitudinally magnetized,and therefore there willbenodistribution of matterexceptattheir ends.Now thebars areallterminated oneach sidebythesurface ofthebody, andconsequentlythe whole magneticeffect isrepresented byacertainsuperficial distribution ofnorthern andsouthernmagneticmatter. It onlyremains todetermine theactual form ofthisdistribution; but, forthesake ofsimplicityinexpression,itwillbecon- venient tostatepreviouslythefollowing definition, borrowed fromCoulomb's writingsonelectricity:— 471. Ifanykind ofmatter bedistributed overasurface, the superficial densityatanypointisthequotient obtained by dividingthequantityofmatter onaninfinitelysmall element ofthesurface intheneighbourhoodofthatpoint, bythearea oftheelement. 472. Todetermine thesuperficial densityatanypointinthe case atpresent under consideration, let cobethearea ofthe perpendicularsection ofaninfinitelythinuniform barofthe solid, with oneendatthatpoint. Then, ifibetheintensity ofmagnetizationofthe solid, icowill be,asmaybereadily shown, the" strength"ofthebar-magnet. Hence atthetwo ends ofthebarwemustsupposetobeplaced quantitiesof northern andsouthern imaginary magneticmatter eachequal to10).Inthedistribution overthesurface ofthegiven magnet, thesequantitiesofmatter must beimaginedtobespreadover theobliqueends ofthebar.Now if6denote theinclination ofthebartoanormal tothesurface throughoneend,thearea ofthatend wallbe7,,and therefore inthat part ofthe cos^^ surface wehave aquantityofmatterequaltoicospreadover anarea^.Hence thesuperficial densityis icos6. objectofinteresting experimental and theoretical investigations byCoulomb, Blot, Green, andEiess. XXIV.] AMathematical Theory ofMagnetism. S&l This expression givesthesuperficial densityatanypoint, P,of thesurface, and itsalgebraic signindicates thekind ofmatter, providedtheangledenoted by6betaken between the externalpartofthenormal, andalinedrawn fromPin thesame direction asthat ofthemotion ofapointcarried from thesouthpoletothenorthpole,ofaportionclose to P,oftheinfinitelythinbar-magnetwhich wehave been con- sidering. 473. Let itberequired,inthelastplace,todetermine the tiredistribution ofmagneticmatternecessarytorepresent thepolarityofanygiven magnet. Wemayconceive thewhole magnetized mass tobedivided intoinfinitelysmallparallelepipeds byplanes paralleltothree planesofrectangularco-ordinates. Leta,/3,7denote the threeedgesofoneoftheseparallelepipeds havingitscentre at apointP{x,y,z).Let idenote thegiven intensity,andI,m,n thegivendirection cosines ofthemagnetizationatP.Itwill follow from thepreceding investigationthatthepolarityofthis infinitelysmalluniformly magnetized parallelepiped maybe represented byimaginary magneticmatter distributed over its sixfaces insuchamanner that thedensitywillbeuniform overeach face,andthat thequantitiesofmatter onthesix faces willbeasfollows :— —il ./37,and il .fiy ;onthetwofacesparalleltoYOZ; —im .ry2,andim.jct; onthetwofacesparalleltoZOX; —in .a/3,andin .a/3 ;onthetwofacesparalleltoXOY. Now ifweconsideradjacent parallelepipedsofequal dimen- sions, touchingthesixfaces oftheonewehavebeen consider- ing,weshould findfromeach ofthem asecond distribution of magnetic matter, tobeplaced uponthatoneofthose sixfaces which ittouches. Thus ifweconsider thefirstface^y,orthat ofwhich thedistance fromYOZ isx— ^7. ;weshall have a second distribution uponitderived from aparallelepiped,the co-ordinates ofthecentre ofwhich arex—OL,y,z; andthe quantityofmatter inthissecond distribution willbe Ia+'^(-a)\0y. 862 AMathematicalTheory ofMagnetism. [xxiv. This, added tothatwhich wasfound above, gives d{il). .^ d{iT) ^ forthe totalamount ofmatter uponthis face. Again, the quantityinthesecond distribution ontheother face, /Sy,is equal to f., .d(il) ]^ ^ -\il+-^.ayI3y,dx andtherefore thetotalamount ofmatter onthisface willbe Bydetermininginasimilar waythefinalquantitiesofmatter ontheother faces oftheparallelepiped, wefindthatthetotal amount ofmatter tobedistributed over itssurface is id{I \dx'^^+ii^+±^iaffy.dxdydz Now astheparallelepipedsintowhich weimaginedthewhole mass divided areinfinitely small, wemaysubstitute acon- tinuous distribution ofmatterthrough them, inplaceofthe superficialdistributions ontheir faces which have been de- termined;and inmakingthis substitution, thequantityof matter which wemustsupposetobespread throughthein- terior ofanyoneofthemmust behalfthetotalquantityonits surface, since each ofitsfaces iscommon toitandanother parallelepiped.Hence thequantityofmatter tobedistributed throughtheparallelepiped a/?7isequalto {d(il)d(im) d(m)] ^ Besides thiscontinuous distribution throughtheinterior ofthe magnet,there must beasuperficialdistribution torepresent theun-neutralizedpolarityatitssurface. Ifpdenote thedensity ofthisdistribution atanypoint ;[I],[m], [n]thedirection- cosines, and[i]theintensityofthemagnetizationofthesolid close toit;and X,fi,vthedirection-cosines ofanormal tothe surface, weshall have, asinthecase oftheuniformly magnet- ized solidpreviously considered, p=\i\cos6=\ir\.\+\im'\.^i+[m].v(1). Ifaccordingtotheusual definition of" density," kdenote the XXIV.] AMathematical Theory ofMagnetism. 363 densityofthemagneticmatter atP,inthecontinuous distri- bution throughtheinterior, theexpressionfound above forthe quantityofmatter intheelementa,/3,7,leads totheformula i^^_{<m^d{pt)^dp] {axaydz)^' These twoequations express respectivelythesuperficialdistri- bution, andthecontinuous distributionthroughthe solid, of themagneticmatter whichentirely representsthepolarityof thegiven magnet. The fact that thequantityofnorthern matter isequaltothequantityofsouthern intheentire distri- bution, isreadilyverified byshowingfrom these formulae, as may readilybedonebyintegration,thatthetotalquantityof matter isalgebraically equaltonothing. 474. Ifthere beanabrupt changeintheintensityordirec- tionofthemagnetization fromonepartofthemagnetizedsub- stance toanother, aslightmodification intheformulaegiven above willbeconvenient. Thuswemaytake acasediffering verylittle from agiven case,butwhich, instead ofpresenting finite differences intheintensityordirection ofmagnetization onthetwosides ofanysurface inthesubstance ofthemagnet, hasmerely verysudden continuous changesinthevalues of those elements :wemayconceive thedistribution tobemade moreandmorenearlythesame asthegiven distribution, with itsabrupt transitions, andwemaydetermine thelimit towards which thevalue oftheexpression (2)approximates,andthus, although accordingtotheordinaryrules ofthe differential calculus thisformula fails inthelimiting case,wemaystill derive thetrue result from it.Itisvery easily shown inthis way, that, besides thecontinuous distributiongiven bythe expression (2)appliedtoallpointsofthesubstance forwhich itdoesnot fail,there willbeasuperficialdistribution ofmag- jpeticmatter onanysurface ofdiscontinuity ;and that the lensityofthissuperficialdistribution willbethe difference between theproductsoftheintensityofmagnetizationinto lecosine oftheinclination ofitsdirection tothenormal, on {hetwosides ofthesurface. 475. This result, obtained bytheinterpretationofformula I)"intheextreme case,might have been obtained directly romtheoriginal investigation, bytakinginto account the 364 AMathematicalTJieory ofMagnetism. [xxiv. abruptvariation oftheraagnetizationatthesurface ofdis- continuity,as(§472)wedidtheabrupttermination ofthe magnetizedsubstance attheboundaryofthemagnet, and re- presentingtheun-neutralizedpolarity which results, byasuper- ficial distribution ofmasfnetic matter. Chapter TV.—DeterminationoftheMutual Actions between any Given PortionsofMagnetizedMatter. 476. Thesynthetical partofthetheoryofmagnetismhas foritsultimateobjectthedetermination ofthe total action between twomagnets, when thedistribution ofmagnetismin each isgiven. Theprinciples accordingtowhich thedata of such aproblem maybespecified have been alreadylaiddown (§§459...62),andwehave seen that, with sufficient data in any case. Coulomb's laws ofmagneticforce aresufficient to enable ustoapply ordinarystaticalprinciplestothesolution oftheproblem. Hence theelements ofthispartofthetheory mayberegardedascomplete,andwemayproceedtothe mathematical treatment ofthesubject. 477. Theinvestigationsofthepreceding chapter, which show ushowwemayconventionally represent anygiven mag- net,initsagency uponother bodies, byanimaginary magnetic matter distributed on itssurface andthroughitsinterior; enable nstoreduce theproblemoffindingtheaction between anytwomagnets,totheknown problemofdeterminingthe resultant oftheattractions orrepulsionsexerted between the particlesoftwogroupsofmatter, accordingtothelawofforce which ismetwith souniversallyinnatural phenomena. The direct formuloeapplicableforthisobjectaresoreadilyobtained bymeans oftheelementary principlesofstatics, and sowell known, that itisunnecessarytocitethem here,andwemay regard equations (1)and(2)ofthepreceding chapter (§473) assufficient forindicatingthemanner inwhich thedetails of theproblem maybeworked outinanyparticularcase. The expressionforthe" potential,"andother formulae ofimportance inLaplace's method oftreatingthissubject,aregivenbelow (§482),asderived from theresults expressedinequations (1) and(2). XXIV.] AMathematical Theory ofMagnetism. 365 478. Theprecedingsolution oftheproblem, although ex- tremely simple andoften convenient, must beregardedasvery artificial, since inittheresultant action isfound bythecom- positionofmutual actions between theparticlesofanimaginary magnetic matter, which arenotthesame astherealmutual actions between thedifferentpartsofthemagnets themselves, althoughtheresultant action between the entiregroupsof matter isnecessarilythesame asthe real resultant action between theentire magnets. Hence itisverydesirable to investigateanother solution, ofaless artificial form, inwhich therequiredresultant actionmaybeobtain edbycompounding therealactions between thedifferentpartsintowhich wemay conceive themagnetstobedivided. Theremainder ofthe chapter,after somepreliminary explanations and definitions, willbedevoted tothisobject. 479. The "resultant magneticforce atanypoint"isan expressionwhich willvery frequentlybeemployedinwhat follows, and itistherefore ofimportancethat itssignification should beclearlydefined. For thispurpose,letusconsider separatelythecases ofanexternalpointintheneighbourhood ofa.magnet,andapointinspace which isactually occupied bymagneticmatter. (1)Theresultant force atapointinspace,void ofmagnet- ized matter,istheforce thatthenorthpoleofaunit-bar(ora positiveunit ofimaginary magnetic matter),ifplacedatthis point, wouldexperience. (2)The resultant force atapointsituated inspace occupied bymagnetized matter,isanexpressionthesignificationofwhich issomewhatarbitrary.Ifweconceive themagneticsubstance toberemoved fromaninfinitelysmallspace round thepoint, theprecedingdefinition would beapplicable; since, ifwe imagineaverysmallbar-magnettobeplacedinadefinite positioninthisspace, the force uponeither endwould be determinate. Thecircumstances ofthis case aremade clear byconsideringthedistribution ofimaginary magnetic matter requiredtorepresentthegiven magnet, without thesmall portion wehave conceived toberemoved from itsinterior; which will differ from thedistribution thatrepresentsthe 'ntiregiven magnet,inwantingthesmallportionofthe 366 AMathematicalTheory ofMagnetism. [xxiv. continuous interior distributioncorrespondingtotheremoved portion, andinhavinginstead asuperficial distribution onthe small internal surface boundingthehollowspace.Ifwecon- sider theportion removed tobeinfinitely small, thew^ant of thesmallportionofthesolid magnetic [imaginary]matter will producenofinite effect upon anypoint ;butthesuperficial distribution attheboundaryofthehollowspacewillproduce afinite forceupon anymagnetic poiutwithin it.Hence the resultant forceuponthegiven point round which thespace was conceived tobehollowed, mayberegardedascompoundedof twoforces, oneduetothepolarityofthecomplete magnet, and theother tothesuperficial polarityleft freebytheremoval ofthemagnetizedsubstance*. Theformer componentisthe forcemeant bytheexpression"theresultant force atapoint within amagnetic substance," when employedinthepresent paperf. 480. Theconventionallanguageandideas with reference to theimaginary magnetic matter, explainedabove(§§463...75), enable ustogivethefollowing simplestatement ofthedefini- tion, includingboth thecases which wehavebeenconsidering. •Iftheportion removed bespherical and infinitely small, itmay be proved that theforce atanypoint withinit,resulting from thefreepolarity ofthesolid atthesurface bounding thehollow space,isinthedirection of thelines ofmagnetization ofthesubstance roundit,and isequal to---. Thistheorem (due toPoisson)willbedemonstrated atthecommencement of theTheory ofMagnetic Induction, because weshall have toconsider the "magnetizing force" upon anysmall portionofaninductively magnetized substance astheactual resultant force thatwould exist within thehollow space thatwould beleft iftheportion considered were removed, and the magnetism oftheremainder constrained toremain unaltered. +Ifweimagine amagnet tobedivided intotwoparts byanyplane pass- ingthrough theline ofmagnetization atanyinternal point, P,and ifwe imagine thetwopartstobeseparated byaninfinitely small interval, anda unitnorth pole tobeplaced between them atP,theforce which thispole would experienceis"theresultant force atapoint, P,ofthemagnetic sub- stance." This isthemost direct definition oftheexpression thatcould have been given, and itagrees with thedefinition Ihave actually adopted; butI have preferred theexplanation andstatement inthetext, asbeing practically more simple, andmore directly connected with thevarious investigations in which theexpressionwillbeemployed. [Noteadded June 15,1850.—Some subsequent investigations onthecom- parisonofcommon magnets andelectro-magnets have altered myopinion, that thedefinition intLe'.text istobepreferred; and Inow believe the definition inthenote topresent thesubject inthesimplest possible manner, and inthat which, fortheapplications tobemade inthecontinuation of thisEssay,ismost convenient::" thewhole.] XXIV.] AMathematicalTheory ofMagnetism. 367 The resultantmagneticforce atany point, whether inthe neighbourhoodofamagnetorinitsinterior, istheforce that aunitofnorthern magneticmatter wouldexperienceifitwere placedatthatpoint,and ifallthemagnetized substance were replaced bythecorrespondingdistribution ofimaginary mag- netic matter. 481. Thedetermination oftheresultant force atanypoint is,asweshall see,much facilitated bymeans ofamethod first introduced byLaplaceinthemathematical treatment ofthe theoryofattraction, anddevelopedtoaveryremarkable extent byGreen inhis"EssayontheApplicationofMathematical AnalysistotheTheories ofElectricityandMagnetism" (Not- tingham, 1828), andinhisotherwritingsonthesame andon alliedsubjectsintheCambridge PhilosophicalTransactionsy and intheTransactions oftheRoyal Society ofEdinburgh, Laplace'sfundamental theorem issowellknown that itis unnecessarytodemonstrate ithere;but forthesake ofre- ference, thefollowingenunciation ofitisgiven. Theterm otential," defined inconnexion withit,was firstintroduced Green inhisEssay (1828).Itwasatalater date intro- duced independently byGauss, and isnow invery general use. Theorem(Laplace).—Theresultant forceproduced byabody, oragroupofattractingorrepelling particles, uponaunit particle placedatanypoint P,issuch thatthedifference be- tween thevalues ofacertain function, atanytwopointspand pinfinitelynear P,divided bythedistancepp'yisequaltoits componentinthedirection ofthelinejoining pandp. Definition (Green).—This function, which, foragiven mass, hasadeterminate value atanypoint P,ofspace,iscalled the potentialofthemass, atthepointP. Itfollows fromLaplace's general demonstration, that,when thelawofforce isthat oftheinversesquareofthedistance, thepotentialisfound bydividingthequantityofmatter in anyinfinitelysmallpartofthemass, byitsdistance from P, andaddingallthequotientssoobtained. 482. Thesame demonstration isa^^pjicabletoprove,in virtue ofCoulomb's fundamental laws ofmagnetic force, the metheorem with reference toanykind ofmagnetthatcanm 368 AMathematicalTheory ofMagnetism. [xxiv. beconceived tobecomposedofuniformly magnetized bars, either finite orinfinitely small, puttogetherinanyway, that is,ofanymagnetother than anelectro-magnet ;andthein- vestigation,inthepreceding chapter,oftheresultingdistribu- tionofmagneticmatter thatmaybeimaginedasrepresenting inthesimplest possible waythepolarityofsuch amagnet, enables ustodetermine atonce, fromequations (1)and(2)of §473, itspotentialatanypoint. Tims ifVdenote thepoten- tialatapoint P,whose co-ordinates aref, tj,f,and ifdS denote anelement ofthesurface ofthemagnet,situated ata point whose co-ordinates are[x], [y], [z],wehave,bythepro- positionenunciated attheendof§480,— d(il)d(im) d(in) whereAand[A]arerespectivelythedistances ofthepoints x,y,z and\x,y,z\from thepoint P,andaregiven bytheequations [Ar=(?-M)^+ {n-[2/]y+(r- wr- Thedouble andtriple integralsinthe firstandsecond terms ofthisexpressionaretobetakenrespectivelyoverthewhole surface boundingthemagnet,andthroughouttheentiremag- netized substance. Since, asiseasily shown, thevalue ofthat portionofthetriple integralinthesecond member which cor- respondstoaninfinitelysmallportionofthe solid contain- ing (f, 77,f),when thispointisinternal, isinfinitely small, itfollows thatthemagneticforce atanyinternalpoint,as defined in§479,isderivable from apotential expressed by equation (3). 483. Theexpressionsfortheresultant force atany point, and itsdirection, maybeimmediatelyobtained when the potentialfunction hasbeen determined, bythe rules ofthe differential calculus. Thus, ifFhasbeendetermined interms oftherectangular co-ordinates, f, 77,f,ofthepoint P,the three components, X,Y,Z,oftheresultant forceonthispoint willbegiven,invirtue ofLaplace'sfundamental theorem enunciated in§481,bytheformulae, i-_dV^_dV^_dV,,.^-"^' ^""^' ^"""S?^^' IXIV.] AMathematicalTheori/ ofMagnetism. 3G9 where thenegative signsareintroduced, because thepotential isestimated insuch awaythat itdiminishes inthedirection along which anorthpoleisurged.Ifwetake theexpression (3)forF,andactuallydifferentiate with reference tof, 77,f under theintegral signs,weobtainexpressionsforX,Y,and Zwhichagreewith theexpressionsthatmighthave been obtaineddirectly, bymeans ofthe firstprinciplesofstatics (see§477), andthus thetheorem isverified. Such averifi- cation, extended soastobeapplicabletoabody actingaccord- ingtoanylawofforce, constitutesvirtuallytheordinaryde- monstration ofthetheorem. 484. The formulae ofthepreceding paragraphsareappli- cable tothedetermination ofthepotential andtheresultant force, atanypoint, whether within themagnetizedsubstance ornot,accordingtothegeneraldefinition of§480. The case ofapointinthemagnetized substance, accordingtothecon- ventional second definition of§479, cannotpresentitself in problemswith reference tothemutual action between two actual magnets.This casebeingtherefore excluded, wemay proceedtotheinvestigationsindicated in§478. 485. Inthemethod which isnow tobefollowed, themag- netized substances considered must beconceived tobedivided intoaninfinite number ofinfinitelysmallparts, andtheactual magnetismofeachpartwillbetaken intoaccount, whether in determiningthepotentialofthemagnetatagivenexternal point,orininvestigatingthemutual action between twomag- nets. Inthe firstplace,letusdetermine thepotential dueto aninfinitelysmall element ofmagnetized substance, and for thispurpose wemaycommence byconsidering aninfinitely thin, uniformly magnetizedbaroffinitelength.Ifmdenote thestreno^th ofthebar,and ifNand8beitsnorth andsouth poles respectively,itspotentialatanypoint, P,willbeaccord- ingto§§465and481,mm LetAdenote thedistance ofthepointofbisection ofthebar fromP,and theangle between this lineandthedirection of thebarmeasured from itscentre towards itsnorthpole. Then, ifabethelengthofthebar,theexpressionforthepotential becomes 24i. 370 AMathematicalTheory ofMagnetism. [xxiv. f 1 1m il- ^(A'-aAcos6+iay^ (A'+aAcos6+Ja')tJ Byexpandingthisinascending powersofa,andneglectingall theterms after thefirst,wefindforthepotentialofaninfinitely smallbar-magnet,macos6 A'' Ifnowwesuppose anynumber ofsuch bar-magnetstobe puttogethersoastoconstitute amassmagnetizedinparallel lines, infinitelysmall inallitsdimensions, thevalues of6and A,andconsequentlythevalue of—r^,willbeinfinitely nearly thesame forallofthem, andtheproductofthis intothesum ofthevalues ofmaforallthebar-magnetswillexpressthe potentialoftheentire mass. Hence,ifthe totalmagnetic moment bedenoted by fi,thepotentialwillbeequalto Now ifweconceive thebars tohave beenarrangedsoasto constitute auniformly magnetized mass, occupyingavolume <f),weshould have(§461),fortheintensityofmagnetization, i=^'Hence if(f>denote thevolume ofaninfinitelysmall element ofuniformly magnetized matter, and itheintensityof itsmagnetization,thepotentialwhich itproducesatanypoint P,atafinite distance fromit,willbe i6.cos6 A'' whereAdenotes thedistance ofPfromanypoint, E,within theelement, and6theangle between EPandalinedrawn through E,inthedirection ofmagnetizationoftheelement, towards thesideofitwhich hasnorthernpolarity. 486. Letusnowsuppose theelement^tobeapartofa magnetoffinite dimensions, ofwhich itisrequiredtodeter- mine thetotalpotentialatanexternalpoint,PLetf,rj,fbe theco-ordinates ofP,referred toasystemofrectangular axes, andletx,y,zbethose ofE.Weshallhave A^=(?-«^r+('/-?/)^+(r-^r; I 3btiv.] AMathematical Theory ofMagnetism. 371 and,ifI,m,ndenote thedirection-cosines ofthemagnetization T-r /I1^—^ v—yK—^ ,at^,cosO=l^-^+m^-^+n^-^. Hence theexpressionforthepotentialoftheelement JSbe- INow thepotentialofawhole isequaltothesum ofthepoten- tials ofallitsparts, andhence,ifwetake</>=docdydz,wehave, tbytheintegral calculus, theexpression forthepotentialatthepoint P,duetotheentire magnet.* [§§487. ..494addedSeptember,1871.] f77t [487. Theexpansionofthisinascending powersof- ,- ,- , where^^=V(f+^'+n, isnecessarily convergentforallspaceoutside theleastspherical surface with theoriginofco-ordinates forcentre, enclosingthe wholemagnet. Tofindit,wehave first toexpand il(f—^)+im(ji—y)-\-in(X—z) byTaylor's Theorem, inaseries ofascending powersofx,y,z, which isnecessarily convergentordivergent accordingas V(^'+3/'+/)isless orgreater thanVCf+^'+D-Thus, forthepartofYdependingonil,wefind ^^^(-^)'^""i.2.5.S.i.2...J /J-'^'^^'^-y-"---(^)' where2SX denotes summation from tooorelativelyto integers 5, t,u.Hence, remarkinof that -,=— 71:- ,and°r^ afr putting *From theform ofdefiuition given inthesecond footnote on§479, for themagnetic force ataninternalpoint,itmaybeshown that theexpression (5),aswellastheexpression (3),isapplicable tothepotential atanypoint, whether internal orexternal. Thesame thingmaybeshown byproving, as may easily bedone, that theinvestigation of§487does not failorbecome nugatory when(|, rj,^)isincluded inthelimits ofintegration. 24—2 372 AMathematical Theory ofMagnetism. [xxiv. jjj(s'^+t—+^7)x'^z'^dxdydz=[5, f,u] (7) subjecttotheexceptionthatterms ofthe firstmember involv- ing x'^jory~^,or2"^aretobeomitted, wehave Each term ofthisexpansionisasolidharmonic function of f,Tj, 5"[Thomson andTaifs NaturalPhilosophy, App.B.(6), and(sr),(14)(15) (21)]. 488. Neglectingallterms ofhigherorders than thesecond, andputting x,y,zforf,97,f,wehave, asanapproximateex- pressionforthepotentialataverydistantpoint {x,y,z), Lx+My +Nz~ {^+y''+z^)i ^A{2x^~y^-z^)+B{2y^-z'^-x'') +Ci2z^-x^-y^) +S{ayz+hzx+cxi/) {x"^+y^+z^)i'" where X,if,N,A,B,C,u,h,careconstants(dependingonthe magnetismofthemagnet, andtheposition relativelytoitof theaxisofco-ordinates) given bytheequations L=fffildxdydz, M—Jffimdxdydz, N=fffindxdydz (10), A=fffilxdxdydz, B—fffimi/dxdydz, C=fjfinzdxdydz -j a=fff{imz +iny)dxdydz, b=fff{inx +ilz)dxdydz, c=fjj{ily +imx)dxdydz y'' 489. Ifweput K=^{U +M'+N) (12), , ^LxMyNz ,^^.and cos^=^- +^-+i^- (13), KrKrKr^ ' inthe firstterm of(9)itbecomes KcosO/,,.—^(1^)' which isthe firstapproximate expressionforthepotentialof themagnetataverydistantpoint,andagreeswith therigorous expression (§485)forthepotentialofaninfinitelysmall uni- formly magnetized magnetattheoriginofco-ordinates, having itsmagnetic momentequaltoK,and itsdirection ofmagnetiza- tionspecified bythedirection-cosines Hence K,given by(12)and(10),isdefined asthemagnetic XXIV.] AMathematical Theory ofMagnetism. 373 moment ofthegiven magnet;andthedirection(15)isreadily provedtofulfil thecondition stated in§§439,440 asthe definition ofamagnetic axis, determinate indirection but (§444)left tillnowindeterminate astoitspositioninthe magnet.Itistoberemarked thatthevalues ofX,M,Ngiven by(10)areindependentofthepositionoftheoriginofco- ordinates, anddepend onlyonthepositionsoftheco-ordinate axesrelativelytothemagnet. 490. Letnowtheaxes ofco-ordinates beturned tobringone ofthethree intoparallelismwith thedirection ofthemagnetic axis(15). CallingthisOX,andusingthesamesymbols, x,y,z, I,m,n,forco-ordinates anddirection-cosinesrelativelytothe new axes,wehave, instead of(9)and(10), r-2A.,|2a.9'2\f{x'^+y^+z-') A{2x^-y-- z^}+B{22f-z^-x^)+G{2z^-x^-i/)+2{ayz+hzx+cxy) K=JJJildxdydz ;Jfjimdxdydz= ;JJJindxdydz=...(17), withequations (11)unchanged. 491.Secondly,lettheaxis ofxbetransferred fromOXto theparallellinethrough anypointforwhichhe.-Qv ^=j^>y=x^^^)- Thevalues oftheintegralsforthenewaxescorrespondingto hand careeach zero, asisreadilyseenfrom(11)and(17). Hence, alteringthenotationy,ztocorrespondtothenew axes, wehave y^Kx ^A(2x2_y2_ ^2)^jg(2y2_^2_ a;2)+(^(2^2_g^_y2)+2ay^ (X2+ 2/2+22)3 (a;2+y2+^2)t withjjj{inx-\-ilz)dxdydz=Q\fJJ{tly+imx)dxdydz=(20), and(11)inotherrespects unchanged. Now for 2y^-z'- x\and2^'-x""-y\\ wemaywrite I(20), -i(2^^-2/^-^)-l-|(2/^-^^), and-1(2^^-2/'-^^) -f(y-/)J atransformation which, simpleasitis,hasanimportant signi- ficance in"spherical harmonics." Hence ifweput "=hflli^i^^~i»^y-inz)dxdydz,andp-^^jjj{im.y-inz)dxdydz (21), (19)becomes 374 AMathematicalTheory ofMagnetism. [xxTV. ^,_Kx ^a(2a^^y^^/)^0(y^^,^)^2ayz (x'+f+^f {x'+f+z'f 492.Thirdly,shifttheoriginfrom tothepoint aK^=4 (23) inOX;that istosay,for ocsubstitute ic+-^.By(21)and (17)wehave JJJ(2ilx—imy—inz)dxdydz= ;/?=^JJf{{my-^inz) dxdydz (24) ; and(22)becomes y_Kx ^^(f-z'^) +2ayz ^ar+f+z'f {x'^+f-l-z'r 493.Lastly,turn theaxes F,OZ,roundOXthrough an angle equalto Jtan-^^ (26). RelativelytoOX,OY,OZinthis finalposition wehave(17) and(24)unchanged,and ffSiimz +iny)dxdydz=0,fff{inx +ilz)dxdydz=0,fff{ily+imx)dxdydz=(27) ; and(25)becomes reduced to F=''^ ^^(a-+^ro/-^ 494. This isthesimplest expressiontotheseconddegreeof approximationforthedistantpotentialofamagnet having any irregulardistribution ofmagnetism. The axisdetermined by §489(15)and§491(18)isthemagnetic axis,andthepoint initdetermined by§492(23)isthemagnetic centre, ofwhich definitions werepromisedintheaddition to§444.] 495. Theexpression (5)of§486 issusceptibleofavery remarkable modification, byintegration byparts. Thus we maydivide thesecond member into three terms, ofwhich thefollowingisone : il .(f—x)dx III.dydz. Integratingherebyparts,with reference tox,weobtain where thebracketsenclosingthedoubleintegral denote that XXIV.] AMathematical Theory ofMagnetism. 875 thevariables initmustbelongtosomepointofthesurface. If\fi,Vdenote thedirection-cosines ofanormal tothesurface atanypoint [f, t;,^],anddSanelement ofthesurface, wemay takedydz=\ .dS,andhence thedoubleintegralisreduced to '[il]X.dS IP [A]' and, aswereadilyseebytracingthelimits ofthe firstintegral with reference tox,forallpossiblevalues ofySiudzthisdouble integralmust beextended overtheentire surface ofthemag- net.Bytreatinginasimilar manner theother twoterms of thepreceding expressionforV,weobtain, finally, d{il) d{im) d{in) y^jjm-K+[irny.+Un]v^^_jjjd^ ^dz_^^^^^^^ Thesecond member ofthisequationistheexpressionforthe potentialofacertain complexdistribution ofmatter, consisting ofasuperficialdistribution andacontinuous internal distribu- tion. Thesuperficial densityofthedistribution onthesurface, andthedensityofthecontinuous distribution atanyinternal point,areexpressed respectively by[il]\-\-[^m] /m+[in] v,and (d{il) d(im) d{in)\^•/• ^i.-u ^- c 'H\— -H\—^\.Hence weinfer thattheaction of (• \dx dydz thecomplete magnet upon anyexternalpointisthesame as would beproduced byacertain distribution ofimaginary mag- Detic matter, determinable bymeans oftheseexpressions, when theactual distribution ofmagnetisminthemagnetisgiven.* Thedemonstration ofthesame theorem, givenabove(§473), illustrates inavery interesting manner theprocessofintegra- tionbyparts appliedtoatriple integral. 496. Themutual action ofanytwomagnets,considered as theresultant ofthemutual actions between theinfinitelysmall elements intowhich wemayconceive them tobedivided, con- sists ofaforceandacoupleofwhich thecomponentswillbe expressed bymeans ofsixtriple integrals. Simpler expres- *Thisveryremarkable theorem isduetoPoisson, andthedemonstration, as ithasbeenjustgiven inthetext,istobefound inhis firstmemoir onMagnet- ism. Thedemonstration which Ihave given in§473may beregarded as exhibiting, bythetheory ofpolarity, thephysical principles expressedinthe analytical formulae. 376 AMathematicalTheory ofMagnetism. [xxiv. sions forthesame results maybeobtained byemploying a notation forsubsidiaryresults derived fromtriple integration with reference tooneofthebodies, inthefollowing manner :— 497. Letusinthe firstplace determine theaction exerted byagiven magnet uponaninfinitelythinuniformly andlongi- tudinally magnetized bar, placedinagiven positioninits neighbourhood. Wemaysupposetherectangular co-ordinates, f,rj,f,ofthe northpole,andf',77',fofthesouthpoleofthebartobegiven, andhence thecomponents X,F,ZandX\Y',Z\ofthere- sultant forces atthosepointsdue totheothergiven magnet mayberegardedasknown. Then, ifySdenote the"strength" ofthebar-magnet,thecomponentsoftheforces onitstwo poleswillberespectively /3Z,/3r, /3Z,onthepoint (f, 77,f), and -/SZ',-^Y\ -jSZ', onthepoint (f,V,?')• Theresultant action duetothissystemofforcesmaybedeter- mined bymeans oftheelementary principlesofstatics. Thus ifweconceive theforces tobetransferred tothemiddle ofthe barbytheintroduction ofcouples,thesystemwillbereduced toaforce, onthispoint,whose componentsare fiiX-T). ^(Y-T), 0{Z-Z'), andacouple,whose componentsare {0(Z+Z').iiv'v)-^{Y+Y').U^-ni {^(r+r).i(f-r)-^(X+x').K'?-'7')}- 498. LetZ,m,ndenote thedirection-cosines ofalinedrawn alongthebar,from itsmiddle towards itsnorthpole,and ifa bethelengthofthebar,weshallhave ^— ^'=alj 77— 17'==am,^—f'=an. Hence,ifthebarbeinfinitely short, and ifx,y,zdenote the co-ordinates ofitsmiddlepoint,wehave ,dXJdX dXX—X =^-.a6-i- -^-.amH—j—.an,aw aydz ^^ ,^,dY .dY dYY—Y= -^—.al+-Y-'am +-,— .an,dx dydz .^ ry,dZ dZ dZand Z—Z^-^—.al+-y-.am+-^—.an.dx dydz XXIV.] AMathematical Theory ofMagnetism. 377 Multiplyingeachmember oftheseequations by /3,weobtain theexpressionsforthecomponentsoftheforce inthiscase; andtheexpressionsforthecomponentsofthecouplesarefound intheir simpler forms, bysubstitutingfor^— ^',etc., their values givenabove;and,onaccount oftheinfinitelysmall factor which eachterm contains, taking 2X,2F,and2Z,in placeofX+X',r+Y\andZ+Z'. 499. Letusnowsupposeaninfinite number ofsuchinfinitely small bar-magnetstobeputtogethersoastoconstitute amass, infinitelysmall inallitsdimensions, uniformly magnetizedin thedirection{I,m,n)tosuchanintensitythat itsmagnetic moment isfi.We infer, from thepreceding investigation,that thetotal action onthisbody,whenplacedatthepoint x,y^z^ willbecomposedofaforcewhose componentsare (dX, dX dX \ f^W^'-Ty'^^-TzV^ (dY, dY dY \ . 1^ fdZ, dZ dZ \ actingatthecentre ofgravityofthe solidsupposed homo- geneous ;andacoupleofwhich thecomponentsare fjb(Zm—Yn), IM{Xn-Zl), fi{Yl-Xm). 500.Thepreceding investigationenables us,bymeans ofthe integral calculus, todetermine thetotalmutual action between anytwogiven magnets. For, ifwetakeX,F,Ztodenote the componentsoftheresultant force due tooneofthemagnets, atanypoint (x,y,z)oftheother, and ifidenote therintensity and(I,m,n)thedirection ofmagnetizationofthesubstance ofthesecond magnetatthispoint,wemaytakefi=i.dxdydz intheexpressions which were obtained, andtheywillthen expresstheaction which one ofthemagnetsexerts uponan elementdxdydzoftheother. Todetermine thetotal resultant action, wemaytransfer alltheforces totheoriginofco-ordi- nates, byintroducingadditionalcouples ;and,bytheusualpro- cess,wefind, forthemutual action between thetwomagnets. 378 AMathematicalTheory ofMagnetism. [xxiv. aforce inalinethroughthispoint, andacouple,ofwhich thecomponents, Fy(?,H,andX,M,N^aregiven bytheequa- tions (,AZ^.dZ ,.(^^-x],,, —a;(i<-5-+»m-1-+"*^j" )fdxdydz 501. If,inthesecond members oftheseequations, weem- ployforX,y,Zrespectivelytheir values obtained, asindicated inequations (4)of§483,bythedifferentiation oftheexpres- sion(5)forFin§486,weobtainexpressionsforF,G,H,Z, MjNywhichmay readily beputunder symmetrical forms with reference tothetwomagnets, exhibitingthepartsofthose quantities dependingonthemutual action between anelement ofoneofthemagnets,andanelement oftheother.Again, expressions exhibitingthemutual action between anyelement oftheimaginary magneticmatter ofonemagnet, andany element oftheimaginary magneticmatter oftheother, may befound byfirstmodifying byintegration b}^parts,asin§495, from theexpressionswhich wehaveactuallyobtained forFy G,HyL,M,N;andthensubstitutingforX,Y,andZtheir values obtained bythe differentiation oftheexpression (3) of§482, forF. XXIV.] AMathematical Theory ofMagnetism. 379 Itisunnecessaryhere todomore than indicate howsuch other formulae maybederived from thosegiven above; for whenever itmayberequired,there canbenodifficultyin applyingtheprincipleswhich have been established inthis papertoobtain anydesired form ofexpressionforthemutual action between twogiven magnets. §§502and503*—OntheExpression ofMutual Action between twoMagnets bymeansoftheDifferential Coefficients ofa Function oftheir relative Position. 502.Byasimple applicationofthetheoryofthepotential, itmaybeshown thattheamount ofmechanical workspentor gainedinanymotion ofapermanent magnet,effected under theaction ofanother permanent magnetinafixedposition, depends solely onthe initial and finalpositions, andnotatall uponthepositions successively occupied bythemagnetin passingfrom one totheother. Hence theamount ofwork requisitetobringagiven magnet frombeing infinitelydistant from allmagneticbodies intoacertainpositionintheneigh- bourhood ofagivenfixedmagnet, depends solely uponthe dis- tributions ofmagnetisminthetwomagnets,andontherelative position whichtheyhaveacquired. Denotingthisamount by Q,wemayconsider §asafunction ofco-ordinates which fix therelativepositionofthetwomagnets;andthevariation whichQexperiences when this isaltered inanywaywillbe theamount ofworkspentorlost, asthecasemay be,ineffect- ingthe alteration. This enables ustoexpress completely themutual action between thetwomagnets, bymeans ofdif- ferential coefficients ofQ,inthefollowingmanner :— Ifwesupposeoneofthemagnetstoremain fixedduring thealterations ofrelativepositionconceived totakeplace, thequantity Qwillbeafunction ofthelinear andangular co-ordinates bywhich thevariablepositionoftheother is expressed. Withoutspecifying anyparticular systemofco- ordinates tobeadopted, wemaydenote byd^Qtheaugmenta- I*Commnnicated June 20,1850. 380 AMathematicalTheory ofMagnetism. [xxiv. tion ofQwhen themoveable magnetispushed through an infinitelysmallspace d^inanygiven direction, andbyd^Q theaugmentationofQwhen itisturned round anygiven axis, through aninfinitelysmallangle c?(^.Then,ifFdenote the forceuponthemagnetinthedirection ofd^,andLthemoment round thefixed axis ofalltheforcesacting uponit(orthe component,round thefixed axis, oftheresultantcoupleob- tained when alltheforces onthedifferentpartsofthemagnet aretransferred toanypointonthisaxis),weshallhave —Fd^=d^Q,and—Ld<fi=d^Q, since aforceequalto—jPisovercomethroughthespace d^ inthe first case,andacouple,ofwhich themoment isequalto —X, isovercomethrough anangle d(^inthesecond case of motion. Hence wehave 503. Itonlyremains toshowhowthefunction Qmaybe determined when thedistributions ofmagnetisminthetwo magnets andtherelativepositionsofthebodies aregiven. For thispurpose,letusconsiderpointsPandP\inthetwo magnets respectively, and lettheir co-ordinates with refer- ence tothree fixedrectangularaxesbedenoted byx,y,zand X,y,z)letalsotheintensityofmagnetizationatPbedenoted by%and itsdirection-cosinesby I,m,n;and letthecorrespond- ingquantities, with reference toP\bedenoted byi\l\m',n. Then itmaybedemonstrated withoutdifficultythat Id'^ d'\ cZ^^ QHMdxdydzdx'dydz'ii'Y^-^,+Iw!^^,+In^^ ,/A/a/a+ml*7—r-7-\-mm -7—7-7+w?i ,,, ayax dyay ayaz d^\ d''\ +nlT-rr+n7n ,—,- ,+nilj-j-,dzdx dzdydzdz(1). XXIV.] AMathematical Theory ofMagnetism. 881 where, forbrevity, Aistaken todenote{(os—xf+(?/—yY+ {z—zf\^,andthedifferentiations upon-j-aremerelyindicated. Now,byanyoftheordinaryformulae forthetransformation of co-ordinates, thevalues ofx,y,z,andx,y\z,maybeexpressed interms ofco-ordinates ofthepointPwith reference toaxes fixed inthemagnettowhich itbelongs,oftheco-ordinates of thepointP'with reference toaxes fixed intheother, andofthe co-ordinates adoptedtoexpresstherelativepositionofthetwo magnets:and sothepreceding expressionforQmaybetrans- formed intoanexpression involving explicitlythe relative co-ordinates, andcontainingtheco-ordinates ofthepointsP andP'inthetwobodiesonlyasvariables inintegrations,the limits ofwhich, depending onlyontheforms anddimensions ofthetwobodies, areabsolutelyconstant. ThusQisobtained asafunction oftherelative co-ordinates ofthebodies, andthe solution oftheproblemiscomplete. There isnodifficultyinworkingoutthe result bythis method, soasactuallytoobtain either theexpressionsof§500, ortheexpressionsindicated in§501,althoughtheprocessis somewhatlong. [Addition, Dec. 11,1871.—Ifintheformula forQwesupposetheintegration withrespecttox',y,ztobe performed, wehave g=-["rrdxdydz{a%' +pY'+r^%') (2) J-CdJ -00J-GO where a,/?,7areputforil,im,in;and36',Y',^'denote the componentsoftheforce at{x,y,z)duetothesecondmagnet, tobetakenaccordingtothedefinition of§480when{x,y,z) isinthemagnetizedsubstance ofthismagnet. Forsimplicity, without lossofgenerality, suppose a,/S,7tovarycontinuously from finite values inthemagnettozero inspacevoid ofmag- tized substance :and,putting «--ii'. ^--f ^--^(»). I integrate bypartsintheusualmanner(§495). Thus IQ= -Xf/...,..g.|4j)F', 382 AMathematical Theory ofMagnetism. [xxiv. But[§474(2)andPoisson's Theorem] ^+!^+*y=_l (^1+^+^) (4).dxdydz 47r\dx dydzJ Hence, byareverseintegration byparts, ^ ^^LfS /"^^^^3^^^(^^'+YY'+^-^O (5). This isavery important result, asweshall seeinChapter VII. Compare §561.] Themethodjustexplainedforexpressingthemutual action between twomagnetsinterms ofafunction oftheir relative position,hasbeenadded tothischapterrather forthesake of completingthemathematicaltheoryofthe division ofthe subjecttowhich itisdevoted, than foritspracticalusefulness inactualproblems regarding magnetic force, forwhich the most convenient solutions maygenerally beobtained bysome ofthemoresynthetical methodsexplainedinthepreceding partsofthechapter. There is,however, afarmoreimportant applicationoftheprinciples uponwhich this lastmethod is founded which remains tobemade. Themechanical value of adistribution ofmagnetism, althoughithasnot, Ibelieve, been noticed inanywritingshithertopublishedonthemathe- matical theoryofmagnetism,isasubjectofinvestigationof great interest, and, asIhopeonalater occasion* tohavean opportunityofshowing,ofmuchconsequence,onaccount ofits maximum andminimumproblems, which lead todemonstra- tions ofimportant theorems inthesolutions ofinverseproblems regarding magneticdistribution. Chapter Y.—OnSolenoidal andLamellar^ Distributions of Magnetism.^ 504. Inthecourse ofsome researchesuponinverse problems regardingdistributions ofmagnetism, anduponthecomparison *[Chap. VII.... X.below; Dec. 1871.] tCommunicated totheEoyal Society June 20,1850. XXIV.]Solenoidal andLamellar Distributions. 383 ofelectromagnets andcommonmagnets,Ihave found it extremely convenient tomake useofdefinite terms toexpress certain distributions ofmagnetism andforms ofmagnetized matterpossessing remarkableproperties. The useofsuch terms willbeofstillgreater consequenceindescribingthe results ofthese researches, and therefore, beforeproceedingto doso,Ishallgivedefinitions oftheterms which Ihaveadopted, andexplain brieflytheprincipal propertiesofthemagnetic distributions towhichtheyareapplied. Theremainder of thischapterwillbedevoted tothreenewmethods ofanalysing theexpressionsfortheresultant force ofamagnetatanypoint, suggested bytheconsideration ofthesespecialforms ofmag- netic distribution. AMathematical TheoryofElectro-Magnets, andInverse Problemsregarding magnetic distributions, arethe subjectsofpapers which Ihopetobeable tolaybefore the Royal Societyonasubsequentoccasion. [Theyarepublished forthe firsttime inthisvolume :Chaps.VI....X.] 505.Definitions andexplanations regarding Magnetic Sole- noids. (1)Amagneticsolenoid* isaninfinitelythinbarofany form, longitudinally magnetizedwith anintensity varyingin- verselyasthearea ofthenormal section indifferentparts. Theconstantproductoftheintensityofmagnetizationinto thearea ofthenormal section, iscalled themagnetic strength, orsometimessimplythestreugthofthesolenoid. Hence the magnetic moment ofany straight portion,orofaninfinitely smairportionofacurved solenoid, isequaltotheproductof themagnetic strengthintothelengthoftheportion. (2)Anumber ofmagneticsolenoids ofdifferentlengths may beputtogethersoastoconstitute whatis,asfarasregards magnetic action, equivalenttoasingle infinitelythinbarof anyform, longitudinall}^ magnetized withanintensity varying *Thisterm (from aoArii^, atube)issuggested bytheterm "electro-dynamic solenoid"applied byAmpfere toacertain tube-like arrangement ofgalvanic circuits which produces precisely thesame external magnetic effect asispro-duced byordinary magnetism distributed inthemanner deianed inthetext. Theespecial appropriateness oftheterm tothemagnetic distribution ismani- festfrom therelation indicated inthefootnote on§513below, between the intensity and direction ofmagnetization inasolenoid, andthevelocity and direction ofmotion ofaliquid flowing through atube ofconstant orvarying section. 884 AMathematicalTheory ofMagnetism. [xxiv. arbitrarilyfrom oneendofthebartotheother. Hence sucb amagnet maybecalled acomplex magneticsolenoid. Themagnetic strengthofacomplexsolenoid isnotuniform, butvaries fromoneparttoanother. (3)Aninfinitelythin closedring,magnetizedinthemanner described in(1),iscalled aclosedmagneticsolenoid. 506.Definitions andexplanations regarding MagneticShells. (1)Amagneticshell isaninfinitelythinsheet ofanyform, normally magnetized withanintensity varying inverselyasthe thickness indifferentparts. The constantproductoftheintensityofmagnetizationinto thethickness iscalled themagnetic strength,orsometimes simplythestrengthoftheshell. Hence themagnetic moment ofanyplane portion,orofaninfinitelysmallportionofa curvedmagnetic shell,isequaltotheproductofthemagnetic strengthintothearea oftheportion. (2)Anumber ofmagneticshells ofdifferent areasmaybe puttogethersoastoconstitute whatis,asfarasregards mag- netic action, equivalenttoasingle infinitelythin sheet ofany form, normally magnetizedwithanintensity varying arbitrarily overthewhole sheet. Hence such amagnet maybecalled a complex magneticshell. Themagnetic strengthofacomplexshell isnotuniform, but varies fromoneparttoanother. (3)Aninfinitelythin sheet, ofwhich thetwo sides are closed surfaces,iscalled aclosedmagneticshell. 507. Solenoidal andLamellar DistributionsofMagnetism.— Ifafinite magnetofanyform becapableofdivision intoan infinite number ofsolenoids which areeither closed orhave their ends inthebounding surface, thedistribution ofmagaet- isminitissaid tobesolenoidal, andthesubstance issaid to besolenoidally magnetized. Ifafinitemagnetofanyformbecapableofdivision intoan infioite number ofmagneticshells which areeither closed or have theiredgesinthebounding surface, thedistribution of magnetisminitissaid tobelamellar,* andthesubstance is saidtobelamellarly magnetized. *Theterm lamellar, adopted forwant ofabetter, ispreferredto"lami- nated"; since thismight beobjectedtoasrather meaning composedofplane XXIV.] Solenoidal andLamellar Distributions. 385 •508. Complex Lamellar DistributionsofMagnetism.—Ifa finitemagnetofanyformbecapableofdivision intoaninfinite number ofcomplex magnetic shells, itissaid topossessacom- plexlamellar distribution ofmagnetism. 509.Complex Solenoidal Distributions ofMagnetism.—Since, bycuttingitalonglines ofmagnetization, everymagnetoffinite dimensions maybedivided intoaninfinite number oflongitu- dinally magnetized infinitelythin bars orrings, anydistribu- tion ofmagnetismwhich isnotsolenoidal mightbecalled a complexsolenoidal distribution;butnoadvantageisobtained bytheuseofthisexpression,which isonlyalluded tohere, onaccount oftheanalogywith thesubjectofthepreceding definition. 510. Pkop.—Theactionofamagneticsolenoid isthesame as ifaquantity ofpositiveornorthern imaginary magneticmatter numerically equaltoitsmagnetic strength wereplacedatoneend, andanequalabsolutequantity ofnegativeorsouthern matter at theother end. The truth ofthispropositionfollows atoncefrom the in- vestigationofChap.III.§§467, 468, 469. Cor. 1.—Theaction ofamagneticsolenoid isindependentof itsform,anddepends solelyonitsstrength andthepositions ofitsextremities. Cor. 2.—Aclosed solenoid exerts noaction onanyother magnet. Cor.3.—The '^resultant force"(definedinChap.IV.§480) atanypointinthesubstance ofaclosedmagneticsolenoid vanishes. 511. Peop.—Ifibetheintensity ofmagnetization, and cothe areaofthenormal section atanypoint P,atadistance sfromone extremity ofacomplex solenoid, andif[ico]and{ico}denote the valuesoftheproduct ofthesequantitiesattheextremity from which sismeasured, and attheotherextremity respectively ;the magnetic action willbethesame asiftherewereadistributionof imaginary magnetic matter, throughthelength ofthebarofwhich thequantityisaninfinitelysmallpaction ds,ofthelengthatthe plates, thancomposed ofshells whether plane orcurve, and isbesides toomuch associated with amechanical structure such asthat ofslate ormica, tobea convenient term forthemagnetic distributions defined inthetext. T.E.. 25 886 AMathematicalTheory ofMagnetism. [xxiv. point P,would he^—ds,andaccumulationsofquantities equalto— [ift)]and[iw]respectivelyatthetiuoextremities. The truth ofthispropositionfollowsimmediately from the conclusions ofChap.III.§469. 512. Pkop.—Thepotential ofaTuagneticshell atanypointis equaltothesolidangle which itsubtends atthatpoint multiplied byitsmagnetic sti^ength*. LetdSdenote thearea ofaninfinitelysmall element ofthe shell,Athedistance ofthiselement from thepoint P,atwhich thepotentialisconsidered, and6theangle between this line, andanormal totheshell dra^vnthroughthenorthpolarside ofdS.Then ifXdenote themagnetic strengthofthe shell, themagnetic moment oftheelement dS -willbeXdS, and (§485)thepotentialduetoitatPwillbe XdS. cosd A' Now— '-r-rr-—isthesolidanglesubtended atPbytheelement dS,andtherefore thepotentialdue toany infinitelysmall element, isequaltotheproductofitsmagnetic strengthinto the solidangle which itsareasubtends atP.Butthepoten- tialduetothewhole isequaltothesum ofthepotentials due totheparts,andthestrengthisthesame foralltheparts. Hence thepotentialduetothewhole shell isequaltothepro- duct ofitsstrengthintothesum ofthesolidangles which all itsparts,orthesolidanglewhich thewhole, subtends atP. Cor. 1.—Theexpression— '-r-^— ,which occurs inthepre- ceding demonstration, being positiveornegative accordingas 6isacute orobtuse,itappearsthatthesolidanglesubtended bydifferentpartsoftheshell atPmust beconsidered asposi- tive ornegative accordingastheir northpolarortheir south polarsides aretowards thispoint. *Thistheorem isduetoGauss(seehispaper"On theGeneral Theoryol Terrestrial Magnetism," §38;ofwhich atranslation ispublishedinTaylor't Scientific Memoirs, vol.ii.).Ampere's well-known theorem, referred tobyGausp. thataclosed galvaniccircuit produces thesame magneticeffect asamagnetic shell ofanyform having thecircuit foritsedge, implies obviously thetruth ol thefirstpartofCor.2below. Lamellar Distributions. 887 Cor. 2.—Thepotentialatanypointduetoamagneticshell isindependentoftheform oftheshell itself,anddepends solely on itsboundingline oredge, subjecttoanambiguity,the nature ofwhich ismade clearbythefollowingstatement :— Iftwo shells ofequal magnetic strength, X,have acommon boundary, and ifthenorthpolarside ofone,andthesouth polarside oftheother betowards theenclosedspace,the potentials due tothem atanyexternalpointwillbeequal ; andthepotentialatanypointintheenclosedspace,due to thatoneofwhich thenorthernpolarityisontheinside, will exceed thepotential duetotheotherbytheconstant 47rX. Cor. 3.—Oftwopoints infinitelynearoneanother onthetwo sides ofamagnetic shell, butnotinfinitelynear itsedge,the potentialatthatonewhich isonthenorthpolarsideexceeds thepotentialattheotherbytheconstant 47rX. Cor. 4.—Thepotentialofaclosedmagneticshell ofstrength X,with itsnorthernpolarity ontheinside,is47rX, forallpoints intheenclosedspace, and forallexternal points ;and for pointsinthemagnetized substance itvaries continuouslyfrom theinside, where itis47r\totheoutside, where itis0. Cor. 5.—Aclosedmagneticshell exerts noforceonanyother magnet. Cor.6.—The"resultant force" asdefined at§§479,480 [polar definition],isequalto,atanypointinthesub- T stance ofaclosedmagnetic shell,ifrbethethickness, orto 47r/,ifibetheintensityofmagnetizationoftheshell inthe neighbourhood ofthepoint, and isinthedirection ofanormal drawn from thepoint throughthesouthpolarsideoftheshell. [The"resultant force"asdefined below in§517,bytheelectro- magnetic definition, iszero atanypointinthesubstance ofa closedmagnetic shell, orofalamellar distributionconsisting ofclosedshells.] Cor. 7.—Iftheintensityofmagnetizationofanopenshellbe finite, theresultant force atanyexternalpointnotinfinitely neartheedgeisinfinitelysmall;buttheforce atanypointin thesubstance notinfinitelynear theedgeisfinite, and isequal to47r2,ifibetheintensityofthemagnetizationintheneigh- 25—2 388 AMathematicalTheory ofMagnetism. [xxiv. bourhood ofthepoint, and isinthedirection ofanormal through thesouthpolarside. 513. Prop.—Adistribution ofmagnetism expressed by K^jA7)at(x,y,z)}^issolenoidal ifand isnotsolenoidal unless, -j-+-J--^^=zO.dxdydz The condition that agivendistribution ofmagnetism,ina substance offinite dimensions, maybesolenoidal, isreadily deduced from theinvestigationsof§473,bymeans ofthepro- positions of§§510and511. For, ifthedistribution ofmag- netism besolenoidal, theimaginary magneticmatter bywhich thepolarityofthewhole magnet mayberepresentedwillbe situated attheends ofthesolenoids, according to§510,and therefore(§507)willbespreadovertheboundingsurface. On theother hand,ifthedistribution benotsolenoidal, thatis,ii themagnet bedivisible into solenoids, ofwhich some, ifnot all,arecomplex;therewill,accordingto§511,beaninternal distribution ofimaginary magnetic matter intherepresenta- tion ofthepolarityofthewholemagnet. Hence itfollows^ from§473 that ifa,/3,7denote thecomponentsofthe intensityofmagnetizationatanyinternalpoint {x,y,z),the equation^+^+^=...(I.)dxdydz expresses thatthedistribution ofmagnetismissolenoidal+. *Wherea,^,7,whichmaybecalled thecomponents, parallel totheaxes co-ordinates, ofthemagnetization at{x,y,z),denote respectively theproducts0. theintensity intothedirection cosines ofthemagnetization. tTheanalogy between thecircumstances ofthisexpression andthose ofth< cinematical condition expressed by"the equation ofcontinuity" towhich thi motion ofahomogeneous incompressible fluid issubject,issoobvious that itii scarcely necessary topointitout.When anincompressiblefluid flows througl atube ofvariableinfinitely small section, thevelocity (orrather themeai velocitj^)inanypartisinversely proportional tothearea ofthesection. Henc< theintensity and direction ofmagnetization, inasolenoid, according toth< definition, aresubjecttothesame lawasthemean fluid velocity inatube witl anincompressiblefluid flowing throughit.Again,ifanyfinite portion of 5 mass ofincompressiblefluid inmotion beatanyinstant divided intoaninfinite number ofsolenoids (that is,tube-likeparts), byfollowing thelines ofmotion ^ thevelocityinanyoneofthese parts will, atdifferent points ofit,beinversely proportional tothearea ofitssection. Hence theintensity and direction o magnetizationinasolenoidal distribution ofmagnetism, according toth* definition, aresubject tothesame condition asthefluid- velocity and itsdirec tion, atanypoint inanincompressiblefluid inmotion. Itmayberemarked thatbymaking aninvestigation ontheplanof§473toexpress merelyth< condition that theremay benointernal distribution ofimaginary magneti' XXIV.] Lamellar Distributions. 389 514?. Pkop.—Adistribution ofmagnetism [(a,yS,7)at{x,y,z)] islamellarif,and isnotlamellar unless, adx+/3dy+7dzisthe differential ofafunction ofthreeindependentvariables. Let-^beavariable which hasacertain value foreach of theseries ofsurfaces, bywhich themagnet maybedivided intomagneticshells;sothat,if -v/rbeconsidered asafunction ofX,y,z,anyoneofthese surfaces willberepresented bythe equation -^{x,y,z)=11{a) ; andtheentire series willbeobtained bygivingtheparameter n,successivelyaseries ofvalues eachgreaterthan thatwhich precedesitbyaninfinitely small amount.Accordingtothe definition ofamagneticshell(§506), thelines ofmagnetiza- tionmust cutthese surfacesorthogonally; and hence, since a,/8,7-denotequantities proportionaltothedirection cosines ofthemagnetizationatanypoint,wemust have d^ djr dAlr^^' dxdydz Letusconsider themagneticshellbetween twoofthecon- secutive surfacescorrespondingtovalues oftheparameterof which theinfinitelysmall difference isot.The thickness of thisshell atanypoint {x,y,z)willbe fd^dyd^\h' [dx''^ dy''^dzV Now theproductoftheintensityofmagnetization,intothe thickness ofthe shell, must beconstant for allpointsofthe matter, theequation -:,-+^+^^=isobtained inamanner precisely similaraxaydz toamode ofinvestigating theequationofcontinuity foranincompressible fluid, nowwellknown, which isgiven inDuhamel's Cours deMecanique, andinthe Gamhridge andDublin Mathematical Journal, vol. ir.p.282. Thefollowing very remarkable propositionisanimmediate consequence oftheproposition that"a closed solenoid exerts noaction onanyothermagnet" (§510, Cor. 2above),in virtue oftheanalogy here indicated. "Ifaclosedvessel, ofanyinternal shape, becompletelyfilled withanin- compressible fluid, thefluid setintoanypossiblestate ofmotion, andthevessel held atrest;and ifasolidmass ofsteel ofthesame shapeasthespace within thevessel bemagnetized ateach point withanintensity proportionalandina directioncorresponding tothevelocity and direction ofthemotion atthe corresponding point ofthefluid atanyinstant;themagnet thusformed will exercise noforceonanyexternal magnet." 390 AMathematicalTheory ofMagnetism. [xxiv. sameshell; andhence, sincewisconstant, and since a,^,7 denotequantitiessuch that(a^+0^-h 7^)^istheintensityof magnetizationatanypoint,wemusthave (a-+/3- +7¥p...,. fdjf^d^d^y'^^'^^^^^' whereF(^fr)denotes aquantity which isconstant when^jr isconstant. Thisequation, andthetwoequations (b),express alltheconditionsrequiredtomake thegivendistribution lamellar. Bycombining themweobtain thefollowing three, which areequivalenttothem :—^ andhence, ifJF{yjrjdyjrbedenoted by^,wehave «=2'^=|. -=S(")' where ^issome function ofx,y,and z.Hence thecondition thatamagnetic distribution(a,^,7)maybelamellar, issimply that (xdx+^dy+r^dzmust bethe differential ofafunction oJ threeindependent variables. Theequationstoexpressthis are obtained intheirsimplest forms byeliminatingthearbitrary function ^bydifferentiation;andareofcourse dzdy dy^^_(\ dxdz dydx~ Cor.—Itfollows from the firstpartoftheprecedingin vestigationthatequations {h)expressthat thedistribution,i notlamellar,iscomplex-lamellar. Byeliminatingthearbitral function-^from thoseequations (which merely expresstha adx+^dy+<ydzisintegrable byafactor), weobtain thewell knownequation asthesimplest expressionofthecondition thata,y9,7mus,(iii.). XXIV.]Lamellar Distrihutiom. 391 satisfy,inorder thatthedistribution whichthey represent may becomplex-lamellar;andwealsoconclude that ifthisequa- tionbesatisfied the distribution must becomplex -lamellar, unless eachterm ofthe firstnumber vanishesbyequations (III.) being satisfied, inwhich casethedistributionis,aswe have seen, lamellar. 515.Theresultant force atanypointexternal toalamellarly- magnetized magnet will, accordingto§512(Cors. 2and4), depend solely upontheedgesoftheshells intowhich itmaybe divided bysurfacesperpendiculartothelines ofmagnetization (orthebands intowhich those surfaces cutthebounding surface),andnotatallontheforms oftheseshells, within the bounding surface, norupon anyclosed shells ofwhichpartof themagnet may consist; andthe resultant force atany internal pointmay (§512, Cors. 2,4,and7)beobtained by compoundingaforce depending solely onthose-edges,with a force inthedirection contrarytothat ofthemagnetization ofthesubstance atthepoint, andequaltotheproductof47r intotheintensityofthemagnetization.Foreither anexternal oraninternalpoint,theresultant forcemaybeexpressed by means ofapotential, accordingto§480;andthevalue ofthis potential maybeobtained bymeans ofthetheorems of§512, inthefollowing manner :— Letussupposealltheopen shells, that istosayallthe shells cutbytheboundingsurface ofthegiven magnet, toberemoved, andaseries ofshellshavingthesameedges, andthesamemagnetic strengths, andcoincidingwith the bounding surface, substituted forthem; and, forthesake ofdefiniteness, letussupposeeach ofthese shells tohave its northpolarsideoutwards, and tooccupyapartofthesurface forwhich thevalue of<^isgreater than atitsedge. Thewhole surface willthus beoccupied byaseries ofsuperimposed magnetic shells, constitutingacomplex magneticshellwhich willproduceapotentialatanyexternalpointthesame asthat duetothewhole ofthegiven magnet; and, atanyinternal point apotential, which, togetherwith thepotentialdue to theclosed shells roundit,ifthere areany,and(§512, Cor.2) together with theproductof47rintothesum ofthestrengths ofanyopenshellshavingitbetween them andtheirsuperficial 392 AMathematicalTheory ofMagnetism. [xxiv. substitutes, willbethepotential duetothewhole ofthegiven magnetatthispoint. Now ifd^denote thedifference between thevalues of^at twoconsecutive surfaces ofthe series, bywhich wemaycon- ceive thewholemagnettobedivided intoshells,itfollows, from theinvestigationof§514,thatthemagnetic strengthof theshell isequaltod^.Hence ifAdenote theleast value of<f>atanypartofthebounding surface, and <f>besupposed tocorrespondtoapointinthe surface, thestrengthofthe complex magnetic shell, found byaddingthestrengthsofall oftheimaginedseries ofshellssuperimposedatthispoint, wiD be(fi—A;and ifPbeaninternalpoint, andthevalue of </> atitbedenoted by (<^),thesum ofthestrengthsofallthe shells between thatwhichpasses through Pandthatwhich correspondstoA,willbe((/>)—A,from which itmay be demonstrated*, that,whether(0)be>or<A,andwhatever bethenature oftheshells, whether allopenorsomeopenand some closed, thequantitytobeadded tothepotential dueto theimagined complexshellcoincidingwith thesurface ofthe magnettofindtheactualpotentialatP,is47r{((j))—A].Now, fromwhatwehave seen above, itfollows that thepotential atanypoint P,due toanelement, dS,ofthiscomplexshell is—— -TTj:ifdenote theangle which anexternal normal, oranormal throughthenorthpolarside ofdS,makes with a linedrawn fromdStoP;andAthelengthofthis line. Hence the totalpotentialatP,due tothewholecomplex shell, is equalto ' {<!>-A}coseds A' inwhich theintegrationincludes thewhole boundingsurface of themagnet. Hence, ifVdenote thepotentialatP,wehave thefollowing expression, accordingasPisexternal orinternal,— {(jy-A}cosedS A'II orII -//|*-^|,r"tt^w)-j|. *Seesecond footnote on§479above, andCors. 2,3,§515below. XXIV.]Lamellar Distributions. 393 These expressions maybesimplifiedifweremark that, forany- external point, rrcos^_ andthat, foranyinternalpoint. II'-A'=-4-: (sinceistheangle between the lineAand theexternal normal through dS).Wethus obtain, foranexternalpoint, '-}} A^~~ andforaninternalpoint, 4>.cos6dS(V.). -f+47rij,), Cor. 1.—Thepotentialsattwopoints infinitely near one another, even ifonebeinthemagnetized substance andthe other beexternal, differinfinitelylittle;forthevalue of '^.cos6dS II II'-atapoint infinitely near thesurface andwithinit,isfound byadding—47r(0)tothevalue ofthesameexpressionatan externalpoint infinitely neartheformer. Cor. 2.—Ifthevalue of '(j).cos6dS A^ bedenoted hj—Qforanyinternalpoint, x,y,z\and if (a),(^), (7)denote thecomponentsoftheintensityofmagneti- zation, andX,Y,Zthecomponentsoftheresultantmagnetic force atthispoint (that is,accordingtothedefinition inthe second foot-note on§479,theforce atapointinaninfinitely small crevasstangentialtothelines ofmagnetizationatx,y,^), wehave dxdx4<7r{y) ,(VI. 'heresultant ofthepartial components,—47r(a),—47r(/8), •(7),isaforceequalto47r(i) actinginadirection contrary 394 AMathematical Theory ofMagnetism. [xxiv. tothat ofmagnetization, and this,compoundedwith there- sultant of dQdQdQ dx^dy^dz^ which depends solelyontheedgesoftheshells, givesthetotal resultant force attheinternalpoint.Wethus seeprecisely how thestatements made atthecommencement of§515arefulfilled. Cor. 3.—Itisobvious, bythepreceding investigation,that dQdQdQ dx^dy'dz arethecomponentsoftheforce atapointinaninfinitelysmall crevassperpendiculartothelines ofmagnetizationatx^y,z. 516.Ananalyticaldemonstration oftheseexpressions may beobtained byapartial integrationofthegeneral expression forthepotentialinthecase ofalamellar distribution, inthe following manner :— Inequation (5)of§486,which, aswasremarked inthefoot- note, expressesthepotentialforanypoint, whether internal or external, let^,-^,and-^besubstituted inplaceofi7,im, andinrespectively;and, forthesake ofbrevity,let p^^ bedenoted byA :then observingthat.3=-^,and sofor thesimilar terms; wehave Dividingthesecond member intothree terms, integratingthe firstbypartscommencingwith thefactor-~dx,andsoforthe other terms;weobtain V=d-r d-r d \\\TJy^'^Tn'^'^''^-^^''^y y©, dz' , where thebrackets which enclose thedouble inteofral denote XXIV.]Lamellar Distributions. 395 Now, forany that ithasreference tothesurface ofthebody. setofvalues ofx,y^z,forwhich -r-isfinite,wehave, asiswell known, d'ld'^d'^ (o);AAA dx'"*df"*"dx" andconsequently,ifthepoint f,rj,fisnotinthespacein- cluded bythetriple integralintheexpressionfor V,each element ofthisintegral,andtherefore alsothewhole, vanishes. Inthecontrary case,thesimultaneous values ic=f,y= 7],and z=^willbeincluded inthelimits ofintegration, and, asthese values make -rinfinitely great,theequation (c)will failforone element oftheintegral, althoughitstillholds forallelements correspondingtopointsatafinite distance from(f, 77,f).Hence, if(<^)denote thevalue assumed bythefunctioncj)atthispoint, wehave d'l c^^i d^l d^l d' d^l where thelimits ofintegration maycorrespondtoanysurface whatever which completelysurrounds thepoint (f, 77,f).Now itiseasily proved (asiswellknown)thatthevalue of 444' dxdydzdafdy'^dx^ , is—47r,when(f, 97,f)isincluded inthelimits ofintegration; andtherefore thevalue ofthetriple integral,intheexpression forF,is—47r(<^). Hence, accordingasthepoint (f, tj,f)is external orinternal with reference tothemagnet,thepotential atitisgiven bytheexpressions orrrr /^a ^a ^a ii rrr /^a^a^a n+4>Tr{<t>)(VII.). Itmaybeproved that theforce derived from apotential having thesame 896 AMathematicalTheory ofMagnetism. [xxiv. Theseagreewith theexpressionsobtained above in§515;the same doubleintegralwith reference tothesurfacebeinghere expressed symmetrically bymeans ofrectangularco-ordinates. 517. Thevalue of (f>atanypointinthesurface ofthemagnet, which, asappearsfrom thepreceding investigations,isallthat isnecessaryfordeterminingthepotential due toalamellar magnetatanypointnotcontained inthemagnetized substance, may, accordingtowell-knownprinciples,bedetermined by integration,ifthetangential componentofthemagnetizationat every pointofthemagnet infinitelynear itssurface begiven. Itappearstherefore that, ifitbeknown that amagnetis lamellarly magnetized throughoutitsinterior, itissufficient toknow thetangential componentofitsmagnetizationat every point infinitelynear thesurface, ortohaveenoughof data fordetermining it,without anyfurtherspecificationre- gardingtheinterior distribution than that itislamellar, to enable ustodeterminecompletelyitsexternalmagneticaction. This conclusion isanalogoustoaconclusion which maybe drawn, forthecase ofasolenoidal distribution, from theex- pressionobtained in§482, forthepotentialofamagnetofany kind. For,from thisexpression, wehave, accordingto§513, thefollowinginthecase ofasolenoidal distribution : F=//(h-hml3 +ny)dS A.(VIII.); from which weconclude, thatwithout further data reorardinsr theinterior distribution than that itissolenoidal, itissufficient toknow thenormalcomponentofthemagnetizationatevery point infinitelynear thesurface toenable ustodetermine theexternal magneticaction. Yet, although analogouscon- clusions arethusdrawn from these twoformulae, theformulae themselves arenotanalogous,astheformer(thatof§482)is applicabletoalldistributions, whether solenoidal ornot,and showsprecisely how theresultantmagneticaction will in general depend ontheinterior distribution besides thenormal expression (VII.)(1)asforexternalpoints, is,foranyinternal point, theforce atapoint within aninfinitely small crevass perpendicular tothelines of magnetization ;asitiseasily shown thatthedifferential coefficients of47r(0) are therectangular components oftheforce atsuch apoint due[§7(5)]tothefree contrary polarities onthetwosides ofthecrevass. XXIV.] AMathematicalTheory ofMagnetism. 397 magnetizationnearthesurface, accordingtothedeviation from beingsolenoidal which itpresents;while theformulae of§515 merely expressafactwith reference tolamellardistributions, andbeing only applicabletolamellar distributions, donot indicate the effect ofadeviation frombeing lamellar, ina distribution ofgeneralform. Certain considerationsregard- ingthecomparison between commonmagnetsand electro- magnets, suggested byAmpere's theorem that themagnetic action ofaclosed galvaniccircuit isthesame asthat ofa "magneticshell"(asdefined in§506)ofanyfigure havingits edgecoincident with the circuit, ledmetoasyntheticalin- vestigation [§554 below]ofadistribution ofgalvanism through the interior and atthe surface ofamagnet magnetizedin anyarbitrary manner, fromwhich Ideduced formulae forthe resultant force atanyexternal orinternalpoint, givingthe desired indicationregardingeffect ofadeviation frombeing lamellar, onexpressions which, forlamellar distributions, de- pend solely onthetangential componentofmagnetizationat points infinitelynear the surface. Thesegalvanic elements throughoutthebody, from theaction ofwhich theresultant force atanyexternalpointiscompounded, produceeffects which arenotseparately expressible bymeans ofapotential, and therefore, althoughofcourse when thethreecomponents X,Y,Zoithetotal resultant force have been obtained, they willbefound tobesuch thatXdx+Ydy+Zdz isacomplete differential, theseparate infinitelysmall elements ofwhich these forces arecompounded byintegrationwith reference tothe elements ofthemagnet, donotseparately satisfy such acon- dition. Hence theinvestigationdoesnotlead toanexpression forthepotential ;butbymeans ofitthefollowing expressions forthethree componentsoftheforce atanyexternalpoint,or atapointwithinanyinfinitelysmall crevassperpendicularto thelines ofmagnetization, havebeen obtained*:— *Theexpression Xdx+Ydy+Zdz willnotbeacompletedifferential for internal points unless thedistribution ofmagnetism belamellar, since, forany internal point, X,Y,Zdiffer from therectangular components ofthe"resultant force," asdefined in§479,bythequantities 47ra, 47r/3, 4:Try, respectively, and since(§483)the"resultant force," forallpoints, whether internal orexternal, isderivable from apotential. (SeePostscript to§517.) (IX.).398 AMathematicalTheory ofMagnetism. [xxiv. z=///...,..j^^(|-|)-£^'(|4;)) r=///«,.,{tf(f-|)-t5(|-f)} .=//j«,..(t^g-|)-'-^'(f-l)C [Postscriptto§517,Fov. 17,1871.—Theseexpressions,to beprovedin§518 forexternalpoints, maybetaken asa definition for"resultant force"atpointsinthemagnetized substance. Theyaresimplified byputting dy_d^_doL^dy_ d/3^da_"j dydz"'dzdx'dxdy\(a), andw/3—my=11^ ly—not=F,men—1/3=W) which, with x\y',zsubstituted forx,y,z\ u\v',w'forii,v,w\ and X,y,zfor^,rj,^;reduces them to with thesymmetricalforms forYand Z.Now observe that isthe^/-componentoftheresultant force at{x,y,z)duetoa distribution ofimaginarymatterthroughthemagnet andover itssurface, having wfordensityatanyinteriorpoint {x,y,z)j andWforsurfacedensityat[x,y,z] ;andfortheother terms of (6), etc.,consider correspondingdistributions(v,V),and(u,U); andtherefore instead of(h),etc.,write j^^dN_dM Y=^-— Z=—-— (c) dy dz^dzdx'dxdy.^ denoting*by *This notation hasbeenintroduced toagree with thatusedbyHelmholtz in corresponding formulae with reference toVortex Motion. Itistoberemarked XXIV.] AMathematical Theory ofMagnetism. 399 Lthepotentialofdistribution{u,U)M „ „ „ {v,r)} {d)N „ „ „ {to,W) sothati:=jjj^^L+ jj^^M=etc.,F=etc.(e); iand,byPoisson's theorem, where, asisnow usual, ^-^+-r-^+-y-^isdenoted byv^The second members of(/)vanish forallpointsexternal tothe magnet,because there ic=0,v=0,w=0.Now forsimplicity supposethemagnetizationtodiminishgradually,notabruptly, tozero attheboundaryofthemagnet. Thesecond terms of theexpressions (d)forL,M,iVwilldisappear,andbydiffe- rentiations andsummation wehave fdu'dxidw\Tri,1 , dLdMdN fffW'^W d^)^ dx^dy'^dz.JJ] D t(a)show that£+f+S=^ ^)' andtherefore -y-+^—+-^=0 in),axaydz Howeverquickthegradationfrom finite values ofu,v,w within themagnet,tozerothroughexternalspace,tliisequa- tionholds, andtherefore itholds inthelimit,when themag- netization comes toanendabruptlyattheboundary. To prove Qi)directly from theexpressions (e),with thesurface- terms included, willbefound agoodexercise forthestudent. From(c)bydifferentiations, andapplicationof(/)and(^), wefind f^+y+f=oik)dx dydz dZdY,dXdZ,dVdX . \ dy dz dzdx dxdy 3rinvirtue of(a)r(0 <]:Z__dY_ /dyd^\dX_dZ_ (da dy\dY_d^_A f^_^\ dy dz \dy dz)*dzdx~ \dz dx)^dx dy'' \dx dyjj thatthequantities u,v,Wj TJ,V,Wthus introduced fulfil theequations (1)and (2)of§539. They represent thecomponentsoftheinternal and superficial distributions ofelectriccurrents, intheelectro-magnetic representative (§554)of thegiven polar magnet. 400 AMathematical Theory ofMagnetism. [xxiv. Thecorresponding propertiesofK,Y,%, ifthese denote thecomponentsofthe"resultant force"asdefined in§§479, 480,are[see§473(2)and§483] ^.^+i^=_47rf—+"^+^1 (m)dx dydz \dx dydz)^'^ dy dz^dzdx'dxdy^ Theseequations,aswell as(^^)and(^),holdthroughallspace, thevalues ofa,y8,7beingzero inevery partofspacenotcon- taining magnetizedmatter. Some ifnot allofthe differential coefficientsappearinginthem become infinite when themag- netization variesabruptlyfrom oneside totheother ofany surface, buttheinterpretation presentsnodifficulty. Taking forinstance thecasewhen themagnetization,finite uptothe boundaryofthemagnet, comes toanendabruptly there, let X^andX^^denote thevalues ofXatpoints infinitelynear oneanother outside and inside theboundary; andsimilarly forr,Z,%¥,%,Wehaveby§7(5),§517(c)and(g),and §473(1), X,-X,,=47r(nF-7/iTr), Y,-T,,=^ir{lW -nV), Z,-Z^=A7r {mU-l V)(o) andX,-X„= 4i7rpl, Y,-Y,,=47rpm, ^^-^^^=4:7rpn\, . wherep=lu+mv+nwJ^ By(a)wehave nV-mW= I(m^-\-ny)—(m^+n-)a=l(la+mjS+ny)—a. Hence, withthenotation of(p), (0)becomes X-X^=4.7r{lp-a), Y-Y^=i7r(mp-P), Z-Z^^^tt {np-y) (q). Inafoot-note to§517above itwasstated thatthevalues of X,Y,Zdiffer fromwhat inthispostscriptIcallX,Y,5Sby quantities equalto47ra, 47ry8, 4!7ry, respectively;astatement which isnodoubt tobeproved directly bycarefully examining themeaningoftheintegralsof§518 forinternalpoints.We maynowverifyitbytakingthedifference between(k)and (m),andthedifferences between(Z)and(n).Ifintheseweput X-ae-47ra=P, r-Y-47ryS=ft Z-S^-^wy=R, ^,. dPdQdR^theygive ^+-^+"5^= wAMathematical Theory ofMagnetism, 401 dR_dQ^^ dP__dR^^ dQ_dP^^ dydz^dzdx'dxdy These lastthreeequationsshow that dx^ dy'dz wherei/rifnotzero isafunction ofa?,3/,z-.andthe firstthen becomes^^^_ dx''*"df"^ (/^^~ Thisequation must holdthroughallspacewhen there isno abruptvariation ofmagnetization ;and, asi/rmust vanish at aninfinite distance from themagnetinanydirection, wemust (§206above)therefore(whetherthere areabruptvariations ornot)have^—^. Theproofmaybeillustrated forabrupt variations, bytakingthedifferences ofequations (g)and['p), which show thatK[X-^- 47ra),-(Z-ae- 47ra),,=0;(7etc,Zetc.) ; or P-P.^=0,Q,-Q^=0, R-R^=0; whichprovethat"^t"'^u~^} thesuffixed accentsdenotingvalues forinfinitelynearpoints onthetwosides ofthesurface ofabrupt change. Weconclude thatthroughallspace Z=3e+47ra,F=|^+47r^,Z=^+4<7ry (r); which, forspace unoccupied bymagnetized matter, give (what weknewbefore) x=x, r=^, z=^. Forspacewithin themagnet,itwasshown in§479 that theforce(^,^,5S)istheresultant forceexperienced byaunit poleinacrevasstangentialtothelines ofmagnetization. From this,and(r),itfollows that, aswasasserted in§517,theforce (X,F,Z)istheresultant forceexperienced byaunitpolein acrevassperpendiculartothelines ofmagnetization. Ofthese twodefinitions of"resultant force"forspacewithin amagnet, theformer, assuitable toaj^olar magnet (§549),willsome- times becalled the''polar definition," andthe latter, assuit- T.E. 26 402 AMathematical Theory ofMagnetism. [xxiv. able foranelectromagnet,the'' electromagnetic definition," for thesake ofbrevity.] 518. Theinvestigation bywhich Ioriginallyobtained the expressions (IX.of§517) is,with reference togalvanism, precisely analogoustotheinvestigationin§473with refer- ence toimaginary magneticmatter. Itcannot begivenwith- outexplanations regardingtheelements ofelectro -magnetism which would exceed thelimits ofthepresent communication*; butwhen Ihadonce discovered theformulae Ihadnodiffi- cultyinworkingoutthesubjoined analyticaldemonstration of them forthecase ofanexternalpoint, which ispreciselyanalo- goustoPoisson'soriginal investigation (givenin§495above) oftheformula of§482. Equations (3)and(4)of§§482and483 lead toexpres- sions forthecomponentsoftheresultant force atanypoint intheneighbourhoodofamagnet. TakingXonly (since theexpressionsforthethree componentsaresymmetrical), wehave Now ifthefactor ofdxdydzinthesecond member ofthis equationbedifferentiated with reference to^,anexpression isobtained which does notbecomeinfinitely greatforany values ofx,y,zincluded within thelimits ofintegration,since thepoint (f, ?;,f)isconsidered tobeexternal inthepresent investigation. Hence thedifferentiation with reference toI maybeperformed under theintegral sign; and, since d-r- d-rA A wethusobtain :jjjdxdydjidi dx' ^^HJ'^^dxdy^^ dxdzW' *[Note, Nov. 1871.—Itisgiven in§554,below.] +Ifthepoint (^, rj,f)beeither within themagnet orinfinitely near it,th factor ofdxdydzinthisintegralisinfinitely great forvalues of(x,y,z)include Hfv, I.]AMathematical Tlieory ofMagnetism.403 Now, forallpointsIdeluded within thelimits ofintegration, wehave, fromLaplace'swell-knownequation, m li_ therefore Dividingthesecond member into four terms, andapplyingan obviousprocessofintegration byparts,wededuce -.J c?-rdjrd-7~ d-ir I JJJ [dydydzdzdxdydxdz\ Modifyingthedoubleintegral byassuming,initsdifferent terms, dydz—IdS; dzdx=7ndS\ djody=ndS, andalteringtheorder ofalltheterms, ^veobtainX= within thelimits ofintegration ;and itmaybedemonstrated thatthevalue ofa partoftheintegral corresponding toanyinfinitely small portionofthemagnet infinitely nearthepoint (^, rj,^)isingeneral finite, andthat itdepends onthe form ofthisportion, onitsposition with reference totheline ofmagnetization through (t, t;,^),andontheproportions ofthedistances ofitsdifferent parts from thispoint.Itfollows that ifthepoint (^, 77,f)beinternal, and ifaportion ofthemagnet round itbeomitted from theintegral, thevalue oftheintegral willbeaffected bytheform oftheomittedportion, however small itsdimensions may be,andconsequently thecomplete integral hasnodeterminate value ifthe point (^, 77,^)beinternal. Hence although, aswehave seen above(§§482, •183), jj\dxdvd.\f±^^'^^y4\ hasinallcases adeterminate value, which, bythedefinition{§479),iscalled thecomponent paralleltoOXoftheresultant force at{^, t],i"),theexpression jjhH{S4-4\ hasnomeaning when(^, 77,f)isinthesubstance ofthemagnet. 26—2 404 AMathematical Theory ofMagnetism. [xxiv. Hll"y'-[i&.-tAfdy dx\ dz\dx dz) -[//|^™''-^^)-'i('^-'"T^' Thisexpression, when theindicated differentiations areactually performed upon— ,becomes identical with theexpressionfor Xattheendof§517,andtheformulae which itwasrequired toprove aretherefore established. 519. Thetriple integralsinthese expressionsvanish inthe case ofalamellardistribution, invirtue oftheequations (HI.) of§514;andwehavesimply Z=x=- r=-- r\^md^ -^(ly-n.)ds']! m7)-^(ma.1/3)ids i(h-n^) dy{njS—my)dS,(X.) Tointerpret theseexpressions,letusassume, forbrevity, U=n0-my; V=ly-noi;W=^7noL -II3...(XI-) From thesewededuce 771W—nV=OL—l(?a+m/3+ny)=^A nU-lW=p^m\loL+ml3 +ny)=p\ CXTI); ZF—mZ7=7—n(Za+771/3+717)=7j wherea^,^^,%denote therectangular componentsofthe tangential componentofthemagnetizationatapoint infinitely near thesurface. Conversely,from theseequations wededuce U=nfi^-my/, V=l%-m/, F=ma^-Z^,...(XIII.) Now the direct datarequiredforobtainingthevalues oi X,Y,andZ,bymeans offormulae(X.),aresimplythevaluer ofZ7,V,Watallpointsofitssurface.Equations (XII.)show thatwith these datathevalues ofa^,jS^,7,maybecalculated andagain, equations (XIII.) show converselythat ifa^,/?,,7 |] XXIV.] AMathematical Theory ofMagnetism, 405 begiventherequireddata fortheproblem maybeimmediately deduced. Weinfer thatthenecessary and sufficient data for determiningtheresultant force ofalamellarmagnet,atany externalpoint, bymeans offormulae(X.),areequivalenttoa specificationofthedirection andmagnitudeofthetangential componentoftheintensityofmagnetizationatevery point infinitelynear thesurface ofthemagnet;andweconclude, aswedidin§517from averydifferentprocessofreasoning, thatbesides these data, nothingbutthat itislamellarthrough- outneedbeknown oftheinterior distribution. 520. The closeanalogy which exists between solenoidal and lamellar distributions ofmagnetism havingledmetothenew formulae which havejustbeengiven,itoccurred tomethata formula(orformulae, ifitwerenecessaryhere toseparate the cases ofinternal andexternalpoints),forsolenoidal distribu- tionsanalogoustotheformulae(VII.)of§516 forlamellar distributions mightbediscovered. Taking ananalytical view oftheproblem (the synthetical view, althoughitselfmuch more obvious, notshowing anyveryobvious wayofarriving ataformula ofthedesiredkind),Iobserved thattheformula ^' . .^isdeduced from thegeneral expressionforthe potential byapartial integration performed uponfactors in- volving a,/3,7,anddependingontheintegrabilityofthe function adoo+jSdy+jdjz,insured bytheequations d0_dr/_^dy_dci_doL_d/3_ dzdy'dxdz'dydx' foralamellar distribution;andIendeavoured tofindacorre- sponding mode oftreatment forsolenoidal distributions, to consist ofapartial integration, commencingstillwith factors involving a,^,7,butdepending nowuponthesingle equation dx dl3dy^. . ^+%+^=''('')' instead ofthethreeequations requiredintheformerprocess. After some fruitlessattemptstoconnect thisequation with theintegrabilityofsome function oftwoindependent variables, Ifelluponthefollowing investigation, whichexactly answered myexpectations:—//' 406 AMathematical Theory ofMagnetism. [xxiv 521. Invirtue ofthepreceding equation (a),wemayassum( a=dH dydG dz'dF_dH dzdx'^S-f--(^i^-). where F,G,Harethree functions toacertain extentarbitrary These functions Ihave since found, have fortheir mostgenera expressions dz doida' dxj d^' "-^Ij-M^-S)*dy d^ dz.(XV.) where-xjrdenotes anabsohitely arbitrary function; and th( indicatedintegrationsareindefinite, with thearbitraries whici theyintroducesubjecttotheequations (XIY.). Thedemonstration oftheseequationsfollows immediate! from theresults obtained bydifferentiatingthethreeequation; (XIV.)with reference tox,y,andzrespectively. Thesimplesi final forms forF,G,andHarethefollowing, which arede duced from thepreceding byintegration:— ^^y^^T adz)+d^ dy H= ij(ady-^dx)+ '^^^.(XVI). Makingsubstitutionsaccordingtotheformulae(XIV.)fo a,y5,7inthegeneral expressionforthepotential,Avehave JJJ [\dy dzJdx\dz dxJdy\dx dy/dz] Dividingthesecond member into sixterms, andintegratin* rlTT eachbyparts, commencing uponthefactors such as-7-dy weobtain anexpression,with atriple integral involvingsi: terms whichdestroyoneanother twoandtwobecause c XXIV.] AMathematicalTheonj ofMagnetism. Wl propertiessuch as ,1 ,1 d^A_dA ^ dydx~ dxdy^ and besides, adoubleintegral, which maybereduced inthe usualmanner toaforminvolving dS,anelement ofthesurface. Wethus obtain, finally, V=[\\\{mH-nG)^^HnF-lR)^^ (XYII.). 522. The second member ofthisequation expressesthe potentialofacertain distribution ofmagnetisminaninfinitely thin sheetcoincidingwith thesurface ofthebody ;thetotal magnetic moment ofthemagnetismintheareadSbeing [(iiiH- nGf+[nF-IHf-\-{lG- mFf}^ dS, and itsdirection cosinesproportionalto mH-nG, nF-lH, IG-mF. Nowwehaveidentically, I[mH-nG)+m {nF- IE)+?i{IG-mF)= ; andhence the direction ofthisimaginary magnetizationat every pointofthesurface isperpendiculartothenormal. It follows thatwehave found adistribution oftangential mag- netism inaninfinitelythinsheetcoincidingwiththebounding surface whichproducesthesamepotentialatany point,in- ternal orexternal, asthegivensolenoidalmagnet. [Itisre- markable that theimaginary tangential magnetizationthus found[depends (§523)upontheno7inal componentofthe actualmagnetization infinitelynearthesurface;sothat, besides thisnormalcomponent, nothing need beknown oftheactual magnetization exceptthat itissolenoidal. Compareconclu- sionof§519.] 523. The conclusion of§522maybearrived atsyntheti- callyinaveryobvious manner, bytakingintoaccount the propertyofasolenoid stated in§510, accordingtowhich itappearsthatanytwosolenoids ofequal strength,with the same ends, producethesame force atanypoint whether inthe magnetized substance ofeither, ornot. For itfollows from 408 AMathematical Theory ofMagnetism. [xxiv. this,thatwhen amagnetisdivisible into solenoids with their endsonitssurface, wemaybyjoiningthetwoends ofeach solenoid byanyarbitrarycurve onthis surface, andlaying a solenoid ofequal strength alongthis curve, obtain aseries of solenoids, constituting bytheirsuperposition, atangential distribution ofmagnetisminaninfinitelythinsheetcoinciding with thebounding surface, whichproducesthesame resultant force atanyinternal orexternalpointasthegiven magnet. Itisnot,however, easytodeduce from thissynthesisaformula involvingtherequisite arbitraryfunctions toexpress asuper- ficial distributionsatisfyingtheexistingconditions inthe mostgeneral manner. Theanalytical investigation given above, supplies,inreality,acompletesolution ofthisproblem. Itmayberemarked thatthesolecondition which F,andH considered asfunctions oftheco-ordinates, x,y,Zjofsomepoint inthesurface ofthemagnet,andtherefore functions oftwo independent variables, mustsatisfyinorder that(XVII.) may express correctlythepotentialatanypoint,is— .(dH dG\^ [dFdH\^Ida dF\,^ ^^/wTTTN X,y,andzofcoursebeing supposedtosatisfytheequationto thesurface;and itmaybeproved, byademonstration inde- pendentoftheinvestigation which hasbeengiven,that the second member of(XYII.)hasthesame value foranyfunc- tions F,G,Hwhatever, which aresubjecttothis relation. [Postscript,Dec. 7,1871,andJan. 6,1872.—Inasmuch asthe second member of(XVIII.)is(§473(1)),thesurfacedensityof theimaginary magnetic matter, representingthepolarityofthe givensolenoidalmagnet, wemayeliminate theidea ofmagne- tization, andsoarrive atthefollowing remarkable theorem :— Letpbethedensityatanypointofasuperficialdistribution ofmatter onasurface>S^,whichmaybeeither aclosed surface oranopen shell, therebeingasmuchnegativematter asposi- tive inthewholedistribution, and letF,G,Hbeanythree quantitiessuch that (dHdG.(dFdH\^ (dGdF\ ,„.„, XXIV.] Electromagnets. 409 thepotentialofthis distribution, that istosayM^t-, is correctly expressed bytheformula(XVII.). When >S>isa closed surface thisexpressionholds forthespacewithin>S^,as well asforexternalspace. From theremark with reference to (XVII.)and(XVIII.)attheconclusion ofthesection inthe original nownumbered§523, itappearsthat thevalues of F,G,Hgiven by(XVI.), although expressingthemostgeneral solution of(XIV.),arenotthemostgeneral expressionsfor functions F,G,Htosatisfy (XVII.);and that instead of F,G,Hin(XVII.) wemaysubstitute FJrF\ G+G',H+H' where F,G,Haregiven by(XVI.) andF\G\H'areany three functions ofx,y,z,which, over thewhole surface S, satisfytheequation l('^^^^] +^(^^^)+n('^-—] =(XX) \di/dzJ \dz dxJ\dx dy)"*^^ The surface distribution oftangential magnetization specified byF',G\H'inaccordance with theexplanationsof§522, consists ofclosed solenoidslyingonthesurface/S^.] Chapter VI.*—OnElectromagnets. 524. Oersted'sdiscoveryofthemutual forces between magnets andconductorscontainingelectric currentsgaverise tothescience ofelectromagnetism.Itwassoon found that there arealsomutual forces between different conductors and between differentpartsofthesame conductorconveying electric currents :andvariousveryremarkableelectro-magenetic phenomena were observed bydifferentexperimenters,ofwhich themost remarkable arethecontinuous rotations ofportions ofconductors roundmagnets andofmagnetsround conductors, * {Note, October 1871.—This chapter waswritten twenty-two years ago,and haslaininmanuscript ever since, because Ihadnotsucceeded infinding time towrite asequel oninverse problems.Itisnow printed from theoriginal manuscript with onlyafewverbal alterations, and itwillbefollowed inthis volume (Chap. IX.)bythelong-projected article oninverse problems, ofwhich something wascommunicated totheBritish Association atitsOxford Meeting of 1847, butnotpublished exceptinthevery short abstract contained inthe Beport ofthatmeeting.] 410 AMathematical Theory ofMagnetism. [xxiv. discovered byFaradaytoresult, incertain circumstances, from theirmutual actions. Thelaws towhich allthese actions are subject were firstcompletely investigated byAmpere. His experimentsarethefoundation, andtheconclusions which he deduces fromthem constitute theelements, oftheMathematical TheoryofElectromagnetism. Asacomplete andsatisfactory account ofthese researches istobefound inAmpere's original papers*, andasuccinctexpositionofthemathema- ticalpartoftheinvestigations,inMurphy'sTreatise onElec- tricity,theresults willbeconsidered asfully established, and those ofthem which arerequiredinthepresent essaywillbe quoted. 525. LetPandP'bepointsintwoconductors, ofwhich the lateral dimensions areverysmallcomparedwith thedistance PP']letaandabethelengthofinfinitelysmall elements ofthese conductors, with their centres atthepointsPandP respectively,andterminatedbyplanes perpendiculartothe directions oftheconductors;letPP'bedenoted byr;let^ and 6'denote theanglesatwhich thedirections ofthecon- ductors atPandP'areinclined tothelinePP';and let be theangle between twoplaneseachpassing through PP\ and respectively containingthedirections oftheconductors. Thus, ifthere beelectrical currents inthetwoconductors, theywill mutuallyactand react with asystemofforce which isthe same aswould result frommutual forces, inlinesjoiningall theinfinitelysmall arcsaoftheone,and cr'oftheother, given inamount(attractions reckonedpositiveandrepulsions nega- tive)bythefollowingformula :— '^'^— (2sin6sin&cos^—cos6 cos$')f *"Sur laTheorie Mathematique desphenombnes electro-dynamiques." Collection ofsix"Memoires" ofdates 4thand20thDecember 1820, 10thJune 1822, 22ndDecember 1823, 12thSeptember and21stNovember 1825. PubHshed intheMemoires oftheFrench Academy, 1827. +[Note,Oct.1871.—Intheoriginal manuscript theformula stands '^'^'^^ (sindsin^cos-Jcos6cos6'). Ihavedoubled itssecond member toavoid theinconvenient distinction between "electro-dynamic" and"electro-magnetic" units towhich initsoriginal form, (theform inwhich Weber used itinhissystem ofabsoluteunits,)itleads. See below, §531.] XXIV.] Electromagnets. 411 where 7and7'denotequantitiesinvariable invalue forthe same twoconductors, with thesame electrical currents flow- ingthrough them. Thesequantities (7,7')arethenumerical measures ofthestrengthsofthecurrents. 526. Again,letNbethenorthpoleofaninfinitelythinuni- formlyandlongitudinally magnetized bar,and8itssouthpole: letPbeapointinaconductor, and letaand7denote thesame asbefore, with reference tothisconductor. LetNPandSP bedenoted byAandA'respectively; and lettheangles betweenNPand crandbetween JV>SiPandabedenoted by <f> and<j>respectively. There willbesuch amutual action between themagnet andthegalvanicarcathateach will experienceaforce, theresultant oftwo forcesthroughPper- pendicular respectivelytotheplanesofNPandcr,andofSP and a,giveninamountbythefollowing expressions respec- tively:— m .7<r.,,m .70-.,, 'sin(p,and—inrsm (f>. Thedirections ofthese forces, upontheelement, ifthedirection ofthecurrent befrom east towest, and ifNandBbeeach north ofP,willbe;—theformerobliquelyordirectly down- wards, andthe latter,—upwards. [Formnemonicprinciplesee below, §547.] Themagnetwdllbeacted uponasifapoint inthepositionofP,rigidlyconnected withit,experienced two forcesequal andoppositetotheforces ofwhich theaction onaiscompounded. 527. Let[x,y,z)denote themiddlepointoftheelementor, and {x\y\z')themiddlepointoftheelement a,according toordinary rectangularco-ordinates. Let alsoI,m,nbethe direction cosines oftheformer element, and V,m\n'those of thelatter; quantitieswhich willbeallpositive when the current ineach element isinasimilar direction tothat of apointmoving from theorigintowards thespace between the positive partsoftheco-ordinateplanes. Theexpressionfor theforcebetween theelements interms ofthese data, willbe !2(/r+mw^+nwO[(a;-a:02+(>/-y02+(^;-g^)S]-3[Z(x-a;0+m(y-yO+?t(g-^0][r(a;-a;0+m^(y-?/^)W(z-^0]} {{x-x'Y-V{y-y'f +(z-z'f]'^ 528.Again,iff,77, i^denote theco-ordinates ofaunitnorth 412 AMathematical Theory ofMagnetism. [xxiv. pole,andoo,y,zthose ofaninfinitelysmall element o-ofan electric current ofstrength 7,inadirection(?,m, ?i),themutual action willbe 7cr.sin (jb 7(7.sin andwillbeinalineofwhich thedirection cosines are m{^-z)'-n{7j-y )n{^-x)-l{i;^z) l{rj-y)-m(^-x) Asiii<j>' Asm<f)* Asin</) 529. Hence thecomponentsoftheforceexperienced bythe element ofelectric current aregiveninmagnitude and direc- tionbythefollowing expressions:— y(T[m(^-z )-n(7}-y)] y(T[n(^-x )-l{^-z)] fya\l{r)-y)-m{^-x) A''A^'A^ Iftheaxes ofco-ordinates besochosen thatwhenOX is from south tonorth, andOYfrom east towest,OZwillbe vertically upwards,theseexpressionswillbeapplicable,as farasregards signs,tothedirection oftheaction which the electric arcexperiences;and itwould benecessarytochange thesignofeach, tomake themapplicabletothedirection of theforceuponthepole. 530. Theseexpressions,sincetheyinvolveI,m,nonlyline- arly,show thatagalvanicarco;ofstrength 7,inthedirection I,m,n,producesthesame effect either upon another arc, or uponamagnet,asthree arcsparalleltotheaxes ofco-ordinates, each ofthesamestrength, 7,andoflengths respectively equal toal,am,an. 531.The factor 7beingtaken asthenumerical measure of thestrengthofthecurrent inthecircuit ofwhich aisanarc, theunit ofstrengthforanelectric current maybedefined in thefollowing manner :— Ifagalvanic current, inaconductor ofinfinitelysmall section, besuch thatthemutual action between anyinfinitely small arcofit,andaunitmagnetic poleheld inadirec- tionperpendiculartothelengthofthe arc,ataunit ofdis- tance,isnumerically equaltoatheinfinitely smalllength oftheelement, thestreugthofthecurrent isunity. XXIV.] Electromagnets.413 Orinthefollowing manner :— Ifagalvaniccurrent inaconductor ofinfinitelysmall section besuch that theaction between twoinfinitelysmall portionsofitinlinewith oneanother and atadistance unityfrom oneanother, isnumerically equaltotheproductof thelengthoftheelements, thestrengthofthecurrent isunity. 532. Ifwhat iscalled"anelectric current" beinreality thetransference ofmatteralongtheconductor inwhich itexists, the"strengthofthecurrent"numerically measured inthe manner which hasbeenexplained,willdepend uponthequantity ofthismatter transmitted inagiven time; andaunit oftime maybechosen, accordingtotheunit ofelectricalquantity which isadopted,sothat thequantity 7,measured asabove explained bytheelectro-magneticaction oftheconductor, may benumericallythequantityofelectricity which flows across anysection ofitinaunit oftime. 533. Inacontinuous current,thisquantityisofcourse the same forevery section; and, asitisimpossiblethatacontinu- ousstream ofelectricitycanemanate from onebody,andbedis- chargedintoanother, thecurrent must bere-entering,orevery continuous current must formwhat istermed"aclosed circuit." Itisfoundbyexperimentthatwhatever bethedimensions or material ofthedifferentpartsoftheconductoralong which thecurrent flows, provided alwaysthedimensions ofthesection besmallcomparedwith thedistances through which the electro-magneticaction isobserved, thequantity 7hasthesame value for allpartsofit;andeven intheplaces where the electro-motive forceoperates,ashasbeenshown byFaraday, asintheliquidofanyordinary galvanic battery,orinacon- ductor inmotion intheneighbourhoodofamagnet,theelectro- magneticeffects areobservable andprobablytoexactlythe samedegree;sothat itwouldprobablybefound that agal- vanic circuitconsistingofabatteryofsmall cells arrangedin acircular arc,andawirecompletingthecircuit byjoiningthe poles, would producethesameelectro-magneticeffects atall points symmetricallysituated with reference tothe circle, irrespectivelyofthepartofthe circuit, whether the cells or thewire; provided alwaysthat thedistances considered be great compared with either thedimensions ofasection oftheI 414 AMathematical Theory ofMagnetism. [xxiv. wire, orofajiyofthe cellsmade byplanes perpendicularto theplaneofthecircle, throughitscentre. 534.Hypothesis ofMatterflowing.—Inthetheoryofelectro- magnetismitisquite unnecessarytoadopt anysuchhypothesis asthis,howeverprobableorimprobableitmaybeasanulterior theory; and allthatwecould introduce asdepending uponit isthat, foralinear circuit ofvaryingsection ormaterial, the quantity 7isthesamethroughoutthe circuit, andthat all finite circuitspossessingcontinuous currents arenecessarily closed;two factswhich cannot beassumed apriori,butwhich areinrealityestablished bysatisfactory experimentalevidence. 535. DivisionofElectromagnetsinto three Classes—Linear, Superficial, andSolid.—Ifallthedimensions ofanysection of theconductoralong which thecurrent iscommunicated be infinitely small, thecompletecircuit constitutes what willbe called alinear electromag^net. When theelectric currents areconfined toashell ofwhich thethickness isinfinitely small, andwhen theyarecontinu- ouslydistributedthrough it,ordistributed throughitinsuch amanner asnottosatisfythecondition bywhich alinear electromagnetisdefined, the entire groupofthecomplete circuits constitutes what iscalled asuperficial electromagnet [orsurface-electromagnet]. When electric currents are'soarrangedastofillanysolid portionofspace, thegroupofthecompletecircuits constitutes asolidelectromagnet. Itisclear that, inpractice, electromagnets maybetreated as linear, orsuperficialifthequantities which oughttobein- finitely small, aremerely verysmall comparedwith thedimen- sions ofthemagnets, andwith thedistances atwhich the electro-magneticaction aretobeobserved; andagain,ifwires, orlinear currents ofanykind, bedisposed upon anysurface orthrough any space,sothat thedistances between those which areadjacentaresmall comparedwith thedimensions ofthe circuits, orofthe curves, orwith thedistances at which themagneticactions aretobeobserved, thegroupmay beconsidered asconstituting practicallyasuperficialelectro- magnet;andasolidelectromagnet maybecomjDosedofagroup ofgalvanicwiressimilarly arranged throughasolidspace. xxiY.] Electromagnets.415 536. Linear Electromagnets. —Alinearelectromagnetiscom- pletely specified when theform oftheclosed curve ofthe current, and7,thestrength,ai'egiven. Irrespectivelyofanytheory,theterm"electric current"will often bemade useof;butastheterms, literally interpreted, implyatheory which, tosaythe least,isdoubtful, itmust be borne inmind thattheyarenottobeinterpreted literally,and thattheyareonlyused inthisessay occasionallyforconveni- ence;andespeciallybecause ofthealmost universal usewhich ismade ofthem bywriters onthesamesubject. Theterm "galvanism"willoften beused todenote theagencytowhich thephenomena presented bycontinuous electric currents are due,andquantity ofgalvanisminalinear conductor willbe measuredaccordingtothefollowingstandard :— Thestrengthofthecurrent inalinearelectromagnetinto thelengthofanypartoftheconductor inwhich itexists,is thequantityofgalvanisminthatportion. Thetermintensitywillbeused with reference tolinear cun-ents, accordingtothefollowingdefinition :— Theintensityofthegalvanisminanypartofalinear electro- magnetisequaltothestrengthofthecurrent, divided bythe areaofthesection oftheconductor. Hence inalinear conductor ofwhich thesection isnotuniform throughout,theintensityofthegalvanismwillvary inversely asthesection fromoneparttoanother oftheconductor*. 537.Superficial Electromagnets. —Def.Thequantityofgal- vanism onanysmallportionofthesurface, divided byitsarea, isthesuperficial intensityofthegalvanismatthatpoint. Ifthesuperficial intensity-]- andthedirection ofthegalvanism isgiven atevery pointofagiven surface, thespecificationofthe superficial electromagnetiscomplete.There are,however, certain conditions towhich such aspecificationissubject, and anarbitrary specification,notsatisfying them, willnotcorre- spond toanypossible superficial electromagnet. Thefounda- * {Note, Oct. 25,1871.—^Ileave this section exactly asIfind itintheold manuscript, under protest that Idonotnowapprove ofthemode inwhich the word"galvanism" isused intheterms which itproposes. Where these terms occur henceforth itisbecause Ihave notinvariably altered themanuscript to substitute more convenient modes ofexpression. ] t[In1871weshould rather saysurface-intensity than superficial intensity.] 416 AMathematical Theory ofMagnetism. [xxiv. tion ofthese conditions isthefactthatnoincompletecircuit can existpermanently,from which itfollows that allthe currents, orthecontinuoussuperficialflux ofelectricitycon- stitutingasuperficial electromagnet, must beresolvable intoa groupofclosedgalvaniccurrents. This willlead toacondition which must besatisfied atevery pointofapurely superficial electromagnet, andagain,acon- dition which must besatisfied attheboundary,ifthesurface benotclosed. Themathematicalexpressionofthese conditions willbegivenlater. 538. SolidElectromagnets. —Def. Theintensityofthegal- vauism atanypointwithin asolidelectromagnetisthequan- tityofgalvanisminaspaceofinfinitelysmall dimensions round thatpoint,divided bythevolume ofthespace. Thecomplete specificationofasolidelectromagnetwillbe theexpressionoftheintensityanddirection ofthegalvanism atevery pointofit. Hereagainthere willbeconditions tobesatisfied bythe specification,toexpressthefactthat allthegalvanismconsists ofagroupofclosed circuits. 539. After thesepreliminary explanations wemayenterupon aregular analyticaltreatment ofthesubject ;commencingwith investigationsoftheconditions towhich thedistribution of galvanisminsolidandinsuperficial electromagnetsissubject. Let u,V,wdenote thecomponentsoftheflux atanypoint {x,y,z)within asolidelectromagnet ;and, ifthere bebesides asuperficialdistribution ofgalvanismonthebounding surface, letZ7,F,TTbethecomponentsofthesuperficialflux atthe point [x,y,z)when thispoint belongstothesurface. These quantities mustsatisfythefollowing conditions, inorder that thegalvanism expressed byu,v,w,U,V,Wmayconsist ofa groupofclosed circuits :— dudvdw_^,- V dxdydz forevery point [x,y,z)ofthemagnet, and dU^dV^dW^(dm.__dm\_ dx dydz\dy dz)^' XXIV.] Electromagnets. 417 forevery point {x,y,z)ofthesurface ofthemagnet,thedirec- tion-cosines ofanormal tothesurfacebeing denoted by I,m,n. 540. Todemonstrate these conditions, letusconsider anin- finitesimal tubularportionofthemagnet bounded bystream- lines*, andbythesurface ofthemagnet,ifanyofthese lines cut it.Letthestream-lines thus considered beinfinitelynearone another, sothattheportionofthemagnetcontained bythem maybearingofinfinitelysmall section, cutornotasthecase may be,bythesurface ofthemagnet. The conditions tobe satisfied with reference tothisportionofthemagnet are,that theintensityofthegalvanicstream ateachpoint must be inversely proportionaltothearea ofsectionperpendicularto thestream-lines ofgalvanism ;andthat iftheringbecutby thesurface ofthemagnet,theincomplete galvanicarcthus existingwithin themagnet must becompleted alongthesur- face. Since thewhole bodymaybedivided intoportionsofthis kind,wehave acondition foreveryinternalpoint, andbyex- pressingthatthesuperficialdistribution U,V,Wmust besuch astocompletecircuits forthegalvanic arcs, ofwhich theends areinthesurface, thecondition towhich U,V,Waresubject isobtained. Toinvestigatethecondition foru,v,w,consider aninfinitely smallparallelepiped 0/87,ofwhich thecentre isat[x,y,z), andtheedges respectively paralleltoOX,OY,OZ,and let thegalvanicarcs intowhich thewhole magnetisdivided be supposedtobeofsections sosmall thataninfinite number ofthem willpassthroughthisparallelepiped. The condition tobeexpressedwillbethat thesum oftheproductsofthe intensities intothesections atone setoftheends ofthese arcs shall beequaltothesum ofthecorresponding products attheother setofends. Thesums oftheseproductsforall theendswhich lieonthetwofaces/5,7,ofwhich thedistances fromFO^are x— ^2f,a?-fJaarerespectively equalto («-iag/37,and(«+iag^7 * [Note, Oct. 25,1871.—This term(itsintroduction isIbelieve due to Rankine)isnowmuch used inwritings onhydrokinetics.Itissubstituted for"Hnes ofgalvanism," which Ifindinmyoldmanuscript.] T.E. 27 418 AMathematical Theory ofMagnetism. [xxiv andsimilarly,forthefaces7,a,weobtain thesuras and, fora,ft(w- ij-^Jol^,and(^+27^)«/5. Nowwhenu,v,and luare allpositive, one setoftheends thegalvanicarcs will lieonthethree faces oftheparallele pipedofwhich thedistances from theco-ordinateplanesar< respectivelyic—JOf,y—l/S,z—iy;andtheother three face willcontain theother setofends,andwemust therefore have ("-^^£)^T+("-4^I)y^+('"-i^S)"^ 541. Toinvestigatetheconditions forthesurface ofth body,itmayberemarked that ifthere werenogalvanicarc fromwithin, terminated atthe surface, there mightben superficial galvanism, andthatanysuperficial galvanismthen could bemust constitute agroupofclosed circuits;buttha when there areinteriorgalvanicarcs ofwhich theends li onthesurface, thesuperficialdistribution must completeth circuits forthem, besidescontaining anyarbitrarydistributio ofclosed circuits. Hence, ifPandP'betwopointson band ofthesurface between twolines ofsuperficial galvanisr infinitely nearoneanother, ySand/8'thebreadths oftheban atthesepoints, and/and/'thesuperficialintensities ofth *Itisscarcely necessary toremark that this isthesame asthe"equation( continuity,"forthemotion ofanincompressible fluid, ofwhich thevelocityf anypoint [x,y,z)istheresultant ofu,v,w.Thecondition that asmuch flui leaves theparallelepiped a^yasenters it,inaunit oftimewould lead topre cisely thesame investigation asthat ofthetext{seeDuhamel's Cours c Mecanique, orCambridge andDublin Mathematical Journal, 1847, p.282). TV electrical matter whichmaybeimagined tobeflowing through thebody,mu.' notbecome accumulated, norleave adeficiencyinanypart. [Note, Jan. 1872. When thiswaswritten, upwardsoftwenty years ago,tl investigation ofthe"equation ofcontinuity" here referredto,adaptedfroi Fourier, wasbut littleknown,] :xiv.] Electromagnets. 419 alvanism; thevalues oftheproducts I.pandF.^'must differ yanamount equaltothesumofthestrengthsoftheinterior resofwhich theends lieontheband between Pand P'. Tow ifdsdenote thelengthofanelement oftheband, the umofthestrengthsofalltheinterior arcshavingtheir ends athispartoftheband willbe (lu+mv-{-nw).^.ds ndtherefore ifPandP'besituated atthetwoextremities fds,wemusthave r^'-1/3=(la+mv+nw)^ds (1); r,ifthesymboltidenote differentiation performed with refer- iicetovariations alongthesuperficialstream-line through P, ri{Il3) ==(lu+7nv+nw)I3ds (2). J^owlet(^besuchafunction ofx,y,zthattheequation <i>=k(3), ithdifferent constant valuesgiventok,shallrepresent any 3tofsurfacescuttingthesurface ofthemagnet alongthe tream-lines; that istosay(asthedirection cosines ofthe tream-line areproportionaltoU,V,TF),let(/>beanyfunction atisfyingtheequation ^S+^|+4t=«w- Lnd,because thestream-line liesonthesurface, wehave lU-]-mV+nW =(5). IfKbethedifference ofthevalues ofkforthetwobounding tream-lines onthetwosides ofthebandthroughPwhich we avebeenconsidering, wereadily obtain, forthebreadth ofthe and,thefollowing expression:— K-s-lJH-s-'Sy-cg-sjF(6). Equations (4)and(5)with u'+v'+w' =r(7), ^polved forU,F,W,give 27—2 420 AMathematical Theory ofMagnetism. [xxi U= F= W=\dydz) (l^^n^ \dz dx#^ d4> \dxi-i- dy(8), K (9). where, invirtue of(6) Andfrom(8)wehave— andtherefore,ifweput wehaveaxaydz d^ dx d4 dy=m-\-B{mW-nV) =Tlm-vn{nU-lW) .(10). ^^=Tln+B{lV-mU)^ Differentiating (9)alongthestream-line, wehave b(/^) Kl(H Hence pds a'dsB'ds' HKdxds'^ dyds'^ dzdsj'"^^ Now ds ds ds.(12) da Usingthese in(11)andthenputtingforU-^etc.,theeq valent formulae ,^-H-^,etc.,andmakinguseofequate Electromagnets.421 :;)wefind Qds dxdydz l{d(j>(dndm\d<i)fdldn\d^fdm dl\\.^. ^B\dU\dy~'Tz)'^'Jy\dz dx)'^dz\d^ dy)]^^' "or-r, 1?,-?ysubstitute their values by(10): then,ifwe axdydz 3mark that .(dndm\ ,fdldn\ /dm dl\ „,^.. ince this isthecondition thatafactor, \,maybefound achthatX(Idx+mdy+ndz)isacomplete differential; we \btain h»='-"»')S-e)+"'-''){£-|)-<"'- lence equation (2)becomes dJJdVdW ,Txr Tr\fdndm\ 542. Goroll. Thecondition tobesatisfied bythequantities 7,F,Wwhichexpressthedistribution ofgalvanismina uperficial electromagnetisthefollowing\— ^_dU,dV dW^.w_ v\(^^^~\ dxdy dz^ ^\dy dz) 543.Thesecond member oftheseequationsisbroughtto mothersymmetrical form(simplerforsomeapplications), by groupinginorder of?7,F,TF,addingtoitsixbalancing terms, dxdy dz dx dydz 422 AMathematicalTJieory ofMagnetism. [xxiv. andobservingthat ,dl ,dm .dn ^ , L^r+*^ —1—l-w-r-=0,etc.ax ax ax Thus for(2)of§539wehave , ,dUdVdW ^jj[,dl .dl dl\ axay dz \dxdy dzj .^j.f,dm ,dm•dm\ .-nrfidn dn dn\ ,,^^ 544.Themutual actions betweenelectromagnetsandcom- monmagnets,orbetween anypartofanelectromagnet andothei partialorcomplete electromagnetsorcommonmagnets, maj bedeterminedbymeans oftheexpressionsof§§525—529: andwhen thedata aresufficient, theapplicationofelementarj staticalprinciplesleads tothesolution ofanyproblemthai canbeproposed. Themode ofspecifyingthe distributior ofgalvanisminanelectromagnet, explainedin§§531—539 leadsimmediately, bymeans ofAmpere's formulagiven above §§525, 527, toproper expressionsforthemutual actior between anytwo solidelectromagnets bymeans offoui definiteintegrals representingthepartsofthatcomponeni duetothemutual actions ofthesolidandsuperficial partso their distributions ofelectric current. 545.Asimilarsyntheticalsolution oftheproblemofdeter miningthemutual action between anelectromagnet and { common magnet,isobtained byfirstinvestigatingaformuL forthemutual action between anelement ofagalvaniccircuit andaninfinitelysmall magnet, which may bedone atonci bymeans oftheformulae of§§526, 529,and thesynthesi ofamagnet explainedin§§461, 462,andthenapplyin< statical principlestoderive formulae forthecomponents (hot! offorce andcouple)ofthemutual action. Itissufficiec here toindicate themethod ofproceeding,forsuchproblems andunnecessarytowrite down theformulae, which, infact may always, when wanted, bewritten down atonce from th' formulae ofthepreceding chapters, accordingtotheprinciple which have beennowexplained. Thus, write down th< formulae fortherectangular componentsoftheforce exertet bytheelectromagnet (w,v,w,U,F,TF,§539),uponapositiv* iv.] Electromagnets, 423 unitpole, accordingtotheformulae of§529;andforthecom- ponentsofcouple which would begiven bytransferringthecon- stituent forces from their supposedlinesthroughtheelements ofelectric current, toparallellinesthroughthemagnetic pole. Itwillbefound that intheintegrals thecomponentsofcouple disappear, and thus isprovedCor. 5of§549;that the resultant force isinalinethroughthepole. Theexpressions forthecomponentsofthis force are,asmereinspectionofthe formulae of§529proves,identical with those of§517, (b).I proceedtopropositions regarding electromagnetic force, the importanceofwhich willappear from theapplication made of them insubsequent investigations. 546.Proposition. —Theaction ofaninfinitelysmallplane closed circuit onanelement ofanother circuit, oronanother complete electromagnetormagnetofanykind, isthesame aswould beproduced byaninfinitelysmallmagnet,inthe sameposition, with itsaxisperpendiculartotheplaneofthe circuit*.[The proofiseasily worked outfrom theformulae of §§483,485, 529.] *[Note added Jan.1872.]—Hence Ampere's theory ofmagnetism, according towhich, magnetization ofsteel orload-stone, orsoft iron, oranyother polar magnet (§549)consists ofelectric currentscirculating round themolecules of themagnetized substance inplanes perpendicular tothedirections ofmagneti- zation. From twenty tofi.ve-and-twenty years ago,when thematerials ofthe present compilation wereworkedout,Ihadnobelief inthereality ofthistheory (compare §602);butIdidnotthenknow thatmotion istheveryessence ofwhat hasbeenhitherto called matter. Atthe1847meetingoftheBritish Association inOxford, Ilearned from Joule thedynamical theory ofheat,andwasforced to abandon atoncemany, andgradually from year toyearallother, statical preconceptions regarding theultimate causes ofapparentlystatical phenomena. Inapaper communicated totheEoyal Society ofLondon, 10thMay 1856,under the title"Dynamical Illustrations oftheMagnetic andtheHeligoidal Rotatory effects ofTransparent Bodies onPolarized Light," [Art.xciii. ofEeprint of Mathematical andPhysical Papers (Vol. ii.)]after proving that thehehgoidal property shown bysyrup,oilofturpentine, quartz crystals, etc., isduetoa right orleft-handed asymmetry intheconstituent molecules, Imade thefollow- ingstatement regarding thenature ofmagnetism:— ''The magnetic influence onlight discovered byFaraday depends onthe "direction ofmotion ofmoving particles. Forinstance, inamediumpossess- "ing it,particles inastraight line parallel tothelines ofmagnetic force, dis-"placed toahelixround thislineasaxis,andthenprojected tangentially with ''such velocities astodescribe circles, willhave different velocitiesaccording as "their motions areround inonedirection (thesame asthenominal direction of "the galvanic current inthemagnetizing coil), orinthecontrary direction. But "the elastic reaction ofthemedium must bethesame forthesamedisplace- "ments, whatever bethevelocities anddirections oftheparticles ;that istosay, ''theforces which arebalanced bycentrifugal force ofthecircular motions are "equal, while theluminiferous motions areunequal. The absolute circular "motions being therefore either equal orsuch astotransmit equal centrifugal 424 AMathematicalTheory ofMagnetism. [xxiv. Cor. 1.Themagnetic moment oftheinfinitelysmallmagnet whichproduces thesamemagneticeffects asaninfinitelysmall planeclosed circuit isequaltothegalvanic strengthofthe circuit, multipliedintotheplaneareawhich itencloses. 547. RuleforDirections. —Themagnet must besoheld relativelytothecurrent which itrepresents,that ifthecircuit and itbeplacedatthecentre oftheearth, with itsplaneinthe earth'sequator, andwith thecurrentgoing round from east to west, thenorthpolarside ofthemagnetshall betowards the earth's North Pole. [Mnemonic principle:—Remember that if terrestrial magnetismwereduetocurrents intheearth'scrust, theirgeneraldirection would be"thewayofthesun;" that is tosay,from east towest.] 548. Cor. 2.Themagneticaction ofalinearelectromagnet (§535) [thatistosay,agalvaniccircuit inaninfinitelythin conducting ring]ofanyform isthesame asthat ofauniform magneticshell(§506) ofanyshape havingitsedgecoincident with thecircuit, andhavingitsmagnetic strength numerically equaltothegalvanic strengthofthe circuit. The rule for "forces totheparticles initially considered, itfollows that theluminiferous •'motions areonlycomponents ofthewhole motion; andthat alessImni- "niferous component inonedirection, compounded with amotion existing in "themedimn when transmitting nolight, givesanequal resultant tothat ofa "greater luminiferous motion inthecontrarydirection compounded with the ''same non-luminous motion. Ithink itisnotonlyimpossible toconceive any "other than thisd^mamical explanation ofthefactthat circularly polarized light "transmitted through magnetized glass parallel tothelines ofmagnetizing "force, with thesamequality, right-handed always, orleft-handed always,is "propagated atdifferent rates according asitscourse isinthedirection oris "contrarytothedirection inwhich anorth magnetic poleisdrawn; butI "believe itcanbedemonstrated thatnoother explanationofthat fact ispossible. "Hence itappearsthatFaraday's optical discovery affords ademonstration of "the reality ofAmpere's explanation oftheultimate nature ofmagnetism ;and "gives adefinition ofmagnetizationinthedynamical theory ofheat. The "introduction oftheprinciple ofmoments ofmomenta ('the conservation of "areas') intothemechanical treatment ofMrEankine's hypothesis of'molecular "vortices,' appears toindicate alineperpendicular totheplane ofresultant "rotatory momentum('the invariable plane')ofthethermal motions asthe "magnetic axis ofamagnetized body, andsuggests theresultant moment of "momenta ofthese motions asthedefinite measure ofthe'magnetic moment.' ''Theexplanation ofallphenomena ofelectro-magnetic attraction orrepulsion, "and ofelectro-magnetic induction,istobelooked forsimplyintheinertia and "pressureofthematter ofwhich themotions constitute heat. Whether tbis "matter isorisnotelectricity, whether itisacontinuous fluid interpermeating "the spaces between molecular nuclei, orisitself molecularly grouped;or "whether allmatter iscontinuous, andmolecular heterogeneousness consists in "finite vortical orother relative motions ofcontiguous partsofabody;itis "impossibletodecide, andperhapsinvain tospeculate,inthepresentstate of "science." XXIV.] Electromagnets. 425 directions is,that ifthecircuit beheld sothat inanypartofit tliecurrent isfrom east towest, then apointcarried inacircle round that partofthegalvanicarcnorthwards above itand southwards below it,willcuttheshellthrough from itsnorth polartoitssouthpolarside. 549. Cor. 3.Acommon magnet [orapolarmagnetasIshall henceforth callanything magnetizedafterthemanner ofaload- stone orasteelmagnet] maybefound which shallproducethe same action asanygiven complete electromagnet, uponother magnetsofeither kind, oruponanyportionofanelectromagnet [orarcofanelectriccircuit]. Cor. 4.The distribution ofordinary [orpolar] magnetism which producesthesame force, accordingtothe"electro- magneticdefinition"(§517),asagiven electromagnetis indeterminate. [Because anylamellar distribution consist- ingofclosed shellsmay (§512, Cor.6)besuperimposedon adistribution ofmagnetismwithoutalteringtheresultant forceelectromagneticallydefined in§517. Compare below §§584—588.] Cor. 5.Themutual action between amagnetic pointorpole, thatis,anendofaninfinitelythinuniformly andlongitudin- allymagnetized bar,andacomplete electromagnet,isinaline throughthatpoint. [Compare §§526, 545.] 550. Cor. 6.Thedefinition(1)of§479andthedefinition ofthepotentialwith thepropositionsonwhich itisfounded, as setforth in§§481,483maybeappliedwithout alteration loan electromagnet,asfarasregards pointsexternal totheconduct- ingmatter through which theelectric currentspass. 551.Withregardtointernalpoints,thedefinitiongivenin §517 fortheresultant forcerequires noconventional under- standingofananalogouscharacter tothatwhich wasmade in thecaseofpointsinthesubstance ofcommonmagnets, andset forth inthetextandinthesecond foot-note of§479.Wecan- not,asinthecase ofacommonmagnet, supposeaportiontobe cutfrom thesubstance ofanelectromagnet, withoutderanging themagnetic condition oftheremainder. Ifweimaginea space hollowed out inthesubstance ofanelectromagnet, wemustsupposesuch arrangements made that thevacancy 426 AMathematicalTheory ofMagnetism. [xxiv. shallonly deflect, notinterruptthe electric currents. Ifa smallspherical portion,forexample, becutfrom anelectro- magnet,theremaybeeither agradualdeflection ofthecurrent through somespace round thepartcutout;ortheinterrupted circuits maybecompleted byacondensation ofelectric cur- rentonthesurfaceboundingthehollow. But itissatisfac- torytoknow that theresultantmagnetic force atanypoint within such ahollowspaceisinfinitelylittle affected bythe supposeddeflection ofthecurrents, when thespaceisinfinitely small. This follows from thecomparisonofsimilar circum- stances forsimilar hollows ofdifferent dimensions, which shows that thedisturbinginfluence isinsimple proportionto thelinear dimensions ofthehollow. Or,simply takingthe triple integralsof§§545,517 (b)or(c),andusing them fora' point, P,within theconducting substance, weseeinamoment thatthepartofeachintegral belongingtoanysmallspace round Pdiminishes inproportiontothe linear dimensions ofthis spacewhenmadeinfinitelysmall withoutchangeofshapeorof position relativelytoP.Hence there isnonecessityforhollow- ingoutaspaceintheelectromagnetoroffurtherconsidering thecomplicatedcircumstances referred toabove, andthere- sultant force atanypointwithin orwithout anelectromagnetis theforcewhichmaybe simplydefined astheforceexpressed by theformulae of§528,accordingtothemodes ofspecification andprinciples explainedin§§536,537, 538,545;[thatisto say,simplytheformulae(6)of§517]. 552. Ifanelectromagnetconsist ofanumber ofconductors which whenputtogetherfitclose tooneanother, without touching,orofasinglewire ofarectangularorhexagonal section, rolled upwith thedifferentpartsofthewirenottouch- ingoneanother, butlyingclosetogethersoastobeseparated byspaces infinitelysmallcomparedwith thelateral dimensions ofthewire;theprecedingdefinition oftheresultant force at anypointofthemagnetconsidered asasinglesolid electro- magnetwillgive sensiblythesame resultant force atneighbour- ingpointswhether inthesubstance oftheconductor orinthe interstitialspace. [Addition and correction, Oct. 27,1871.—Buteven ifthe spaces between the differejit circuits, ortheneighbouring por- I mi I* b€XIV.] Electromagnets. 427 ions ofone circuit constitutinganordinaryartificial electro- magnet,benotinfinitelysmall orbeinfinitely great compared ththesections oftheconductors, thevariation offorce from ointtopointbetween twoneighbouring portionsofcircuit will besmall incomparisonwith thewhole forcegenerally, pro- dedthattheratio ofspace occupiedtowholespacewithin the ounds oftheelectromagnetbegreatincomparisonwith the ratio ofthediameter ofthewire tothediameter ofasection oftheelectromagnetacross allthecircuits orwires. This is easily provedfrom(c)of§517. Consideration ofthecorre- sponding gravitationalcase isinstructive. Inthe firstplace forsimplicity ;consider agreat spherical space, >Si,ofradius, R, withagreat number, n,ofequal homogeneous spheresofvery small radius, r,anddensity, p,distributed withaverage homo- geneousness through it,soastogiveanaverage density equal to--jsT' ^t tl^®boundaryofStheresultant force willbe jti approximatelytowards thecentre andequalto 47r7ir^p„ andatdistance osfrom the centre, itwillbeapproximately towards thecentre andequalto 47rnr^p Thegreatestdeviation from these approximationswould be produced bytakingoneofthesmall constituentspheresfrom agreat distance, andbringingitintocontact with thepoint attracted, which would introduce aforceamountingto 47r and therefore would producebutasmall difference oneither V themagnitude orthedirection oftheresultant force if-is nv small incomparisonwith^.Generally,foranygroupof moleculesattracting accordingtotheNewtonian law, ifthepro- ductofthedensityintothediameter ofamolecule beverysmall incomparisonwith theproductofmeandensityintodiameter ofthewhole;themasses ofthemolecules mightbeexpanded 428 AMathematicalTheory ofMagnetism. [xxiv. into the interstices soastocontinuously occupythewhole volume ofthewholegroup,withoutproducing anywhere more thanaverysmall changeintheresultantforce.] 553.Asuperficialdistribution ofelectric currentsgives the same normal component,butdifferenttangential components, fortheresultant magneticforce atpoints infinitelynear iton itstwo sides. Thetangential componentatoneside isfound bycompoundingwithaforceequal andparalleltothetangen- tialcomponentforce attheother side,aforceperpendicularto thestream-lines andequalto47r7,if/denote the surface intensityoftheelectric stream. [These propositionsareeasily provedfrom thesurface term oftheexpression (b)of§517, appliedtothepresent subject accordingto§551. Theyare infactproved byequations (o)of§517. Equations (p)ofthe same sectionexpressinsymbolsthewell-knowncorresponding propositioninrespecttoasuperficialdistribution ofmatter acting accordingtotheinversesquareofthedistance, which in words is,—that thetangential componentisthesame, for points infinitelynearoneanother onthetwo sides ofthesur- face,butthenormal componentsdifferby4i7rp,ifpdenote the surface density.] 554. Original investigation q/"§517(IX.) referredtoin§518. "Glasgow College^ 7thNovember, 1849.—YesterdayIfellupon *'atrain ofsynthesisandanalysisofgalvanic distributions "which Ithink willaddmuch consistence andsymmetryto "thewhole firstpartofmypaperonmagnetism (aportionof "the firstpartwascommunicated onthe21st ofJunelast,by "Colonel Sabine, totheRoyal Society), and itwillhelpmein " gettingtowork towrite outthematter Ihavehad solong "inhand. Itoccurred tometotreatgalvanicdistributions "accordingtotheanalogyofChapter III.,'Ontheimaginary "magneticmatter bywhich thepolarityofamagnet maybe "represented [§§463—475above];' thus, a,yS,7,beingthe "componentsoftheintensities ofmagnetizationat{x,y,z), "consider Ampere's imaginarycurrents rounddxdydz. We *'have strengthofcurrent round OX,alongfacesdxdy, dxdz^ "dxdy,anddxdz,=adx, "Consider allthepartial currentsparalleltoOX.Wehave :xiv.] Electromagnets. 429 —pdydx, alongoneofthedydzfaces(that whichcorresponds toa;,y,z-{-dz), and^dzdx alongoneofthedzdx faces(that whichcorrespondsiox,y\-dy,z). "The coincident face, dydx,ofacontiguous elementary- parallelepiped has +(^+^dz)dydz andthecoincident facedzdx ofanothercontiguous parallel- epipedhas Hence(asinChapter III.) theshare fortheelement dxdydz,ofgalvanism paralleltoOX, is, (f-|)^^^^^"- /dy \dx ^and But atthesurfacedy SoforsharesparalleltoFandOZwefind ofthemagnetthere isunneutralized galvanism. Hence, besides theinternal distribution wehave asuperficial distribution;andtheshare toasuperficialele- ment dshas, Ifind, foritscomponents paralleltoOX,OY, OZ,thefollowing:— — (/3?i—ym)ds —{yl— OLn)ds—{am—pi)ds andweverifythat these arethecomponentsofacurrent in thesurface byobservingthat I(/3w—ym)-^m {yl—an)+n(am—^l)—0; Z,m,nbeingthedirection cosines ofanormal. "This concludes theanalogueofChapterIII. "LetX,F,Zbethecomponentsoftheforce atanexternal pointP.Wehave", [formulaIX.of§517,which need not erepeated here]. "Since thepotential method cannot beappliedwhere galvanic elements orincompletecircuits areconsidered, thefollowing 430 AMathematicalTheory ofMagnetism. [xxv. "istheanalogueof[§495]thesection inChapter IV.,where a, "second oranalytical (Poisson's original) demonstration isgiven"oftheequivalenceofacertain determined distribution ofima- " ginary magneticmatter tothegivendistribution ofmagnetism." [Here follows inthemanuscript memorandum, theinvestigation (§518 above), which wascommunicated totheRoyal Society, June 20,1850,andpublishedintheTransactions.'] XXY. OnthePotentialofaClosed Galvanic CircuitofanyForm. [From theCambridge andDublin MatheTuatical Journal, 1850.] Theobjectofthefollowingnote istopointoutanextremelyin- teresting applicationoftheprinciples explained byProfessor De Morganinthepreceding paper [oftheCambridge andDublin Mathematical Journal, 1850, on"Extension oftheWordArea''], which occurred tomeinconnexion withthedetermination ofthe potentialofanelectromagnetinterms ofthesolidangleofacone. 555. Ithasbeenshown byAmperethataclosedgalvanic circuit inare-entering curve ofanyform producesthesame magneticaction asany infinitelythin sheet ofsteel, having thiscurve for itsedge,wouldproduceifuniformly andnor- mally magnetized. Now theresultant force ofamagnetatany pointmaybeexpressed,after themanner ofLaplace,interms ofthedifferential coefficients ofa"potential function," and therefore thesamepropositionistrue foraclosedgalvanic circuit *.When this isknown tobetrue, foreither acommon or anelectro-magnet,thefollowingdefinition maybelaiddown:— *Inother words, thequantity ofloork necessary tobring amagnetic pole fromanyposition intheneighbourhood ofaclosed galvaniccircuit toanyother position doesnotvarywith theform ofthecurve along which itisdrawn from onepoint totheother. There ishowever oneremarkable difference between the case ofanelectromagnet andthat ofanygiven steelmagnet. Inthecase ofan electromagnet, although thequantity ofwork doesnotvarywith thepath, yet ithasdeterminately different values according asthepathliesononeside,oron another ofanypart ofthegalvanic wire circuit, oraccording totheconvolutions round anypart ofthewhewhich itmaybearbitrarily chosen tomake. Hence arises themultipHcityofvalues ofthepotential atanypointintheneighbour- hood ofanelectromagnet noticed below. Yetforanyoneform ofamagnetized sheet ofsteel ofthekind described inthe text, agreeing,intheaction which it produces onallpoints notinitsown substance, with theelectromagnet, the potentialisperfectly determinate without amultiplicityofvalues;andthe difference inthetwocases isaccounted forwhenweconsider that themagnetic potentials atanytwopoints infinitely near oneanother, ontwosides ofthe sheet ofsteel, differby47r7,where 7isaconstant such thatywisthemagnetic moment ofany infinitely small areawofthesheet. Theagreementinthe magnetic circumstances ofthetwocases fails forallpointsinthesubstance of themagnetized steel. [Compare §515,Cor.2.] I.]Potential ofaClosed Oalvanic CircuitofanyForm. 431 556.DefThepotentialatanypointintheneighbourhood ofamagnetisthequantityofworknecessarytobringaunit north-pole (orthenorth-poleofaninfinitelythin uniformly andlongitudinally magnetized unit-bar) fromaninfinite distance tothatpoint. Todetermine thepotentialatanypoint due toagiven closedgalvanic circuit, letusimagineamagnetizedsheet ofsteel(theform ofthesheet isarbitrary, provided only that itsedgecoincide with thecurve ofthegalvanic circuit), whichaccordingtoAmpere producesthesamemagnetic action, andconsequentlythesamepotential,asagivenclosed circuit, tobedivided intoinfinitelysmall areas. Then itiseasily demonstrated, onthemostelementary principlesofthetheory ofmagnetism,that thepotentialsatanypoint, P,produced by these areas, areproportionaltothe solidangles which they subtend atP;thetruesignofthepotentialofanysmall area beingobtained byconsideringthesolidangleaspositive,ifthe sideoftheareacontaining northpoles,ornegative,iftheother side,betowards P.Hence thepotentialofthewhole sheet of steel, atanypoint P,isproportionaltotheentire solidangle which itsubtends atP;andconsequentlythepotentialofa closedgalvanic circuit, atanypoint P,isequaltoaconstant (which maybetaken asameasure ofthestrengthofthegal- vanism, orasitisoften termed, the"quantity"ofthecurrent) multipliedintothe solidangleofthecone described bya straightlinealways passing through P,and carried round the circuit. Inallcases, exceptthose inwhich thegalvaniccircuit iscontained inoneplane, there willbepositionsofPforwhich thiscone willbe"autotomic"; and inmany cases, especially themostcommonpractical case ofanelectromagnet,inwhich thecircuit consists ofdouble ormultipleconcentrichelices, with their ends connected, orofasinglewirewrappedina complex manner round abodyofsomeirregular shape,soas toconstitute mostcomplicated curves ofdouble curvature, there willbenoposition ofthepointPforwhich thecone is notexcessively autotomic. The solidangleofsuch acone, or thearea enclosed byitsintersection with aspherical surface ofunitradius, havingforcentre itsvertex, maybedetermined inamannerpreciselysimilar tothatwhich hasbeenexplained 432 AMathematicalTheory ofMagnetism. [xxv. byProfessor DeMorganforplane self-cutting curves, without anyambiguityastothecircuit bywhich thecurve, when self- cutting*,istobedescribed, since theactualgalvaniccurrent isinadeterminate circuit, and itsprojection, bytheconical surface, onthesurface ofthesphereistobedescribed bythe projectionofapointmoving alongtheelectric conductor, either inthesame direction asthecurrent, orintheopposite,accord- ingtotheconvention wepleasetomake. There ishowever a source ofambiguity whichreallyaffects theevaluation ofthe solidangleofacone, orofthearea ofanygivencircuit de- scribed inadeterminate manner onaspherical surface, and givesrisetoamultiplicityofsolutions oftheproblem, arising from thecircumstance that ofallthe"primary parts" (onlytwo innumber ifthecircuit benotself-cutting)into which the sphericalsurface isdivided bythecurve, there isnoreason for choosing one,more than another, asazerospace (oraspacecorre- spondingtothespaceexterior toaclosed circuit inaplane) "f. 557.When thevalue ofthearea, accordingtoanyoneof these solutions, hasbeen obtained, alltheothers maybede- duced, byaddingtoitorsubtractingfrom itanynumber of times thearea ofthewholesphericalsurface. Hence themost general expressionforthesolidangleofacone described ina determinate manner,is (7= <7^-}-4i7r, where<t^denotes anyonevalue and ianypositiveornegative integer.Iftoogreatapositiveortoosmall anegativevalue begiventoi,allthe" primary spaces"ofthesphericalsurface willbepositiveorallwillbenegative;and therefore ifwe wish toobtainonlythose solutionsaccordingtowhich some portionofthesphericalsurface isconsidered aszero orexternal tothecircuit, alimited numberonly (notexceedingthenumber *Seenoteontheword "circuit" inthepreceding paper [oftheCambridge andDublin Mathematical Journal, year 1850, p.140]. tThus,ifthegiven curve heacircle ofthesphere, described inagivendirec- tion,and if6denote theangular radius measured from thatpole0,which would benorth ifthedirection ofdescribing thecircle werefrom west toeast; thearea ofthecircuit is+27r(1-cos6)ifthespace ontheother sideofthecircle from be considered asthezerospace, but itwould be-27r(1+cos6)ifthespaceinwhich Oissituated weretaken aszero, orexternal tothecircuit. Ingeneral, thearea ofacircuit notself-cutting, onaspherical surface, willbeeither oneofthetwo partsintowhich thespherical surface isdivided, with thesign+ ,ortheother part,with thesign-. Iv.]Potential ofaClosed Galvanic Circuit ofanyForm, 483 primary partsintowhich thesphericalsurface isdivided by thecircuit) ofvalues foriaretobeadmitted. Thephysical problem, however, requiresnolimitation totherangeofvalues thatmaybegiventoi:for,ifwetakeanytwopathstothe pointPfromaninfinite distance, such thatthespace between them isonce crossed bythegalvanic circuit, thepotentialatP will differ by47r7accordingasitisestimatedbyonepathor bytheother; and therefore, bytaking (forthesake ofsim- plicityintheconception)differentpathstothepointPwhich goround acertainportionofthegalvaniccircuit once, twice, three times, fourtimes, etc.,inonedirection, andagaindiffer- entpathswhichgoround thesameportionofthewire once, twice, three times, fourtimes, etc.,inthecontrary direction, we obtain, accordingtothedefinition, aninfinite number ofvalues ofthepotentialatthepoint P,which aresuccessively expressed bytheformula V=V^-\-MTrfy, whenwegiveithevalues1,2,3,4,etc.,andagainthevalues —1,—2, --3,—4, etc.;v^beingthepotentialestimated bya path,which makes none ofthose convolutions. 558. Hence weseethat, tofindthegeneral expressionfor thepotentialatapointintheneighbourhoodofanelectro- magnet, wemayfirstchoose some determinatepath froman infinite distance tothepoint P,andinvestigatethevalue of thepotentialforit,which maybeused asthevalue ofv^in thepreceding expression.Ifaninfinitestraightline inany direction, terminated atthepoint P,bethepath chosen, the determinatepotentialwillbefound byconsidering,asthepor- tionexternal tothecircuit, theprimary portionofthespherical surface described fromPascentre, which iscutbythis line. Hence,ifwemark thisprimary portionwithazero,thenumber withwhich anyotherprimary partistobemarked, according toProfessor DeMorgan's rule, willbegotbydrawingalineto anypointwithinit,fromanypoint 0,intheexternalprimary part,andcountingthenumber oftimes itiscutbythecurve; every time itiscutfromrighttoleft(withreference toaperson walkingfrom 0,along it,ontheconvex surface ofthesphere) beingcounted as+1,andeverytime itiscutintheother T.E. 28I 434 AMathematical Theory ofMagnetism. [xxv. direction, as—1;andthealgebraical sum taken. When the number foreachprimary parthasbeen thus determined, the sum oftheareas ofthedifferentprimary parts, each multi- pliedbyitsnumber(positiveornegative,asthecasemay be), willbetherequiredarea ofthecircuit;andthepotentialat thecentre ofthespherewillbeobtained bymultiplyingthis by7,thestrengthofthegalvaniccurrent. The absolutesign ofthepotentialthusdetermined maybereadily shown tobe correct,ifweagreetoconsider thepotentialduetoterrestrial magnetismasonthewholepositiveforpositions north, and negativeforpositionssouth ofthemagnetic equator; since, asiswellknown, currents round theearth, proceedingonthe whole from easttowest,wouldproduce phenomenasimilar to theactual phenomenaofterrestrial magnetism. 559.Asanexample,letusconsider aconductingcircuit which consists oftwelve complete spiresofahelix, andaline Fig. 1. alongtheaxiswithtwoperpendicular portions connectingits extremities with those ofthehelix. Theaccompanying diagrams representtheprojections, byradii, ofthe circuit, onaspherical 1 .]Potential ofaClosed Galvanic Circuit ofanyForm. 435 surface intwodifferentpositions, viewed ineach casefrom the interior ofthesphere. Inthecase illustrated byfig. (1),thecentre ofthesphereis nearlyinalinewith theaxis ofthehelix, onthesidetowards thenorthpole*ofthehelix, anddistant from itbyabout half thelengthofthe axis. Inthecase illustrated byfig.(2),the centre ofthesphereisinaperpendicular throughapointof the axis, distant byabout one-fourth ofitslengthfrom the northpoleofthehelix, and isatabout thesame distance from thenearestpartofthehelix, asinthecase offig.(1);andthe curve onthesphericalsurface isshown inthediagram,accord- ingtoMercator'sprojection with thegreatcirclecontaining theaxis ofthehelix asequator -f".Ineachdiagramtheinner sideofthesphericalsurface isshown. 560. The radii ofthespheres being supposedtobeequalin thetwo cases,ifwedenote theircommon value byr,and if Aj^andA^betheareas ofthesphericalcurvesrepresentedin thediagrams,thezero orexternalportions onthespherical surfaces beingtaken asthosewhich become infinite intheplane diagrams,thevalues ofthepotentialatthecentre ofthesphere willbe ^1 AA 7-p,and7-^% respectively,foranypaths fromaninfinite distance which do notlieroundanyportionofthegalvanic wire, norbetween any ofthespires. The areaA^willbedetermined(inaccordance with Pro- fessor deMorgan'sruleJ)byfindingtheareas ofthe"primary *Theends ofthehelixwhich would berepelled from thenorth andfrom thesouth respectively bytheearth's magnetic action are, intheordinary vague useoftheterm "pole," called thenorth andsouth poles oftheelectro- magnet. tThediagram wasactually drawn bytracing upon acylindrical surface the shadow ofahelix oftwelvespires, |in.indiameter and4in.inlength, pro- duced byaluminous point intheaxis ofthecylindrical surface;theaxis ofthe helix being held intheplane through theluminous point perpendicular tothe axisofthesurface. Onaccount ofthenarrowness oftheband occupied bythe diagram, thecylindrical surface verynearly coincided with thespherical surface, which instrictness ought tohave received theshadow. After theshadow was thus traced, thecylindrical surface wasunbent intoaplane. tInfig. (1),allthearrow-heads which arenecessaryforrendering deter- minate the"balances" fortheprimary parts aregiven; andthenumbers ex- pressing thebalances aremarked forthe first sixprimary parts, commencing 28—2 436 AMathematical Theory ofMagnetism. [xxv. parts,"markedsuccessivelywith thenumbers 1,2,...upto12, multiplyingeach areabythecorresponding number, andtaking thesum oftheproducts. TheareaA^willbesimilarlyde- termined byfindingtheareas oftheprimary partsinfig.(2), multiplyingeachbythepositiveornegative number withwhich itismarked, andtakingthealgebraic sumoftheproducts. with theoutermost. Infig. (2),allthearrow-heads which arenecessary tomake thediagram represent determinately aclosed circuit areindicated, exceptina fewplaces where thespaces aretooconfined foradmitting ofthisbeing done in aclearmanner;andthe"balances" ofalltheprimary parts aremarked with numbers, except intheinstance ofaverysmalltriple primary part,which is marked with three dots(...)instead of+3. Glasgow College, March 25,1850. XXVI. [January, 1872.] Jhapter VII.—OntheMechanical Values ofDistributions ofMatter*, andofMagnets. 561. Preliminary proposition. —Thework againstmutual repulsions accordingtotheinversesquaresofthe distances, requiredtoproduce anychangeinadistribution ofmatter,is equaltotheaugmentationwhich itproducesinthevalue of theintegral /.QO /•CO /.CO r>2 f-dxdydz(1)m I^BiereEdenotes theresultant force atcc,y,z, I^KThis isanobvious conclusion from thefollowing investiga- tion forthemutualpotential energy (§503,Addition ofdate 11thDecember, 1871)oftwodistributions ofmatter;or,asfor brevity wemaycallthem, twobodies. *"Matter" ishereused conventionally andmerelyforbrevity,todenote asubstancefulfilling theconditions bywhich "imaginary magneticmatter" (§463)isdefined; thatis,substance ofwhich anytwosmall portions repel oneanother mutually with aforce equal totheproductoftheir quantities divided bythesquare ofthedistance between them. Either orboth quan- titiesmaybenegative, andthenegative productofunlike masses indicates attraction. Notbeing inanyway occupied with Kinetics atpresent, we suppose thisimaginary matter toremain where itisplaceduntilwepleaseto moveit;sothat a"distribution" ofitmaybesupposedtobeeither arigid bodyoraflexible body, oraflexible andcompressible body, held atrestby thenecessary force, except when wesupposeittomove; andthenweper- form work, positive ornegative, uponittowhatever amount isnecessaryto produce, irrespectively ofinertia, thesupposed motion againstorwith the forces resulting from attraction orrepulsion, which theportionsofthematter movedexperiences. Alltheformula and conclusions areapplicabletoreal matter, gravitating according totheNewtonian law,ifwesubstitute attrac- tion forrepulsion, that istosay,change thesigns ofeachformula forforce orwork, andexclude negative matter. Inapplicationsofgravity, therefore, instead ofthe"mechanical value" or"potential energy"ofadistribution oftheimaginary magnetic matter, wehave an"exhaustion ofenergy" (Thomson and Tait's Natural Philosophy, §549)inadistribution ofreal matter. 438 AMathematicalTheory ofMagnetism. [xxvi. Letpbethedensityatanypoint {x,y,z)ofoneoftheses bodiesM\and letVbethepotentialatthesamepoint,du( totheotherbody M'.Thendenoting byQthemutualpoten- tialenergyofthetwo,wehave Q=[ [rpV'dxdydz ....(2). WehavebyPoisson's theorem, " 4^7r\dx dydzJ where X,F,Zdenote thecomponentsoftheforce at{x,y,z,]• due tothebody M.Thisequation (asitalsoexpresses Laplace's theorem forspace containing none ofthematter o: M^since therep= ;)holdsthroughout space. Hence for(2^ wemaywrite «=;ii/:/:/:(s-f-f)^^*^'-<-'>-Hence byintegration byparts Q=^rrr(XX+rr+Z^Odxdydz...(4), where X'Y'Z' denote thecomponentsoftheforce at{x,y,z,) duetoM\ Letnow thesecond bodyconsist ofadistribution ol matter coincident with the firstand similar toitthroughout, but letthewholequantityofmatter inthesecond bodybe infinitelysmall andbedenoted bydm,that ofthe firstbeing denoted bym :weshallhave X'=—XY'=—Y Z'=—Z. 771'm'm Instead ofQwritenowdE.Wehave This formulaexpressesthequantityofworkrequiredtoadd dmsimilarlydistributed toadistribution malreadymade. Oursupposedmatter beingnotsubjecttothelawofimpenetra- bility,wemight simply supposethedistribution ofdm,precisely similar tothatofm,tobegivenataninfinite distance andtobe moved againsttherepulsionofminto coincidence :thework IVI.]Mechanical Values ofDistributionsofMatter. 439 requiredisthatwhich isdenoted bydE. Sofar itisnot necessarytosuppose dminfinitelysmall. But ifdmbein- finitely small, theworkrequiredtobringitininfinitelysmaller partsfrom infinite mutual distances intothesupposed position ofcoincidence with thedistribution ofm,would involveonly aninfinitelysmallamount oftheseconddegreeofinfinitesimals, onaccount ofthemutual influences ofthedifferentpartsof dm.Hence theformula(5)representstheworkrequiredto augmentthesupposeddistribution frommtom+cZm,by bringing altogether from astate ofinfinite diffusion the in- finitesimalportionofmatterdm;andtherefore theintegralof thisformula from tomisthewhole workrequiredtobuild upthedistribution mfrominfinitelydiffused matter. Now, with reference tothevariation ofm,each ofX,F,Zvaries in simple proportiontom,andtherefore thetriple integral may bedenoted byGm^, sothatwehave dE=^-r-Gmdm, 4i'7r whichgives E^lcm'. OTT Finally eliminating Cwehave ^= -^f_ f_fdxdydz^X'+Y'+Z') (6). Theprecedingdeduction oftheformula(4)from(2)mutatis mutandis allows ustocome back tothefollowing important alternative formula E=jrffpYdxdydz (7). J—ooJ—OoJ —00 The directproofofthisformula byintegrationwith reference tom,commencingwithanexpressionfordEderived from(2) isobvious. 562. The forces atpoints similarlysituatedrelativelyto similar bodies, areproportionaltothelinear dimensions ofthe bodies, andtotheir densities incorresponding places. Thevalues of(1)forsimilar bodies aretherefore asthe fifth powersofthe linear dimensions, and asthesquaresofthe densities. Hence ifahomogeneous rectangular parallelepipedI 440 AMathematical Theory ofMagnetism. [xxvi. bedivided into i^equal andsimilarparts, andthesepartsbe separatedtoinfinite distances from oneanother, thewhole value oftheintegral (1)forthescatteredpartsisequalto-^of itsvalue fortheundividedbody.Itfollows that ifafinite body bedivided intoaninfinite number ofinfinitelysmall parts, andthesepartsbeseparatedtoinfinite distances from oneanother, thevalue oftheintegral (1)forallthepartsbe- comes aninfinitelysmallquantityofthesame order asthe squareofthediameter ofoneoftheparts. Hence theintegral (1)relativelytoafinite bodyordistribution ofmatter, composed ofultimately homogeneouscontinuous substance, expressesthe workrequiredtobuild itupoutofinfinitelysmallparts having thesamedensity (oranyotherdensitynottooinfinitely great) andgivenatinfinitely greatdistances fromoneanother. 563.Acomplete analyticalview ofthecircumstances con- templatedin§5()2is,asisgenerallythecase, easier than the quasi- elementary method, involvingintricacies oflanguageand perplexitiesof"compound proportion,"towhich, astheonly alternative touttervagueness,"popular" expositionsarecom- monlyrestricted. Atanypoint {x,y,jz,)letVbethepotential andX,Y,Zthecomponentsofforce duetoabodyM\and let mbeitsmass. Consider asimilar distribution ofmatter of 9'-fold densityatcorresponding points,and ofp-foldlinear dimensions. Themass ofthisbodywillbe'p^qm,and its potential andforce-componentsatthepoint correspondingto (a?, 2/, -g-,)willbe p\V, pqX,pqY, pqZ. Hence ifweput E=-^r rr(X'+Y'+Z')dxdydz,OTTJ-ooJ-ooy-oo that istosay,ifEdenote themechanical value ofthe distri- bution if,themechanical value ofthesupposedsimilar distri- bution ofaltered dimensions willbe 564. Considering now similarmagnetsofdifferent dimen- sions, whetherpolarmagnetsorelectro-magnets, weseefrom the fundamental formulae(§§482, 483, 486, 544)thattheforces atcorresponding pointsareindependentofthelinear dimensions. THPv.I.] Mechanical Values ofPolar Magnets.441 andareequal,with equalintensities ofmagnetization,when polar magnetsarecompared,orwith intensities ofelectric currents inversely proportionaltothelinear dimensions ofthe bodies when electro-magnetsarecompared. Hence thevalues oftheintegral (1)of§561forsimilar magnetsaresimply pro- portionaltotheirvolumes;provided that,whenpolarmagnets arecomparedtheir intensities ofmagnetizationareequal,and when electro-magnets,theintensities oftheir electric currents areinversely proportionaltotheir linear dimensions. Farther whenpolarmagnetsarecompared,thepropositionholdswhether thepolarortheelectro-magneticdefinition(§517)ofresultant forcethroughinteriors isadopted. Butanelectro-magnetcan- notbesimplydivided intoparts infinitelysmall inalltheir dimensions each ofwhich isanindependent electro-magnet ; andtherefore thefurther consideration ofelectro-magnets must bedeferred, while weusethedivisibilityofapolar magnet asserted in§447, toinvestigatethemechanical value ofa distribution ofpolar magnetism,after themanner of§562. 565.Atanypoint {x,y,z,)let3^denote theresultant force due toapolar magnet ;thedefinition of§480being adopted when(^x,y,z,)isinthesubstance ofthemagnet. Theprelimi- nary proposition (§561)isimmediately applicable,andshows thattheworkrequiredtoproduce anychangeintherelative positionofasetofmagnetsisequaltotheaugmentationof rrr^d^<iydz w. Hence(§564)w^hen auniformly magnetized magnetisof such ashapethat itcanbedivided into similarparts,the mechanical value ofthewhole issimply equaltothesum of themechanical values oftheparts ;[aremarkable contrast to thecorresponding proposition (§562)relative toahomo- geneousdistribution ofmatter].Inother words, thework requiredtoseparatetoinfinitely great mutual distances any number ofparts,each similar tothewhole, ofauniformly mag- netized magnet, iszero. Itfollows that ifaninfinite number ofinfinitelysmall magnets,each distributedthroughafinite volume ofspace,with their magneticaxesparallelandwith equal sums ofmagnetic moments inequalfinite portionsof 442 AMathematical Theory ofMagnetism. [xxvi. thatspace, nowork willberequiredtocondense orrarefythe distribution withoutalteringtheproportionsofmutual dis- tances, orthedirection ofthemagnetic axesrelativelytothe lines ofthese distances;providedthat thecondensation is never pushedsofarastobringtheconstituents within dis- tances notinfinitely greatincomparison withthelinear dimen- sions oftheconstituentmagnets. This lastprovisoisunne- cessary when theconstituents areuniformly magnetized,allwith thesameintensityofmagnetization, andaresoshapedthat when broughtintocontact inthesupposedcondensation they fittogetherandform awhole, similar inshapetoeachpart. 56Q. Consider nowabarorcylinderofuniformly andlongi- tudinally magnetized substance, terminated byplanes perpen- dicular toitslength ;and let idenote theintensityofthe magnetization.This limit isapproximatelyreached when the lengthofthebar isverygreatincomparisonwith itsgreatest transverse diameter. Thecorrespondingdistribution ofimagi- nary magnetic matter consists(§473) ofdistributions of positiveandnegative matter, ofsurfacedensityionthetwo terminalplanes. The resultant force atpoints infinitelynear theedgeofeither oftheseplanesisinfinite; butnotwith- standing this,itiseasily provedthatthevalue oftheintegral (1)isfinite. Ifwesupposethebartobeatfirstinfinitely shortandtobegraduallyincreased inlength,thevalue ofthe integral (1),expressingtheworkrequiredtodraw thetwo terminalplanesasunderagainsttheir mutual attraction, increasescontinuouslyfrom zero toalimitingvalueequalto twice thevalue ofthecorresponding integralforeither ofthe terminalplanesalone. Hence, because forsimilar bars the values oftheintegralare(§565)asthevolumes ofthebars,it follows that forbars ofsimilar cross sections theintegralhas valuesproportionaltothecubes oftransverse dimensions and independentofthelengths, provided onlythat thelengthof each barconsidered isvery greatincomparisonwith its greatesttransverse diameter. Hence,ifanypolar magnetbe divided intoinfinitelythinbars*alongitslines ofmagnetiza- *Byaninfinitely thin bar, Imean abarofwhich thetransverse diameters areallinfinitely small incomparison with thelength. IVI.] Mechanical ValuesofPolarMagnets.443 tion,and ifthese barsbeseparatedtoinfinite distances from oneanother, thewhole value oftheintegral (1)becomes in- finitelysmall*. ^^K567.Hence ifmagnetizedsubstancegivenininfinitelythin ^Krs atinfinitely greatdistances fromoneanother beputto- gethersoastoform apolar magnet,thevalue ofintegral (1) forthismagnet expressestheamount ofworkwhich wasspent inthusbuildingitup.Neglectingthen the(unknown) mecha- nical value ofthematerial, supposed givenininfinitelythin permanently magnetizedbars atinfinitely greatdistances from oneanother, anddefiningthemechanical value ofamagnetas theamount ofworkrequiredtobuild itupofsuch materials, weseethat this isexpressed bytheintegral (1)of§565. 568. Thevalue oftheintegral (1)(§565)iszero,when the magnetconsists ofclosed solenoids; because, inthiscase(§510 Cors. 2and3)3^=forevery point.This result mightatfirst sight appear erroneous, because afinitepositive amount ofwork isrequiredtocutupafinite closed solenoid into barsand separatethem toinfinite distances from oneanother. But itis verified byremarkingthat ifeach such bar,beingoffinite transverse dimensions, issplitupintoinfinitelythin bars,work isgained byallowingtheseinfinitelythin bars torepelone another toinfinite mutual distances;and that thewhole amount ofwork thusgainedisexactly equaltowhatwasspent inreducingthesolenoid toseparatefinite bars. Orvarythe process bysupposingafinite solenoid tobefirstsplitupinto aninfinite number ofinfinitelythin solenoids;then thesum oftheinfinitely greatnumber ofinfinitelysmall amounts of workrequiredtobreak theseinfinitelythin solenoids intobars andseparatethebars toinfinite mutual distances, isinfinitely small. Inshort theexplanationoftheapparent difficultyis contained in§566. 569. Itisonlyforamagnet consistingofclosed solenoids that id iseverywherezero. Foreveryothermagnet,the *But ifeach ofthese bars bedivided into lengths comparable with its transverse dimensions, and ifthese parts beseparated todistances from one anotherinfinitely great incomparison with their dimensions, theintegral (1) acquires afinite value which isequal totheamount ofwork necessary to produce thisseparation. i 444 AMathematicalTheory ofMagnetism. [xxvi integral (1)of§565hasconsequentlyafinitepositivevalue This Ishallnowprovetobealwayslessthan /•00 /-co /.00 27r I / /i^dxdydz J—ooy—00-—00 (where idenotes theintensityofmagnetization), exceptinthe extreme caseofamagnet consistingofclosed shells, when the limitingvalue isreached. Asinthepostscriptto§517, let,foranypoint {x,y,z),B denote theresultant forceaccordingtotheelectro-magnetic definition, andX,Y,Zitscomponents ;a,/3,7thecom- ponentintensities ofmagnetization ;id (stillasin§565)the resultant forceaccordingtothepolardefinition;and^,^,5^. 17,itscomponents and itspotential,sothat ^=-f'l^=-f.^=-f(^)- Let^denote thevalue oftheintegral (1)of§565;andE thecorresponding integralofthe electro-magneticresultant force; that istosay,let (2B=~*^ Wdxdydz (3),OTTJ —ooj—ocJ—00 The formulae(r)ofthepostscriptto§517, with(2)ofthe presentsectiongive Use this in(4);follow theusualprocessofintegration by parts,whichgives remark that[§473(2)]J+g+g=-p (6), wherepdenotes thedensityoftheimaginary magneticmatter which wesubstitute forthegiven magnet (whenthepolardefi- 1 ] Mechanical Values ofPolar Magnets. 445 •nition isused fortheforce throughthespace occupied byit); andremark thataccordingtothealternative formula(7)of§561, (B^^if rrpVdxdydz (7). J—ccJ-coJ—» /•CO /"OO /"OO wefind^=^-2(!B +27r /pdxdydz\ J—COJ—00J—30 /•OO /•XfCO andtherefore (B+E==27rj/{''dxdi/dz (8).J—00J—XJ—00 NowEhasalwaysapositivefinite valueexceptfortheextreme caseofamagnet consistingofclosed shells, when itiszero, because (§512cor.6),^=inthiscase forevery point whether inthesubstance ofthemagnetornot.Hence theproposition isproved. 570. ForX'+F'4-Z^take, invirtue of(c),§517, jr(dN__dM\ Y(dL_dN\ ^(dM_dL\ \dz dyJ \dz dx) \dx dyj' andintegrate bypartsafter themanner of§518,butwith infinities forlimits. Wethus find 1ff-r:,r-rfdZ dY\^JdX dZ\^JdY dx\^,^, orby§517(0 ^=hi I(dxdydz{Lu +Mvi-Nw)... (10), J—COJ—zoJ—» This,which istheanalogueto(7)of§569,wasdiscovered for fluidmotion byHelmholtz, andgiveninhispaper onVortex Motion(Crelle's Journal^ 1858, or,translation byTait, Philo- sophical Magazine, 1867, second halfyear). Lastly,substitut- ingforu,V,wtheir valuesby(a)of§517,andintegrating againbypartsasbefore, wefind ^=jrrrdxdydz{aX-^^Y-\'r^Z).,.{\\),J—00J—00''—00 Theanaloguetothis is[compare §503(2)], (B^-if rrdxdydzia^ +^'^+y^)... (12). J—ST./—coJ—00 Theaddition ofthese twoformulae verifies(8)of§569. 571. Inamemorandum-book under date Oct. 16th, 1851, I findthefollowing statement:—"Iconcluded thatthevalue of 446 AMathematical Theory ofMagnetism. [xxvi. "acurrent inaclosed conductor, leftwithout electromotive "force, isthequantityofwork thatwould begotbyletting "alltheinfinitelysmall currents intowhich itmaybedivided " alongthelines ofmotion oftheelectricity cometogether "from aninj&nite distance, andmake itup.Each ofthese "'infinitelysmall currents' isofcourse inacircuit which is "generallyoffinitelength.Itisthesection ofeachpartial "conductor andthestrengthofthecurrent initthatmust be " infinitelysmall." Amemorandum ofprinciples andformulae provingthisstatement hadbeen written afewdays previously (Oct. 13th, 1851). Asomewhatamplifiedstatement ofthe principle was firstpublished,butwithout theformulae, in1860, inthesecond edition ofNichol'sCyclopcBdia (Article"Magnet- ism,DynamicalRelationsof"). Thoughthesubjectdoes not belong properlytothepresent volume, Iappendinfoot- notes theoriginal memorandum*, andanextract from Nichol's *Memorandum, Oct. 13,1851.—Eefers first toanerroneous temporary conclusion which ledmetothink "that thevalue ofacurrent inaclosed "conductor willbeeffected bysteelmagnetsinitsneighbourhood." "From "this Iwasshaken alittle byFaraday's finding {Exp. Res. §1100)that steel "does notdosowellassoftiron," etc. [inrespecttoelectro-magnetic induc- tion], "and Isoon saw that Imust have fallen intosome mistake. . .. "Imade outthetrue state ofthe case. This istheexplanation. Let "-7--rrdt.y bethequantity ofwork done intime dt,bybringing asteel "magnet towards agalvanic current, kept up,say,byabattery. Then C, "the electromotive force due tothechemical action, willbeincreased by " -3--7-. Hence ifkbetheresistance inabsolute measure dsdt ^dEds dsdt7=—I— ; 'sothat ifwdtdenote thework \^^ dsdt) dsdt\™^^wdt=dt, "and ifMdtbethemechanical equivalent ofthechemical action (increased "onaccount oftheincreasedcurrent), wehave C{^^dsTtJMdt=Cydt-. K 'Lastly,ifHdtbetheheat developed, wehave JHdt=l'fdt=.^ ^L£^ dt;K 'andtherefore JB.dt=ivdt+Mdt. XXVI.]Mechanical ValuesofElectro-Magnets.447 CyclopcBdia*, containingtheamplifiedstatement. Definingthen thedynamicalvalue ofanelectro-magnetasthequantityof **We conclude that thework actually spent, together with themechanical **equivalent ofthechemical action, together produce exactly anequivalent "ofheat,andtherefore noother effect. Hence themechanical values ofthe "current and ofthemagnet together arenotaltered. Ontheother hand, "lettwopure electro-magnets bebrought towards oneanother. Adopting ^notation corresponding totheformer wehave ^„^ ^dEds, ^,dEds w=z-j--j-W\ dsdt',dsdt' dsdt"7=,,7'='h' ^dEds \2/ ,dEdsyM=Cy= JH=ky"^+k'y'-^=^- -/+J^ .'^M-^M'^ 2w. K ft "Hence [Jdenoting Joule's equivalent]there ismore heat evolved than "-{M+M'+w) byyw,and therefore themechanical value oftwo cur-J J "rents isdiminished byjwdtinthetime dt.'' * ^'Electricityinmotion.—Ifanelectric current beexcited inaconductor, "and then leftwithout electro-motive force, itretains energy toproduce heat, "light, andother kinds ofmechanical effect, and itgradually fallsin strength "until itbecomes insensible, asisamply demonstrated bytheinitial experi- "ments ofFaraday andHenry, onthespark which takes placewhen agal- "vanic circuit isopenedatanypoint, andbythose ofWeber, Helmholtz, and "others ontheelectro-magneticeffects ofvarying currents. Professor W. "Thomson hasshown how themechanical value ofalltheeffects thatacur- "rent inaclosed circuit canproduceafter theelectro-motive force ceases, "may beascertained byadetermination, founded ontheknown laws of "electro-dynamic induction, ofthemechanical value oftheenergy ofacur- "rent ofgiven strength, circulating inalinear conductor(abent wire, for "instance)ofanyform. Todothis,itmay beremarked, inthe first place, "that acurrent, once instituted inaconductor, andcirculatinginitafter "the electro-motive force ceases, does sojustasiftheelectricity had inertia, "and willdiminish instrength according tothesame, ornearly thesame, "laws asacurrent ofwater orother fluid, once setinmotion and leftwith- "out moving force, inapipe forming aclosed circuit. Butaccording to "Faraday, whofound thatanelectric circuit consisting ofawiredoubled on "itself, with thetwoparts close together, gives nosensible spark when "suddenly broken, incomparison with that given byanequal lengthofwire "bent intoacoil, itappears that theeffects ofordinary inertia either donot "exist forelectricityinmotion, orarebutsmall compared with those which, "inasuitable arrangement, areproduced bythe'induction ofthecurrent "'uponitself,' Inthepresentstate ofscience itisonly these effects that "can bedetermined byamathematical investigation ;buttheeffects ofelec- "trical inertia, should itbefound toexist, willbetaken intoaccount by "adding aterm ofdeterminate form tothe fully determined result ofthe "present investigation which expresses themechanical value ofacurrent in "alinear conductor asfarasitdepends ontheinduction ofthecurrent on "itself. "The general principleoftheinvestigationisthis—Iftwoconductors, "with acurrent sustained ineachbyaconstant electro-motive force, be 448 AMathematicalTheory ofMagnetism. [xxvi. workspecifiedinthestatementquotedabove inthetext,we have inequation (5)aproofthe firsthitherto pubhshed,ofthe assertion intheextract from Nichol'sCyclopcedia quotedinthe foot-note, that thedynamicalvalue ofacurrent inaclosed circuit maybecalculated bytheformula(4).For letopen magneticshells(§§506,548)besubstituted forthe" infinitely small currents"referred tointhepreceding statement, sup- posedfirsttobeintheir actualpositionsintheelectro-magnet composedofthem;and letthese shells beseparatedtoinfinite distances from oneanother. Itiseasily proved byconsidera- tions ofinfinitesimalsanalogoustothosefullysetforth in §566,thatwhen theshells arebroughttoinfinite distances from oneanother, thevalue ofEbecomes zero;and, therefore, as thesecond member of(5)remains constant, thevalue ofE before thecircuits wereseparated,isequaltotheaddition of value which CIEexperiences duringtheprocessofseparation, "slowly moved towards oneanother, andthere beacertain gain oficork on "the whole, byelectro-dynamic force operating during themotion, there "will betwice asmuch asthis ofwork spent bytheelectro-motive forces "(for instance, twice theequivalent ofchemical action inthebatteries, should "the electro-motive forces bechemical)overandabove thatwhich they "would havehad tospendinthesame time, merely tokeepupthecurrents, "iftheconductors hadbeen atrest, because theelectro- dynamic induction "produced bythemotion willaugment thecurrents; while ontheother "hand, ifthemotion besuch astorequire theexpenditure ofwork against "electro -dynamic forces toproduce it,there willbetwice asmuch work "saved offtheaction oftheelectro-motive forces bythecurrents being dimin- •'ished during themotion. Hence theaggregate mechanical value ofthe "currents inthetwoconductors, when brought torest, willbeincreased in "the onecasebyanamount equal tothework done bymutual electro- •'dynamic forces inthemotion, and willbediminished bythecorresponding "amount intheother case. Thesame considerations areapplicable to "relative motions oftwoportions ofthesame linear conductor (supposed "perfectly flexible). Hence itisconcluded that themechanical value ofa "current ofgiven strength inalinear conductor ofanyform, isdetermined "by calculating theamount ofwork against electrordynamic forces, required "todouble itupon itself, while acurrent ofconstant strengthissustained in "it.Themathematical problem thus presented leads toanexpressionfor "the required mechanical value consisting oftwo factors, ofwhich, one is "determined according totheformanddimensions oftheHneofthecon- "ductor inanycase, irrespectivelyofitssection, andtheother isthesquare "ofthestrength ofthecurrent. Themechanical value ofacurrent ina "closed circuit, determined onthese principles, maybecalculated bymeans "ofthefollowing simple formula, nothitherto published:— "where JRdenotes theresultant electro-magnetic force atanypoint {x,y,z). "This expressionisvery useful inthedynamical theory ofmagneto-electrio "machines and electro-magnetic engines."—From Article "Magnetism, "Dynamical Kelations of,"Nichol's Cydopadia,edit. 1860. XXVII.] HydrO'hinetic Analogy,44<9 that istosay,isequaltotheworkspentineffectingthis process. 572. Equation (5)expressesthefollowing veryremarkable proposition. Thesum ofthedynamicalvalues ofanelectro- magnet andofanycorrespondinglamellarpolar magnetis equalto27rmultipliedintothesum ofthesquaresofthe intensities ofmagnetizationofallpartsofthelatter;thetwo species ofdynamical value understood, beingthose defined in §571and§567. XXVII.{Jan. 1872.] Chapter VIII.—Hydro-kinetic Analogy, 573.Thehydro-kinetic analogyfortheforce ofapolar magnet seems tohave been firstperceived byEuler. Itre- quiresthesuppositionofgenerationandannihilation offluid inplacesofpositive andnegative magnetic polarity,ifwe adoptfor"the resultant force" inthemagneticsubstance the definitionproperforapolarmagnetlaiddown in§479;unless welimit the field offorce considered, toplacesvoid ofmag- netized matter, whether external tothemagnetorinhollows within it.Thus,ifweconsider allspaceasfilled withan incompressiblefrictionlessliquid initiallyatrest,and ifat certainpoints, lines, surfaces, orvolumes, weassume more liquidofthesamedensitytobecontinuously generated,and atthesame time inotherplaces liquidinequal quantitytobe continuously annihilated, thevelocityoftheresultingfluid motion would bethesame indirection andmagnitudeasthe resultantmagneticforce due toadistribution ofmagnetism presentingunneutralizedpolarity, positive (ornorthern)inthe places ofthe fluidanalogue where there isgeneration,and negative (orsouthern)intheplaces where there isannihilation. Thereis,however, nointerest inpursuingtheconsideration of thisextension ofthehydro-kinetic analogy through spaces occupied bymagnetized matter, involvingasitdoesthestrained supposition ofthegeneration and annihilation ofmatter iii spaces through which theliquidisperfectlyfreetomove. 574.Ontheother hand, thehydro-kinetic analogylimited to spaces unoccupied bymagnetizedmatter isperfectly satisfactory, T.E. 29 450 AMathematicalTheory ofMagnetism, [xxvrr/ asfarasitgoes. Let allthesespaces beoccupied byincom- pressible liquid,and letthemagnetized matter bereplaced by arigidbody perforatedsoastoconstitute aninfinitely numer- ousgroupofinfinitelyfinetubesfulfillingthefollowingcon- ditions :—Divide thewhole surface ofthemagnetintoinfinitely small areasinversely proportionaltothemagnitudesofthe normalcomponentforces across them whether outwards or inwards. Because the surfaceintegralofthenormal com- ponent force forthewhole surface ofthemagnetiszero, the number ofthese infinitesimal areas inthatpartofthesurface where thenormalcomponentforce isoutwards must beequal tothenumber intheremainder ofthesurface. Now topass tothe fluidanalogue;instead ofthemagnetsubstitute a rigid body perforated from each oftheinfinitesimal areas in thepartofthesurface where thenormalcomponentforce ispositive, byasingletunnelthroughtooneoftheareas in theotherpartofthesurface. Letthere beinthe firstplace apistonineach ofthese tunnels ortubes, andapplyforce to ituntil itmoves withsuchavelocitythatthevelocityofefilux atoneendandinflux attheother isnumerically equaltothe normalcomponentofthemagneticforce toberepresented: andwhen thiscondition hasbeen once reached letthepistons become dissolved intoperfect liquid homogeneouswith the rest. The solidwith itsperforations remainingarigidtubular system, theliquidwill continue forevercirculating through thetubes andthefreeexternalspace:and itsmotionthrough allexternalspacewillbesuch that thevelocityiseverywhere ofthesame magnitude andinthesame direction asthere- sultant magneticforce inthecorresponding position relatively tothemagnet. Theproofofthisproposition*is;—thataccord- ingtoawell-known hydro-kinetic theorem, themotion ofthe liquid must beeverywhere"irrotational" [Yortex Motion, §59{e)\andthat ifthenormal componentfluidvelocity,or normalcomponentforce inthemagnetic analogue,begiven *Allthehydro-kinetic terminology andpropositions used intheremainder ofthisvolume arefully explained, with demonstrations when necessary,in theportion already published (intheTransactions oftheRoyal Society of Edinburgh, April 1867andDec. 1869) ofapaper on"Vortex Motion," with thecontinuation ofwhich Iamatpresent occupied.References toitare givenwhennecessary tojustify anyoftheassertions inhydro-kinetic subjects made henceforward. I VII.] Hydro-kinetic Analogy, 451 overthewhole surface, the fluidmotion ormagneticforce is determinate throughallexternalspace (§591,Theorems 1 and2).Thepermanenceofthe fluid motionfulfilling the same condition follows atonce from theconstancyofthe circulationthrougheachperforation [Vortex Motion, §59(<?)], consequent uponthe frictionless character which weassume i^hefluid topossess. |^P675. Intheprecedingstatement nocondition hasbeen imposedastothepairsofaperturesinthesurface oftherigid bodysubstituted foramagnet,which are tobeconnected throughtheinternal tubes;nosuch conditionhaving been necessary,because wesupposedtheaperturesover thewhole surface tobeinversely proportionaltothemagnitudeofthe normalcomponentforce. Thestatement maybevaried thus :— take allthatpartofthesurface forwhich thenormal component force isoutwards, anddivide itinanymanner intoinfinitesimal areas. From eachpointintheboundaryofanyoneofthese areas, draw alinethroughexternalspacetill itmeetsagain, asitwillmeetagain,thesurface ofthemagnet. Bydoing this foreveryinfinitesimal area oftheboundarytraversed outwards, acorresponding area, where thenormalcomponent force isinwards, isfound, andthewhole remainder ofthe surface isthusdivided intoareascorrespondingtothose chosen inthe firstpart.Letthepairsofcorrespondingareas becon- nected byinternal tubes. Theremainder ofthestatement maybeapplied without alteration tothis tubular arrange- ment. The fluidanaloguethus constructed, willhave the peculiarity,thateachportionoffluid circulates foreveralong onecircuit(thatistosay,closedcurve). 576. Thehydro-kinetic analogyisbothmore completeand moresimple,itisinfactperfectly complete,andtherefore per- fectly simple,ifinstead ofasin§479adoptingthedefinition properforapolar magnet (§549),weadoptthe"electro- magnetic definition" (§517 and postscriptto§517),forthe resultant force atanypointinthesubstance ofthemagnet, whether itbeapolar magnetoranelectro-magnet. The resultantforce defined" electromagnetically" forthespace occupied hythemagnet, andtheresultant magnetic forceaccord- ingtotheunambiguous definition forspacenotoccupied hythe 29—2I 452 AMathematical Theory ofMagnetism. [xxvii. magnet, agree everywhereinmagnitude and direction with the velocityinapossiblecaseofmotion ofanincompressible liquid fillingallspace. Toprovethis itisonlynecessarytoremark thatthesolecondition thatX,F,Z,maybethevelocity-com- ponentsinapossiblecase ofmotion ofanincompressible fluid, isthattheyfulfil theequationofcontinuity dXdYdZ^Q,dx dydz* andwehave seen(§517) that d^dYdZ^^dxdydz throughout thesubstance ofthemagnetaswell asthrough externalspace,ifX,F,Zdenote componentsofthemagnetic force. Thecomponent intensities ofelectric current inthe electro-magnet producingthisforce are[§517(a),(Q] }_(dZ_dY\ }_fdX__dZ\ }_fdY_dX\ 47r\dy dzJ' 4!'ir\dz dx)'47r\dx dy)' 577. Thisproposition,which Ifound more than twenty years agoasanobvious deduction frommyformulae forelectro- magnetic force, publishedintheTransactions oftheRoyal SocietyforJune 1850(§§515—518above),ispurelykine- matical. Since thattime ithasacquiredaninterest which it didnotthenpossessforme,invirtue ofHelmholtz's splendid discoveryofthedynamicallaws ofvortex motion*. Ihadnot known more than thatthedistribution of'*electro-niagnetic" forcethroughthesubstance ofthemagnet,aswell asthrough externalspace, correspondedtoapossibledistribution ofmotion inacontinuousincompressiblefluidfillingallspace,andhad noclue totheconsequencesofleavingafrictionlessliquidto itself, with such amotion once established init.ByHelm- holtz'stheory,itisdemonstrated thatthefluidmotion alters so astoalways remain therepresentativeoftheelectromotive force duetoanelectro-magnet continuouslyvariedaccordingtothe followinglaw. Lines offluid matter whichinitiallycoincided with thelines ofelectric current intheelectro-magnet initially *Crelle's Journal, 1858, and (Tait's translation) Philosophical Magazine, July 1867, ^BEv; II.] Hydro-kinetic Analogy. 453 replaced bythe fluid, however theychangeinthesubsequent motion, always mark thelines ofelectric current which must beconstituted toproducethealteredelectro-magnet ;andthe whole amount oftheintensityoftheelectric currentcrossing anyareabounded byanyclosed curvepassing always through thesame fluidparticlesremains constant. Itisunnecessary, however, toenternowonthewidehydro-kinetic subject thus indicated;althoughIcannot butrefer toHelmholtz's theorem ofvortex motion, notmerely onaccount ofitsintrinsicbeauty, butbecause Ihave found itofgreatvalue inassisting meto realize thepurelykinematic representationofelectro-magnetic forcewhich fluidmotion affords. Thegeneral hydro-kinematic analogy,andthedynamicsoftheirrotationally moving portions ofthefluid, astheyservedmeprimarily twenty-four years ago ininvestigatingtheinverse problems,willbefurther considered inthefollowing chapter. 578. Thehydro-kinetic analogyisvaluable inthemathe- maticaltheoryofelectro-magnetismasleadingtoasetof theoremsrespecting magneticforces produced byelectric cur- rents, precisely analogoustothose theorems ofGreen'srespect- ingforces due tocentresacting accordingtotheNewtonian law,which Ideduced in1841 from ananalogy with the "Uniform motion ofheat inhomogeneous bodies," bythe investigation formingthe firstpartofthisvolume(§§1—4 above). Thefollowingtheorems I—III.areparticularcases ofthegeneral propositionof§576,andrequire nofurther demonstration. 579. Theorem I,—(Compare §594below.)—Consideringall spaceasoccupied byanincompressiblefrictionlessliquid,let /Sibeaclosed surface, which(tofacilitate conceptions) maybe supposedtobeconstituted ofaperfectlyflexible andextensible membrane. Atfirst letthere benomotion oftheliquidin anypartofspace, andthen letanymotion whatever bearbi- trarily givento>Si,subject onlytothecondition ofnotaltering thevolume enclosed byit.Themotion which isgiventothe liquidwillbeeverywhereirrotational ("Vortex Motion," §16 and§60),and will therefore becontinuously expressible throughoutexternalspace byapotential;and continuously expressible, likewise, throughtheinternal space:butthere 454 AMathematical Theory ofMagnetism. [xxvii. willbeadiscontinuitya,tS;onthetwo sides ofwhich the velocity-potentialmust differ byanamountequaltoP,the impulsive pressurewhich would have tobeappliedto >Sftopro- duce theactual motioninstantaneously from rest. Divide S into infinitelynarrow bands bylinescorrespondingtoequal values ofP,andineach ofthese bands letanelectric current BP circulate ofstrength equaltoy—whereBPdenotes thediffer- ence ofthevalues ofPatitstwoboundaries. Themagnetic forceproduced bythedistribution ofelectric currents thus con- stituted, willagreeinmagnitudeanddirection with thefluid velocityinthehydro-kinetic analogue. ThispropositionIused inacommunication totheBritish Association atOxford, in June 1847,"OntheElectric Currents bywhich thePhenomena ofTerrestrial Magnetism maybeproduced;" and itisreferred tointheabstract ofthatcommunication (now reprintedin §§602,603below), whichappearedintheyearly volume. It wasprobablyoneoffivepropositions which Iwrote toLiou- ville intheSeptember following (see§589below). 580.Corollary.—Intheelectro-magnetic analoguethedirec- tion ofthe electric current isperpendiculartotherelative tangentialmotion oftheliquid onthetwosides ofS,andthe surfaceintensityoftheelectric current isequaltotherelative tangential velocitydivided by47r. 581.Example,—LetBbekeptofconstantfigure, and letthe motion giventoitbepurely translatory. Theliquidwithin itwillmove asifitwere arigid body. Hence theinterior velocity-potentialwillbeUx, ifUbethevelocity, and ifits direction beparalleltotheaxis ofx.Hence ifweconsider a solid carriedalong throughafrictionlessliquid ;determine the velocityand direction, relativelytothe solid, oftheliquid gliding alongeachpartofitssurface;andconstruct theana- logoussurfaceelectro-magnet accordingtotherule of§579; this distribution ofelectric currents willproduceauniform field offorce, ofintensity Uthroughoutthespaceenclosed by thesurface onwhichtheyaredistributed, and willproducea resultant force ateveryexternalpoint, agreeinginmagnitude and direction with theabsolutevelocity which theliquidis compelledtotake inmaking wayforthe solid. Theanalytical VII.] HydrO-hinetic Analogy.455 expressionofthisvery interesting theorem iscontained in(IX.) of§517,appliedtothecaseinwhich .=£,^=0,7=0. >82.Theorem II.(Includesthecase§581ofTheorem I.)— Letanymotion ofrotation begiventoarigidbodyinanin- finite incompressible liquid. Themagnetic analogueconsists ofauniform current traversingthevolume oftherigidbody inlinesparalleltotheaxis ofrotation, andofintensity equal totwice theangular velocity;with the circuit completed superficially bythesurface distribution constructed according totheruleof§581. Theresultant force ofthecompletedsolid andsuperficial electro-magnet (§535)thusformed willagree everywhereinmagnitudeanddirection with theabsolute velo- cityofthematter, whether solid orliquid,inthekinematic analogue. Theanalytical expressionofthistheorem(ifwe taketheaxisofthesolid's rotation fortheaxisofx)ishadby puttingin(IX.)of§517 a=0,^=-f^,7=fy. 583. Theorem III—Consider afixedrigid ring, having,for simplicity,butoneperforation, and therefore giving duplex continuitytothespaceexternal toit.Letthewhole ofthe externalspace beoccupied byanincompressiblefrictionless liquidinastate ofcyclic motion, with theringforcore. Take anysurface Sbounded bystream lines. This isnecessarilya surface ofduplex continuity enclosingthering.Ononeof thestream linesformingacircuit of>Sf,take ipointscorre- spondingtoinfinitelysmall differences ofthevelocity potential, each anexactsub-multiple-ofthe'' cyclic constant," or "whole circulation"[k).Throughthesepoints drawequi- potentiallines on/Sf,which therefore willeach cutperpendicu- larlyallthestream lines on8.Ineach oftheinfinitely narrow bands intowhich Bisthus divided(constituting a geometricalcircuit which crosses allthestream linecircuits), letanelectric current ofstrength 7—. circulate. Theresulting electro-magneticforce willbezero atevery pointwithin S,and willbeequal to,andinthesame direction as,thefluidvelocity 456 AMathematical Theory ofMagnetum. [xxviii inthespaceexternal toS.Thisinteresting andimportant propositionisperfectly analogoustothatwhich isgiven by Green forsurface distribution ofelectricity andtheresulting electric force inArticle 12ofhisEssay (towhich reference is made inThomson andTait's NaturalPhilosophy, §507,under thedesignation*'reducible caseofGreen'sproblem"). XXVIII.[ifov. 1871.] Chapter IX.—Inverse Problems. 584. Inverseproblemsofmagnetismareproblemsinwhich thedata areofmagnetic force, and itisrequiredtofind distri- butions ofmagnetismorofelectric currentsbywhich thegiven force canbeproduced. Theyfallunder twoclasses :—I.Those inwhich theforce isgiven^for every pointofspace:—and II, Those inwhich theforce orsome componentoftheforce is given through someportionofspace, whether volume, surface, orline;—and itisrequired, under certain limitations orcondi- tions, tofinddistributions ofmagnetismorofelectric currents bywhich thegivenforce canbeproduced. Acomplete and unconditional solution ofevery problemofClass I.is,aswe shallimmediately see,always easilyfound. 585. Class I.—First case, polar definition (§479andPost- scriptto§517)ofresultantforce adopted.Inthiscase the magneticforce isexpressible bymeans ofapotential, and therefore themostgeneralform ofdatais;—giventhepotential atevery pointofspace.LetVbeitsvalue at{x,y,z),sothat if3C,^,5Sdenote thecomponentsofthemagnetic force, ---^'3»=^".^--#«• Ifa,/9,7denote therectangular componentsoftherequired magnetization, wehave ^+^+I=4^fe+^+-^)[§-17(^)repeated], and Gt,y3,7maybeanyfunctions whatever which fulfil this equation.Then asaparticularsolution wehave ,__1_^ r._l_dV ._±dV^"^irdx' '^"^irdy'^~ ^irdz^^' XXVIII.]Inverse Problems. 457 Letnow a",ff\7"denote anythree functions whateverfulfilling thefollowing equation:— f:v^/:+*r-=o(3).ax aydz Thecompletesolution oftheproblem is, a=a'+a",^=^'+/3",7=7+7^ W- Thearbitrary part a'y^\7',ofthissolution consists ofany distribution ofmagnetization agreeing everywhereinintensity anddirection with the velocity and direction ofapossible motion ofanincompressiblefluid throughallspace. When thegivenfunction Yissuch that itsfirstandsecond diffe- rential coefficients dv^ydVd^VdVgF dx'dx''dy'df'dz'dz' areeverywhere finite, there isnothing more tobesaidinrespect totheprecedingsolution;butwhen the first differential co- dV efficients -r- ,etc.,though themselves everywhere finite, vary anywhere abruptlyintheir values, aninterpretationofasuffi- cientlyobvious character becomesnecessarytodeduce the solution from theprecedingformulae. Ortheform ofsolution maybevaried byintroducingtheproperformulae[§473(1)] forsurface-distributions oftheimaginary magnetic matter at thesurfaces ofdiscontinuity. 586. Class I.—Second case, electro-magnetic definition adopted. Inthiscasetheforce, though expressible bymeans ofapoten- tialthroughout every portionofspacefreefrommagnetized matter,isnotsoexpressible throughthesubstance ofthemagnet. Hence thedatamust betheintensity anddirection ofthere- sultant force atevery pointofspace ;butthese data arenot altogether arbitrary inasmuch asifX,Y,Zdenote thethree rectangular componentsoftheforce, dJCdY dZl ^ rnwt*^ ,t\ .1-1 ri«+^+S=°[§517Wrepeated]. Hence theproblem is;—given X,F,Z,eachanyfunction of(x,y,z),butsubject toequation {k)of§517;itisrequired !1 458 AMathematical Theory ofMagnetism, [xxviii. tofindthreequantities a,A7suchthat .fdydfi\_dZ_dY.(^_dy\_dX_dZ (^_^\_^'^ _^ \^^dzj~ dydz' \dzdxj~dzdx' ""Xdx dyj dxdy [§517(0repeated]. Ofthisproblemthegeneralsolution is wherei/rdenotes anyarbitraryfunction of{x,y,z).Forsim- plicity wehavesupposedthat there isnoabruptvariation in thegivenvalues ofX,Y,Z.Theproperformulae tosuitthe caseofabruptvariations fromoneside toanother ofanysur- face, areeasilyfound. 587.Remark thatthearbitraryfunctions a",yS",7",inthe solution(4)of§585express anysolenoidal distribution what- everwith thesolenoids allclosed;andthatthearbitrary part '>^inthesolution(5)of§586expresses anylamellar distribu- tion-whatever with theshells allclosed. 588.Remark alsothatthedistribution ofimaginary magnetic matter derivable(§473)from thesolution of§584,and of electric current derivable(§554)from thesolution of§585, areeach determinate, andthat itisonlythedistribution of magnetizationwhich isaffected bythearbitrary partofthe solution ineither case. 589. Class II.—Forthepresentitisenoughtoconsider the following typical problemsofthis class. Given the force through spaceexternal toagivenclosed surface B\required thedistribution ofimaginary magnetic matter, orofelectric currents, orofmagnetization ;each distribution confined toan infinitelythinlayerofmatter coincident with thissurface :and toinvestigatethedeterminacyofthesolution ineach case. With reference totheseproblems,Ifindaleafofmanuscript written inFrench, indorsed:—"Fragmentofdraft ofletter "toM.Liouville, written ontheFaulhom, Sunday, September *'12,1847, andposted ontheMondayorTuesday week after, "atMaidstone. The letter hasnotbeenpublished yet,although "inSept. 1848 Iunderstood fromM.Liouville inParis, thathe "had itforpublication. Probablyithasfallen aside and is "lost[?inconsequenceofthedisturbed state ofParis atthat I I VIII.]Inverse Problems, 459 "time], which Ishouldregret,asitcontains myfirst ideas, <'andphysical, especially hydro-dynamical, demonstrations of "thetheorems Iamnowabout towrite outforpublicationin "my paperonmagnetismfortheEoyal Society, fromrough "drafts written inAugust 1848. W.T.Oct29th, 1849." The"now" hasbeen deferred until thepresent time, November 20th, 1871. Iamobligedtowrite frommemory,as Ihave notbeen able torecoveranyofthose roughdrafts. I have added importantdetailsinvolving new ideasregarding polycyclic*fluid motion, formuch ofwhich, asforthewhole terminologyofmultiple continuity,Iamindebted toHelm- holtz's paperonVortex Motion. 590. First, with reference tothedata,itmust beremarked that theforcebeing byhypothesis due topolar magnetsor electro-magnets altogetherwithin 8,cannot begiven arbitrarily throughthewholespaceexternal tothat surface. Itmay indeed bereadily proved from aremarkable andimportant propositionduetoGauss, tobefound inThomson and Tait's Natural Philosophy, §497,that ifthepotentialweregivenfor anyclosed surface, lying altogetherexternal toS,whether enclosing 8ornot,and ifnotenclosing 8,enclosing anyportion ofexternalspace however small, theforcewould bedeterminate throughoutthewholespaceexternal to8.Thesamemaybe provedif(insteadofthepotential)thenormalcomponent force weregivenoveranysurface whatever, external to8,andnot enclosing it,oroveranysimplycontinuous surfaceenclosing8. Atpresent, however, two casesonlyshall beconsidered :— thepotential givenoverthewhole surface of ;Sf(Case 1),and thenormal forcegivenoverthewhole of8(Case 2). 591. Preliminary Theorems 1—5.—Theorem 1(Discovered byGreen). Thepotential being given arbitrarilyover S,the resultantforceisdeterminate throughallexternalspace, anda determinate distributionofmatter over S,acting accordingtothe inversesquare ofthedistance, maybefound which shallproduceit. Theorem 2.—Thenormal component force being given for S, theforceisdeterminatethroughallexternalspace, andadetermi- *"Vortex Motion," §60 (2). 460 A3Iathematical Theory ofMagnetism. [xxviii. imte distribution ofmatter overSacting accordingtotheinverse square ofthedistance maybefoundwhich shallproduce it,pro- vided thatSissimplycontinuous. [Compare §207.] Theorem 3.—Thepotential being arbitrarily given forS,sub- jecttothecondition that itsintegral amount forthewholesurface iszero;orthenormal component force being arbitrarily given for S,subjecttothecondition that itsintegral amountforthe whole surfaceiszero; theforceineach case isdeterminatethrough allexternalspace, andadeterminate distributionofelectric currents overSmaybefound which shallproduce it,provided that inthecaseinwhich thenormal componentforce isgiven, thesurface Sissimplycontinuous. Theorem 4.—If8becomplexly continuous, let(7^,C^,G^, etc., bemutuallyirreconcilable closed curvesencircling it,whether incontact withit,orinthespaceexternal toit.Ifthecon- tinuityisn-fold, there arensuch circuits. Thenormal com- ponent force being given arbitrarily for S,subject onlytothe condition that itsintegral amountforthewholesurface iszero; andanarbitraryvalue/c^, /c^, etc.,being given forthe integral ofthetangential component force round eachofthe circuitsC^,C^,etc.: theresultantforceisdeterminatethrough thewholespaceexternal toS,andadeterminate distributionof electiic currents overSmaybefound which shallproduceit Theorem 5.—WhenSiscomplexly continuous, nodistribu- tionofmatter over itcanbefoundtoproduce force throughex- ternal space fulfillingtheconditionsofTheorem 4,when the valuesofthecyclicconstantsk^,k^,.,,areallfinite;butifinfin- itelythin sheetsofmatter beintroduced asbarriersclosingallthe apertures ofS,adeterminate distribution ofmatter onthese sheets andoverSmaybefoundwhich shallproducethatforce through allthespaceexternal toS,excepttheinfinitelysmallparts ofit occupied bythebarriers. 592. DemonstrationsofTheorems 1—4.—Toprove Theorem 1,letthewholespacewithinSandthewholespaceexternal to 8,beoccupied byhomogeneous incompressible liquid,but let there beaninfinitelythinvacuousspace separatingtheexternal from theinternal fluid. Letequal impulsive pressuresbeap- XXVIII.]Inverse Problems. 461 pliedinopposite directions, totheliquidsurfaces onthetwosides ofthisvacuousspace, equal everywheretothegivenvalue of thepotentialatthecorresponding positioninS,ofthemagnetic problem,thepressure beingreckoned aspositive when itis outwards fromSontheexternalliquid, andinwards fromSon theinternalliquid. Themotion willbeirrotational throughout each portionofthefluid;andthe initialvelocity-potentialsin portionsofthe fluidinfinitelynear oneanother onthetwo sides ofS,willbeequaltothegiven magnetic potential. Hence(§7)thegiven potentialover ^Sfwould beproduced byadistribution ofmatter over S,havingitssurfacedensity everywhere equaltothevelocityofseparation (reckoned negative when there isapproach)ofthetwo fluid surfaces divided by47r*.By"velocityofseparation"ismeant the dijBference ofthenormal componentvelocities onthetwosides ofS. 593. DemonstrationofTheorem 2.—With thesame hydro- kinematicarrangementasin§592,lettheboundaryofthe fluid external toSbeimpulsively pressedsoastoproduce instantaneously anormalcomponent velocity equaltothe given normalcomponent magneticforce. And letthebounding surface ofthefluid within Sbesimultaneouslyacted on,with apressure equal andoppositetothatwhichproducesthespeci- fied effect ontheexternal fluid. Themotiongeneratedis irrotationalthrough eachportionofthefluid,andthepotentials onthetwosides ofS,areeachequaltothepotentialatSof thedistribution offorcethroughexternalspace, which hasfor itsnormalcomponent thegiven value forevery pointofS,the densityofthedeterminate distribution ofmatter over8which wouldgive that external distribution offorceis,asin§592, equaltothevelocityofseparationoftheliquid surface, dividedby47r. 594. Demonstration ofTheorem 3(compare §§579, 580).— Letthewhole ofspace becontinuously occupied byhomo- geneous incompressible liquid, without anyvacuousspaceatS; *This ismerely ahydro-dynamical proof ofGreen's celebrated theorem thatadistribution ofmatter, acting according totheinverse square ofthe distance, over asurface Smaybefound determinately, which shall produce anyarbitrarily given potential overthewhole ofS. 462 AMathematicalTheory ofMagnetism. [xxviii. and, asimmediaterecipientfortheaction offorce, imagine S toconsist ofaperfectlyflexible and extensible membrane, separatingtheinternal from theexternal fluid. Apply per- pendicularlytothismembrane animpulsive pressure which shall produceanormal component velocity equaltotheex- ternal normal componentforce determinable from thegiven potential accordingtoTheorem 1,when itispotentialthat is given,orequaltothegivennormalcomponentforcewhen itisforce that isgiven. Themotion isirrotationalthrough- outeachportionofthe fluid; and thenormalcomponent velocities onthetwo sides of8^areeverywhere equaltoone another;butthetangential motions ofthefluids, andtherefore thevelocity potentials,areunequal onthetwo sides. Inthe former casethevelocity potentialintheexternal fluidinfinitely near8orinthelatter case, thenormal component velocityof the fluid oneach side of8hasspecifiedvalues. Ineither casethedeterminate distribution ofexternal forcefulfillingthe specified condition at8,whether astopotentialorastonormal component,isproduced (§§579, 580)byadeterminate dis- tribution ofelectric currents on8,fulfillingthefollowing specification. The direction ofthe electric current istobe everyw^here perpendiculartothedirection oftheslipinthe fluidanalogue ;andthesurfaceintensityofthecurrent istobe equal tothevelocityoftheslipdividedby47r. 595. Demonstration ofTheorem 4.—Letthesamehydro- kinematicarrangementsasthose inthedemonstration of Theorem 3bemade, andinaddition leteachapertureof /S'be temporarily stopped byaperfectlyflexible and extensible membrane, introduced merelyasarecipientfortheaction offorce. Let8beimpulsively pressedsoastoproduce anormalcomponent velocity equaltothegivennormal component force, and letuniform impulsive pressures equal respectivelytok^,k^,k^,etc.,besimultaneously appliedtothe barriers. Theconstancyofthedifierence, k,ofthepotentials betweencontiguous portionsoffluidonthetwo sides ofeach barrier, securesequalityinthetangential component velocities, and therefore no"slip" between them. Suppose then the barriers annihilated. Thedeterminate motion thusproduced isirrotational throughout eachportionofthefluid,and itfulfils IVIII.]' Inverse Problems. 463 inthespaceexternal to8preciselytheconditions which, when magneticforce issubstituted forfluidvelocity,arethosespeci- fiedintheenunciation ofTheorem 4.Hence adeterminate distribution ofcurrents over/S',answeringtothesamespeci- fication asthat ofTheorem 3,producesforce inthespace external to8which fulfils ourpresent conditions, andthus Theorem 4isdemonstrated. 596. DemonstrationofTheorem 5.—Lettheaperturesof8be stopped bymaterial sheets offinite thickness. Imaginethe matter ofthese sheets tobeliquid, homogeneouswith that occupyingtherestofspace, andcontinuous with theliquid sup- posedtooccupytheinterior of8.Theboundaryofthewhole ofthisliquidisasimplycontinuous closed surface, consisting ofthepartof8notcovered bytheaddition ofthesupposed barriers, andthetwosurfaces ofeach ofthese barriers. Let8' denote thatpartofthesurface of8',and letB^, J5/,B^,^/, etc.,denote thesurfaces ofthebarriers. Asinthedemonstra- tions ofTheorems 1and 2,lettheexternal fluid beseparated from theinternal byaninfinitelythinvacuousspaceoverthe whole bounding surface, and letpressureactsoastoproduce agiven normalcomponentintheexternal fluidnext to8]zero potentialintheexternal fluid next toB^,B^,etc.;potentials equaltok^,k^,etc.,intheexternal fluid next toB^,B^,etc.; andeverywhere equal potentialsinportions infinitelynear oneanother, oftheexternal and internal fluids. Asinthe demonstrations ofTheorems 1and 2,itisseen that there isadeterminate distribution ofmatter over thewhole bound- ingsurface which shallproducethegivennormal component force over8,potentialzero forB^,B^, etc.,andpotentials ATj,K^,etc., forB^,B^,etc. Ifnow the barriers bemade infinitely thin, sothatB^andB^shallbeinfinitelynear one another, andB^^BJinfinitelynear oneanother, and soon; theprescribedconditions are fulfilled bythedistribution of matter determined forthelimitingcase thus reached. The distribution ofimaginary magneticmatter onB^,B^,B^,B^\ etc.,maybeexplicitly determined bythefollowing simplecon- siderations. Consider aninfinitelysmall column ofthefluid betweenB^andB^,bounded byany cylindricalorprismatic surfacecuttingthesurfaces B^B^\atright angles,andenclos- I 464 AMathematical Theory ofMagnetism. [xxviii. ingequal infinitelysmall areas onthese surfaces. Thedensity ofthefluid being unity,themass ofthiscolumn willbeAt, iftdenote thethickness ofthespace betweenB^andB^,and Athearea ofeither endofthecolumn. Thismass isacted onbyanimpulse k^A,because byhypothesis oneend ofit experiences, duringtheinitiating impulse, animpulsive pres- sureequaltok^perunit area,andtheother, zeropressure. Hence thevelocity acquired bytheinfinitesimal column is --.Letndenote thenormalcomponent velocityoftheex- ternal fluid,which isequalforpoints infinitely nearoneanother onthetwo sides ofthebarriersupposed infinitelythin. The velocityofseparationofthefluid surfaces oneach sideoiB^,and thevelocityofapproachofthefluid surfaces oneach sideofB^ willbeeachequalton-\--f.Hence thematter tobedis- tributed over thetwo surfacesB^,B^willberespectively, +J—(w+-tM.As-iisinfinitely great,theflnite term nmay beneglected,andtherefore thedensities onthetwo surfaces are+-r^.These are(§472) preciselythedensities ofthe positiveandnegative magneticmatterrepresentingthe free polaritiesonthetwosides ofamagneticshell(§506)ofstrength j-^.Thethickness imay,ofcourse, bedifierent indifferent partsofthe shell, asisallowed inthegeneraldefinition [§506(1)]ofamagneticshell. Theprescribeddifference of potentials, k^,reckoned fromB^throughtheexternal fluid to J5j,isverified by§512,cor. 3. 597. Purely analytical proofsoftheorems, including Theorem 1andTheorem 2above, aretobefound inThomson and Tait's NaturalPhilosophy, AppendixA.(e),and§317,Example (3), andareincluded in§§206,207above [compare §§709—716 below]. These references supplyalso allthat isnecessaryto eliminate allhydro-dynamicalconsiderations from thepreced- ingproofsofTheorems3,4,and 5.Itherefore confinemyself onthepresentoccasion tothehydro-dynamical proofs now given;butremark that theanalytical proofsarevaluable in I VIII.]Inverse Problems. 465 respecttophysicalscience asshowingthat ineach case the integral extended throughexternal spaceisanabsolute minimum [com- pare §758below] subjecttotheconditions prescribedinthe enunciations oftheseveral cases, andthatthevalue ofthesame integralfortheinternalspaceisalsoaminimumsubjecttothe conditionsspecifiedintheseveral demonstrations givenabove. From this,with§§567,571above,itfollows thatthedynamical value ofthedeterminate distribution ofimaginary magnetic matter onthesurface>Si,whichproducesatthat surface the prescribed potentialofTheorem 1,orthegiven normal com- ponentforce ofTheorem 2,islessthan that ofanydistribution ofimaginary magnetic matter notconfined tothat surface, but stillproducingover itthesamepotentialorthesame normal component force; and that theelectro-magnetic dynamical value ofthedeterminate distribution ofcurrents onSwhich producesatthat surface theprescribed potentialorthepre- scribed normal componentforce ofTheorem3,islessthan that ofanydistribution ofcurrents notconfined toS,but stillpro- ducingthesamepotentialorthesame normal componentforce overthat surface. 598. Topassfrom adeterminate distribution ofimaginary magnetic matter, oradeterminate distribution ofelectric currents, toadistribution ofmagnetizationwhich shallpro- duce thesame resultant force, isaswehave seen(§587)an indeterminateproblem,even iftheforce isgiven throughout space.Stillmore istheproblemindeterminate iftheforcebe giveninonlyonepartofspace,and itisrequiredtofindadis- tribution ofmagnetizationintheremainder ofspacewhich shallproducethat force. Tofindthecompletesolution ofthis problemwith theproper arbitrary functions, wemay proceed either from thedeterminate distribution ofimaginary magnetic matter of§591,Theorems 1and 2,orfrom thedeterminate distribution ofelectric currents of§591,Theorems 3and 4,on theboundingsurface. Our firststeptowards thecomplete solution shall betofind,from adeterminate distribution of imaginary magnetic matter, orfrom adeterminate distribu- T.E. 30 466 AMathematicalTheory ofMagnetism. [xxviii. tion ofelectric currents, onasurface S,distributions of magnetization, confined tothis surface, which shallproducethe given external force. 599. Divide thewholesuperficialdistribution ofimaginary magnetic matter intoaninfinite number ofequal parts,irre- spectivelyofsign. Asin§523, join positive andnegative partsinpairs chosenarbitrarily, byarbitrarycurves allinthe surface>Si,andlaysolenoids ofequal strengths alongthese curves. Thus onthesurface >Siadistribution oftangential magnetizationtoacertaindegree arbitraryisobtained, which shallproduce throughexternalspaceadeterminate distribution ^ofmagneticforcefulfillingtheprescribedsurface condition. Acomplete representationofwhat isarbitraryinthissolution consists ofanydistribution whatever ofclosed solenoids, each whollycoincident with S.Anysuch distribution ofmagneti- zationmay (§510, Cor.2)besuperimposed ononefulfillingthe prescribed condition withoutviolatingthisfulfilment. 600. Toproceedfrom surface distribution ofcurrents to surface distribution ofmagnetism; (whichifSissimply continuous canbedone always,but ifSiscomplexlycon- tinuous canonly bedonewheneverystream linebounds anarea on>S';)divide 8byelectric stream lines intoan infinite number ofbands ofsuch breadths astogive equal strengthsofcurrent inthem. This division mustbeginand end inpoints which forthepresentIcallpoles. There must therefore beatleast twopoles, andtheremaybeany number, odd oreven, greaterthan two. ThesepolesIcall north orpositive when theelectric currents inthebands en- circling them areinthedirection inwhich thehands ofa watch, placed uponthemfacing outwards, would move. All thepolesmaybenorthpolesorallsouthpoles,orsomemay benorth andsome south. Commencingwithanyoneofthe poles,substitute amagneticshellpassing throughitandlying altogether onS,foreachbandencirclingit.Ifthewhole sur- facecanbethus exhausted thethingisdone. Ifnot,take next apoleontheunexhaustedportionofsurface andfollow againthesame rule; and soonuntil foreachinfinitelythin band ofcurrent, amag^netic shell hasbeen substituted. Thus I XXVIII.]Inverse Problems. 467 wehave(§508)acomplex magneticshell instead ofthedistri- bution ofcurrents. Unlike theresult of§599, this result is determinate, involving, however, onearbitraryconstant. The solutions thusobtained, differing accordingtotheorder inwhich thetwoormorepoleshavebeen taken, are,each ofthem, fully determinate. The difference between anytwo ofthem is clearlyauniformmagneticshell ofdeterminate strengthcoin- cident withthewhole ofS.Thegeneralsolution comprehend- ingthemall,oranycombination ofthem, ishadbytaking any oneofthem andsuperimposing uponitauniform magnetic shell ofarbitrary strength,coincident with thewhole ofS. Thisarbitrary partofthegeneralsolution beinga"closed shell"(§512, Cor. 5)exercises noresultant force through either external orinternalspace. 601. Considerlastly,thegeneral problemoffinding magne- tization onandwithinanyclosedsimplycontinuous surface S, which shallproducethedeterminate external distribution of force(§591,Theorems 1and2)duetoanyarbitrarily given poten- tialorarbitrarily givennormalcomponent force, foreveryexter- nalpoint infinitelynear S,with, ofcourse, thecondition thatthe surfaceintegraloverthewhole ofSofthegiven potentialorof thegiven normalcomponentforce iszero. InTheorems 1,2, and3of§591,provedin§§592, 593,and594,wehave seen that adeterminate distribution ofimaginary magnetic matter, oradeterminate distribution ofelectric currents, over S,may befound which shallproducethespecifiedexternal distribution offorce. And in§§599and600wehave seenhow inany casewhen asurface distribution, either ofmagneticmatter or ofelectric currents hasbeen found, wecanfindsyntheticallya surface distribution ofmagnetization which shallproducethe same external force;thismagnetization being purely tangential, involving anarbitraryfunction when derived fromimaginary magnetic matter, andbeing purely normal, involvinganarbi- trary constant when derived from distribution ofcurrents. Thecompletesolution ofthepresent problemisobtained by firstassuming arbitrarily anydistribution ofmagnetization whatever within S,which maybealtogether bodily magne- tizationspread throughtheinterior, oraltogethersurface mag- aetization, whethertangentialornormal oroblique, infinitely 30—2 468 AMathematical Theory ofMagnetism. [xxix close totheinside of>S',orinpart bodily magnetization,and inpartsurfacemagnetization ;thenfindingthe external potentialornormal componentforce atpoints infinitely near S,due tothismagnetization, accordingasitispoten- tialornormal componentforce that isgiven ;then subtract- ingfrom thegiven potentialornormal componentforce the potentialornormal componentforcedue tothearbitrarily assumedmagnetization;andlastly, finding (atpleasure either) atangentialoranormal distribution ofmagnetization onS which shall produce potentialornormalcomponentforce equaltothedifference. Thesurface-magnetization thus found, compoundedwith the arbitrarily assumedmagnetization,is themostgeneraldistribution ofmagnetizationwithinSwhich canproduce,atexternalpoints infinitelyclose toS,thegiven potentialorthegiven normal componentforce. XXIX.—OntheElectric Currents hywhich thePhenomenaof Terrestrial Magnetism mayheproduced. [From theReport oftheBritish Association fortheMeeting of1867 inOxford.] 602. Itisawell-known theorem,firstdemonstratedby Green, that theaction ofamass ofanynature inattracting anexternal point,mayberepresented bymeans ofadistribu- tion ofmatter ofthesame kind overthesurface ofthebody ; that istosay,thatacertain distribution ofmatter over the surface ofabodymaybedetermined, which willproduce exactlythesame force, whether ofgravitation,ofmagnetism, orofelectricityasresults from thebodyitself. Thus, by applyingthistheorem tothecase inwhich theforce considered isthat ofterrestrialmagnetism, weseethatacertain distribu- tion ofimaginary magnetic matter maybefound which would produceallthephenomena ofterrestrial magnetismobserved atthesurface oftheearth oraboveit,exceptthose which are due toatmosphericorexternal sources ofmagnetism,ifany such exist. Thisproposition, althoughofgreattheoretical interest, cannot beentertained asexpressingaphysical fact; forthere areonlytwoways inwhich wecanconceive internal sources ofterrestrial magnetismtoexist.Wemayeither imagine,asGilbert did,theearth tobewhollyorinparta 'XXIX.]Terrestrial Magnetism.469 magnet,such asamagnetofsteel, orwemayconceive ittobe anelectro-magnetwith orwithout acoresusceptibleofin- duced magnetism.Inthepresentstate ofourknowledgethis secondhypothesisseems tobethemoreprobable [?Feb. 4, 1872]; andindeed wehavenowmanyreasons forbelieving that theexistence ofterrestrial currents, producing whollyor inpartthemagnetic phenomena,isaphysicalfact. [The *'earth currents" which render thelocalization ofafault ina submarine cable sodifficult, certainlycontribute totheresult- antmagneticforce observed attheearth'ssurface.] Connected with this itbecomes aninteresting question, whether mere electric currents couldproducetheactual phenomenaobserved. Ampere's electro-magnetic theoryleads ustoanaffirmative answer, butananswer which must beregardedasmerely theoretical;for itisabsolutely impossible [compare §546, foot-note]toconceive ofthecurrents which hedescribes round themolecules ofmatter, ashavingaphysicalexist- ence. The idea ofanelectro-magnetiswhatnaturally pre- sents itselfwhen weendeavour toimagine apossibleelec- tricaltheoryofterrestrial magnetism ;andthequestion which now occurs isthis :—Can themagnetic phenomenaatthe earth's surface, andaboveit,beproduced byaninternal dis- tribution ofclosedgalvaniccurrents occupyingacertain limited space below thesurface? Theansweris,thatwhatever bethe formandmagneticcontents oftheearth, thesame force asthat which itexerts upon anyexteriorpointmay actuallybepro- duced bymeans ofadistribution ofclosed electric currents onthesurface. Ihave arrived atthis result with theaidof Ampere's theoryoftheclosed circuit, bymeans ofthetheorem ofGreenalready mentioned, andbyananalogoustheorem of which aphysical demonstration maybegiven byconsiderations connected with fluid motion. Thestepsintheanalytical pro- cessofdeterminingtherequireddistribution ofclosed currents areasfollows :— • 603. LetVbethemagnetic potential, accordingtoGreen s definition, atanyexteriorpoint P;daanelement ofthesur- face;Athedistance from c^o-toP; lym,nthedirection-cosines ofthenormal atda: 470 AMathematical Theonj ofMagnetism. [xxix.. I.Findp,sothat[[^=F. II.Find (7*sothat-v-^+-^^+-j^=forinternalpoints, .jdU^ dU^dU,,.ndUand I-J-+m-j-+7i-,-=pa,tthesurface, or-y-=p,dec ay az dv III. Construct onthesurface a"mapofthevaUies ofU" Ifwires belaidalongthelinesround thesurfacecorrespond- ingtosufficientlycloseequidifferentvalues ofU,asindicated bythismap, and ifcurrents ofequal intensitybemade to circulate through them (each beingaclosedcurve), theelectro- magneticforce that will result, uponexternalpoints,willbe thesame astheforce ofterrestrial magnetism. Theexplicitsolution ofthisproblemisvery easy,when the bodyconsidered isasphere ;asisactuallythecase, toasuffi- cient degreeofapproximation,with reference totheEarth. Thus, ifthepotentialatthesurface begiven bytheequation F=F.+F,+r3+etc., whereY^,Y^,etc.,maybecalculated foranylatitude, bymeans oftheGaussian constants[andadenote theradius ofthe spherical surface], wereadilyfind[Thomson andTait's Natural Philosophy, App.B.(52)] Hence wehave themeans ofconstructinganelectro-magnetic model oftheearth, which would exhibit allthepeculiarities that canbeexpressedinamapconstructed uponGauss's theory. * {Note, Jan. 17,1872.—This function issuch that itssurface vahie is equal tothesuperficialfunction Pof§579,multiplied by47r.] I.,.._... I^PChapter X.Magnetic Induction. OntheTheory ofMagneticInduction inCrystallineandNon- CrystallineSubstances. »XXX.[From thePhilosophical Magazine, March 1851.] 604. Poisson, inhismathematicaltheoryofmagneticinduc- tion,founded onthehypothesisof"magneticfluids"moveable within theinfinitelysmall"magneticelements" ofwhich he assumes magnetizablematter tobeconstituted, does notover- look thepossibilityofthesemagneticelements beingnon- sphericalandsymmetrically arrangedincrystalline matter; andheremarks, thatafinitespherical portionofsuch asub- stance would, when intheneighbourhoodofamagnet,act differently accordingtothedifferentpositionsintowhich it mio^ht beturned with itscentre held fixed*. But"such a circumstance nothaving yetbeen observed-[-,"heexcludes the consideration ofthestructure which would lead toitfrom his researches, andconfines himself inhistheoryofmagneticin- duction tothecase ofmatter, consistingeither ofspherical magnetic elements, orofnon-symmetrically disposedelements ofanyforms. Itiseasytoconceive themodification which he would have introduced into hisformulae tomake themapplic- able toacrystallinestructure such ashedescribes;but, so farasIamaware, nowriter hashithertoattemptedtomake thisextension ofPoisson's mathematicaltheoryofmagnetic induction. Now, however, when arecentdiscoveryofPliicker's hasestablished thevery circumstance, theobservation ofwhich waswantingtoinduce Poisson toenter uponafulltreatment ofthesubject,theimportanceofworkingoutamathematical * ["Thesubstance ofahomogeneoussolid iscalled isotropic when aspheri- "cal portionofittested byanyphysical agency exhibits nodifference in "quality however itisturned. Or,which amounts tothesame, acubical "portioncutfromanypositioninanisotropic body exhibits thesame qualities "relatively toeach pair ofparallel faces. Ortwoequal andsimilar portions "cutfromanypositionsinthebody notsubject tothecondition ofparallelism " (§675)areundistinguishable from oneanother. Asubstance which isnot "isotropic butexhibits differences ofqualityindifferent directions iscalled "eeolotropic." —Thompson andTait's Natural Philosophy, §676.] t"Mdmoire surleMagndtisme enMouvement." {Mem. deVInstitut, 1823, vol. vi.Paris, 1827.) Forquotations from thisandthetwopreceding memoirs ofPoisson, showing histheoretical anticipation ofthediscovery ofmagne- crystallic action, seetheAppendixtothis article. 472 AMathematical Theory ofMagnetism. [xxx. theoryofmagneticinduction isobvious. Ontheother hand inthepresentstate ofscience, notheory founded onPoisson's hypothesisof"twomagneticfluids" moveable inthe"mag- netic elements"could besatisfactory,asitisgenerallyadmitted thatthetruth ofanysuchhypothesisisextremely improbable. Hence itisatpresentdesirable thatacomplete theoryofmag- netic induction incrystallineornon-crystallinematter should beestablishedindependentlyofanyhypothesisofmagnetic fluids, and,ifpossible, uponapurely experimentalfoundation. With thisobject,Ihave endeavoured todetach thehypothesis ofmagneticfluids from Poisson'stheory, and tosubstitute elementary principlesdeducible from itasthefoundation ofa mathematical theoryidentical with Poisson's inallsubstantial conclusions. Inthepresent communication Ishall state these principles,andpointoutwhat modifications ofthemmaybe required byamorecomplete experimental investigationofthe subjectthan hasyetbeenmade; and, adopting them tem- porarilyasaxioms ofmagnetic induction, Ishallgivean account ofsomeimportant practicalconclusions deduced from them, bymathematicalreasoning which Iproposetopublish onafuture occasion. Someexplanations and definitions areprefixedtoshow the significationinwhich certainextremely convenient terms and expressions, occasionally employed byFaraday andother writers, willbeused inwhat follows. 605.Definition.—Theforceatanypoint due toamagnetis theforcewhich itwould exert onthenorthpoleofaninfinitely thin, uniformly andlongitudinally magnetizedbar ofunit strength placedatthatpoint*,ifitexperienced noinductive action from thelattermagnet. Definition.—The totalmagnetic force atanypointistheforce *"Iftwo infinitely thin bars beequally, andeach uniformly andlongi- tudinally, magnetized, andif,when anendofone isplaced ataunit ofdis- tance fromanendoftheother, themutual force between these ends isunity, themagnetic strength ofeach isunity." {Philosophical Magazine, Oct,1850, pp.241, 242.) The definition ofmagnetic force inthetext willagree pre- cisely with thedefinition of"magnetic force inabsolute measure"adopted bytheRoyal Society, in its"Instructions formaking observations on terrestrial magnetism," if,inthedefinition ofaunit bar, theunit oflength understood beone foot,andtheunit offorce, aforce which, ifacting on agrain ofmatter, would inonesecond oftime generate onefootpersecond ofvelocity. (See Admiralty Manual ofScientific Inquiry, pp. 16,33,37.) I.] MagneticInduction. 473 which thenorthpoleofaunitbar-magnet wouldexperience from allmagnets which exertanysensible action onit,ifit producednoinductive action onanymagnetorotherbody. Or, The totalnnagneUc forceatanypointisthequotientobtained bydividingtheforceexperienced byeitherpole, placedatthat point,ofaninfinitelythin bar,uniformly andlongitudinally magnetizedtoafinitedegreeofintensity, bytheinfinitely small numerical measure ofthemagnetic strengthofthebar; and itsdirection isthat oftheforceexperienced bythenorth poleofthebar. Definition.—Any spaceatevery pointofwhich there isa finitemagneticforce iscalled "afield ofmagnetic force;" or, magnetic being understood, simply"afield offorce;" or,some- times, "amagneticfield." Definition.—A"line offorce" isalinedrawnthroughamag- netic field inthedirection oftheforce ateachpoint through which itpasses ;oralinetouched ateachpointofitselfbythe direction ofthemagneticforce. Definition.—A"uniform field ofmagneticforce" isaspace throughout which thelines offorce areparallel straight lines, andtheintensityoftheforce isuniform. Definition.—Asubstance magnetizedsothat theintensity anddirection ofmagnetizationateachpoint (§462)arerepre- sented bythediagonalofaparallelogram,ofwhich thesides representtheintensities and directions atthesamepointin twootherdistributions,issaid topossessadistribution of magnetism which istheresultant ofthese twosuperimposed, oneontheother. Itisdemonstrated byPoisson, that theforce atanypoint duetoaresultant distribution ofmagnetismistheresultant of Itmay beremarked, that this unit offorce willbethefraction-ofthe weight, inany locality, ofonegrain ofmatter,ifgdenote thevelocity acquired inonesecond byafalling bodyinthat locality ;and that itis therefore very nearly ^^~oftheweight, inanypart ofGreat Britain or Ireland, ofagrain. {Addition, May 30,1872,—Theunits ofmass andlength nowadopted arethegramme andthecentimetre. As32-2 feet isequal to 981-6 centimetres, wemay take 982asthenumber ofabsolute kinetic units offorce, intheapparent force ofgravity ononegramme ofmatter inthese latitudes.] 474 AMathematicalTheory ofMagnetism. [xxx the forces thatwould beproducedatthesamepointifth« componentdistributions existedseparately. 606. AxiomsofMagneticForce. I.Allmechanical action which amagnet experiencesii virtue ofitsmag^netism isduetoother ma<?nets *. II.Theaction betweenanytwomagnetsismutual. III.Thewhole actionexperienced byanymagnetisth( mechanical resultant oftheactions which itwouldexperience| from allthemagnetsinitsneighbourhood,ifeach acted on i asiftheothers wereremoved, thedistributions ofmagnetisn inthetworemainingunaltered. 607.LawsofMagnetic InductionaccordingtoPoisson'sTheory I.When agiven body, susceptibleofinductivemagnetizatior (whetheritbeferromagneticordiamagnetic),isplacedinthe neighbourhoodofamagnet,itbecomesmagnetizedinamannei dependent solely onthefield offorcewhich itismade tooccupy, II.Superposition ofMagneticInductions.—^Different magnets placed simultaneouslyintheneighbourhoodofaninductivel}? magnetizable (ferromagneticordiamagnetic) bodyinduce init adistribution ofmagnetismwhich istheresultant ofthe different distributions thatwould beinduced bytheseparate influences ofthedifferentmagnets,each initsownposition, with theothers removed. 608. The firstofthese twopropositions merely impliesthat anymagnet,whether anelectro-magnet,oramagnet consistingof magnetized substance, whichproducesateachpointofacertain spacethesame"force"asanother magnetofanykind, would producethesame inductive effect onamagnetizablesubstance occupyingthatspace. Everythingthat isknown ofinductive action isconsistent with it;and itis,Ibelieve, universally admitted asanaxiomaticprinciple. 609. Thesecondproposition, which asserts themutual inde- pendenceofsuperimposed magnetic inductions, isequivalentto anassertion that, iftheforce atevery pointofamagneticfield bealtered inacertain ratio, themagnetizationofasubstance placedinitwillbealteredproportionately.This isundoubtedly *Thisprinciple appears, from hisdiscovery that thephenomenaofterres- trialmagnetism areproduced bytheearth acting asagreat magnet, tohave been firstrecognised byGilbert. V I ] Magnetic Induction. 475 notaprincipleofuniversalapplication.Itisnotapplicable tosteel, nortothesubstances ofwhich natural magnetsare composed; nor, ingeneral,tosubstancespossessinginany degreethatpropertyofresisting magnetizationordemagnetiza- tion, called byPoisson"coercive force," invirtue ofwhichthey canpermanentlyretain magnetism. Neither isit,asJoule's experiments,andthemore recentexperimentsofGartenhauser and Miiller demonstrate, applicabletosoft iron, exceptasan approximatelawofthemagnetization when themagnetizing force doesnotexceed certain limits ofintensity. But, that it isveryapproximately,ifnotrigorously,fulfilled inthemagneti- zation ofallhomogeneoussubstances ofveryfeeble inductive capacit)^,anddestitute of"coercive force"(asallknown diamag- netics and allferromagneticswhich contain noiron ornickel, oronlyverysmallproportionsinchemical combination, appear tobe), is,Ithink, extremely probable. Thefoundation ofa complete theoryofmagneticinductionrequiresanexperimental investigationofthelawsaccordingtowhich the"coercive force"actsinvarious substances, andofthevariation ofinduc- tivecapacity producedinsoftiron,and itmaybeinother sub- stances, byactualmagnetization. Thefollowing conclusions, being mathematical deductions from thelaws stated above, are liable tomodification, accordingtothedeviations from those lawswhich actualexperiments maypointout :— 610. 1.Thedetermination oftheconditions ofmagnetic induction inabodyofanykind inanycircumstances maybe made todependonaknowledgeofthe state ofmagnetization induced inahomogeneous sphereofthesame substance, placed inauniform field ofmagneticforce. 2.Ahomogeneous sphereofanysubstance placedina uniform field offorcebecomes uniformly magnetizedinparallel lines withanintensity which isindependentoftheradius of thesphere. [Toprove this,imagineauniformly magnetized sphereof substancehavinginfinite "coercive power."Letaspherical portion beremoved from itsinterior. The resultant force atanypointinthehollow willbe(§§479,473)thatdue to" imaginary magneticmatter"orfreepolarity,asitmaybe properly called, ontheouter andinnersphericalsurface bound- I 4)76 AMathematical Theory ofMagnetism. [xxx ingthemagnetized matter which isleft. Thesurface density ofthepolarityatanypointofeither surface willbeequalt( icos6,ifidenote theintensityofthemagnetization and6the angle between thedirection ofmagnetizationandtheradium throughthepointconsidered. The distribution ononealone o thespherical surfaces, accordingtoaveryelementaryresult o spherical analysisstated above inafoot-note on§479(andprovec intheappended foot-note*),isparalleltothedirection ofmag- *Tofindtheresultant duetoonesuch distribution ofmatter onaspheri calsurface, imaginefirst asolid material globe ofuniform volume- density f throughout. ByNewton's theorems fortheattraction ofauniformspheri- calmass, acting according tohislaw oftheinverse square ofthe dis- tance, theresultant force atanypoint within thesubstance willbetowards thecentre, andequal to-~multiplied bythedistance oftheattracted point from thecentre oftheglobe. Consider nowtwoequal globes, oneofuni- form positive matter andtheother ofuniform negative matter ofthesame density, theformer repelling andthe latter attracting aunit ofpositive matter (asintheelectric andmagnetic applications oftheNewtonianlaw). Letthem beplaced with their centres Cand C",atanydistance dpart less than thesum oftheir radii, and firstimagine their materials toco-exist in thespacecommon tothetwospherical volumes, each acting asiftheother were away. The resultant force AtanypointPwithin this space willbe found bycompounding aforce equal to—^CPwith aforce—^C'P, in thedirection fromPtowards C",and therefore, according totheparallelo- gram offorces, willbeinthedirection PDparallel toCC,andwillbeequal to-^CC. This(asthepositive andnegative matters inthespacecommon tothetwospheres neutralize oneanother)istherefore theresultant force at P,duetouniform distribution ofpositive andnegative matter inthetwo meniscuses formed bythenon-coincident portions ofthetwospheres. Now letCCbecomeinfinitely small, andpinfinitely large, anddenote byithe product pCC, which wemaysuppose tohaveanyvalue weplease. Thetwo meniscuses become acontinuous superficialdistribution ofmatter over a single spherical surface, having forsurface-density rcos0, atanypoint where theinclination ofthenormal tothediameter through CC isd.The re- sultant force isparallel tothisdiameter and ofconstant value equal to-=- o throughout theentire spherical space.Asimilar investigation gives theresultant magnetic force atanypointin theinterior ofauniformly magnetized ellipsoid ;butinthiscase itiscon- venient toconsider components ofmagnetization andofforce inthedirec- tions ofthethree principal axes. Thus ifa, /S,7bethecomponentsof magnetization, and X,^,Zthecomponents ofthemagnetic force according I .]Magnetic Susceptibility differentindifferentdirections. 477 netization, andequalto—^;andtherefore thetwobalance one o another forevery pointwithin thesupposedhollowspace. The resultant force istherefore zerothroughoutthis space. Replacing nowthemagnetizedmaterial inthehollowspace,let theuniformly magnetized hollow sphere beplacedinauniform field offeree, andinstead of"coercivepower,"letitssubstance beendowed with such inductivesusceptibilityineachpart ofit,thatbyinduction itshall remainuniformly magnetized. Themagnetizingforceactually experienced byanyspherical portionofitisthesame asifthesurrounding substance were removed. Hence differentequal spherical portionsofthe wholerequire equalinductivesusceptibilitiestokeepthem equally magnetized; and aswemay supposethesespherical portionstobeassmall asweplease,itfollows thattheinduc- tivesusceptibility must beequal throughout,andthat ifthe substance beaeolotropicitsquality must bethroughout similarly related totheforce ofthe field. Conversely,theinductive magnetization experienced byaglobeofhomogeneoussubstance devoid of"coercive power" whenplacedinauniform field of force,must beuniform and inparallel lines.] 3.Ifthespherebeofisotropic substance, thelines ofits tothepolar definition, wefind y_47r^a^_ 47ri3/3 y_47rC7 *~3'^~~3~' ^~~3~' where ^^,\^,^Cdenote thethreeelliptic integrals which appear in(6)of §23, above, each with thefactor\J{l-e'^)s/{l-e'^) retained. These expres- sionsdepend onlyontheproportions oftheaxes, andtherefore theresultant force iszero inthehollow space left,when from auniformly magnetized ellipsoid anysimilarellipsoidal portion with principal axes inthesame direc- tions isremoved. Hence thedemonstration ofthe text proves thatan ellipsoid ofhomogeneous substance, susceptible ofmagnetic induction, becomes uniformly magnetized when placed inauniform field offorce. Anobvious extension of§626, below, gives thefollowing equations fordetermining a,/3,7,thecomponents ofthemagnetization, interms ofF,G,H,thecom- ponents oftheforce ofthe field, /z, /*', /t"theprincipal susceptibilities, and {I,m,n), {V,m',n), {I",w", n")thethree principal inductive axes, all specified with reference tothedirections ofthethree-principal axes offigure (l+^/^)^a +(l+^fi\vip+fl+^fj,\ny=,m{Fl+Gm +7Iw) \ 478 AMathematical Theory ofMagnetism. [xxx. magnetizationareinthesame direction asthelines offorce inthe field intowhich itisintroduced, andtheintensityof magnetizationisequaltotheproductofaconstant(which may becalled theinductivecapacity ofthesubstance) intotheinten- sityofthemagnetizingforce. [Forobvious reasons Inowpreferadifferent definition of inductivequality ;and forthesake ofbrevityIprefertheone wordsusceptibilitytothetwo"inductivecapacity."Instead ofthepreceding definition, therefore, Ishall henceforth adopt thefollowing:— Definition1.—Themagnetic susceptibility ofanisotrojncsub- stance istheintensity ofmagnetization acquired byaninfinitely thinbarofitplaced lengthwiseinauniform fieldofunitmag- neticforce. And Iadd;— Definition2.—Themagnetic susceptibility,inanydirection of ancBolotropicsubstance isthelongitudinal component intensity of magnetization experienced byaninfinitelythinbarcutfromthe substance inthat direction, andplaced lengthwiseinauniform field ofunitforce^ 4.Ifthespherebeofcrystalline substance, thelines ofits magnetization maynotingeneral beinthesame direction as thelines offorce ofthe field intowhich itisintroduced;and theyarenotsoifthesphere, when freetoturnround itscentre, isobserved tobenotinequilibrium. 611.Definition.—Aprincipalaxisofmagneticinductionofa substance isaline init,such thataspherical portion when in- troduced, with that lineparalleltothelines offorce, intoa uniformmagnetic field,becomesmagnetizedinthedirection of those lines. Definition.—Aprincipalinductivecapacity ofasubstance, or theinductivecapacity ofasubstance inthedirectionofaprincipal axis, isthe coefificient bywhich theintensityofthemagnetiz- ingforcemust bemultipliedtoobtain theintensityofmag- netization when aspherical portionisintroduced intoauniform magnetic field, with aprincipalaxisparalleltothelines of force. i 612. 5.Anysubstance hasthrough every pointofit,three principalaxes atright anglestooneanother;and iftheindue- XXX.] Turning Motiveexperienced byCrystal. 479 tivecapacitieswith reference tothree suchaxesbedifferent, no other linethroughthesamepointisaprincipalaxis*. 6.Iftheinductivecapacitieswith reference totwoprincipal axesthrough anypointofahomogeneoussubstance beequal, everyline intheplaneofthese two, orparalleltoit,isaprin- cipal axis,andtheinductivecapacitieswith reference toall theseprincipalaxes areequal. 7.Iftheinductivecapacities with reference tothreeprincipal axesthrough anypointofasubstance beequal, everyline throughthesubstance isaprincipal axis,andtheinductive capacities with reference toalldirections areequal ;orthe substance isdestitute ofmagnecrystallic properties. 613. 8.Aspherical portionofanyhomogeneous substance, supportedinauniformmagneticfield insuch amanner that it canturnfreelyinanymanner round itscentre which isimmove- able,cannot beinequilibriumunless aprincipalaxisbeinthe direction ofthelines offorce. Ifthethreeprincipalinductive capacities beunequal,thebodywillbeinstable-f*^ equilibrium withtheprincipalaxis ofgreatestinductivecapacity,orinun- stableequilibrium with either ofthetwootherprincipal axes, in thedirection ofthelines offorce. Ifthetwo lessprincipalin- ductivecapacities beequaltooneanother, thebodywillbein stable-)-^ equilibrium withtheprincipalaxisofgreatestinductive capacityinthedirection ofthelines offorce, orinunstable 3quilibrium with thesame axisperpendiculartothelines of force. Ifthetwogreater principalinductivecapacitiesbeequal booneanother, thebodywillbeinstable Jequilibriumwith the planeofthecorresponding principalaxesparalleltothelines offorce, orinunstableequilibriumwith thatplane perpen- dicular tothelines offorce. *Such,itmay beexpected,willbethemagnetic circumstances inthecase )fanytransparent substance which belongs totheoptical class of"biaxal crystals ;"and itsthree principal axes ofmagnetic induction willbethe ihree rectangular axesdeduced bySirDavid Brewster from the"optic axes,"mdknown intheundulatory theory astheprincipal axes ofelasticity ofthe nedium inwhich theundulations arepropagated. +^'2Inonerespect theequilibrium might besaid tobeneutral rather than stable, since every position intowhich thebodymaybeturned round the stable axis isaposition ofequilibrium. XIntworespects theequilibrium might besaid tobeneutral;since every position intowhich thebodymaybeturned round thedirection ofthelines )fforce isaposition ofequilibrium, andevery position intowhich itmaybe ;urned intheplane ofthestableprincipal axes isaposition ofequilibrium. 480 AMathematicalTheory ofMagnetism. [xxx 614. 9.Ifaspherical portion,ofvolume a,ofasubstance o which thethreeprincipalinductivecapacitiesareA,B,andC beheld inauniformmagneticfieldwhere theintensityofthe force inabsolute measure isR,withthethreeprincipalaxesofin- duction inclined tothedirection oftheforce atanglesofwhicl: thecosines arerespectively Z,m,n^itwill receive astate o: magnetizationwhich istheresultant ofthree states ofuniforn: magnetization ;oneofintensity A .Rl,inthedirection ofth€ firstprincipalaxis;asecond ofintensity B.Rm,inthedirectior ofthesecondprincipalaxis;andathird, ofintensity C .Rn,ir thedirection ofthethirdprincipalaxis;and itwillexperience aturning action, ofwhich themechanical definition isacouple ofmoment a.R\{mV [B-Of+n'r{G- ^)^4-ZW{A-Bf]^...{\\ inaplaneofwhich thedirection cosines* with reference tothe threeprincipalaxes arerespectively mn(B-^C)nl{G-A) lm{A-B)D'D'D^^' whereDdenotes thesquareroot ofthesumofthesquaresoi thenumerators ofthese three fractions, orthethird factor oi thepreceding expression. 615. 10.Ifthesphere beinfinitely small, and ifitbeputinto auniform' ornon-uniform field offorce, theentire action which itexperiences,whether directivetendencyortendencytomove fromonepartofthefield toanother, isdefinedbythefollowing proposition:— Thequantityofmechanical workwhich isrequiredtobring thebodyfrom apositionwhere theintensityoftheforce isR, and itsdirection cosines with reference tothethreeprincipal inductive axes I,m,n,toaposition where theintensityofthe force isR\and itsdirection cosines with reference tothethree principalinductive axes intheirnewpositionsV,m\n\is equalto 1(7{{Ar+Bm"+Cn")R"-(AF+Bm'+Cn')R'] (3). 11. IfA=B=G,thisexpressionbecomessimply ^aA (R'^—R^),and thepropositionisequivalenttothemathe- *Orthecosines oftheinclinations ofaperpendiculartotheplane,totiM three axes.-^^ I."] Ferromagnetics andDianiagnetics. 481 ticalexpressionofFaraday'slawregardingthetendencyto placesofstrongerorofweaker force, offerromagneticordia- magnetic non-crystalline substances, onwhich some remarks [reprinted, §§647—668below]arepublishedinthePhiloso- phical MagazineforOctober 1850. 616. 12. If,without movingitscentre, theballbeturned so that itsthreeprincipalaxes shallsuccessively beinthedirection ofthelines offorce (thefieldbeing non-uniform, butthebody infinitely small),itwillineachposition experienceaforce inthe lineofmostrapidvariation ofthe*'force ofthefield;"butthe magnitudeoftheforce will ingeneraldiffer inthethreeposi- tions, being proportionaltoA,B,andGrespectively*.If *Thus aballcutoutofacrystal ofpure calcareousspar,which tends to turnwith itsoptic axisperpendicular tothelines offorce, andwhich tends asawhole fromplaces ofstronger towards places ofweaker force, would experience this latter tendency lessstrongly when theoptic axis isperpen- dicular tothelines offorce thanwhen itisparallel tothem;since, accord- ingto§612ofthetext, thecrystal must have greatest inductive capacity or (thelanguage inthetext being strictly algebraic when negative quantities ireconcerned)leastcapacityfordiamagnetic induction perpendiculartothe optic axis. Iamnotaware that this particular conclusion hasbeen verified oyanyexperimenter; but Iaminformed(Oct. 25,1850) byMrFaraday, :hathefinds apiece ofcrystalline bismuth toexperience adifferent"repul-non"according asitisheldwith itsmagnecrystallic axisalong orperpen- licular tothelines offorce inanon-uniform field;therepulsion being less intheformer casethan inthelatter, which agrees perfectly with theconclu- dons ofthe text, since, asaball ofIjismuth would tend toplaceitsmagne- jrystallic axisalong thelines offorce, that axismust, according to§612,be :heprincipal axis ofgreatest inductive capacity, or,bismuth being diamag- letic, theaxis ofleastdiamagnetic capacity. Itisright toadd, that what, according tothetheory explainedinthe :ext,must bethecorrect explanation ofthepeculiar phenomena ofmagnetic nduction depending onmagnecrystallic properties, was clearly stated inthe *orm ofaconjecture byFaraday inhis22d Series(2588)inthefollowing erms:—"Orwemight suppose that thecrystalisalittlemore aptformag- 'neticinduction, oralittle lessaptfordiamagnetic induction, inthedirec- 'tion ofthemagnecrystallic axisthan inother directions. But, ifso, it 'shouldsurely show***inthecase ofdiamagnetic bodies, asbismuth, a 'difference inthe degree ofrepulsion when presented with themagne- ' crystallic axisparallel andperpendicular tothe lines ofmagnetic force '(2552); which itdoesnotdo." (Eead before theKoyal Society, December7, 1848.) The failure ofthe firstexperiment (2552) todetect this difference of iction neednotbewonderedat,whenweconsider howminute itmust probably )e;andtheconjecture, apparently abandoned atthetimebytheauthor for vant ofexperimental support, maybeconsidered asfully established byhisown subsequent experimental researches. [The following appeared inthePhilosophical Magazine for1851, second half- •ear,under thetitle"Magnecrystallic PropertyofCalcareous Spar":—] Extract fromletter totheEditors. Glasgow College, Nov.7,1851.—****Inthepassage, asoriginally published(line4frombeginning offoot-note), theword"more" occurred inthe )lace of"less." Themistake waspointed outtomelastApril byProfessor T.E. 31 482 AMathematicalTheory ofMagnetism. [xxx each ofthesequantitiesbepositive, theforce onthe ball ii eachpositionwillbeinthedirection inwhich theforce ofth( field increases;ifanyoneofthesequantities benegative,th( forceontheballwhen thecorresponding principalaxis isinth< direction ofthelines offorce, willbeinthecontrarydirectioB orthat inwhich theforce ofthefield decreases mostrapidly. 617. 13.IfA,B,and (7beallpositive,thebodyiscalled ferro magnetic ;iftheybeallnegative,itiscalleddiamagnetic. N< substance hasasyetbeenfound tohavesome ofthequantitie A,B,Cpositive,andothersnegative. 618. 14.Iftheinductivecapacitiesbevery small, allthepre cedingconclusions willbeapplicabletotheactionsexperiencec bybodies inair(ferromagnetic),orinanymagnetizableflui( ofeitherferromagneticordiamagnetic inductivecapacity, pro vided, instead oftheabsolute inductivecapacitiesofthesub stance ineach case,weuse forA,B,and G,orforth " principalinductivecapacities"intheverbal enunciations theexcesses oftheabsoluteprincipalinductivecapacitieso thesubstance, above theinductivecapacityofthefluid. 619.Curious experiments mightbemadebymeans ofavary ingfield offorceoccupied byamagnetizable fluid,andaball c crystallinesubstance allowed tomovefreelyinthelineofmos rapidvariation oftheforce. Iftheinductivecapacity (whethe positiveornegative)ofthefluidbeintermediate between th Stokes, andIimmediately requested youtocorrectit,which youaccordinglydi byanintimation inthe"Errata." When theperplexity occasioned bytb mistake isremoved,itisobvious toanyonereading thepassage carefully, thf themistake itselfwasonlyaslipofthepen, asattheconclusion ofthesentenc itisasserted thatacrystal ofpure calcareous sparmusthave the''leastcapacit fordiamagnetic induction, perpendicular totheopticaxis," This conclusion isverified byDrTyndall, who describes experiments, in paper publishedinyour September Number, bywhich itappears thatthe dit magneticinductive capacityofcalcareous sparinadirection parallel tott opticaxis istoitsdiamagnetic inductive capacity perpendiculartotheopticax. as57to51.— Iremain, gentlemen, your obedient servant, William Thomson. [Wehave also received acommunication onthis subject fromMrTyndal who inreference toanote received byhimfrom Prof. Thomson, writes i follows:—"Ihave only tosaythat thefacts areprecisely what they arehei "stated tobe.Previous towriting theremarks inquestion, Ilooked to tl: "Errata, butnot itseems with sufficient attention, forProfessor Thomson's co: "rection escaped me.Notonlydoourresults agreeinprinciple, butthesaa '*substance andform ofsubstance which Professor Thomson hadreferred to i "illustration ofhistheory wasunwittingly examined bymeinBerlin, and tl «'exact result which hehadtheoretically predictedarrived atbywayofexper'ment."—Edit,] ^Hx.]Poissons Anticipation ofMagnecrystallic Quality. 483 greatestandtheleast oftheabsoluteprincipalinductivecapa- cities ofthesubstances, theball willbeurgedfromplacesof weaker towards placesofstrongerforcewhen itsaxis of greatestinductive capacityisplaced alongthelines offorce, andinthecontrarydirection when theaxis ofleast inductive capacityisplacedinthesame direction. Itwould beeasytoadjustthestrengthofasolution ofsul- phateofiron soastosatisfythiscondition foraferromagnetic crystalline substance; butthere mightbegreat difficultyin demonstrating byexperimenttheexistence ofthe forces, on account oftheir feebleness. Appendix. . Quotations fromPoissonregarding MagnecrystallicAction. 620."laforme desClemenspourraaussi influer "surcette intensity;etcette influence aura celadeparticulier, " qu'elleneserapaslameme endessens differens. Supposons, J'*parexemple, queles^l^mens magnetiquessontdesellipsoides "dont lesaxes ont lam^me direction dans toute I'etendue "d'unmemecorps,etquececorpsestunesphereaimanteepar "influence, danslaquellelaforce coercitive estnulle; les ;"attractions ourepulsions qu'elleexercera au-dehors seront i"differentes dans lesensdesaxes deseselemens etdans tout f'*autre sens;ensorteque,siTon faittourner cettespheresur 1'-^elle-meme, sonaction surunmemepoint changera,eng^n^ral, \'engrandeuretendirection :mais, sileselemensmagnetiques r'sontdesspheresdediametres egaux ouin^gaux,oubien s'ils *''s'^cartent delaformespherique,maisqu'ilssoientdisposes \''sansaucuner^gularitedans I'interieur d'uncorps aimantepar \'influence, leurs formes n'influerontplussurlesresultatsqui J"d^pendrontseulement delasomme deleursvolumes, comparde \"auvolume entier dececorps*,etquiseront alors lesmemos en f'•'tout sens. Cedernier casestceluiduferforg^,etsansdoute i"aussi desautrescorpsnon cristallises danslesquelsona ;'observe lemagn^tisme:mais ilserait curieux dechercher si i'lepremiercasn'auraitpaslieulorsquecessubstances sont [Thiserrorwascorrected byPoisson himself inasubsequent memoir.] 31—2 484 AMathematicalTheory ofMagnetism. [xx: "cristallisees;onpourraits'enassurer parFexp^rience,soit e " approchant"ancristal d'uneaiguille aimantee, librement su; "pendue,soitenfaisant osciller depetites aiguillestailldc "dans descristaux entoute sorte desens etsoumises aTactic "d'un tres fortaimant."—Pp.258,259,M^moire surlaTh^or duMagnetisme, parM.Poisson. LuaI'Acad^mie desScience le2Fevrier, 1824. Mem, deVInst 1821-22. Paris, 1826. "laforme deselemens etleurspositions parrappoi **auxplansfixes descoordonn^es x,y,z,peuventinfluer si "I'etat magnetiquedeJ.,etsurlesattractions our^pulsior " qu'ilexerce audehors. IIpourrait memo arriver quecett "influence nefutpaslameme entout sens,ensorteque,si . "^taitunesphere homogene,etqu'onfittourner cecorpssai " d^placersoncentre etsans rienchanger auxforces ext^rieure "oualafonction F,lesactionsmagnetiques deAchangeraier "neanmoins engrandeuretendirection. Cecassingulie"quenousavonsdejaindiquedans lepreambule deceMemoir* "nes'^tantpasencorepresenteaI'observation, nous I'excluror "denosrecherches, quant apresent,etnous aliens, enconst "quence,determiner lesrelationsquidoivent exister entre q "/3', 7'*,etlesquantites a,,/3^,xt,pour qu'iln'aitpaslieu —Ibid.p.278. 621. Thefollowing explanation mayserve togiveanidea ( Poisson's mode oftreatingthesubjectofthelastquotation, an toshow therelation itbears tothetheoryofwhich anoutlin hasbeengivenabove. Asphereofanyhomogeneous magnetizablesubstance bein placedinauniform field offorce, intensity R,letthedirectio oftheforcemakeangles whose cosines areI,m,nwith thre rectangularaxes fixedrelativelytothesubstance;and let { P,7bethecomponentsoftheinducedmagnetization.Poisso deduces, from hishypothesisofmagnetic fluids, equations *Componentintensities ofmagnetization. tComponentsofthemagnetizing force. XTheproductsofthe firstmembers ofPoisson's three equations inp.2'J ofhisfirstMemoire, intok,theratio ofthesum ofthevolumes ofthemagnet elements tothewhole volume ofthebody,arerespectively equal tothe thrt components oftheintensityofmagnetization (a,^,7);and if^,B,etc.,I taken todenote thevalues oftheproductsofkintoPoisson's coefficients P, i, etc., respectively, theequations inthetextcoincide with those ofPoisson, cxx.] TheoremofPrincipal Axes Demonstrated. 485 ;hareequivalenttothefollowing:— a={Al+Fm+C"n)B\ p^{A"l +Bm+C'n)R\(4), 7={A'l-YB"m+ Cn)R J reA,B,etc.,arecoefficients depending solelyonthenature )fthesubstance. Theseequationsarededucible from the Lxioms andthehypothetical principleofthesuperpositionof }nagnetic inductions, stated above, without thenecessityof eferringatalltothehypothesisof"fluids." Allthatremains i)fPoisson'stheoryisconfined tothecase ofnon-crystalline 'natter, with reference towhich itisprovedthatA,B,andG nustbeequaltooneanother, andthateach oftheother six I'-oefficients must vanish; andthere isnothingtoindicate the )ossibilityofestablishing anyrelations amongthenine co- efficients which must hold formatter ingeneral.Ihave found hatthefollowing relations, reducingthenumber ofindependent ;oefficients from nine tosix,must befulfilled, whatever bethe lature ofthesubstance:— 5"=C;C"=A\A"=B'(5), hedemonstration[added below, §622] beingfounded onno mcertain orspecial hypothesis,butontheprinciplethat a phereofmatter ofanykind, placedinauniform field offorce, .ndmade toturnround anaxis fixedperpendiculartothelines >fforce, cannot beaninexhaustible source ofmechanical effect. Ultheconclusions with reference tomagnecrystallicaction enun- iated intheprecedingabstract arefounded onthese relations. [622. Demonstration:January1872.—Because the field of orce isuniform thedynamicalactionexperienced bythemag- letizedsphereifofunitvolume consists simplyofacouple §499)whosecomponentsare (0n-ym)B, {yl-an)B, {am- ^l)R (6), xpressions which show thattheaxis oftheresultantcoupleis )erpendicular to{I,m,n).Nowrememberingthattheaxes of o-ordinates arefixedrelativelytothesubstance, supposeitto )eturned, carrying YandOZwith itround theaxisOX, hrough aninfinitesimalangle dcj)-,and let </>deaote theangle •etween theplaneYOX andtheplaneofOXand(I,m,n). 486 AMathematical Theory ofMagnetism. [xxx Thework donebythemagnetizedsubstanceduringthismotioi willbe (my- 7i}3)Rdcj) (7), which, ifweputI—cos6,m—sin cos(j>,n=sin6sin<^,anc use(4),becomes JJ{sin^cose{A'cos<p-A"sin(p)+sin^d[{C-B)sin cos+B"cos^ -C"sin20]}d0...(8) Integratingthisexpression from (/>=to=27r,wefind fortheintegral amount ofwork done duringarevolution rounc OX. But thismust bezero, foravoidance ofthe"perpetua motion*," since thebodyisbrought back toitsprimitive positioi andphysicalcondition attheendofthemotion; andtherefon B"=C.Similarly, byturningthebodyonceround the axii OY,weprovethatG"—A',andbyturningitroundOZw( provethatA"=B'.Thus areestablished thethree relation; between theco-efficientsexpressed byequations (5)above. 623. Tofindasymmetrical expressionforthework done ii anyinfinitesimal rotation, remark thatwhen Iisconstant w( have ,,_c?m_dn ^nm' Hence (my—n/3) dcj)=ydn+^dm. Hence by(7)andcorresponding expressionsforthework don( ininfinitesimal rotations, round YandOZ,wefind forth( whole work, dQ,donebyanyinfinitesimal rotation whatever dQ=R{adl+^dn-\-ydm) (9). Usinginthis fora,/3,y,theirexpressions by(4),aslinea: functions of7,m,n,andlookingtotherelations(5)establishec between the coefficients, weseethatdQis a,completediffer ential ofaquadraticfunction ofI,m,n,asifthese were three independentvariables;andtherefore byintegration Q=l{Ar+Bm""+On''+2amw+"ihnl+2c?m)R\ ..(10), wherea,6,cdenoterespectivelythevalue ofeither members o thethreeequations (5).Hence bydifferentiation andcompari sonwith(4),. ^_\_dq r._ldQ ^_1^ n.^ ""'Bdr ^~Rdm' ^"Rdn^^^' andQ=l{al-\-l3m-^yn)R (12). *Seebelow, §670,footnote. XXXI.] Magnetic Permeability andAnalogues. 487 This isnecessarily equaltotheexhaustion ofenergy (Thomson nidTait's NaturalPhilosophy, §549)inlettingtheglobule come from anyplaceofzeromagnetic force, into itsactual positioninthesupposed magneticfield. Compare §732,and §722(70) bis,and§503(2). 624.Theelementary theoryofthetransformation ofquad- ratic functions shows how,when A,B,G,a,b,careknown for anyonesetofthreerectangularaxes inthesubstance, wecan finddeterminately byaidofthesolution ofacubicequation,a setofthreerectangularaxessuch that ifwetakethem foraxes ofX,Y,Z,thecoefficients ofm7i, nl,Imwillvanish inthe transformedquadratic function, andweshould havesimply Q=l{Ar+Bm'+Cn')E' (13), and OL^AIR, ^=BmR, y=CnR (14). Hence thepropositionsof§§612, 613, 614.] XXXI. Magnetic Permeability, andAnaloguesinElectro-static Induction, ConductionofHeat, andFluid Motion. March 1872. 625. Supposingthe coefficients A,B,C,and a,b,cof §§621—624, (5)and(10),tobeknown foraparticularset ofaxes inasubstancesusceptibleofmagnetic induction, letit berequiredtofind itssusceptibilityformagnetizationinany givendirection. Letasphereofthesubstance beplacedina uniform field offorcehaving components F,G,Hparallelto theaxes ofco-ordinates. By§623(11)wehave forthecom- ponentsofmagnetization a^AF-\-cG +bH\ l3==cF+BG+aH\ (1); y=bF+aG-{-CH\ anddenoting byitheintensityoftheresultantmagnetization, andI,m,nitsdirection-cosines, ^=V(a^+/3^+7') (2), ^=«,^=^, n=l(3). Conceive nowaninfinitelythin barofthesubstance, ofany length alongthelines ofmagnetization,toberemoved. The agneticforce inthehollowspacewillbecompoundedofthe I 488 AMathematicalTheory ofMagnetism. [xxxi. force ofthe field{F,G,H)andtheforceduetothefreesurface- polarityofthesphere ;andtherefore(§610, 2,foot-note)ifwe denotebyX,Y,Zitscomponents, wehave o o o Itisthiswhich isthemagnetizingforceactually experienced bythebarinitspositionaspartofthesphere. Themagnet- ization induced byitisofintensity V(a^+/3^+7^),and isin thedirection ofthebar'slength. Hence themagnetic suscep- tibilityofthesubstance inthedirection(3)ofthisbar is ^/(X'+Y' +Z')^^^• Tofindthemagnetic susceptibilityinanydirection{I,m,n) explicitlyinterms oiI,m,nandtheco-efficients A,B,G, a,b,c, allthat isnecessaryistoeliminatea,/S,7,X,F,Z,F,(r,H from(0)bymeans ofthenineequations (1), (3), (4).The algebraic process requiredinvolvesonlythesolution ofthe three linearequations (1)forF,G,H.Thesimplifiedsolution giveninthefollowingsection mayberegardedasalgebraically equivalenttoanexpressionoftheprecedingdirect solution in terms ofsymmetricalfunctions oftheroots ofacubicequation. 626. Tosimplifylettheaxes ofco-ordinates bechosen in thedirection ofthethreeprincipalaxes(§611)ofmagnetic susceptibility.Thismakes a=0,6=0,c=0,andwehave a=AF, ^=BG,y=CH(6), X=(l-i^)i^, F=(l-^)(^, Z={i-'^^)e(7). Hence by(5), (3),and(2)wehave, forthemagnetic suscepti- bilityinthedirectionI,m,n, ^ (8), ^3 3^3 627. The coefficients denoted in(9)byX,fi,varethethree principal magnetic susceptibilities,asweseebyconsidering thecases inwhich(I,m,n)coincides with theaxes ofco- L,] Magnetic Permeahility andAnalogues. 489 ordinates. Byequations (9),conversely,fortheinductive mag- netization ofasphere when itsprincipal susceptibilities \, //-,v aregiven, wefind ^=_4_, B=—^, C=-_4_...(10). 1+T^l+T'' ^^T" 628. IntheexpositionofFaraday's greatelectro -static discovery, givenabove(§§36—50),Ipointedoutaperfectly closeanalogy between themathematical theories oftheelectro- polarinduction which hefound tobeexperienced byinsulators inafield ofelectric force, oftheinductivemagnetizationof ferromagnetics, air,anddiamagnetics, and oftheconduction ofheatthroughaheterogeneoussolid. Thisvolume willend with afourthanalogy (§§751—763, below),inwhich itwill beshown thatpreciselythesame lawsandmathematical ex- pressionsareapplicabletotheflow ofafrictionless incom- pressible liquid, throughaporoussolid ofinfinitelyfinetexture, when themotion oftheliquidisthroughoutirrotational(or such asmaybeproducedfrom restbyanymotiongiventothe boundaryoftheliquid). Thesingularcombination ofmathe- matical acuteness, withexperimentalresearch andprofound physical speculation,whichFaraday, thoughnota"mathe- matician," presented,isremarkablyillustrated byhisuseof theexpression, conducting power ofamagnetic medium forlines offorce,referred tointhefoot-note to§44,above. Theana- logue correspondingtoconducting powerofasolid forheat, or, asitisshortly called, "thermalconductivity," is,inelectro- static induction, the" specificinductive capacity"ofthe di-electric; inmagnetismitisnotwhat hashitherto been calledmagnetic inductivecapacity,—aqualitywhich isnegative indiamagnetics, but itisFaraday's"conducting powerfor lines offorce;" and inhydrokineticsitis(§753,below)flux perimit area, perunitintensityofenergy. Thecommon word "permeability" seems welladaptedtoexpressthespecific qualityineach ofthefouranalogous subjects. Adoptingit wehave thermalpermeability,asynonymforthermal con- ductivity; permeabilityforlines ofelectric force, asynonym fortheelectro-static inductivecapacityofaninsulator; mag- neticpermeability,asynonymforconducting powerforlines 490 AMathematicalTheory ofMagnetism. [xxxi. ofmagneticforce;andhydrokinetic permeability,aname for thespecific qualityofaporous solid, accordingtowhich, when placedinamovingfrictionlessliquid,itmodifies theflow. 629.Tofindtherelation between what hasbeen called above magnetic susceptibility andmagnetic permeability,consider a bodywithnointrinsicmagnetization (§698,below) surrounded byairinamagneticfield. LetAbeanyinfinitesimal area of itssurfacecutting perpendicularly oneofthethreeprincipal inductive axes ofthesubstance initsneighbourhood.Let^ bethenormalcomponentofthemagnetization induced inthe substanceinfinitelynearA;and letiV",N'bethevalues of thenormal componentforce atexternal andinternalpoints infinitelynearA,thelatteraccordingtothepolardefinition (§517, Postscript). Wehave[§473(1),and§7] N'=N-4<7r^(11). Letnowfjubethemagnetic susceptibilityinthedirection ofthe normal, sothat(§610, 3,definition2)wehave ^=/^i\^' (12). Eliminating ^from thisby(11),wehave iV'=N—^TTfiN' , andtherefore N^ , ,^„. -^=l+47r/^ (13). Hence(compare §44,above)1+47r/xisthemagnetic permea- bilityofthesubstance inthedirection ofitsprincipalaxis perpendiculartoA.Thusweseethat ifjju, fju,ix'denote the threeprincipal magnetic susceptibilitiesofasubstance, and -57, -st',-cr"itsprincipal magnetic permeabilities, wehave OT=1+47r/^,ot'=1+47r/,ot''=1+47r/i" (14). 630. Experiment hashithertogiven but little accurate know- ledgeofthemagnetic susceptibilitiesofdifferent substances. Comparisonsofthesusceptibilitiesofdiamagneticsand feeble ferromagneticswith oneanother andwith that ofironhave beenattempted ;buttheonlydetermination inabsolute measure hitherto made orevenattemptedisthat ofThalen* foriron. Hefound themagnetic susceptibilitiesofdifferentspecimens tobeverydifferent. Thegreatest susceptibilitywhich hefound *"Kecherches surlesproprietes magnetiques dufer."' ParT.E.Thalen. Extrait desactes delaSooi^te Eoyale desSciences d'Upsal.Serie iii*.T. iv.Upsal, 1861. WrajXI.] Magnetic Permeability andAnalogues.491 asinsome specimensofthebest soft iron,andamounted to about 45."Coercive force/' thelaws ofwhich areatpresent wholly unknown, exists toagreat degreeinallvarieties of ironand steel, includingthesoftest iron;andvariesverymuch inthesamespecimenwith itsstate oftemper.Itcomplicates excessively every investigation regardingtheinductivequali- tiesofironand steel. Ontheother hand(and particularly now thattheBritish Association hasgiventoexperimenters standards ofelectric resistance inabsoluteelectro-magnetic measure*, andimportantcontributions towards thegeneral practiceoftheabsolutesystem)itisavery easy thingto measure, withsomedegreeofaccuracy,theabsolute value of theinductivequalityofsubstances destitute ofcoercive force. (Allfluids arenecessarilyso;and, asstated in§609, itispro- bable that alldiamagnetics, and allhomogeneous substances offeebleferromagnetic quality,arenearly so.)Asyetnosuch measurement hasbeenmade, but itistobehopedthat before longsomeexperimenterwilltakeupthesubject. 631. Thal^n's number, 45,gives, accordingto(14),1+47rx45, orabout 566forthepermeabilityofthebest softiron. Ithas been stated thattheinductivesusceptibilityofcobalt isgreater than that ofsoftiron,butthisseems tobebynomeanscertain; andIbelieve itiscertain that allother substances hitherto experimented onarelesssusceptiblethan iron. Thepermea- bilities ofallferromagnetics exceedunity,butonlybyvery small fractions, exceptthefew so-called magnetic metals, or substancescontaining them inlarge proportion.Itisalso remarkable thatnosubstance hasbeen discovered forwhich thepermeabilityfalls short ofunity bymore than avery minute fraction, asisshown bytheextreme feebleness ofthe forces due todiamagneticinduction inallcases which have been hitherto observed. Ifweknew something instead of nothingofthemolecular theoryofmagnetic induction, we shouldprobablyseethatthepermeabilityofevery substance must bepositive. *British Association Committee onElectric Measurement, appointed first intheyear 1860, andreappointed after thatfrom year toyear.Areprint ofitssuccessive Eeports collected isbeing made bytheCommittee, with permission oftheCouncil oftheBritish Association, and willsoon beready forpubhcation inaseparate form. [Pubhshed in1873byE.andF.N.Spon, London, under the title of"ReportsofElectrical Standards," edited byProf. F.Jenkin, F.R.S., LL.D.] 492 AMathematicalTheory ofMagnetism. [xxxil. XXXII.Diagrams ofLinesofForce ;toillustrate Magnetic Permeability. [May 29,1872.] 632.The differentialequationforlines offorce invoidspace resulting from theNewtonian law isalways integrable when thedistribution issymmetrical round anaxis, aswas first shown inanarticle "On theEquationsofMotion ofHeat referred toCurvilinear Co-ordinates"intheCambridge Mathe- matical Journal, Nov.1843[Art.ix.ofmy" EeprintofMathe- matical andPhysical Papers,"Vol. I.University Press, Cam- bridge, 1882.]thus :—Inthecase ofsymmetry round anaxis, take forco-ordinates xalongtheaxis ofsymmetry, andyper- pendiculartoitinanyplane throughit.LaplaceandPoisson's equation becomes d'V d'VIdV,^^+^+y^="^^^- Thereforethroughvoidspace, d'Vd'VIdV^^ dx^dy"^ ydy The differential equationofthelines offorce is dV. dV. --^ax—j-dy=0. ay dx^ This, invirtue of(1),isrenderedintegrable bythefactory,and therefore theintegral equationofthelines offorce is -^/r=const.,) ^^"'"/(2^^^^"^£^^)=W^' Forexamplelet V=—^^ 3,-Fx(3), {x'+ff^ sothat thedistribution offorce isthat ofauniform field, of intensity Fydisturbed bythepresenceofaninfinitesimal magnet,ofmagnetic moment/a,placed with itsmagneticaxis paralleltothelines oftheundisturbed force.Wefind ^=r^^+4^2/^ W' (^+2/T which,ifweput-~=a^,and~-=b^(5), .2_7.2 ay gives f=h'-—-^-~(6); (x'+yy or,resolved fora?,^^\{w:_r^j~y^\C^)* Ibii.]nacccExamples ofLinesofForce. 493 1naccount ofthedoublesignoftheradical in(6)wemay, without lossofgenerality, suppose aalways positive ;andthe branches ofthecurvescorrespondingtonegativevalues ofthe radical willthencorrespondtothecase inwhich themagnet isplacedinthepositioninwhich, ifitwererigidly magnetized, and freetoturn,itsequilibrium would beunstable. Inthese branches, which forbrevitywillbecalledexflected, y^isevery- wheregreaterthan6^;while inthebranchescorrespondingto magnet placedinpositionofstableequilibrium, which will 'ecalledinflected, 'ifiseverywherelessthan 6^Ofthe Eadius ofCircle=a. Fm. 1. annexed woodcuts*, fig.1representstheentire series ofboth setsofbranches forallpositivevalues of6^;fig.2thewhole series ofinflected branches;fig.3thewhole series ofexflected branches; andfigs. 4,5,6,7selections from thetwo sets to illustrate inductive influences ofsphericalbodies ofvarious qualities, placedinauniform current ofincompressiblefriction- lessliquid,orinuniform fields ofelectric ormagneticforce. *From photographsoflarge-scale diagrams calculated from equation (7), anddrawn fortheNatural Philosophy Class intheUniversity ofGlasgow about twenty-three years agobyMr.D.Macfarlane, toillustrate fluidmotion andtheallied subjects ofphysical mathematics, 494 AMathematical Theory ofMagnetism. [xxxil. Thetwodouble pomtsshown infigs. 1,2,and4correspondto Fm. 2. thepairsofequalroots2/=3-^>2/=~"17^>which thetwo quintics Eadius ofCircle=a. Fig. 3. V3havewhen h=-^=1*375.Acircle{fig. 4)described from the originascentrethroughthese doublepoints,and therefore IExamples ofLines ofForce. 495 having -^^forradius, cutsperpendicularlyeach oftheinflected curves, excepttheonegiven by6=—^,which itcutsthrough thedoublepointsatanglesof+tan"' l-6Xa Fig. 5. 496 AMathematicalTheory ofMagnetism. [xxxii. Oftheexflected curves(fig. 3),thatgiven by6=consists ofacircle ofradius a,havingitscentre attheorigin, together Radius ofCircle =a. a FlQ. with thepartsoftheaxis ofxexternal tothat circle, each doubled;orthesamecircle, together with thepartoftheaxis ofXwithinit,doubled. a Fig. 7. IHIiT.] Examples ofLines ofForce. 407 Fig.4representsthelines ofelectric force intheneighbour- hood ofanunchargedinsulated metalglobe placedinauniform field ofelectric force. Italsorepresents (§631)without sensible distinction thelines ofmagneticforce intheneighbourhoodof aglobeofsoftiron inauniform magneticfield.Fig.6repre- sents thestream lines ofafrictionlessincompressible liquid passingafixedsphericalobstacle. 633. Toinvestigatetherelation ofthelines offorce inthe neighbourhoodofasolidglobeofanyferromagneticordia- magnetic homogeneousmaterial destitute ofintrinsic magnet- ism,putintoauniformmagnetic field, with oneofthethree principalaxes(§611)ifthesubstance benotisotropic, placed paralleltothelines offorce :—Let -crbethepermeabilityofthe substance(§629)andrtheradius oftheglobe.Theinduced magnetization being (§610) uniform, andparalleltothelines offorce ofthe field, itsactionthroughexternalspacewill (§610, foot-note) bethesame asthat ofaninfinitelysmall magnetatitscentre. Hence, usingthenotation of(5)in(3), andinstead ofadmittingthenegative signforthe radical, takingtheproper diamagneticformula byitself,wehave V^^F\ 7-7—23-2^I (ferromagnetic) (external) ^ ^!^_\ff]^^ > ^=i-'^\{7^'']i~n(<Jiamagnetic)...(8), forthepotentialinexternal spaceduetothemagnetismofthe ^lobeandtheuniform force ofthe field. Throughoutthein- ternalspacetheforce is(§610, foot-note) uniform, and its potential must beoftheform Cx.Choosing Gsothat atthe surface ofthesphere (radius r)theexternal andinternalpoten- tialsshallbeequal,wefind (internal)F=\F[-^—2\x (ferromagnetic) F=-iFf^+2)X. ..(diamagnetic).(9). t?rom thisand(8)wefind, fortheforce atanypointinthe T.E. 32 [XXXII (external) and498 AMathematical Theory ofMagnetism. axisof00, ( ( a^\ \ X=i^(l+-^j (ferromagnetic) X=Fll3) (diamagnetic) f { d^\ \X=i^fl — J-gj(ferromagnetic) X—F\\-\-\-^ (diamagnetic) Forpointsintheaxis of ocinfinitelynearoneanother x=r and(§629) wehave X(external) _X(internal) Hence, by(10)and(11),(internal)(10) .(11). (>*?) OT2-^ ('-^;) 2+(ferromagnetic) (diamagnetic) ^3(12); ...(13).or(resolvingforr) r=aa/2(1--1)(ferromagnetic) ^=^^ 2(1-1)(diamagnetic) Forgreatvalues ofctwehave r= -3-9(1+— )approximately (14). Hence forsuch values ofctasthose discovered insoftironb} Thalen(§631above) thevalue ofrwould beonlygreater b} about-g^partthan thatshown infig.4.The circles showr infigs.5and7were described with radii chosen atrandom Bymeasuring them inproportiontoaineach case, Ifine thepermeabilitiesoftheinductively magnetized globes,whose influence onthelines ofmagneticforce isrepresentedinthose diagrams,toberespectively2*8and -48. XXXIII.]Attraction ofFerromagnetics. 499 XXXIII. OntheForces experienced bySmall Spheres under Magnetic Influence ;andonsome ofthePhenomenapre- V^^sented byDiamagneticSubstances. ^^"[FromtheCambridge andDublin Mathematical Journal, May 1847.] 634.Thecircumstance thatamagnet*attracts smallpieces 3firon, istbephenomenonofmagnetism which was firstob- served;andananalogous action, presented byrubbed amber, Srstdrew attention tothephenomenaofelectricity. Now it lassincebeen discovered thatnomutual attraction orrepulsion oetween twobodies canresult frommagnetisminone,unless :heother bealsomagnetized,andthatnoelectric force can 3xist unless eachbodybeelectricallyexcited. Hence itap- pears thattheforcesoriginallyobserved aretheconsequences )fatemporary magneticorelectric state induced inaneutral oody,whenplacedintheneighbourhoodofamagnetorofan ilectrifiedbody. Inthefollowing paperthelawofsuchphenomenawith •eference tomagnetism fisconsidered. Itiseasily shown lowever that,bytakingi=lintheformulae obtained below, hecorrespondingresults forsmall insulated conductors, elec- rified byinfluence, maybeobtained, althoughthephysical oroblems areentirelydistinct. 635.Wemaycommence byconsideringthecase ofasmall phereofsoft iron, orofanyother substancesusceptibleof aagnetic induction; and itiseasily shown that theformulae ^pressingtheresults maybeappliedtothecase ofasmall ubebymerely alteringthevalue ofacertain coefficient;and ngeneraltothecase ofasmallportionofmatter ofany orm, such that inwhatever wayitbeturned, theresultant ,xisofmagnetization,forthewhole mass,shall coincide with ihedirection ofthemagnetizingforce. * Originally apiece ofmagnetic iron-ore orloadstone. Thetermmaynow •eapplied toanymass possessing permanent magnetism, andmayeven be xtended toagalvanic wire ofanyform. tThishasnotbeenmade thesubject ofaspecial investigation byany Titer, sofarasIamaware, although thenature oftheresult, inthecase fmagnetism, appears tobeentirely understood byMrFaraday. Thus, :om§2418 ofhisExperimental Researches [quoted below, inthetext(§646)] ^emight infer that asmall sphere orcube ofsoftironwould insome cases be urged along, andinothers obliquely ordirectly across thelines ofmagnetic orce;" andthat allthephenomena would resolve themselves into this, that ichaportion ofmatter, when under magnetic action, tends tomove from laces ofweaker toplaces ofstronger force. 32—2 500 AMathematical Theory ofMagnetism. [xxxiii. 636. Itiswellknown[andprovedin§609above]tliat if asmall homogeneous sphereofsoft iron, orofanyother substancesusceptibleofmagnetic induction, beplacedinthe neighbourhoodofamagnet,itwillbecomeuniformly magnet- ized, throughoutitsmass, with anintensity numericallyex- pressed bymultiplyingthemagnetizing force, byacoefficient independentofthedimensions ofthesphere. Thus ifRdenote theresultant force ofthemagnet,ortheforce that itwould exert uponanimaginaryunit ofmagnetism,attheposition occupied bythesphere,ofwhich wesupposethedimensional tobesosmall thatRhassensiblythesame value anddirection* throughout; and ifkbetheintensityoftheinduced magnetism: wehave -i^• W' where iisaproperfraction(nearly equaltounityforsoftiron) depending onthecapacityofthesubstance formagneticin- duction. 637. IftheforceRwererigorouslyconstant inmagnitude anddirectionthroughoutthewholespaceBoccupied bythe*? sphere, then there would benoresultingforcetendingtomove^ thesphere ;as,forexample, wemayconceive ittobe,without committing anappreciable error, inthecase ofaballofiron oj anyordinarydimensionsmagnetized bythe terrestrial force Intheinvestigationwhich follows weshall therefore have tc consider thesmall variation ofRthroughthespace 8,but although consideringthe effect ofthissmall variation incaus- ingamovingforceuponthemagnetized sphere, wemayneglect thedeviation fromrigorous uniformityofmagnetizationwhich itwillproduce. 638. LetX,T,Zhe thecomponents ofR atthe-point (x,y,z) whichmaybetaken asthecentre ofthesmallsphere. Atan} point (os+f), (y+g),(z+h),inthesphere, weshall have,foi thecomponentsoftheresultant forceduetothemagnet, ^'^-d^J'^ly^^-dz^^ „dZ, dZ dZ, ^^-dxf-'^j^^d.^' XXXIII.]Attraction ofFerromagnetics. 501 Byconsideringtheeffects ofthese forces upontheelements(as forinstance thin bars, inthedirection ofmagnetization)into which themagnetized sphere maybesupposedtobedivided, it iseasily shown[§500above],ashasalsobeendonebyPoisson, thatthecomponentsoftheresultingforce onthesphereare givenbytheequations F=dX dxKa .m+dX dx dx'Kcr .I-\—f- dy Ka .+—r.fccr.m+ dy dZ dZdz dY dzII^P dx dydz where aisthevolume ofthesphere, andI,m,nthecosines oftheangles made bythedirection ofmagnetizationwiththe axes.Now since thisdirection isthat oftheforce i^,wehavel+-Kcr .n. KG .n, Ka.n. 1=Xm—R'n=R Lence, since k= -,—i .R,wehave 47r FSi 47rx«+r:¥+zf)'dx dy 4f7rX— 4-Y~+Z^\dxdydzJJ(2)- 139.Now ifRbeduetoanymagnet,ortoaclosed galvanic current, Xdx+Ydy+Zdz isnecessarilyacomplete differential, andtherefore wehave dY^d_Z dZ^dX dX^dY dzdy'dx dz^ dydx Modifyingthesecond members of(2)bymeans oftheseequa- tions,wefind U^f..dX ..dY „dZ\ Zi 4f7r(3). F: (ydXdY^dZ\^SidR \dx dx dxJ4}'7r'dx Si f^,dX.^,dY .r.dZ\ Si^dR Vdy dy dyJ ^^^^(x^^Y-^+z"^]47r Vdz dz dzJ47r dy —aR— 4i7r'dz,(4). 502 AMathematical Theory ofMagnetism. [xxxiii.j From thesewededuce Fdo)+Gdi/+Hdz=^a.BdB=df^a.R^y..(5), whichexpresses fullytheresult ofequations (4). 640.Theinterpretationofthisresult shows thatasphereol softiron isurgedinthedirection inwhich themagnetizing force increases mostrapidly;thecomponentsoftheforce in different directions being expressible bythedifferential coeffi- cients ofthefunction—cri^l Thus insome cases itmay actuallybeurgedacross thedirection ofthemagnetizingforce. Forinstance,ifaballofsoftironbeplaced symmetricallywith respecttothetwopolesofahorse-shoemagnet, andatsome distance fromthelinejoining them,itwillbeurgedtowards this lineinadirectionperpendiculartoit,althoughthemagnetizing force isparalleltoit;orifthemagnetizingforce bedue toa straight galvanic wire, aball ofsoftiron willbeattracted to- wards thewire, althoughtheforce onanimaginary"magnetic point" isperpendiculartoaplane throughitandthewire. 641. Thepositionsofequilibriumofasmallsphereacted uponbythemagneticforces alone, willbepointsintheneigh- bourhood ofwhich B!^isstationaryinvalue, orpoints where d(R"^)=0.This condition issatisfied byeitherR=0,oi dR=0. Hence thespherewillbeinequilibriumatpoints where theresultantmagnetizingforce vanishes;where itisa maximum orminimum;orwhere itisstationaryinvalue. 642.Apositionofstableequilibriumwillbesuch thatR^ diminishes ineverydirection from it;andhence,ifthere be anypoint,external tothemagnet,atwhich theresultant force hasamaximum value, itwould beapositionofstableequi- librium forasmall ball ofsoft iron,andanyotherpositionoi equilibriumisessentiallyunstable. 643. AccordingtoMrFaraday'srecent researches,itap- pearsthat there areagreatmanysubstancessusceptibleol magnetic induction, ofsuch akind that forthem thevalue of the coefficient iisnegative. These hecallsdiamagneticsub- stances, and, indescribingtheremarkable results towhich hisexperiments conducted himwith reference toinduction indiamagnetic matter, hesays:"allthephenomenaresolve I XIII.] Repulsion ofDiamagnetics.503 themselves into this, thataportionofsuch matter, when under magnetic action, tends tomove fromstrongertoweakerplaces orpointsofforce*." This isentirelyinaccordance with the result obtained above;and itappearsthat thelawofallthe phenomenaofinduction discovered byFaradaywith reference todiamagnetics maybeexpressedinthesame terms asinthe case ofordinary magnetic induction, bymerely supposingthe coefficient itohave anegativevalue-j*. 644. Inthecase ofadiamagnetic sphere,theconsideration ofthestabilityorinstabilityofequilibriumindifferentposi- tions, isextremely interesting. Thus, atapointwhere R^is aminimum, asmallsphereofdiamagneticmatter willbein stableequilibrium ;and this isactuallythecase atanypoint forwhich theforce vanishes;even ifwetake intoaccount the weightofthesphere,itisreadily shown that stablepositions ofequilibrium mayexist. Thus ahollowcylindricalbar- magnet (ifsufficiently powerful),heldwith itsaxis vertical, wouldsupportasmalldiamagnetic sphereinapositionof stableequilibriumatapointinthe axis, alittle below the lower end ofthemagnet. For, consideringdifferentpointsin theaxis,weperceivethatthere isonebelow thelower end(at adistance =-^,if«,theradius ofthecylinder,bevery great comparedwith itsthickness, andverysmall comparedwith its length,and ifthedistribution ofmagnetismbeuniform)at which theresultant force isamaximum. If,onmovinga smalldiamagnetic sphere upwardsfrom thisposition, wearrive atapoint where theforceurgingitupwardsisgreaterthanthe weight, andthen let itmovefreely from rest,itwill oscillate about apositionofstableequilibrium.Itwillprobablybe impossible ever toobserve thisphenomenon,onaccount ofthe difficultyofgettingamagnet strong enough,andadiamagnetic substancesufficiently light,astheforces manifested inallcases of diamagneticinduction hitherto examined areexcessivelyfeeble. *Experimental Researches, §2418. tThelawofinduction inamass ofanyform, whether ofmagnetic or diamagnetic matter, may bestated asfollows :—LetBbethemagnetic force upon apoint within aninfinitely small spherical surface, described round apointPinthemass, resulting from themagnetismofallthematter external tothis surface. The intensity ofthemagnetism atPisequal to %viR, and itsdirection isthat oftheresultant forceB. 504 AMathematicalTJteory ofMagnetism. [xxxiii. 645.Averycurious phenomenon might readilybeobserved, accordingtotheresultsgiven above, byplacing twobar-mag- nets,with similarpoles,intheneighbourhoodofaball ofsoft ironallowed tomove inahorizontalstraightline(ok,suspended insuch amanner thatanymotion which cantakeplaceisin acircle ofconsiderableradius). Thus ifapole, S,ofabar- magnet which wemay regardforsimplicityasverylongand thin, beheld intheneighbourhood, theball willbedrawn towards thepoint A,inwhich aperpendicular from >Simeets thelineofmotion, andAwilltherefore beapositionofstable equilibrium.Ifnowapole >S",ofanequally powerful magnet, bepresentedandheld atanequaldistance in>Si^produced, A willbecome anunstableposition; and iftheballbeplacedin itslineofmotion, atanydistance fromAlessthan—r^,itwill berepelledfromA,althougheither magnetalone would cause ittomove towards thispoint, 646. Theresult obtained above affords thetrueexplanation ofthephenomenonobservedbyFaraday,that a-thin baror needle ofadiamagnetic substance, whensuspended between thepolesofamagnet, assumes apositionacross thelinejoin- ingthem. Forsuchaneedle hasnotendencytoarrangeitself across thelines ofmagneticforce;but, aswillbeshown[§684, below]inafuturepaper,ifitbeverysmall compared withthe dimensions anddistance ofthemagnet (asisthecase, forinstance, withabarofanyordinary dimensions, subject onlytotheearth's influence), thedirection itwillassume, when allowed toturn freely about itscentre ofgravity,willbethat ofthelines of force, whether thematerial ofwhich itconsists bediamagnetic, ormagneticmatter such assoftiron :butFaraday'sresult is duetotherapiddecrease ofmagnetic intensity round thepoles ofthemagnet, and totheleno;th oftheneedle, which iscon- siderablecomparedwith thedistance between thepolesof themagnet;and isthusexplained bythediscoverer himself. (§2269 ofhisExperimental Reseaixhes.) "The cause ofthe " pointingofthebar,oranyoblong arrangementoftheheavy "glassisnow evident. Itismerelyaresult ofthetendencyof ''theparticlestomove outwards, orintothepositionsofweakest XXXIV.]Attractions andRepulsions. 505 "magneticaction*. Thejointexertion oftheaction ofallthe " particles bringsthemass intotheposition which, byexperiment, IIfound tobelongtoit." IStPeter's College, il/ay 13,1847. lXIV. Remarks ontheForcesexperienced byInductively Magnetized FerromagneticorDiamagnetic Non-Crystalline rSubstances. \ [From thePhilosophical Magazine, October 1850.] ,Theremarkable lawlaiddown byFaradayin[§2418 ofhis Experimental Researches'\hisMemoir ontheMagnetic Condition ofallMatter[Transactions Royal Society, 1846, p.21,orPhil. Mag.Vol. xxviii., 1846],thatasmallportion ofdia^nagnetic matterplacedintheneighbourhood ofamagnet experiences a 2yressure urgingitfrom places ofstronger towardsplaces ofweaker force,isasimple conclusion, derived from themathematical solution oftheproblemofdeterminingtheactionexperienced by asmallsphereofmattermagnetized inductively,andacted uponinvirtue ofitsinducedmagnetism.Withoutentering upontheanalytical investigation, which willbefound in[§§634—646above]apaper"On theForcesexperienced bysmall SpheresunderMagneticInfluence;andonsome ofthePhe- nomenapresented byDiamagnetic Substances*[•,"Ishall, inthe present communication, stateandexplain briefly theresult, and pointoutsomeremarkable inferences whichmaybedrawn from it, 647. LetPbeanypointintheneighbourhoodofamagnet, and letP'beapointataninfinitelysmall distance, which maybedenoted bya,from P.LetRdenote theforcewhich a"unit northpoleJ"ifplacedatPwouldexperience, or,as itiscalled, "the resultantmagneticforce atP;"and letR! *Theextreme feebleness ofthediamagnetic action onaccount ofwhich anysmall sphere orcube ofthematter willexperience very nearly thesame force asifalltherestwereremoved, seemsfully tojustify thisexplanation. tCambridge andDublin Mathematical Journal, May 1847. Ij:That is,theend ofaninfinitely thin uniformly and longitudinally magnetized bar of"unit strength" which isrepelled onthewhole fromthe north bythemagnetism oftheearth; "unit strength" being defined bythe following statement :— Iftwoinfinitely thin bars beequally, andeach uniformly and longitu- dinally, magnetized, andif,when anend ofone isplacedataunit(an inch, forexample) ofdistance from anend ofthe other, themutual force between these ends isunity; themagnetic strength ofeach isunity. The forcei\,defined inthetext,isofcourse equal and oppositetotheforce thata"unit south pole" would experienceifplaced atP. 506 AMathematical Theory ofMagnetism. [xxxiv. denote thesame with reference toF'.Then,ifasmallsphere ofanykind ofnon-crystalline homogeneous matter, naturally unmagnetic,butsusceptibleofmagnetization byinfluence, be placedatP,itwillexperienceaforce ofwhich thecomponent alongPP' is 2a where a-denotes thevolume ofthesphere,andAacoefficient dependingonthenature ofthesubstance. This coefficient, A [agreeingwith theA,B,or(7of§§614—618appliedtoan 3 isotropic substance]hasavalue alittle lessthan—forsoft iron,and ithasverysmallpositivevalues forallferromagnetic substances containinglittle ornoiron. 648. Ifitbetrue, asIthink itmust be,that theforces experienced bydiamagnetic substances areoccasioned bythe influencing magnet magnetizing theminductively*, andacting uponthemwhen somagnetized, accordingtotheestablished laws ofthemutual action oftwomagnets,theprecedingresult willhold for allnon-crystalline matter; and toapplyAto adiamagneticsubstance itwillbeonlynecessarytogiveita negativevalue. [From §630,§628(14),and§627weseethat theextremenegativevalueconceivablyadmissible is—o—• Thus forevery substance, whetherferromagneticordiamag- 3 3 netic,Aisbetween -\--r—and—o~ •] • 649.Tointerprettheresult of§647,wemayremark, thatby theelementary principlesofthedifferential calculus asapplied tothevariation ofaquantity dependingonthepositionofapoint *Themost natural explanation ofthephenomena which hehad dis- covered issuggested hyFaraday inhisoriginal paper onthesubject, and itisconfirmed bytheresearches ofsubseqent experimenters, especially those ofEeich andWeber, whohavemade experiments toshow thatadiamagnetic substance, under theinfluence oftwomagnets, will actupon oneinvirtue ofthemagnetization which itexperiences from the other. Theextreme feebleness ofthepolarity induced indiamagnetic substances isproved by Faradayinaseries ofexperiments forming thesubjectofhislastcommunica- tion totheRoyal Society; inwhich anattemptismade, byvery delicate means, totesttheinduced current inahelix due tomagnetization or demagnetization ofadiamagnetic substance which itsurrounds, butonly negative results areobt-ained. :xiv.] Attractions andRepulsions. 507 7->'2__p2 Inspace,itmaybeshown that thefraction isgreater when thepointP'ischosen inacertain determinate direction fromPthan inanyother;that itisofequal absolute value, butnegative,ifP'bechosen intheoppositedirection;and that itvanishes ifP'beinaplane throughPatright angles tothelineofthose two directions. Hence itfollows that the resultant forceuponthesmallsphereisalong that line, inone direction ortheother, accordingasAispositiveornegative, daccordingly wedraw thefollowingconclusions :— (1)Asmallferromagnetic sphereintheneighbourhood of amagnet,willexperienceaforceurgingitinthatdirection in which the"magnetic force"increases mostrapidly. (2)Asmalldiamagnetic sphere,intheneighbourhoodofa magnet,willexperienceaforceurgingitinthat direction in which themagnetic forcedecreases mostrapidly. (3)The absolute magnitudeoftheforce inanycase in which thedistribution ofmagneticforce intheneighbourhood ofthemagnetisknown, isthevalue which theexpression in §647obtains whenwegivethevalue foundbymeans ofthedifferential calculus, forapoint P'ataninfinitelysmall distance PP' inthedirection ofthemostrapidvariation ofthe magneticforcefromP,theactualpositionoftheball. 650. Itisdeservingofspecial remark, thatthedirection of theforceexperienced bytheballhasnorelation tothedirection ofthelines ofmagneticforcethroughthepositioninwhich it isplaced. Themathematicalinvestigation thus affords fullcon- firmation andexplanationoftheveryremarkable observation made byFaraday (§2418 ofhisExperimental Researches), that asmallsphereorcube ofinductively magnetizedsubstance isin some cases" urged along,and inothersobliquelyordirectly across thelines ofmagneticforce." Itisinfactvery easyto imagine,oractuallytoconstruct, arrangementsinwhich there- sultant forceexperienced byaball ofsoft iron, orofsome diamagnetic substance,isperpendiculartothe lines ofthe magnetizingforce. Forinstance, ifaballofsoftironbeplaced symmetrically withrespecttothetwopolesofahorse-shoe magnet, andatsome distance from thelinejoining them,itwill 508 AMathematicalTheory ofMagnetism. [xxxiv. beurged towards this line, inadirectionperpendiculartoit, andconsequently perpendiculartothe lines ofmagnetizing force inthespaceinwhich itissituated;andaball ofbis- muth, orofanyotherdiamagnetic substance, similarly situated, would experienceaforce inthecontrarydirection. Oragain, ifaball ofanysubstance beplacedintheneighbourhoodof along straight galvanic wire,itwillbeurged towards orfrom thewire (accordingasthesubstance isferromagneticordia- magnetic)inaline atright anglestoit,andconsequently cutting perpendicularlythe lines offorce, which arecircles with their centres inthewireand inplanes perpendicular toit. 651. Theprecedingconclusions enable ustodefineclearly thesense inwhich theterms"attraction" and" repulsion" maybeappliedtotheaction exerted byamagnet onaferro- magneticandadiamagnetic body respectively. Asmallsphere offerromagnetic substance, placedintheneighbourhoodof amagnet, experiencesingeneral, aforce;buttheterm attrac- tion, accordingtoitsderivation, means aforce towards; and ifweapplyitinany case,wemust beable tosupplyanob- jectforthepreposition. Now, inthiscasetheforce istowards placesofstronger"magneticforce;"andhence theaction experienced byaferromagneticballmaybecalled anatti^ac- tion ifweunderstand towardsplaces ofstronger force.Places ofstrongerforce aregenerallynearer themagnet thanplaces ofweaker force, andhence smallpiecesofsoft iron are generally urged,onthewhole, towards amagnet (inconse- quenceofwhich nodoubt theterm "attraction" cameoriginally tobeapplied):but, aswillbeseen below, this isbynomeans universallythecase;balls ofsoftironbeing,insome cases, actually repelledfrom theinfluencing magnet;andtheterm "attraction" canonlybeuniversally used with reference to ferromagnetic substances, ontheunderstandingthat itis towardsplacesofstrongerforce. Theterm"repulsion,"the reverse of"attraction," may, accordingtothesameprinciples, beapplied universallytoindicate theforce withwhich asmall diamagnetic sphereisurgedtowardsplacesofweaker force, or repelled from places ofstronger forxe. 652. Thefollowing passage, containingastatement ofprin- I XXXIV.] ExperimentalIllustrationsofFaradaysLaw. 509 ciplesonsome ofwhichFaradayhimselflaysmuchstress, but which have not, Ithink, beensufficiently attended toby subsequent experimenters,isquoted from the article inthe Maihematical Journalalreadyreferred to.[Here comesquota- tionof§Q4<Qabove.] G53. Itmaybeadded tothis, that thetendencyofabar, whether offerromagneticorofdiamagnetic substance, inauni- form field ofmagnetic force, totake thedirection ofthelines offorce, depends ontheeffect ofthemutual action oftheparts inalteringthegeneral magnetizationofthebar,and iscon- sequentlysoexcessivelyfeeble foranyknowndiamagnetic substance thatthemost delicateexperiments would inallpro- babilityfailtorender itsensible*. 654.Faraday's law, stated atthecommencement ofthese remarks, may beillustrated bysomeverycuriousalthough extremely simple experiments, which Ishallnow describe briefly -f-. 655. Thespecial apparatus requiredismerelyalong light arm(Ihave used oneabout four feetinlength ;butamuch shorter rod,ifsuspended byafiner orbyalongertorsion- thread, would have answeredequally well) suspendedfrom a "torsion-head" bymeans ofaveryfinewire, orthread ofun- spunsilk fibres attached to itnear itsmiddle;andacase round itadaptedtoprevent currents ofairfromdisturbingits equilibrium, butallowingitsufficientangularmotion ina horizontalplane.Asmall ball ofsoftiron isattached toone endofthearm(orhung from itbyafinethread, which, for thesake ofstabilityinmanyoftheexperiments,asforinstance, experiments 2and3described below, must notbetoolong), andacounterbalance isadjustednear theother end soasto make thearm horizontal. Ifonlyasmall angular motion beallowed tothearm, thepathoftheball willbesensibly straight, andwemayconsider that,bythearrangement which *Avery briefcommunication onthissubject was laidbefore tbeBritish Association atthemeeting of1848, and ispublishedintheReportforthat year,under thetitle"On theEquilibrium ofMagnetic orDiamagnetic Bodies of anyform, under theInfluence ofTerrestrial Magnetic Force."[Art.xxxiv. Vol. i. ofmyEeprint ofMathematical andPhysical Papers, University Press. 1882.] +These experiments were shown, inillustration oflectures onmagnetism intheNatural Philosophy Class intheUniversity ofGlasgow, during the Session 1848-49. 510 AMathematical Theory ofMagnetism. [xxxiv. hasbeen described, theball isallowed tomove withgreat freedom inastraight line,butprevented from allother motion. 656. Inmakingtheexperimentsdescribed below, itiscon- venient tohavetwostopssoarrangedthat themotion ofthe armmaybekeptwithinanydesiredlimits, andmanageable insuch away,thatbymeans ofthem thearmmayberapidly broughttorest inany position. Ingeneral,before com- mencing anexperiment,thearmoughttobebroughttorest near oneend ofitscourse, andkept pressing very slightly upon oneofthestopsbythetorsion ofthewire, which may besuitably adjusted bythetorsion-head, andtheotherstop oughttobepushed away,soastoleave thearm freetomove inonedirection. 657.Experiment1.—Place acommon bar-magnetwith either pole,thesouth, forinstance, near theball ofsoftiron inits line ofmotion, butonthat sidetowards which itisprevented from moving bythestop. Takinganother bar-magnetof considerably greater strengththan theformer, bringitsnorth pole graduallynear thefixed southpoleoftheother, inthe continuation ofthe line ofmotion oftheiron ball.When thisnorthpolereaches acertainposition,thearm willcease topressonthestop,and ifwepushthenorthpolealittle nearerstill,thearm willaltogetherleave thestopandtakea positionofequilibrium,inwhich, after itissteadied (asmay easily bedonebymeans ofthestops),itwillremain stable, althoughthestopsberemovedentirely. If,bymeans ofone ofthestops,theballbepushedtoanydistance farther from themagnetsthan thispositionofstableequilibrium,itwill return towards itwhen left free. Ifitbedrawn alittle nearer bymeans oftheotherstop, and,when left forafewseconds, itbefound tocontinuepressing uponthestop, then, when thestopisremoved, theball willreturn tothatpositionof stableequilibrium. If,however,itbevery slowlydrawn still nearer themagnets, when itreaches acertainpositionitwill cease topress onthestop ;and ifafter this itexperiencethe slightest agitation, orifitbedrawn anynearer, itwillleave thestopandmove uptillitstrikes thenearer magnet,incon- tactwithwhich itwillalmostimmediately come torest. It thusappearsthat there isapositionofunstable equilibrium XXXIV.] ExperimentalIllustrations ofFaraday's Law. 511 fortheballbetween theformer stablepositionandthenearer mao-net. Itiseasytoarrangethetorsion-head sothat the torsion ofthesuspending-threadorwiremayhave aslittle effect asweplease, byfinding, bysuccessivetrials, either of these positionsofequilibrium, subjecttothecondition that, when themagnetsareremoved, thetorsion would notsensibly disturb thearmfrom thepositionsofound. 658. After theexplanationswhich havebeengiven above, it isscarcely necessarytopointoutthat thepositionofunstable equilibrium,determined inthisexperiment,isapoint where themagnetizingforce due tothesouthpoleisdestroyed by that ofthemore distant butmorepowerfulnorthpole ;and thatthepositionofstableequilibriumisonewhere theexcess ofthemagnetizingforce due tothenorthpole, above that which isduetothe lesspowerfulsouthpole,hasamaximum value with reference topointsinthecontinuation, throughthe lesspowerful pole,ofthe linejoiningthetwopoles.Ifthe polesweremathematicalpoints,andthebars solongthat their remote ends couldproducenosensible action onthe ball,the positionofunstableequilibriumwould ofcourse besuch that itsdistances fromthetwopoles would hedirectlyasthesquare rootsofthestrengths ofthemagnets; and,bythesolution ofa mostsimple"maximumproblem,"itmaybeshown that the stableposition would besuch that itsdistances fromthepoles would hedirectlyasthecube rootsofthestrengths. 659. Experiment2.—Place twoequal bar-magnets symmetri- callywith reference totheline ofmotion, with similarpoles atequaldistances ontwo sides, inaperpendiculartothis line, and, tomake thebestarrangement,letthelengthsofthe magnets beinthecontinuations ofthelinesjoiningtheirpoles. Operating bymeans ofthestops,inamaimer similar tothat described forthepreceding experiment,itisreadilyascer- tained that there aretwopositionsofstable equilibriumfor theballatequaldistances ontwosides ofthelinejoiningthe poles, andthatthemiddlepointofthis line isapositionof unstableequilibrium. 660. Here, again,theexplanationisobvious. Thepositions ofstableequilibrium beingsuch that,with reference topoints intheline ofmotion ofthe ball, themagnetizingforce due 512 AMathematical Theory ofMagnetism. [xxxiv tothetwosimilarpolesmaybeamaximum, arereadilyfounc tobeatdistances^-y^onthetwosides ofthelinejoiningth( poles (thelengthofthis linebeing denoted bya),ifthese b( mathematicalpoints, and ifthelengthsofthebarsbesogrea thatthedistantpoles producenosensible effects. 661.Experiment3.—Hold acommon horse-shoe magnetwitl thelinejoiningitspoles perpendiculartotheline ofmotior ofthe ball, and,byasuitable managementofthestopsanc ofthetorsion -head, theexistence ofaforceurgingthebal perpendicularlyacross the"lines offorce"tow^ards themiddle pointofthe linejoiningthepoles, maybeeasilymad( manifest. 662. Experiments ondiamagnetic substances, andonferro magneticsubstancesoffeebleinductivecapacity.—Thepheno mena discovered byFaradayrelative totheaction ofmagnet; onsubstances notpreviously known tobesusceptibleofmag netic influence maybeexhibited withgreateasebymeans o\ theapparatus described above. Small balls ofthe substance.'^ tobeexperimented uponmaybehungfrom oneendofthd balance(theball ofsoftironbeingofcourse removed) byfine^ threads ofsufficientlengthtoallow thearm,which maybe o:4 anysubstancecontaining noiron, tobeoutofreach ofany sensible influence from themagnet employed.There isinthese casesnodifficulty, regardingthelengthofthesuspending-thread ofthekind noticed above[§655]with reference tosoftiron asthemagneticforcesexperiencedarenever strong enougl toproducelateralinstability (that is,awant ofstabilityintbci lineofmotion), even with thelightestofthesubstances ex- perimented on,unless thesuspendingthread befarlongei than isnecessary.IntheexperimentsIhave made,th( threadsbearingthesmall balls have notbeen more thai four orfiveincheslong. Thediameters ofthe balls have been from aquarterofaninch toaninch, oraninch anc ahalf Instead ofsimplebar-magnetsofsteel, which an notpowerful enoughtobeconvenient forthese experiments Ihave used abar electro-magnetofverymoderate power consistingofahelix and softiron core. This core isacylin- derofabout aninch indiameter andafootandahalflong I IV.] ExperimentalIllvstraiionsofFaradaysLaw. 518 withround ends (nearly hemispherical), which, when thecore isinitscentralposition, extend about aninchbeyond the helixoneach side.Bythese means therepulsionofballs of diamagnetic substance, andtheattraction ofvery feeblyferro- magnetic substances, maybeshown withgreat facility. 663. Forexample,Imaymention that Ihavehungasmall apple, whole, byathread three orfour incheslong, and puttingitatfirst atrest,pressing slightly (invirtue oftorsion produced bythetorsion -head mentioned above, §QtbO) upon oneendofthe softiron corepreviouslytotheexcitement of theelectro-magnet,Ihave found that assoon asthegalvanic current isproduced,theappleisrepelled away ;and,bypush- ingforward thesoftiron core, Ihave chased itacross the field throughaspaceoffour orfiveinches. 664. Ihave alsousedthesameapparatustoshow thatabody which isfeeblyattracted inair isrepelled when immersed below thesurface ofasufficiently strongsolution ofsulphate ofiron inasmalltrough,soarrangedthatwhen, bytheforce oftorsion, thebody immersed intheliquidismade to press onaside ofthetrough,the electro-magnet maybe placed withoneendofitscorepressingontheoutside ofthe trough,close tothepointwhere itispressed upon bythe bodywithin.Usingsmallglassballs(which, when empty, exhibit nosensible effects oftheinfluence ofthemagnet),the magneticconditions ofdifferentliquids filling themmaybe easilytested.Faraday'sbeautiful experiments ontherelative magnetic capacitiesofsolutions ofsulphateofiron ofdifferent strengths,orrather, other experimentstoillustrate thesame principles, may beperformedinanextremelyconvenient manner, byfillingaglassball ofthiskind with asolution, hangingitfrom oneendofthearm, and,byasuitable ad- justment oftheweightattheother, immersingitbelow the surface ofanother solution contained inthetrough.Ihave found thatwhenever thedifference ofthestrengthsofthetwo solutions was considerable, the ballimmersed wasattracted orrepelled bytheexternal magnet, accordingasthesolution contained intheballwasstrongerorweaker than thesolution surroundingit. T.E. 33 514 AMathematical Theory ofMagnetism. [xxxiv. OntheStability ofsmall Inductively MagnetizedBodies in Positions ofEquilibrium. 665. Inthepaper [§§634... 646above] publishedinthe Cambridge andDublin Mathematical Journal(referredtoabove). Ipointedoutthat asmall ball ofeitherferromagneticordia- magneticsubstanceplacedintheneighbourhoodofamagnet,and notacteduponbyanynon-magnetic force, isinequilibriumifi1 beinasituation where the"resultant force" (thatwhich was denoted byR)iseither amaximum orminimum, or"stationary' invalue;thatadiamagneticball isinstableequilibriumif,anc notinstableequilibrium unless,itbesituated where theforceL isaminimum inabsolute value; andthat "ifthere bean) " pointexternal tothemagnet,atwhich theresultant force ha^ "amaximum value, itwould beapositionofstableequilibriun "forasmall barofsoft iron,andanyotherpositionisessen " tiallyunstable."Shortlyafter thepublicationofthatpaper ^ Isucceeded inprovingthat theresultant force cannot beami absolute maximum atanypointexternal toamagnet,an(4 consequentlythatnopositionofstableequilibriumforaferro* magnetic ball, perfectlyfreefrom allconstraint, can exist. A havevery recentlyfound that theremaybepoints where th< resultant force isanabsolute minimum withoutbeingzero and therefore theremaybepositionsofstableequilibriun foradiamagneticballnotincluded inthecase ofthe forc< vanishing,noticed intheprevious paper. That case,howevei affords thesimplestillustration thatcanbegivenofthatmos extraordinary fact, that asolidbodymayberepelled by; magnet,ormagnets,intoapositionofstableequilibrium.IJ forinstance, wetake thearrangement (describedforExp.5 §659above)oftwobar-magnets,fixed with similarpoles nea oneanother, wehaveobviously between thesepolesapointwher theresultant force vanishes, andtowards whichconsequentl; asmalldiamagneticballplaced anywhere sufficiently near i would berepelled.Itiseasily shown that, actually unde theaction ofgravity,aball ofdiamagneticsubstance woul( beinstableequilibriumalittle below thisposition, withou anyexternalsupportorconstraint whatever,ifonlyth. magnetswerestrong enough.Itis,however, extremelyim I IV.] Relations toMagnetizingForce. 515 probablethatanyattempttorealize thisbyexperimentwill succeed, since, even inthemost favourable cases, nodiamag- neticrepulsion uponasolid hasyetbeen obtained which at allapproachesinmagnitudetotheweightofthebody.Still wemust consider thatatrue theoretical solution ofthecele- bratedphysical problem* suggested by"Mahomet's coffin" hasbeen obtained, which isnottheleast curious amongthe remarkableconsequencesofFaraday's magneticdiscoveries. OntherelationsofFerromagnetic andBiamagnetic MagnetizationtotheMagnetizingForce, Q^Q. Inthemathematicalinvestigation bywhich theresult stated above wasobtained, itisassumed thatthemagnetization ofthesubstance oftheball ineach case isproportionaltothe magnetizingforce(althoughthisassumption mayofcourse be avoidedbymerely supposing /atohave avalue varyingwith theforce, which willnotaffect either theinvestigationorthe form oftheresult).Itappearstomeveryprobablethat this assumptioniscorrect forallknowndiamagnetic substances, and forhomogeneous feebly ferromagnetic substances; since[§606, AxiomII.]itisequivalenttoanassumptionthatinductive mag- netization ofasubstance does notimpairorinanywayalter itssusceptibilityforfresh magnetization bymeans ofanother magnet broughtinto itsneighbourhood.Thisopinion cannot, however, atpresentberegarded but asamereconjecture, beingasyetunsupported byexperiment.Itisindeeddirectly opposedtothefollowingconclusion towhich M.Plucker arrives, fromsome ofhisexperimentalresearches:—"J'ai deduit de "la cette loig^n^rale,savoir: quelediamagnetismed^croit " plusvitequelemagn^tisme quandlaforce deI'aimant dimi- "nue, ouquandladistance despoles augmentef:"butmany *Itis,Ibelieve, often thought that thisproblemissolved intheexperi- ment inwhich aneedle isattracted into agalvanic heHx held with its axisvertical; butIhave convinced myself that theneedle always touches somewhere onthe sides ofthetube(ifthere beoneroundit)oronthe wire ofthehelix; and Ihave alsoascertained that,when apowerfulhelix is used with, inplace oftheneedle, atin-plate [iron] cylinder, even ifitbevery little less indiameter than theinner cylindrical surface ofthehelix, there isnever stable equilibrium without contact between them. Thephasnomenon ofasolid body, hovering freelyinthe air,instable equilibrium, without anyexternal support orconstraint, hasnever, Iamconvinced, been witnessed astheresult ofany electric ormagnetic experiment. +Quoted from apaper intheFrench Annales deChimie etdePhysique,I 516 AMathematicalTheory ofMagnetism. [xxxiv. ofthecuriousphaenomena fromwhich M.Plucker was led tothis conclusion, andwhich headduces inconfirmation of it,donotappeartometosupport it,butrather tobecon- nected with thepeculiar magneto-inductive propertiesofcrys- talline orquasi-crystallinestructure which hediscovered subsequently*; andwithrespecttothose whichappearat firstsight reallytosupport it,Ihaveconjecturedthatthey mayadmit ofexplanation solelyontheprinciple expressedin Faraday's law,quotedatthecommencement ofthese remarks. Thus, theexperiments uponawatch-glass containing mercury, placedatdifferent distances from amagnet,which show that theresultant forceexperienced bythewatch-glass,in virtue ofitsownmagnetizationasaferromagnetic substance, andthecontrary magnetizationofthediamagnetic mercury, issometimes increased byremovingthewhole toaslightly greaterdistance from themagnet,donotprovethatwhen the magnetizingforce isdiminished theinducedmagnetizationof themercuryisdiminished byagreaterfraction ofitsformer amount than that ofthewatch-glass, bubaremostprobably tobeexplained bythecircumstance thatthe"field offorce" occupied bythemercury andwatch-glass when removed a veryshort distance, issuch thatthemean value ofthediffer- ential coefficient ofthesquareoftheforce, with reference to co-ordinatesparalleltothedirection ofmotion ofthewatch- glass,isgreaterthan themean value ofthesame function, throughthe fieldoccupied when thewatch-glassisincontact with themagnet.Itisofcourseimpossibletogivemore than ageneral explanationsuch asthiswithout somespecific know- ledgeofthedistribution ofmagneticforce intheneighbour- hood oftheactual magnet employed ;butthephaenomena described byM.Pliicker inthis case areundoubtedlyofa kind thatmightbeanticipatedifaverticalbar-magnetbe June 1850, bearing the title, "Sur leMagnetismeetleDiamagn^tisme: parM.Piiicker." This paper appearstobearesume oftheauthor's ex- periment'researches and discoveries regarding magnetic induction, of which detailed accounts have been published invarious communications to Poggendorff's Annalen inthecourse ofthelasttwoyears. *This connexion isrecognised bythediscoverer himself, asisshown by thestatement hemakes atthecommencement of§4ofthepaper already referred to.Yethementions hisexperiments oncylindersofcharcoal as thefoundation onwhich heestablishes, asageneral law, theconclusion quoted inthetext. 'S:3XIV.]Relations toMagnetizingForce. 517 used, especiallyiftheupper pole,overwhich thewatch-glass issuspended,be flat.Anelectro-magnet with, forcore, a hollowcylinderofsoftironopenattheends, would evenrepel asmall ferromagnetic body capableofmoving alongtheaxis, insomepositions, and attract italittle further off,since there would bevariations offorce inthis casepreciselysimilar to thoseexplainedwith reference topointsinthelineofmotion oftheballinExperiment 2,§659above. 667. Themoststriking experiments adduced byM.Plucker tosupporthishypothesis,that"diamagnetismincreases more rapidlythanmagnetism"when themagnetizingforce isin- creased, arethose inwhich theforceexperienced byasmall inductively magnetized bodyinaconstantpositionistested fordifferentstrengthsofthesameelectro-magnet, produced by usingagreaterorlessnumber ofcells intheexciting battery. Attherecent MeetingoftheBritish Association inEdin- burgh,Iventured tosuggestthatachangeinthedistribution ofmagnetic forceintheneighbourhood ofthemagnet, accom- panying anincrease ordiminution inthestrength ofthegal- vanic current, mighthave contributed toproduce someofthe singular phcenomenawhich hadbeen observed;and that there issome considerable changeinthedistributionofforceinthe neighbourhood ofanelectro-magnetwithasoftiron core ina stateofintense magnetization when, forinstance, thestrength ofthecurrent isdoubled, seemsextremely probable whenwecon- \sider thatapiece ofsoftii^on inastateofintensemagnetiza- itioncannot beexpectedtobeasopentofresh magnetization as itwould beifnotmagnetizedinthefirstimtance*. Onthe same occasion Iremarked, thatsomeexperiments madeby MrJoule inconnexion w^ith hisresearches onchangesof dimensionsproducediniron barsbymagnetic influence, ap- pearedtoindicate diminished inductivecapacitiesinstates of intense inductivemagnetization f.Atthat time Iwasnot aware oftherecentexperimentalresearches ofGart( hauser and Miiller onthemagnetizationofsoftiron;but Ihave Isincemetwith anumber ofPoggendorff's Annalen(1850, *[Embodied inArt.xxx.(§§604—624) above.] tPhil.Mag. 1847, vol.xxx. pp.76,225. Also Sturgeon's Annals, Aug. 1840. 518 AMathematical Theory ofMagnetism. [xxxiv. No. 3,publishedlastApril) containinganaccount ofthese researches*, which completelyconfirms thesecondpartoi theconjectureIhadthrown out.Whether ornot,how- ever, thechangeinthedistribution offorce isofsuch akind astoaccount forthephgenomena bywhich M.Pluckersup- portstheconclusion which hasbeenquoted,itisimpossible topronouncewithout acomplete knowledgeofthe circum- stances. Anewperimentumcrucis might bemade bymeans oi anelectro-magnetwithout asoftiron core. 668. InonerespectM.Plucker's views receive aremark- ableconfirmation byJoule andbyGartenhauser andMuUer's experiments,ifitbetrue that ahomogeneous diamagnetic substance isinductively magnetizabletoanextentprecisely proportionaltothemagnetizing force, ordeviatinglessfrom thisproportionalitythan themagnetizationofsoft iron. For ifacomplex bodyweremade upconsistingofadiamag- netic substance(eithersolid orinpowder) andanextremely smallquantityofsoftiron inveryfinepowderorfilings spread uniformly throughit;asmall ball ofthisbodywould, when acted uponbyafeeblemagnetizing force,become onthe whole magnetizedlikeaferromagnetic, andwould beurged fromplacesofweaker towardsplacesofstrongerforce. Ifnow themagnetizingforceweregradually increased, the"resultani magnetic moment"ofthecomplex bodywould atfirst in- crease, then, afterattainingamaximum value, decrease tc zero, afterwhich itwould become"negative,"ortheball would beonthewholemagnetizedlike adiamagnetic, ancl would beurgedfromplacesofstrongertowardsplaces oi weaker force. Such,ifImistake not, isthebearingwhich M.Pluckerexpectsofanycomplexsolidconsistingof £ suitable mixture offerromagnetic anddiamagneticsubstances; butmereexperimentsonsoftiron,such asthose ofJoule and ofGartenhauser and Miiller, donotrender itprobablethai ahomogeneous feebly ferromagnetic substance, containing noiron, oronlyaverysmallquantityand thatchemicallj combined, should have itscapacityforfreshmagnetization *"Ueber dieMagnetisirung vonEiseustaben durch denGalvanischer Strom;von J.Miiller." XXXV.] MagneticCurves. 519 diminished bytheslight magnetization which thestrongest magnetizingforce thatcould beapplied wouldproduce*. »Eow,Gare Loch, Aug, 21,1850. XXXV. Abstracts oftwoCommunications [From theReport ofBritish Association forBelfast, 1852.] pncertain MagneticCurves;withapplicationstoProblems in |K.theTheoriesofHeatElectricity, andFluid Motion. 669.Amethod[§632above], which hadbeengiven bythe author intheCambridge Mathematical Journal, Vol. IV.,Nov. 184!3f,forintegratingthedifferentialequationsofthelines of force inanycase ofsymmetryabout anaxis, isappliedinthis communication tothecaseofaninfinitelysmallmagnet placed with itsaxis direct orreversealongthelines offorce ofauni- form magneticfield. Diagrams [§632 above] containingthe curves drawnaccurately, accordingtocalculations founded onthe result ofthisinvestigation (correspondingtoseries oftenortwelve different valuesgiventotheconstant ofintegration),Avere ex- hibited totheSection. Certainpartsofthese curves were shown inaseparate diagram [§632, fig. 4],asconstituting preciselytheseries oflines ofelectric force about aninsulated sphericalconductor under theinfluence ofadistant electrified body;andtheotherparts,inaseparate diagram [fig. 6],as constitutingthelines ofmotion ofafluidmass intheneigh- bourhood ofafixedspherical solid, atconsiderable distances fromwhich thefluid ismoving uniformlyinparallellines so slowlyastocause noeddies round theobstacle. The circle representingthe section ofthespherical conductor, inthe former ofthesediagrams,cuts theentire series ofcurves at right angles,with theexceptionofone curve, which itcuts throughadoublepointatanangleof45°toeach branch. The circlerepresentingthesection ofthesphericalobstacle inthe latterdiagram, alongwithtwoinfinite double branches consist- ingoftheaxial diameterproduced externallyineach direction, constitutes thelimitingcurve ofthe series shown, and isnot intersected byanyofthem.Aseries ofdiagrams (deduced from *[The lastsentence ofthisarticle iscancelled from thereprint (July 5,1872).] t[Note ofFob. 22,1884. Now republished, constituting Art. ix.ofmy"Re- print ofMathematical andPhysical Papers," Vol. i.1882. W.T.] I 520 AMatheinaticalTheory, ofMagnetism. [xxxv theformer ofthese bydescribingacircle ofthesame size as thatshown init,anddrawing,onasmaller scale, asmuch ofthe curves aslieswithout thiscircle) wasshown asrepresentingthe disturbed lines ofmagneticforce about balls offerromagnetic substance ofdifferent inductivecapacities, placedinauniform magneticfield[oneofthese isshown infig.5of§632] ;and another series, similarlyderived from thelatter(that is,the onerepresentingthelines offluid motion about aspherical obstacle), wasshown asrepresentingthedisturbance caused bythepresenceofdiamagneticballs ofdifferent inductive capacitiesinauniformmagneticfield[oneofthese isshown in fig.7of§632]. These twoseries ofdiagramsarealsoaccurate representationsofthelines ofmotion ofheat inalargehomo- geneoussolidhavingheatuniformly conducted across it,dis- turbed byspherical spaces occupied bysolid matter ofgreater orlessconducting powerthan thematter round them;the twoprincipal diagramsfromwhichtheyarederivedbeingthe corresponding representationsforthecases ofspherical spaces occupied respectively bymatter ofinfinitely greatandinfinitely smallconductivity. Theauthor called attention totheremark- ableresemblance which thesediagramsbore tothose which MrFaraday hadshownrecentlyattheRoyalInstitution to illustrate hisviews regardingtheaction offerromagnetics and diamagneticsininfluencingthe field offorce inwhichthey areplaced;andjustifiedand illustrated theexpression"con- ducting powerforthelines offorce," byreferringtorigorous mathematical analogies presented bythetheoryofheat. OntheEquilibrium ofelongated Masses ofFerromagnetic Sub- stance inuniformandvaried Fields ofForce. The fact, firstdiscoveredexperimentally byGilbert, that a barofsoft iron, heldbyitscentre ofgravityinauniform magnetic field, settles with itslength paralleltothelines of force,isnotexplained correctly when itissaid tobemerely due tothepropertyofmagneticinduction invirtue ofwhich the barofsoftironbecomestemporarilyamagnetlikeapermanent magnetinitspositionofstableequilibrium.Forexactlythe same statement would beapplicabletoarowofsoftiron balls rigidlyconnectedbyanon-magneticframe;yetsuchanan-ange- 1 v.] Equilibrium ofFerromagneticBars. 521 ment would notexperience anydirectional tendency (since no oneoftheballs initwouldexperienceeitKer aresultant force or aresultant couplefrom theforce ofthefield),unless invirtue ofchangesinthestates ofmagnetizationoftheballs induced bytheirmutual actions. Hence themutual action oftheparts ofarowofballs,andasiseasily shown, ofarowofcubes, or ofabarofanykind,must betaken intoaccount before atrue theoryoftheir directional tendencies canbeobtained. The author ofthiscommunication, byelementarymechanical reason- ingfounded onwhat isknown withcertainty regarding magnetic induction andmagneticactiongenerally,shows thatanelongated mass, inauniform magnetic field, tends toplaceitslength paralleltothelines offorce, whether itsinductivecapacity be ferromagneticordiamagnetic, provideditbenon-crystalline, be- cause ifferromagneticitbecomes more, orifdiamagnetic,less intensely magnetized,ifplacedinsuchaposition, than ifplaced with itslengthacross thelines offorce. But forallsubstances, whether ferromagneticordiamagnetic, possessingsolittlecapacity forinduction asanyoftheknowndiamagnetics,thistendency, dependingasitdoesonthemutual action ofthepartsofthe elongated mass, is,andprobablywillalways remain, utterly imperceptibleinexperiment.All directional tendencies in bars ofdiamagneticsubstance which haveyetbeen, andpro- bablyallwhich caneverbediscovered byexperiment,aredue either tosomemagne-crystallic propertyoftheirsubstances, or tothetendencyoftheir ends orothermoveableparts,/rompZaces ofstrongertowardsplaces ofweakerforce,invariedmagnetic fields, ortothese twocauses combined, andinnorespecttothe inductive effects ofthemutual influence oftheirparts. To consider theeffects ofawant ofuniformityofthe force, ina varied field, ontheequilibriumofaferromagnetic bar,the author quoted Faraday'sadmirable statement ofthelawregard- ingthetendencyofaball orcube ofdiamagnetic substance*, andreferred toformerpapers [Arts,xxxiii. andxxxiv. above (§§634—668)],inwhich hehadproved that,whenappliedto non-crystallinesubstancesgenerally,with thepropermodifica- *[SeeFaraday's "Memoir ontheMagnetic Condition ofallMatter," Trans- actions oftheRotjal Society, 1846, page21;or,Philosophical Magazine, Vol. Kviii. 1846.] 522 AMathematicalTheory ofMagnetism, [xxxv. tion forthecase offerromagnetics,itexpresses with admirable simplicitytheresult ofamathematicalinvestigation involving some ofthemostremarkableprinciplesinthetheoryofattrac- tion.From this itwasshown, that ifweconceive aferromagnetic mass tobedivided intoverysmall cubes, each oftheseparts would, ofitself, tendtowardsplacesofstronger force, andthere- forethatthebearingofthewhole mass inavaried field willbe produced partly bythistendency andpartly bythetendencyde- pendingonthemutual inductive influence which alone exists when thefield isuniform. Theauthor thenproceededtoillus- trate these theoretical viewsbyaseries ofexperiments.Insome ofthem asteelbar-magnet wasused,andsmall softironwires, fixed invariouspositionsonlightwooden arms, wereshown tobe sometimes urgedonthewhole fromplacesofstrongertoplaces ofweaker forcebytheirtendencytogetintopositions with their lengths alongthelines offorce. Inothers, aringelectro-magnet, consistingofinsulatedcopper wire, rolledfiftytimes round as closelyaspossibletothecircumference ofacircle ofabout 25 centimetres diameter, fixed inaverticalplaneatright anglesto themagnetic meridian, wasused, andasinglecube ofsoftiron, placedinanexcentricposition onalongnarrowpasteboard tray centrally suspendedinthe field offorcebyunspun silk,was attracted intotheplaneofthering ;butarowofthree orfour cubes placed touchingoneanother inalinethroughtheaxis ofsuspension,settled asfarfrom theplaneaspossible,invirtue ofthetendencyofanelongated mass togetitslength alongthe lines offorce. Twocubesplacedincontact arefound tobe instable equilibriumintheplaneofthering,orinoblique positions,orasfarfrom theringaspossible, accordingtothe greaterorlessdistances atwhichtheyareplacedinthetray, from thepointofsuspension. Anumber ofequalandsimilar bars ofacompositionofwaxand soft ironfilingsofdifferent ferromagnetic strengths, suspended successivelywith their middle pointsinthecentre ofthemagnet,settled invarious positions.Those ofthem which were ofgreatest ferromagnetic capacitysettledperpendiculartotheplaneoftheringoralong thelines offorce;others, with asmallerproportionofiron fil- ings,hadpositionsofstableequilibriumboth intheplaneof theringRndperpendiculartoit;andothers, with astillsmaller Kxvi.]Oscillations ofInductively Magnetized Needles. 523 •portionofironfilings,had their solepositionsofstable equilibriumintheplaneofthering. Thelast-mentioned ex- perimentsillustratedvery curiouslythediminishedproportion borne bytheeffects ofmutual influence ofthepartstothose of anon-uniformityinthe field offorce, insimilar bodies of smaller ferromagnetic capacity. [Comparelasttwosentences of§670below.] IXXVI. Remarquessurlesoscillationsd'aigwilles noncristal- lUsees defaihle pouvoir inductif paramagnetiques oudia- ^magnetiques,etsurd'autres phenomenes magnetiques pro- duitspardescorpscristallises ou7ion cristallises. [Fromthe'Comptes Rendus' oftheFrench Academy, 1854, firsthalf-year.] "Glasgow, le22mars 1854. 670."J'ailuaujourd'hui,dans lesComptes Bendus du25avril deI'annee dernifere, unExtrait detroisMemoires deM.Mat- teucci relatifs aumagnetisme, quirenferment ungrand nombre d'observations interessantes.J'ytrouve laremarque quedes aiguilles prismatiquesdebismuth non cristallise oscillent entre lespolesd'unaimant dansdestemps egaux,lorsmemoqueleurs poidssont differents, quandleurslongueurssont lesmemes. J'ai eulapensde quelapropositionserait encore vraie, lorsm^me que cette derniere condition neseraitpoint remplie, oudumoins en ysubstituant cette autre condition moins absolue :leslongueurs desdifferentes aiguillesnedoivent point depasser unepetite fractiondeladistancecompriseentre lesdeuxpoles deTaimant. *'I1me suffit, pour prouvercetteproposition, deremonter alaraison donnee desI'origine parM.Faraday deTaction eprouvee paruneaiguilledebismuth non cristalliseplacde entre lesdeuxpolesd'unaimant :savoirquecette action estla resultante destendances qu'eprouventtoutes lesparticules de I'aiguille asetransporterdespoints oulaforce magndiqueest laplusintense vers ceuoo ou,elle estlaplusfaihle ;j'appliqueici lath^oriemathematique, presentee pourlapremierefoisdans le Journal deMatheniatiquesdeCambridgeetdeDublin^. *Des forces quiagissent surdepetites spheres soumises adesinfluences magnetiques ;aper9u dequelques phenomfenes present^s par lessubstances diamagnetiques. Cambridge andDublin MathematicalJournal ;mai1847 [§§634... 646above], Voyezaussi unarticle duPhilosophical Magazine, octobre 1850, intitule: "Kemarques surlesforces quiagissent surlessubstances ferromagn^tiques 524 AMathematicalTheory ofMagnetism. [xxxvi. "IIestenefFetd^montr^ dans cetteinvestigationmathdma- tique, qu'en d^signant par fjuuncoefficient exprimantlepouvoir inductif delasubstance(cecoefficient, positif pourlessubstances ferromagnetiquesouparamagn^tiques,etnegatif pourlessub- stancesdiamagnetiques, exprime parfaitementladifference de proprietes,decouverte parM.Faraday,etquiaservi debaseala division detons lescorpsendeux classes, corps paramagndtiques etcorps diamagnetiques) ;paralevolume d'uneparticule du corps; parRlar^sultante desforces magnetiques quis'exercent au point (w,y,z)duchamp magn^tiquedanslequelilestplac^,c'est- a-dire laforcequiagiraitsurunpolemagnetique ^galaI'unitd, ousurI'unite deniagnetisme boreal, oudematihre magnetique imaginaire,oudeflaide magnStique quisetrouverait encepoint. Laforce alaquelleseraeffectivement soumise cetteparticule magn^tisee parinduction sera laresultante des trois forces X,Y,ZdonneesparlestroisEquations [§639(5)above] ^ ^d{R^ ,d(R') d{R') -^=i^^-rf^' ^=i'^"-d^' ^=^'^^[r- "Supposons que I'originedescoordonnees soitplaceeau centre delaligne quijointlesdeuxpolesdeI'aimant, etque I'axe descoordonnees X'OX coincide avec cetaxeduchamp magnetique:lavaleur deBj^seraunminimum aupointrela- tivement auxdiverspointsdelaligneX'OX, etunmaximum relativement auxpointsd'unplan equatorial quiluiseraitper- pendiculaire. On a,d'apres cela,pourdespoints placesaune distance infinimentpetite dupoint 0, E'=R,'+Ax'-Bif- Cz'; Rqrepr^sentelavaleur deRaupoint 0,etA,B,Csont trois quantit^s positives. ''Supposonsmaintenantqu'un petit corps (devolume<r, demasse m,depouvoirinductiffi)soit fix^aTextr^mite d'un bras rectiligneinfinimentlegerOAl(delongueur a),quipuisse semouvoir librement etuniquementautour deI'axeOZ, c'est- a-dire dans leplanYOX, etconstitue ainsi cequ'onnomme un pendule magnetique simple ;1'Equationdesonmouvement sera m^a=Fcos^—Xsm6, oudiamagndtiques non cristallis^es magn^tis^es parinduction" [§§647... 668 above]. XXXVI.]Oscillations ofInductively MagnetizedNeedles. 525 6repr^sentant Tangle MOX. Lesexpressions pre'c^dentes nousdonnentX=fjLo-AxetF=—ficrBy, ^^ommeonag^ometriquement ^^Ba;=acos^,y—asin6, rdquation dumouvement devient ^=_/f^M4.msinl9cosl9. dtm^ Comme I'dquationestindependantedea,nous enconcluons que:lemouvement angulaireestindependant durayon ducercle danslequelils'effectue,ouquelesoscillations dedifferents pen- dules(d^finis comme nous Tavonsfait) autour ducentre du champ magnetiquesont isochrones, Menqueleurslongueurssoient differentes. "Lademi-p^rioded'une oscillation infinimentpetiteest Vm Ixct{A-{-B) ou,sipreprdsenteladensity ducorps, TT^,[A+BY (IIestevident quelesoscillations d'unpendule magnetique infinimentpetitautour d'unpoint quineposs^de aucunepro- prietydemaximum oudeminimummagnetique,seferont dans destemps proportionnelsauxracines carrees deslongueurs,et suivront ainsi lesmemes loisquelependule ordinaire, simple oucompose.) "Ces conclusions sontapplicablesaux oscillations d'un petit corpsd'une nature quelconquenon cristallise. Si//-est positif,c'est-a-dire silecorpsestparamagnetique,lesposi- tionsd'equilibrestablecorrespondronta^=ou6=7r,c'est- a-dire setrouveront' surTaxe. Siaucontraire, fiestnegatif, c'est-a-dire silamati^re estdiamagnetique,lespositions d'equilibrestablerepondronta^=Jttet^=ftt,etsetrouveront dans leplan perpendiculaireaI'axe, dans leplan Equatorial du champ magnetique. "SiTonassemble une serie departiculeslelongdela ligneOM, etsilepouvoir inductif, paramagnetiqueoudia- magnetique,estassez faiblepour qu'ellesn'exercentpoint une influence sensible lesunes surlesautres, chacune d'elles sera 526 AMathematical Theory ofMagnetism. [xxxvi. influenc^e comme sielle ^tait isolee. Mais ila6i^d^montr^ quesielles sontformees delameme substance, leurmouve- mentangulairesera lememe sionlesderangedeleurposition d'equilibredelamemequantity angulaire,etqu'ellesnesoient pasunies Tune aI'autreparunlienrigide. Nous enconcluons quelesoscillations d'uneaiguille (c'est-a-dired'une barre dont lalongueurestunmultipletres-^lev^ desdimensionslat^rales) d'une substance paramagnetiqueoudiamagnetiquenon cristal- lis^e,autour d'unpointfixeplac^aucentre duchamp magnetique, sontinddpendantesdesamasse etdesalongueur,etquela demi-p^rioded'unepetiteoscillation estegaleatt*/.„..V fj,(^A.-f-Jj) "II est clair quelesoscillations d'une barre cristallis^e ounon, seront independantesdesdimensions lat^rales, pourvu quecelles-ci soient tres faibles comparativement asalongueur, etqu'il n'yaitpointd'influence inductive sensible exercde entre sesdiversesparties; et,parconsequent, quediversesaiguilles prismatiquesdelamemelongueur (memesicettelongueurest assezgrande pour quelesconsiderationsprdc^dentessoient inapplicables),etd'une substance semblable etdisposeesembla- blement, soitquellesoitounon cristallisee, oscilleront dans le memetemps, quelquesoitleurpoids. Cen'estqu'ades dif- ferences dansI'arrangementcristallin semblables acelles sui lesquelles M.Matteucci aport^ I'attention, etnonpasades differences depoids, qu'ilfaut attribuer lesvariationsqu'ila observ^es dans lespe^riodesd'oscillations dediversesaiguilles cristallisees dememelongueur. ^'Les limites delalongueurd'uneaiguillenon cristalline oscillant autour ducentre d'unchamp magnetiqueendecjades- quelles onpent appliquerlesresultatsprecedentsavec une suffisante approximation, dependentdesdimensions etdela forme deI'aimant, etenpartTculierdeladispositiondeses poles. Onpentobserverqu'une aiguille paramagnetiqued'une trop grande longueuroscillera certainementplusrapidement quelath^orie neI'indique,etqu'une aiguille diamagnetique oscillera probablementd'autantpluslentement quesalongueur seraplusgrande,sisalongueuresttellequelesEquations pr^- cedentes nepuissent repr^sentersesmouvements avec une rigueursuffisante. TSx Yi.]Oscillation ofInductively Magnetized Needles. 527 "La determination desmouvements debarres cristallines oudemasses d'une formequelconque,dans lescirconstances indiquees parM.Matte ucci,pents'effectuer sans difficult^ en appliquantlatli^orie deI'inductionmagn^tiquedans lescorps cristallins, dont lesdeveloppements mathdmatiquesont6t6 soumis, en1850, a1'AssociationbritanniqueaEdimbourg,et quia^t^publi^e depuisdans lePhilosophical Magazine. On trouvera dans ceMemoire*, etdans ceuxque j'aicit^splus haut, lapreuve quelesph^nomenes dedirectionquepr^sente lebismuth cristalliseplac^entre lespolesd'un aimant, et observes parM.Matteucci, trouvent leurparfaite explication dans latendance quepossedentlesmolecules aseporterdes pointsouI'intensitemagn^tiqueestlaplusgrandeversceux oil elleestlaplusfaible;combin^e avec latendance directrice qui dependdecedernier ^Mment, etqui,ainsi que I'indiquela thdorie, rdsulte d'unein^galite dupouvoirinductif dans les diverses directions d'un cristal. "J'ai lieud'esp^rer quelesraisonnements etlesde- veloppementscontenus dans cesM^moiresparaitrontsuffisants pourm'autoriser aexprimeruneopinioncontraire acelleque M.Matteucci aavancee relativement auxphenomenesremar- quables qu'ilaobserves. "Puisque j'aioccasion deparler dupassage {Comptes Rendus,t.XXXVI.p.743)ouM.Matteucci attribue aM. Tyndallladecouverte d'uneinegalit^ dans larepulsion diamag- n^tique presentee parlescristaux, suivant lapositiondeI'axe ducristal, jecrois n^cessaire defaire remarquer quecette importantedecouverte estdueaM.Faraday. M.Tyndallen rendantcomptedesesrecherches surcesujet (Philosophical Magazine, septembre 1851),cite lestravaux anterieurs deM. Faraday (Boyal Society, novembre1850). Dans leparagraphe 2839 deceMdmoire, M.Faraday^nonce cette loicomme une conjectureenrann^e 1848(§2588) ;mais, fauted'exp^riences Buffisantes,ilnes'yappesantit point:ilrevient surcesujet,a propos dubismuth cristalHs^, dans leparagraphe2839 dece M^moire, etr^ussit ensuite averifier sesprevisions parI'ex- p^rience (§2841). Plus tard, ausujetduspathcalcaire *Sur latheorie derinduction magnetique dans lessubstances cristallisees etuon cristallisees. Philosophical Magazine; mars 1851[§§647... 668above]. 528 AMathematical Theory ofMagnetism. [xxxvi. (§2842),ilditnotamment quesiVaxeoptiqveestd'ahordplace parallelement aVaxemagn^tique, puisperpendiculairement acet axe, lecorpsseraplus diamagnetiquedans lapremiere position quedans laseconde, etindiquelesd^fauts desadisposition par suitedesquelsilnepeutverifier cetteproposition.M.Tyndall, endisposant I'exp^rienceavecplusdeprecautions,rdussit ken donner lademonstrationexperimentale. Dans lacommunica- tionaI'Associationbritannique que j'aicit^eplus haut, j'ai faitremarquer moi-meme, des lemois d'ao^t 1850, qu'ildoit exister des differences dans lespouvoirsinductifs descorps cristallins suivant lesdiverses directions, etquec'^tait lala seuleexplication possibledesph^nomenesdedirection cristallo- magn^tiqued^couverts parPliicker etFaraday,etdans cette occasionjedonnai lesrdsultatsparticuliersaubismuth etau spathcalcaire que1'experienceaconfirmesdepuis.C'est Poisson, lepremier, quiaprevulesphenomenes cristallomag- netiques,dusaune difference dans lespouvoirsinductifs dans lesdifferentes directions d'uncorps cristallise; mais ilne cherchapoint averifier latheoriequ'ilemit alors, parce qu'il neconnaissaitpoint decorps auxquelselleputetreapplicable. Lesexperiencesactuelles deM.PlUcker etdeM.Faradayont etesuggerees parleMemoire quelutPoisson, a1'Academic, le2fevrier 1842. "Quand lepouvoirinductif dessubstances est tel,que lesdiversespartiesexercent une actionmagnetiquemutuelle lesunes surlesautres, onnepeut plussupposer, comme nous Favons fait,queI'aimantagitsurcliaque particule comme sielle etait isoiee. Leferdoux offreI'exempled'une sub- stancepareille (lecoefficientfin'est, pourcecorps, qu'un peu inferieur ^t-)'cette influence mutuelle est icilacause de phenomenes tres-remarquables,surtoutquandonfaitlesobserva- tions surdesmassesallongees. LaNoteci-apresserapportea cettepartie dusujetetauxexperiencesdont elleaeteI'objet. J'ajouteraiiciladescriptiond'uneexperience analogueacelle quefitM.Matteucci avecdescubes debismuth cristallise, fixes aubout d'uneaiguilledesulfate dechaux dont lesclivages plans etaientperpendiculaires alalongueur:dans laposition d'equi- libre stable, cescubes etaient avssirapprochps quepossibledes I VI.] Equilibrium ofFerromagneticBars, 529 polesderaimant. Fixez deux finesaiguillesdeferdouxauxdeux bouts d'unetigedroite enbois(outoute autre substance nonsen- siblementmagn^tique)etperpendiculairementacettetige,sus- pendue parun filaucentre duchamp magn^tique,entre lesdeux poles,etdquilibr^e demaniere kcequeleplandesaiguillesde fersoithorizontal. Silatigeenboisn'estpastroplongue,ellese placera perpendiculairement alalignedespoles,c'est-a-dire que lesaiguilles deferdoux, pouretreendquilibre stable, devront ^treaussi loinquepossibledespolesdeI'aimant. Cette experience peutetre faiteavecfacility, aumoyend'unsinaple aimant d'acier enferacheval. Ler^sultat observe estdualatendance qua chacune desdeuxaiguillesdeferdoux, envertu desactions mutuelles desesdiffdrentesparties, kseplacer parallelement aladirection desforces. Leresultat deM.Matteucci doit ^tre attribud alatendance quepossede chaquecubedebismuth, en vertu desastructure cristalline, aplacersonplandeclivage perpendiculairement aladirection delaforce.' Note.—DeVequilibre desmassesallongeesdesubstances ferromagne- tiques dans deschamps deforce m^gnetiqueconstante etvariable. Lefait,decouvert d'abord experimentalement parGilbert, qu'une barre deferdoux, fixee asoncentre degravite dansunchamp mag- netique uniforme, seplace parallelementaladirection desforces, n'est passuffisammentexplique quandonI'attribue uniquementalavertu inductivequepossedeleferdouxdesetransformer momentanement en unaimant semblable aunaimant permanentdans saposition d'equi- libre stable. Carlamemeexplicationdevraits'appliqueraunerangee despheres deferdoux assemblees aI'aide dejoints nonmagnetiques ; cependant untelassemblage nepresenterait pointdephenom^nede direction(puisqu'aucunedesspheres partiellesnerecevi-aitFaction d'une force oud'uncouple resultantmagnetiques)amoins quelesspheres n'agissentlesunes sur lesautres, etqu'ilneseproduiseainsi des changements dans leur etatmagnetique.IIfautdoneadmettrequ'il s'operedesactions mutuelles dans lesdifferentespartiesd'unerangee despheres oudecubes, ousimplementdansunebarre,siTonveut arriver alavraie theorie desphenomenesdedirection. L'auteur decettecommunication, aI'aide deraisonnements demecan- iqueelementaire fondes surlesprincipeslesmieux etablis deI'induction magnetiqueetdeTactionmagnetiqueengeneral,faitvoirqu'unemasse allongee, ferromagnetique oudiamagnetique, placeedansunchamp magnetique uniforme, tendaseplacer parallelementaladirection des forces, pourvu qu'elle nesoitpointcristallisee :eneffet,quandelleest ferromagnetique,elleestmoins facilementmagnetisee, quandonlaplace T.E. 34 530 AMathematicalTlieory ofMagnetism. [xxxvi. dans laposition ci-dessus, quedans laposition perpendiculaire ;lecon- traire alieuquandelleestdiamagnetique. Mais pour toutes lessubstances, desdeux classes, quipossedent un aussi faiblepouvoirinductifquecertainscorps diamagnetiques connus, cette tendancequiresulte d'actions mutuelles interieures nepeutetre verifieepar1'experience. Toutes lestendances directrices desbarres diamagnetiques quiontetejusqu'ici,etsansdoute toutes cellesqui seront encore decouvertesparexperience,sontdues soitaquelque pro- priete cristallomagnetique,soitalatendance desextremites oudes autresportions mobiles achangerdeplace,demanik^e kcequeles moleculesoccupentlespositionsd'intensitemagnetique minimum, ou acesdeux causes reunies, plutot qu'auxeffets inductifs mutuels. En etudiant leseffets d'une forcemagnetiquevariable surlespositions d'equilibre d'une baiTeferromagnetique,I'auteur citeI'admirable ex- plication donneeparFaraday,delaloirelative auxtendances direc- trices d'unesphereoud'un cubediamagnetiques,etrappelle que precedemmentilafaitvoir, qu'appliquee auxsubstances non cristal- lisees engeneral,avec lesmodifications convenables dans lecasouelles sontferromagnetiques,cette loiexprimeavecuneadmirablesimplicity lesresultats d'un travailmathematique comprenant quelques-unsdes principeslesplusremarquablesd'une theorie deI'attraction. D'aprescetteloi,onvoitqu'en supposant unemasse ferromagnetique divisee encubestr^s-petits, chacune decespartiestendrait d'elle-meme vers lapositiond'intensite maximum, etqu'ainsilapositiondelamasse enti^re, dans lecasd'une forcemagnetique variable, serait dueen partie acette tendance etenpartie auxactions interieures mutuelles quiagissent seules, quandlaforce estconstante. L'auteur acherche a verifier, parI'experience,cesvuestheoriques.IIaemployeunbarreau d'acier formant aimant etdes filsminces deferdoux, fixesdans diverses positions surunetigeenbois;latigeenbois seplagaitdefagon que lesfilsdeferayantleur directionparalleleacelledelaforce,lesmole- cules fussent dans lespositionsd'intensite minimum. Dansuneautre experience, unanneauelectromagnetique,forme defilsdecuivre isoles, roulescinquantefoisautour d'un cercle d'undiametreegala25centi- metres, etait fixedansunplanverticalperpendiculairementaumeridian magnetique ;unsimplecubedeferdoux, place excentriquementsur unplateau decarton mince suspenduasoncentre parun fildesole naturel dans leplandelaforce, etait attire dans leplandeI'anneau; maisunesuite detrois aquatrecubesplacesaucontact alasuite les unsdesautres enlignedroite lelongdeI'axedesuspension,seplacait aussi loinduplanquepossibleenvertu delatendance d'une masse allongee,aplacersaplusgrande dimension parall^lementaladirection delaforce. Deux cubesplacesaucontact etaient enequilibre stable dans leplandeI'anneau oudans uneposition oblique,ou aussi loinque possibledeI'anneau, suivant ladistance variable alaquelle onlesplagaitsur leplateauaupointdesuspension.Des barresegalesetsemblables, formeesparunmelangedecire etde limaille deferdoux etdepuissances diamagnetiques difierentes, suspen- duessuccessivementparleurpoint milieu, sefixaient dansdespositions diverses :cellesquipossedaientleplusgrand pouvoir ferromagnetique '"sXVII.] Elementary ProofsofFundamental Tlieorems. 581 seplagaient perpendiculairement auplandeI'anneau oudans ladirec- tiondesforces;lesaiitres, celles quicontenaient moins defer,avaient leurposition d'equilibrealafoisdans leplandeI'anneau etperpen- diculairement aceplan;etcellesquiencontenaient encore moins, etaient enequilibre uniquementdans leplandeI'anneau. Cesdernieresexperiencesfont voird'une fa^on trfes-remarquablela part qu'ilfautfaire, dans cetordre deplienom^nes, aux actions mutuelles interieures, etenmeme tempsalavariation delaforce [compare original, beinglastsentence of§669].Desmelangesde sable etdelimaille deferdoux, places dans destubes deverre, feraient lem^me eiFetquelesbarreaux dontnousvenons deparleretvaudraient peut-etre mieux dans certains cas. XXXVII.Elementary DemonstrationsofPropositionsinthe Theory ofMagnetic Force. [From thePhilosophical Magazine, April 1855.] 671.Bef1.The lines offorceduetoanymagnetorelectro- magnet,orcombination ofmagnetsofanykind, arethelines thatwould betraced byplacingthecentre ofgravityofavery small steel needle, perfectlyfree toturn about thispoint,in any positionintheirneighbourhood, andthencarryingit alwaysinthedirectionpointed bythemagneticaxis ofthe needle. Remark Exceptinthecase ofsymmetrical magnets,the lines offorce willgenerallybelines ofdouble curvature. Def2.The lines ofcomponentforce inanyplanearethe lines traced byplacingthecentre ofasteel needleany- where inthisplane, andcarryingitalwaysinthisplanein thenearest direction tothatpointed byitsmagneticaxis;that is,thedirection oftheorthogonal projectionofthemagnetic axisontheplane;orthedirection that thesteel needle would point with itsmagneticaxis ifplacedwith itintheplane,and left free toturn about anaxisthroughitscentre ofgravity perpendiculartotheplane. 672.Prop.I.Ifthelineofcomponent magneticforcethrough anypointinaplanebecurved atthispoint,theforce willvary inalineperpendiculartothelineofforce initsplane, increasing inthedirection towards thecentre ofcurvature. LetEABF(Fig. 1)bealineofcomponentforce intheplane 34—2 532 AMathematicalTheory ofMagnetism, [xxxvii. ofthediagram, and letGCDH beanother nearit,eachand all between thembeing curved inthesame direction, thearrow head oneachindicatingthewayanorthpolewould beurged. LetAC,BDbelinesdrawnperpendiculartoallthe lines of componentforcebetween these two. Because ofthecurvature ofthese lines, thelinesAGandBD(whether straightorcurved) must besoinclined tooneanother thattheportion CDcut oflf from the last shall belessthan theportionABcut offfrom the first. Letanorthpoleofaninfinitely thin,uniformly and longitudinally magnetized bar,ofwhich thesouthpoleisata great distance from themagnets,becarried fromDtoGalong thelineofcomponent forcethroughthesepoints,fromGtoA perpendiculartoallthe lines offorce traversed, fromAtoB again alongalineofforce, andlastly, fromBtoDperpendi- cular tothelines offorce. Work must bespenton itin carryingitfromDtoG,andwork isgainedinpassingitfrom AtoB.Then, because nowork iseithergainedorspentin carryingitfromGtoAorfromBtoD,theworkgainedin moving alongABcannot exceed theworkspentinthefirstpart ofthemotion, orelseweshould have[compare §622above]a perpetual developmentofenergy from nosource*, bysimplylet- tingthecycleofmotion berepeatedoverandoveragain:andthe *[Note added March 26,1855.]—Itmight beobjected, thatperhaps the magnet, inthemotion carried onasdescribed, would absorb heat,andconvert itintomechanical effect, andtherefore that there would benoabsurdity in admitting thehypothesis ofacontinued development ofenergy. This ob- jection, which hasoccurred tomesince thepresent paper was written, is perfectly valid against thereason assigned inthetext forrejecting that hypothesis ;butthesecond lawofthedynamical theory ofheat (the principle discovered byCarnot, andintroduced byClausius andmyself into thedy- namical theory, ofwhich, after Joule's law,itcompletes thefoundation) shows thetruereason forrejecting it,and establishes thevalidity ofthe remainder ofthereasoning inthe text. Infact, theonly absurdity that would beinvolved inadmitting thehypothesisthat there iseither more or lesswork spentinonepart ofthemotion than lostintheother, would be thesupposition thatathermo-dynamic engine could absorb heat firom matter initsneighbourhood, and either convert itwhollyintomechanical effect, or convert apart intomechanical effect andemit theremainder intoabodj^ ata higher temperature than thatfrom which thesupplyisdrawn. The inves- tigation ofanewbranch ofthermo-dynamics, which Iintendshortly to communicate totheKoyal Society ofEdinburgh, shows that themagnet (if ofmagnetized steel) does really experienceacoolingeffectwhen itspoleis carried fromAtoB,andwould experience aheating effect ifcarried inthe reverse direction. [See Art. xlviii., partvii.ofmy"ReprintofMathematical andPhysical Papers," (Vol. i.,page 291)]. Butthesame investigation alsoshows thatthemagnet must absorb justasmuch heat tokeepupitstemperature during themotion ofitspoleicith theforce along AB,asitmust emit tokeepfrom rising intemperature when itspoleiscarried against theforce, alongDC. I XXXVII.] Faraday's Lawdeduced fromLawofEnergy. 533 workspent alongDGcannot exceed thatgained fromAtoB,or wemight have aperpetual developmentofenergy fromno Fig. 1. source, merely byreversingthemotion described, andsorepeat- ing.Theworkspent andgainedinthemotionsalongDGand ABrespectively must therefore beexactly equal. Hence the meanintensityoftheforcealong CI)^which istheshorter ofthe twopaths, must exceed themeanintensityofthe forcealong theother; andtherefore theintensityoftheforce increases fromPintheperpendiculardirection towards which the concavityofthelinethroughitisturned. 673. Prop.II.Theaugmentationofthecomponentforce in anyplaneataninfinitelysmall distance fromanypoint, towards thecentre ofcurvature oftheline ofthecomponentforce through it,bears tothewholeintensityatthispointtheratio oftheinfinitelysmall distance considered, totheradius of curvature. If,inthediagramforthepreceding proposition, wesupposeABandCDtobeinfinitelynear oneanother, andeach in- finitely short, theywillbeinfinitely nearlyarcsofcircles with infinitely nearly equalradii. Hence the dilBference oftheir lengths must bear toeither ofthem theratio ofthedistance between them tothe radius ofcurvature. But themean intensitiesalong these lines must, accordingtothepreceding demonstration, beinverselyastheirlengths, andhence the excess ofthemeanintensityinGDabove themeanintensity inABmust bear tothelatter theratio oftheexcess ofthe lengthofABabove thatofCDtothelatterlength ;thatis,as hasbeenshown, theratio ofthedistance betweenABandGD totheradius ofcurvature. 674. Prop.III.The totalintensitydoesnotvaryfromany 534 AMathematicalTheory ofMagnetism. [xxxvii. pointinamagneticfield toapoint infinitelynear itinadirec- tionperpendiculartotheplaneofcurvature ofthelineofforce throughit. 675. Prop.IV.The totalintensityincreases fromanypoint toapoint infinitelynear itinadirection towards thecentre of curvature oftheline offorcethrough it,byanamount which bears tothetotalintensity itself, theratio ofthedistance be- tween these twopointstotheradius ofcurvature. These twopropositionsfollow from thetwothatprecede thembyobviousgeometricalconsiderations. Theyareequivalenttoasserting,that ifX,F,Zdenote the components, paralleltofixedrectangular axes, oftheforce atanypoint whose co-ordinates are(x,y,z),theexpression Xdx+Ydy+Zdzmust bethedifferential ofafunction of three independentvariables. ExaminationoftheActionexperienced hyaninfinitelythin uniformly andlongitndinally Magnetized Bar, placedin anon-uniformField ofForce, with itslengthdirectalonga LineofForce. 67G. LetSNbethemagnetized bar,andST,NTstraight linestouchingthelineofforce inwhich, byhypothesis,itsex- tremitieslie,andPapoint onit,midway between them. The resultant force onthebarwillbetheresultant oftwoforces pullingitsends inthelinesST,NT', Ifthese twoforces were equal (astheywould beiftheintensityofthe field didnot varyatallalongalineofforce, asforinstance when thelines offorce areconcentric circles, astheyarewhensimply due to acurrent ofelectricity passing alongastraightconductor;or ifPwere inasituation between two dissimilarpolessymme- trically placed oneach side ofit),theresultant force would clearlybisect theangle between thelinesTS,T'N,andwould therefore beperpendiculartothebarandtothelines offorce inthedirection towards which theyarecurved;that is(Prop. lY.),would befromplacesofweaker toplacesofstronger force, perpendicularlyacross thelines offorce. Ontheother hand,ifthelineofforce through Phasnocurvature atthis point,ornosensible curvature asfarfrom itasNand^S',the t.., Faraday'sLaw. 535 linesiVTandST willbeinthesamestraight line,andthe resultant force onthebarwillbesimplythe excess ofthe forceononeendabove thatontheotheractinginthedirec- tionofthegreater ;andsince inthiscaseCProp. IV.)there is I novariation oftheintensityoftheforce inthe field ina directionperpendiculartothelines offorce, theresultant force experienced bythebar isstillsimplyinthedirection inwhich theintensityofthe field increases, thisbeing nowadirec- tion coincident with aline offorce.Lastly,iftheintensity increases mostrapidlyinanobliquedirection inthefield, from Pinsome direction between FSandFF\ theremustclearly beanaugmentation (a"component" augmentation) fromF towards F';andtherefore(Prop. IV.)thelinethroughFmust becurved, with itsconcavitytowards F',and alsoa"com- ponent" augmentationfromNtowards S,and therefore the endSmustexperienceagreaterforce than theend -V. It follows thatthemagnetwillexperiencearesultant forcealong some lineintheangleSNF\ thatis,onthewhole fromplaces ofweaker towardsplacesofstronger force, obliquelyacross the lines offorce. 677. Frop.V.(Mechanical Lemma.)—Two forcesinfinitely nearly equaltooneanother, acting tangentiallyinopposeddirec- tions ontheextremities ofaninfinitelysmall chord ofacircle, areequivalenttotwo forcesrespectively alongthechord and perpendiculartoitthroughitspointofbisection, ofwhich the former isequaltothedifference between thetwogivenforces and actsonthe side ofthegreater ;andthe latter, acting towards thecentre ofthe circle, bears toeither ofthegiven forces theratio ofthelengthofthearctotheradius. 536 AMathematical Theory ofMagnetism. [xxxvii. Thetruth ofthispropositionissoobvious aconsequenceof "theparallelogramofforces," that itisnotnecessarytogivea formal demonstration ofithere. 678. Prop.YI.Avery short, infinitely thin, uniformly and longitudinally magnetized needle, placed with itstwoends in onelineofferee inanypartofamagnetic field, experiencesa forcewhich istheresultant ofalongitudinalforceequaltothe difference oftheforcesexperienced byitsends, andanother forceperpendiculartoitthroughitsmiddlepoint equaltothe difference between theforceactually experienced byeither end, andthatwhich itwouldexperienceifremoved, intheplaneof curvature ofthelineofforce, toadistanceequaltothelength oftheneedle, ononesideortheother ofitsgiven position. iV>S^beingthebarasbefore, let/denote theintensityofthe force inthefield atthepoint occupied byN,Itheintensityat ;Sf,e7theintensityatPontheline offorcemidway between ^ and iV,andJ'theintensity p atapoint P',atadistance pPP'equaltothelengthof"^ thebar,inadirectionper- pendiculartothe line of force. Then ifmdenote tl thestrength ofmaornetism ofthebar,mlandmT will bethe forces onitstwo Fig. 3. extremitiesrespectively. Hence bythemechanical lemma, theresultant ofthese forces willbethesame astheresultant ofaforcem{J—T) acting alongthebarinthedirection BN, andaforceperpendiculartoittowards thecentre ofcurvature, bearingthesame ratio toeithermlormT,ortomJ(which istheir mean, and isinfinitely nearly equaltoeach ofthem), asiV>Sitotheradius ofcurvature, or(byProp. II.)theratio of theexcess oftheintensityatP'above that atPtotheinten- sityateither, that istheratio ofJ'—JtoJ",and therefore itself equaltom{J'—J).Thebarthereforeexperiencesaforce thesame astheresultant ofm{I— I'}acting alongitfromS towards if,andm{J'—J)perpendicularlyacross ittowards P', throughitsmiddlepoint. 679. Cor.Thedirection oftheresultant forceonthebar is IXVII.] Faraday's Law, 537 that inwhich thetotalintensityofthe field increases most rapidly ;or,which isthesame,itisperpendiculartothesur- faceofnovariation ofthetotalintensity. Prop.yil.Theresultant forceonaninfinitelysmall magnet ofanykindplacedinamagnetic field, with itsmagneticaxis alongthelines offorce,isintheline ofmostrapidvariation ofthetotalintensityofthe field,and isequaltothemagnetic moment ofthemagnet multiplied bytherate ofvariation of thetotalintensity perunit ofdistance;beinginthedirection inwhich theforce increases when themagneticaxis is" direct," (that is,inthepositionitwould restinifthemagnet were free toturnabout itscentre ofgravity). Cor. 1.Theresultant forceexperienced bythemagnetwill beinthecontrary direction, thatis,thedirection inwhich the total intensityofthe field diminishes mostrapidly, when itis heldwith itsmagneticaxisreversealongthelines offorce of thefield. 680. Cor. 2.Aball ofsoft iron, orofany non-crystalline paramagnetic substance, heldanyhowinanon-uniformmagnetic field, oraballorsmallfragmentofanyshape,ofanykind of paramagneticsubstance whethercrystallineornot, leftfreeto turnabout itscentre ofgravity,willexperiencearesultant force inthedirection inwhich thetotalintensityofthefieldincreases mostrapidly,andinmagnitude equaltothemagnetic moment ofthemagnetizationinduced inthemassmultiplied bythe rateofvariation ofthetotalintensity perunit distance inthe lineofgreatestvariation inthe field. Forsuch abodyinsuch apositionisknown tobeamagnet byinduction, with its magneticaxisdirectalongthelines offorce. 681. Cor. 3.Aballofnon-crystalline diamagneticsubstance heldanyhowinamagnetic field, orasmall barorfragmentof any.shapeofanykind ofdiamagnetic substance, crystallineor non-crystalline,heldbyitscentre ofgravity,but left free to turnabout thispoint, experiencesthesame resultant force asa small steel orother permanent magnetsubstituted forit,and heldwith itsmagneticaxisreverse alongthelines offorce. For Faradayhasdiscovered, thatalargeclass ofnatural substances inthestated conditions experiencenoother action than a 538 AMathematicalTheory ofMagnetism. [xxxvii. tendency from places ofstronger towardsplaces ofweaherforce, quite irrespective ofthedirections thelinesofforcemay have, andhehascalled such substancesdiamagnetics. 682. Cor. 4.Adiamagnetic,heldbyitscentre ofgravity but free toturnabout thispoint, must react uponothermagnets with thesame forces asasteel orothermagnet substituted in itsplace,andheld with itsmagneticaxis reversealongthe lines offorceduetoallthemagnetsinitsneighbourhood. 683. Cor. 5.Anyoneofarow ofballs orcubes ofdiamag- netic substance held inamagneticfieldwith thelinejoining their centres alongalineofforce, isinalocalityoflessintense forcethan itwould beiftheothers wereremoved;butanyone ballorcube oftherow, ifheldwiththelinejoiningtheir centres perpendicularlyacross thelineofforce, isinalocalityofmore intense forcethan itwould beiftheothers wereremoved. 684. Cor. 6.When arowofballs orcubes, orabar,ofper- fectly non-crystalline diamagnetic substance, isheldobliquely across thelines offorce inamagnetic field, themagneticaxisof each ballorcube, orofeverysmallpartofthesubstance,isnearly inthedirection ofthelines offorce, butslightlyinclined from thisdirection towards thedirectionperpendiculartothelength oftherow orbar. Hence, since themagneticaxis ofevery partdiffersonlyalittle frombeing exd^cilj reversealongthe linesofforce,thedirection oftheresultant ofthecouples with which themagnets,towhich thefield isdue, actontheparts oftheroworbarmustbesuch astoturn itslength alongthe lines offorce. 685. Cor. 7.Thepositionsofequilibriumofarowofballs or cubesrigidly connected, orofabarofperfectly non-crystalline diamagnetic substance, freetomove about itscentre ofgravity inaperfectlyuniform field offorce, areeither withthelength alongorwith thelength perpendicularlyacross the lines of force :positionswith thelength along the lines offorce are stable;positionswith thelength perpendicularlyacross the lines offorce areunstable. 686. Cor. 8.Themutual influence and itseffects, referred to inCors. 5,6,7,issoexcessively minute, that itcannotpossibly havebeensensibly concerned inanyphaenomcnathathaveyet IXVII.] Faraday's Law. 539 been observed;and itisprobablethat itmayalways remain insensible, even toexperiments especiallydirected totest it. Fortheinfluence ofthemostpowerful electro-magnets induces thepeculiar magneticcondition ofwhichdiamagneticsare capable,tososlightadegreeastogiverisetoonlyvery feeble, scarcely sensible, mutual forcebetween thediamagnetic andthe magnet ;andtherefore themagnetizinginfluence ofaneigh- bouring diamagnetic, which couldscarcely,ifatall,beobserved onapieceofsoft iron,must beinsensiblysmall onanother diamagnetic. 687. Cor. 9.Allphaenomenaofmotion thathave been ob- served asproducedinadiamagnetic bodyofanyform orsub- stance bytheaction offixed magnetsorelectro-magnets,are duetotheresultant offorcesurgingallpartsofit,andcouples tendingtoturnthem;theforce andcouple acting oneach smallpartbeing sensiblythesame asitwould beifallthe otherpartswereremoved. 688. Cor. 10.Thedeflecting power (observed andmeasured byWeber)with w^hich abarofnon-crystalline bismuth, placed verticallyascore inacylinder electro-magnet (ahelixconveying anelectriccurrent), urgesamagnetizedneedle onalevel with either ofitsends, isthereaction ofatendencyofallpartsofthe baritselffromplacesofstronger towardsplacesofweaker force initsactual field. Thepreceding investigation, leadingtoProps.YI.andVII., isthesame(only expressedinnon-analytical language)asone which was firstpublishedintheCambridge andDublin Mathe- matical Journal, May 1846^.[§§ 638—640 above]. The chief conclusions nowdrawn from it,withparticularsnotrepeated, were stated inapaperentitled"Remarks ontheForcesexperi- enced byinductively magnetized FerromagneticorDiamagnetic Substances," inthePhilosophical MagazineforOctober 1850 [Article XXXIV, above]. Glasgow College, March 16,1855. 540 AMathematical Theory ofMagnetism, [xxxviii. XXXVIII. Correspondence withProfessor Tyndall. Letter toProfessor Tyndallonthe"Magnetic Medium^' andon theEffects ofCompression. [From thePhilosophical Magazine, April 1855.] [Editorial.^—Thefollowingletter wasreceived afewdays ago.Itwas notwritten forpublication,butthesubjectto which itrefersbeingofgeneralinterest atpresent,Iven- tured tosuggesttoProfessor Thomson thedesirableness of havingtheletterprinted.Thisheatonceagreedto.With the exceptionofaparagraph relatingtomatters ofapurely private nature, theletterappearsasIreceived it. John Tyndall. March 24,1855. 2College, Glasgow, March 12,1855. 689.Mydear Sir,—Allowmetothank youfortheabstract ofyourletter onmagnetism, audthecopyofyourletter toMr Faraday, which Ihaverecentlyreceived fromyou,andhave readwithmuch interest. Iam stillstrongly disposedtobelieve inthemagneticcharacter ofthemediumoccupying space, and Iamnotsurebutthatyourlastargumentinfavour ofthe reversebodily polarityofdiamagnetics maybeturned to supportthetheoryofuniversallydirectpolarity. There isno doubt butthat themediumoccupying interplanetary space, andthebestapproximationstovacuum which wecanmake, haveperfectlydecided mechanicalqualities, andamong others, that ofbeingable totransmit mechanicalenergyinenormous quantities (aplatinum wire, forinstance, keptincandescentby agalvaniccurrent inthereceiver ofanair-pump,emits tothe glassandexternal bodies thewhole mechanical value ofthe energyofcurrentspentinovercomingitsgalvanic resistance). Some ofthesepropertiesdiffer but little from those ofairor oxygenatanordinarybarometricpressure. Why not, then, themagnetic property? (ofwhich weknow solittle thatwe have norighttopronounceanegative). Displacetheinter- planetary medium byoxygen, andyouhave aslightincrease ofmagnetic polarityinthelocalitywith adrawinginofthe lines offorce.Displaceitwith apieceofbismuth orapiece ofwood, andaslightdecrease ofmagnetic polarity throughthe XXXVIII.] CorrespondencewithProfessor Tyndall.541 localitytakesplace, accompanied byapushingoutofthelines offorce.Astate ofstrain bycompression may enhance, in thedirection ofthe strain, thatqualityofthesubstance by which itlessens themagnetizabilityofthespace fromwhich it displacesairor"ether;" justasasimilar statemay enhance, inthedirection ofcompression, theaugmenting powerofa paramagneticsubstance. 690.Bythebye,alongtime ago(rather more than ayear aftertheEdinburgh meetingoftheBritishAssociation)Ire- peatedwithmuchpleasure some ofyour compression experi- ments, andfound apieceoffresh breadinstantlyaffected by pressure,soasalwaystoturnthecompressedlineperpendicular tothelines offorce, towhatever form thefragmentwasreduced. Avery slight squeezebetween thefingers wasquite enoughto producethisproperty,oragaintoalter itsoastomake anew lineofcompressionsetequatorially.Irepeateditafewdays agowith thesame results, andgotaball ofbismuth, too,to actsimilarly.Irememberformerly findingthebread attracted asawhole, instead ofbeing repelled,asIexpectedfrom your results. Isuppose, however, thismust have resulted from someferruginous impurities,which itmay readilyhavegot either inthecourse oftheexperimentswithit,orinthebaking. Imean totrythisagain*. 691. Idonotquite admit theargument youdraw fromyour compression experiments regardingthe effect ofcontiguityof particles, because infactweknownothingoftheactual state of themolecules ofastrained solid. Youhavemade outamost interestingfactregardingtheir magnetic bearings ;butexperi- ments areneither wanted, norcanbemade, toshow any sensible effect whatever ofthemutual influence ofarowof smallpiecesofbismuthplacednear oneanother, ortouching oneanother. Itisperfectly easytodemonstrate that itmust besuch astoimpairthe" diamagnetization"ofeachpiece when thelineoftherow isparalleltothelines offorce, andto enhance itwhen that line isperpendiculartothelines offorce, butineach case tosoinfinitesimallyminute adegree,astobe *Prof. Thomson's suppositioniscorrect; pure bread isrepelled bya magnetic pole. Imayremark that Iamatpresent engagedinthefurther examination oftheinfluence ofcompression, andhave already obtained numerous instructive results.—J.T.' 542 AMathematicalTlieory ofMagnetism, [xxxviir. wholly inappreciabletothemost refined tests thathave ever beenapplied. For letthelines offorcebeparalleltotheline shown inthefigure,and actonasteel needle inthemanner thererepresented. Then, whateverhypothesis betrue for n 5« s\ diamagnetism,there isnotadoubt butthateachpieceisacted on,andconsequently reacts, preciselyasapieceofsteelvery feebly magnetized,with itsmagneticaxis reverse tothat ofa steel needle free toturn, substituted forit,would do.Each pieceofbismuth therefore acts asalittlemagnet, havingits polarityasmarked inthediagram, would do.Hence the magnetizingforcebywhich themiddlefragmentisinfluenced islessthan ifthetwoothers wereaway (this beingsuch a force aswould beproduced byanorthpoleontheleft-hand sideofthediagram,andasouthpoleontheright).Itiseasily seen, similarl}^,that ifthelinejoiningthecentres beperpen- dicular tothelines offorce, themagnetizingforce onthespace occupied bythemiddlefragmentisincreased. Corresponding assertions aretrue fortheterminalfragments, althoughthe disturbinof effect willbelessonthem ineach casethan in themiddle one. Hence thedia- magnetizationofeach willbeen- feebled intheformer caseand enhanced inthelatter, bythepre- sence oftheothers. Itfollows, accordingtotheprincipleofsu- perpositionofmagnetizations,that ifthe line oftherowbe placed obliquelyacross thelines offorce, themagneticaxis of eachparticle,instead ofbeing exactly paralleltothelines of force, willbealittle inclined tothem, intheanglebetween their direction andthedirection transverse tothebar.The magnets causingtheforce ofthefieldmust actonthe little dia- magnets,eachwith itsaxisthusrendered somewhatoblique,so astoproduceonitastaticalcouple (asshown bythearrow- heads), andtheresultant ofthecouplesthusactingonthefrag- E[XVIII.] Correspondence withProfessor Tijndall 543 mentswill,when allthese areplacedonaframe, orrigidly connected, tend toturnthewhole mass insuchadirection asto placethelengthofthebaralongthelines offorce. Still, I repeat,this action, althoughdemonstrated with asmuch cer- taintyastheparallelogramofforces,issoexcessivelyfeeble as tobeabsolutely inappreciable. Afragmentofbismuth, ofany shape whatever, held inanyposition whatever inanykind of magnetic field, uniform orvarying mostintensely, onlyexhibits theresultant action ofcouplesonallitssmallpartsifcrystal- line,andofforcesacting always accordingtoFaraday'slawon them ifthefield inwhich itisplacedbenon-uniform. Some phaenomena thathavebeen observed aretobeexplained bythe resultant offorces fromplacesofstrongertoplacesofweaker intensityinthefield, others bytheresultant ofcouples depend- ingoncrystalline structure, andothers bytheresultant ofsuch forces andcouples co-existing;andnone observeddependat allonanyother cause. 692. Igaveaverybriefsummaryofthese views (whichI hadexplained somewhatfully and illustrated byexperiments onparamagneticsofsufficient inductivecapacitytomanifest the effecrts ofmutual influence, atthemeetingatBelfast)asan abstract ofmycommunication, forpublicationintheReportof theBelfastmeetingoftheBritish Association, whereyoumay seethem[§669above] stated, Ihope intelligibly. Theexperi- ments ontheparamagneticsarevery easy,andcertainlyexhibit someverycuriousphaenomena,illustrative oftheresultant effects duetotheattractionsexperienced bythepartsinvirtue ofavariation oftheintensityofthe field, and tothecouples theyexperience when their axes arediverted fromparallelismto thelines offorcebymutual influence ofthemagnetized parts. 693. Ihadnointention ofenteringonthislong disquisition when Icommenced, butmerelywished totryandbriefly point out,thattheassertions Ihavemaderegardingmutual influence aredemonstrable ineverycasewithoutspecial experiment,are confirmedamply byexperimentforparamagnetics,andare absolutely incontrovertible, aswell asincapableofverification byexperiment orobservation ondiamagnetics. —Believe me, yours very truly, William Thomson. Prof. Tyndall. 544 AMathematical Theory ofMagnetism, [xxxviii. OnReciprocal Molecular Induction: Letterfrom Professor TyndalltoProfessor W.Thomson^ F.R.S. [From thePhilosophical Magazine^ December 1855.] EoYAii Institution, Nov. 26,1855. 694.Mydear Sir,—Thecommunication from Professor Weber whichappearsinthepresent number ofthePhiloso- phical Magazine^ hasreminded me,almost toolate, ofyourown interestingletter onthesamesubject publishedintheApril number ofthisJournal. Adesire tofinish allIhave tosay uponthisquestionatpresentinduces metomake thefollowing remarks, which, had itnotbeen forthecircumstancejust alludedto,mighthavebeenindefinitelydeferred. With reference tothemutual action ofarowofbismuthpar- ticles, yousaythat "it isperfectly easytodemonstrate that "itmustbesuch astoimpairthe' diamagnetization'when the "line oftherow isparalleltothelines offorce" (the"must,'' youwillremember,isputinitalics byyourself). From this you infer, that inauniform field offorce abarofbismuth would setitslength alongthelines offorce. Further onitis stated that thisaction is"demonstrated with asmuch certainty "astheparallelogramofforces;"andyouconclude yourletter byobservingthat"theassertions which I[yourself]havemade "are demonstrable ineverycasewithoutspecial experiment,... "andareabsolutely incontrovertible, aswell asincapableo1 "verification byexperimentorobservation ondiamagnetics." Most ofwhat Ihave tosayuponthissubject condensesf itself intoonequestion. Supposingacylinderofbismuth tobeplacedwithin ahelix, andsurrounded byanelectric current ofsufficient intensity: canyou say,withcertainty, what theaction ofeither end 01 thatcylinderwould beonanexternal fragmentofbismuth presentedtoit? Ifyou can, I,formypart,shallrejoicetolearn theprocess bywhich suchcertaintyisattained: but ifyoucannot,itwill Ithink, beevident toyouthat theverb"must" islogically "defective." Weknow thatmasfnetized iron attracts iron :weknow that iviii.] ReciprocalAction ofDiamagnetic Particles. 545 fcgnetizedironrepels bismuth :this, sofarasIcansee,is your only experimental groundforassumingthatmagnetized bismuth repels bismuth, andyetyouaffirm thatanaction deduced from thisassumption"isdemonstrated with asmuch "certaintyastheparallelogramofforces." DoInotstate the question fairly?Ican, atallevents, answer formyearnest wish todoso. Itisneedless toremind one sowellacquainted with the mentalexperienceofthescientificinquirer, thattheveryletters which youattach toyour sketch, page 291[ofPhilosophical Magazine, §691above], maytemptustoanactofabstraction—aforgetfulness ofapossible physical difference between then ofironandthenofbismuth—whichmayleadusverywide of thetruth. Theveryterm" pole"oftenpledgesustoatheoretic conceptionwithout ourbeingconscious ofit.Youarealsowell aware ofthedangerofshuttingthedooragainst experimental inquiryonanunpromising subject ;andwhen youapparently dothisinyour concluding paragraph,Isimply acceptitasa strongwayofexpressing your personal conviction, thattheaction referred toistoofeeble toberendered sensible byexperiment. — Believe me,dear Sir,mosttruly yours, John Tyndall. OntheReciprocal Action ofDiamagnetic Particles: Letterfrom Professor Thomson toProfessor Tyndall. [From thePhilosophical Magazine, January 1856.] Glasgow College, Dec. 24,1855. 695.Mydeae Sir,—Ihavebeen preventeduntilto-day, by apressureofbusiness, fromreplyingtotheletteryouaddressed tomeinthenumber ofthePhilosophical Magazine published at thebeginningofthismonth. Youaskmethequestion,''Supposingacylinderofbismuth "tobeplacedwithin ahelix, andsurrounded byanelectric "current ofsufficientintensity ;canyou say,withcertainty, "what theaction ofeither endofthatcylinder would beonan "externalfragmentofbismuth presentedtoit?" 696. Inanswer, Isaythatthefragmentofbismuth willbere- pelled from either endofthebarprovidedthehelix beinfinitely T.E. 35 546 AMathematical Theory ofMagnetism, [xxxviii, long,orlongenoughtoexercise nosensible directmagneticaction inthelocalityofthebismuthfragment.Icanonly saythis withthesame kind ofconfidence that Icansaythedifferent partsoftheearth'satmosphereattract oneanother. The con- fidence amounts inmyownmind toafeelingofcertainty.In everycase inwhich theforcesexperienced byalittlemagnetized steel needle heldwith itsaxis reversealongthelines offorce, andafragmentofbismuth substituted for itinthesame localityofamagnetic field, have beencompared, they have been found toagree.Inavastvarietyofcases, afragmentof bismuth hasbeenfound toexperiencetheoppositeforce tothat experienced byalittle ball ofiron, thatis,thesame force as alittle steelmagnetheldwith itsaxis reverse tothelines of force;andinnocasehasadiscrepance,orhave anyindica- tions ofadiscrepance, from thislawbeen observed. Ifeel, therefore, inmyownmind acertain conviction, thatevenwhen theaction issofeeble thatnoforce canbediscovered atallon thebismuth byexperimental tests, such inregardtosensi- bilityashave been hithertoapplied,thebismuth isreally acted onbythesame force asthatwhich alittle reversemagnet, ifonlyfeebleenough,wouldexperience when substituted in itsplace. Now there isnodoubt ofthenature oftheforce experienced bythesteelmagnet,orbyalittle ballofsoft iron, inthelocalityinwhich youputthefragmentofbismuth. One endofamagnetizedneedle willbeattracted, andtheother end repelled bytheneighbouringendofthebismuth bar;andthe attraction ortherepulsionwillpreponderate accordingasthe attracted ortherepelled partisnearer. There isthencertainly repulsion when thesteelmagnetisheld inthereverse direc- tion tothat inwhich itwould settle ifbalanced onitscentre of gravity.Ineverycase inwhichanymagneticforce atallcan beobserved onafragmentofbismuth, itissuch asthe steel magnetthus heldexperiences.Therefore Isayitisinthis caserepulsion. But itwillbeasmuch smaller inproportion totheforceexperienced bythesteel magnet,asitwould beif anironwirewere substituted forthebismuth core. Yet in thiscasetherepulsion onthebismuth isvery slight, barely sensible, orperhapsnotatallsensible when theneedle exhibits mostenergetic signsoftheforces itexperiences. Youknow ^^fcxviii.] ReciprocalActionofDiamagnetic Particles. 547 '^©urself, byyourownexperiments, howverysmall iseven the directive agency experienced byasteelmagnet placedacross thelines offorce due tothebismuth core. Youmayjudge howmuch lesssensible would betheattraction orrepulsionit wouldexperienceasaw^hole, ifheldalongthelines offorce; andthen think ifthecorrespondingforceexperienced bya fragmentofbismuth substituted forit,islikelytobeverified bydirectexperimentorobservation. Ithinkyouwilladmit that itis"incapableofverification," aswell as"incontro- vertible" byanycollation oftheresults ofexperimentshitherto made ondiamagnetics. Astotheconcluding paragraphofmy letter which youquote, youdomejustice when yousayyou acceptitasanexpressionofmy" personal conviction that the "action referred toistoofeeble toberendered sensible by " experiment."Iwillnotmaintain itsunqualified application toallthatcanpossiblybedone infuture inthewayofexperi- mental research totestthemutual action ofdiamagnetics undermagneticinfluence. Onthecontrary,Iadmit thatno realphysical agencycanberightlysaid tobe"incapableof "verification byexperimentorobservation;"andIwillaskyou tolimit thatexpressiontoexperimentsandobservations hitherto made, and tosubstitute fortheconcluding paragraphofmy letter thefollowingstatement[§686above], written forpubli- cation threedays later, andpublishedinthesamenumber of theMagazineasthat towhich youcommunicated myletter I{Phil. Mag., April 1855, p.247). "The mutual influence" between rows ofballs orcubes ofbismuth inamagnetic field, "and itseffects"ingivingatendencytoabarofthesubstance toassume aposition alongthelines offorce,"aresoexcessively "minute, thattheycannotpossiblyhave beensensiblycon- "cerned inanyphaenomenathathaveyetbeen observed;and "itisprobablethattheymayalwaysremain insensible, even "toexperiments especiallydirected totestthem." Iremain, mydear Sir,yours very truly, William Thomson. DrTyndall. 35—2 548 AMathematicalTheory ofMagnetism. [xxxix. XXXIX. InductiveSusceptibility ofaPolarMagnet. [March 1872. Nothithertoimhlished.'] 697. Itisprobablethateveryloadstone orsteelmagnet,or polar magnetofanykind, whateverdegreeofintrinsic mag- netization itmay possess,hasalsoasusceptibilityformagnetic induction, accordingtowhich, under theinfluence ofother magnets broughtinto itsneighbourhood,itwillexperience inductivemagnetization temporarily superimposed uponitsin- trinsicmagnetization. Hithertoexperimenthasgivenuslittle ornodefinite knowledge onthissubject,orindeedgenerally ontherelation betweenmagneticretentiveness andmagnetic susceptibility. Waitingformorecomplete experimentalin- vestigationofthemagnetic propertiesofmatter, Ishallassume asatypical magnetic solid, arigidbody possessing anydegree ofintrinsicmagnetizationinany direction, withperfectre- tentiveness; andhavinginductivequalitydefined bythree principal magnetic susceptibilities alongthreeprincipalrect- angularaxes ofinductivecapacity,inanygivendirections throughit.The" rigid polarmagnets"which wehave hitherto considered areintrinsic magnetsofzerosusceptibility; and it nowbecomesnecessarytodefine intrinsic magnetizationfora substance ofwhich thesusceptibilityisnotzero. 698.DefThe intrinsicmagnetizationofabodyisthere- sultant(§605)ofthethree intensities ofmagnetizationfound bycuttingthreeinfinitelythinbarsfrom directions initagree- ingwith itsprincipalinductive axes, andtesting them ina uniformmagneticfield ofairbymeasuringthecouples which they experience when held atright anglestothelines offorce. Beforegoingonwith thegeneral problemofmagneticinduc- tion,wemayconsider thefollowing particularcaseofit,merely asanillustration ofthisdefinition :— 699. Problem.—Asolidsphereofuniform material, having /Lfc, /x',/x"foritsthreeprincipal magnetic susceptibilities,and possessingintrinsicmagnetizationofintensityiinthe direc- tionsspecifiedwith reference totheprincipalinductive axes bythedirection-cosines, I,I',I",isplacediuairwith nodis- turbing bodyinitsneighbourhood;itisrequiredtofind its I XL.] General Problem ofMagneticInduction. 549 actualmagnetization.Let-f,- ?',-|",bethecomponents ofinduced magnetizationinthedirections ofthethreeprincipal axes;therequired magnetizationwillbetheresultant of Pil-S, il'-^', il"-i" (1); andtherefore theproblemissolved when^,f,|''aredeter- mined. From thefootnote to§609, itfollows immediatelythat theresultant force atanypointwithin thespherehasforits components,inthedirections oftheprincipal axes, P -^(^-^-f), -^(tT-r). -^(ir-D(2). Now—f,—f',—^"aretheintensities ofinducedmagnetiza- tiondueseparatelytothese three componentsofmagnetizing force,andtherefore(§610,Def. 2) f=^^(,7_f), |'=^'^(,T-r), r=/^"^(i^"-r)-(3). Solvingthese for|,f',^",wehave I-^il -^il -f^^^ 1^^^^' 1^^^^" 1.4V^^' andtherefore(componentsofthewholemagnetization) il-i= '[,^r-r=—4— r,^•^''-r= -4-77 (5). XL. General ProblemofMagneticInduction. \2Iarch 1872. Nothitherto published.] 700. Thisproblemis(§628) identical withthethreegeneral problems—electro-static induction throughaheterogeneousin- sulating solid,—thermal orelectric conduction throughahetero- geneous conducting solid,—and(proved below, §§751—759) theflowofafrictionless incompressible liquid throughahetero- geneous poroussolid. 701. Let allspace beoccupiedwithmatter ofgiven permea- bilities, CT,ct', tn-",alongthreeprincipalinductive axes{I,m,n), (l\ni, n'), (I",m", ?i"), (§611) through anypoint {w,y,z). 550 AMathematicalTheory ofMagnetism. [XL. Letthere beintrinsicmagnetization (of, y3,7)at{x,y,z) ;and letconstant electric currents bemaintainedhaving u,v,lofor componentsofintensityat{x,y,z) ;subjecttothecondition (§540) du ^dvdw dx' dydz0..(1). I^etf,7;,fbethecomponentsofinducedmagnetizationat (a?,y,z).Then-ci, ct', -cj-", {I,m,n),{l\m\n), (l'\m\n'\ a,/3,7,u,V,w,being givenforevery point {x,y,z),itisre- quiredtofindf,77,^.This isthegeneral problemofmagnetic induction. Inita,^,7areabsolutely arbitraryfunctions of (x,y,z) ;their valuesbeingzero inanypartofspacedestitute ofintrinsicmagnetization:andu,v,warearbitraryfunctions of{x,y,z),subject onlytothecondition(1);their values being zerothroughout anyportionofspace through which there isno electric current. 702. Letjp,Q5i,J^bethecomponentsoftheresultant mag- netic forceaccordingtothepolardefinition(§517, Postscript), calculated from thegivenintrinsicmagnetizationonthesup- positionofnoinducedmagnetism ;andF,G,Hthecomponents oftheunambiguousresultant force(§551)calculated from the givenelectric currents. By§545and§517{m), (n),and {k),(Q,wehave dy wherem dzd£^d^_^d^dx dy dz 0,d_£_^m dz dx 'dxdS dxdy dFda dx dydHdF0, =-(:4f7rp dx d^^drA dydz) dHdyy:=o....(2), dz= ^TTW.(8).^-^-4 u---— =4^-^ dydz'dx dz'dy dx Equations (2)sufiice todeterminejp,CS,|^from thedata or,/?,7,byexpressingthattheyarethedifferential coefiicients ofafunction, andthatthatfunction isthepotentialofadistri- bution ofimaginary magneticmatter having~ (j~+"j~+j^J foritsdensityat{x,y,z),which wedenote])yp.Similarly [] General ProblemofMagneticInduction. 551I"Equations (3)determine F,G,Hbyvirtually expressingthat tbeyarethecomponentsoftheresultantmagneticforcedueto thegivendistribution ofelectric currents{u, v,w),and are thereforedirectlycalculable from thedata bytheformulae Ib)of§517withF,G,i?instead ofX,F,Z. 1703. Letnow I Z=iF+i^, ^=€i+(?,E^^A^ (4). pequantities ^,g,Hsatisfytheequations I-d£dgdH^\ dxdydz"' \dH dG,dFdH,dGdF, ay dz dzdx dxdy andtheseequationssuffice todetermine F,G,Hfully, by virtually expressingthattheyarethesums ofthetwosets of components explicitly expressedinterms ofthedata,bythe formulse referred tointheprecedingsection. Asweshall see immediatelythatwerequire from thedatarespectingintrinsic magnetization and electric currentsnothingbutthevalues ofF,G,H,wemaysimply regardthesequantitiesasexpress- ingthenecessarydata inthisrespect ;and itisimportantto remark thattheyareunconditionally arbitraryforevery point 704. Letnowthepotentialofthedistribution ofimaginary magnetic mattercorrespondingtotheinduced magnetism (?>V>^®denoted by17;that istosay,letVbethefunction of[Xjy,z)whichthroughallspacesatisfies theequation 'd^^-df^'d^~\d~^^d-y^di)^^^' T1 . ^dW^dV rr <^T^/*7\andlet X=-^,^=-^,^=-^(7). We shall seeimmediatelythatourproblemisreduced tothe determination ofthesinglefunction IT;and w^eshall have simple equations [§705 (10)] giving explicitlytherequired componentsofinducedmagnetization f,77, f,interms ofthe differential coefficients ofthisfunction. 705. Let /,T,I"denote thecomponentsoftheresultant ofF,G,H,and^,^\^'\thecomponentsoftheresultant of ^,|9,^,alongtheprincijDalinductive axes.Wehave 552 AMathematicalTheory ofMagnetism. [xl. I=lF+mG+nH, I'=I'F+m'G+n'H, I"=V'F+m"G+n"H\ F=lI+lT+l"I", G=mI+mT+m'%H=nI+nT +n'T' I &=IX+wil+nZ, S'= I'X+m'g+n'Z, ^"=l"X+m"^+n"Z (^^'' Thethreeprincipal magnetic susceptibilities (§629)being OT—1 -cj'—1 -cr"—1 47r'47r'47r' thecomponentintensities ofinducedmagnetization alongthe principal inductive axes(tobedenoted, §712 below, by ^,V,^")are Hence taking components alongtheaxes of(x,y,z),andmulti- plying by47r,wehave 47rf=t^(7+S?)? +^\r+S:')i' +^-(r+g>'')r-z-a^] 47777=.^ (/+^)m+^'(r+S:')m'+^"(/"+Sb")^"-^-il (10). 47^5•=^(/+&)/^ +^'(r+Sb>' +tz7"(r+^'>i''-^-^J 706.These threeequations, together withthethreeequations bywhich ^,^,^might, accordingto§§518, 482, 483,be expressedinterms off, tj,f,suffice todetermine the six unknownquantities f,rj,^,3C,U,5^;but,by(7)and(6)intro- ducing U,wemayeliminate those sixunknownquantities, and obtain asingle equationfortheoneunknownquantity J-J,thus : —Taking -^ofthe firstofthethreeequations (10), -7-ofthe second, and-^-ofthethird, addingandusing (6)and(7),wefind dx dy dz d{F-wll-'Es'I'V- 'nr'Tl") d[G-~wlm-wTm'- w"I"m") d{H- isln-'m'l'n' -w'Tn"] dx dydz Substitutinginthis for^, S>',^"their values by(8),then for^,^,%by(7),and for7,/',I"their values by(8),we haveexplicitlyalinear differentialequationofthesecond order with second member aknown function of[x,y,z)to determine theunknown function 17. General ProblemofMagnetic Induction. 553 The coefficients of ddvd^ d-^r .^,, , -J-,-T—,-7-under thesymbols ,,,,-^arerelated intheordinary symmetrical manner tothecoefficients whichappearinthequadratic function ^[Gr(?aE+7nf+n2)2+c7'(r3^ +m'i+n'2)2+OT"(riE +m"f+n"2)2] (i2) whenexpanded; and itisunnecessarytowritethem outex- plicitly. Asimilar remark isapplicabletothecoefficients of FjG,Hunder differentiation inthesecond member. Denot- ing(12)by^,andthesame function ofF,G,HbyQ,sothat using againthenotation of(8)forbrevity, wehave ©=8^(i^&H^'*"+^"*'").,(13) andg= g^{^r+r^'I"+ZT-'I") (14), weseeatonce thatthedifferentialequation (11)maybewritten short, thus— d^d^ d^dM d^d^_, Pdxd^'^ dyd^'^ dzd%~P' fddQ ddQddQ\ Equations (10), similarlywritten short, areasfollows; dQd® 1(15). ^=#+W-4^^^+^> dQddh dQd(^1 (16). When, bytheintegrationof(15),17isdetermined, equations (16) give explicitly f, 77,f,thecomponentsoftherequired magnetization. 708. Ishall conclude withtwoslightlydifferent demonstra- tions that,provided thepermeabilitiesareeverywhere positive, as(§631) webelievetheymust beforeverysubstance in nature, there isone,andonly one,value of17forevery point {x,y,z)if(15),withanygiven arbitrary function of{x,y,z)for itssecond member, besatisfied forevery pointofspace.The firstdemonstration, towhich Inowproceed,isthemore con- 554 AMathematical Theory ofMagnetism. [xL. venient forthemagneticor(§700)electricsubjectwhich we have had hitherto under consideration;thesecond willbe added onaccount ofconvenience forthehydro-kinetic analogy. 709. First demonstration ofDeterminacy andSingleness.—Let lEt,1^\WC'beanythree realquantities, arbitraryfunctions of {x,y,z).Consider thefunction f=^[^(a>-mr+^'(*'-B?+^"(g:"-m'T]-(i7), andthetriple integral E=rff'^dxdydz (18). J—COJ—00J—CO (Compare §§503,561,206,732,and753—763). Thefunction^ isnecessarily positive, exceptintheparticularcase ofS>=?Bt, S'=Wi\^''=Wi'\when itiszero.Rememberingthat^,^\ ^"arelinear functions of-7— ,—p,—^,withgivenfunc- tions of(x,y,z)fortheircoefficients, applythecalculus of variations toassignU,sothatEmaybeaminimum. Using forbrevitythenotation(7)of§704,wehave Hence, followingtheusualprocessofintegration byparts,ac- cordingtothecalculus ofvariations, wefind forthecondition thatEmaybeaminimum, ^m,±d^ddii_ dxd^'^ dyd^'^'dzd^'^^ ^• Now ifweput which implythat >...(20) andlook toequations (13)and(8)of§707,weseethat^isthe samequadraticfunction of3£—H,|^—jW,%—jS,that(S isof3^,^,%,Hence-^,-^,-^arelinear functions of ^—'^,^—iKl>^—iS ;and ifwedenote by$thesame quadraticfunction ofH,JIX,iSthat^isof3C,|9,%,that is tosay,ifweput i=^(:^w+^'m:'-\-vT"W)(21), I.] Determinacy andSingleness proved.555 wehave ^_j(a_^ dji__d^_d][d^_m_di dXd^d%' d^B d'^dM' d%~~d%dB:"^^' Hence(19)becomes d^dm^ ddm^ ddm^^d^^ d_^liL (^^\ dxd^dyd^'^dzd% dxd%"^ dyd^ dzd^""^^' which, expandedinterms of17,isalinearpartialdifferential equationofthesecond order, withright-hand member a givenfunction of[x^y,z).The fulfilment ofthisequation throughallspaceisthesolecondition whichUmust fulfil to makeEaminimum. Now itispossibletoassign 17soasto makeEaminimum, andtherefore there exists afunction Tof which satisfies equation (23)throughallspace.This isan obvious extension ofTheorem 1,§206. Demonstration 2of §206extended inanobvious mannerprovesthatnofunction differingatanypointfrom onefunction which satisfies(23) throughallspace, cansatisfy (23)throughallspace. Hence thesolution ofthisequationisdeterminate and freefrom all ambiguityormultiplicityofvalues. 710. Theextension of§206, 2,givesthefollowinguseful theorems :—Let 17"beafunction of(cc,y,z)satisfying (23) throughallspace ;letAUbeanyfunction whatever of(x,y,z); letAS, AS)',AS",^(A)bethevalues ofg^,S:',&",^,when AIT issubstituted forV;and letE+AE bethevalue ofE whenU"4-AU issubstituted forIT.Then— Theorem L rrrdxdydz(^^A^+^'=b'^S>'+^"^"^^")^0 (24); J—COJ—00J—00 proved bytheordinary integration bypartsof§199, (a), (b), asextended in§206,Demonstrations 1and2,andnowfurther extended. Theorem IL AE=^E{A) (25). Thisvery important theorem isaninstant consequenceof Theorem I. AsE(A)isnecessarily positive,afunction V,which satisfies (23),hastheuniquecharacteristic thateveryfunctiondiffering from itgivesalargervalue toE. 711. The firstmember of(23)isidentical with the first 556 AMathematicalTheory ofMagnetism. [xL. member of(15).Wemaymake thesecond member of(23) equaltothesecond member of(15),bytaking t"=-1"+^(/"+u'T'+'o"m"+%i"n")(26), where u,b,toareanythreequantities such that *'+J+^=(27).axay dz^ (28);Thisweseeatoncebyremarkingthat 47r^='u^m+^"^'l'+t^"Wl'\ etc. etc. dQand 47r-^f^='gtII+t!j'/7'+'ur'T'l", etc. etc., andtakingaccount of(8)and(5).Hence§§709,710,withthe values(26)forict,Wi\Wi",provethatthere exists afunction T^T satisfyingtheinductiveequation (16)throughallspace ;that thissolution makes thetriple integralE(18)aminimum;that if 17beafunctionsatisfying (15),andAIT"anyfunction whatever, liT+AU substituted for17augmentsthevalue ofEbythe necessarily positivevalue ofthetriple integral found bysubsti- tuting A17 for17;and,therefore, thatnofunctiondiffering from onewhich satisfies(15)canalsosatisfyit. 712.PreliminarytoSecond DemonstratioriofDeterminacyarid Singleness.—First,itwillbeconvenient toputtheinductive equations (11)and(16)intoadifferent form, aform suitable to theuniformreckoningof"resultant magnetic force," accord- ingtothe"electro-magneticdefinition"(§517, Postscript). Remembering (§§702,704)that§,^,|^andX,^,^arethe componentsoftheresultant forces calculatedseparately,ac- cordingtothepolar definition, from theintrinsic andinduced magnetizations respectively, wesee[§517 (r)]that JP+3e+47r(a+f),CBf+^+47r(y3+7;),|^+^+477(7+ ?), arethecomponentsoftheresultant force ofintrinsic and in- ducedmagnetizations together, accordingtotheelectro-magnetic definition. Tothese wemust addF,G,Htofind forthe whole system (ofinducingintrinsicmagnetization and electric currents, andinducedmagnetization)thecomponentsofthe resultantmagnetic force, accordingtotheelectro-magnetic 1 ElectromagneticFormulce. 557 definition. CallingtheseX,Y,Z,andtaking advantageofthe short notation(4),wehave Z=i:+3^+47r(a+K, r=^+|g+47r(/3+77),Z=^+SS+47r(7+r) (29). i^Kkenowcomponentsofforces andofniagnetizations alongthe ^^ncipalinductive axes. Thuswehave S=I+^ +i7r{A+'^),S'=r+&'+4.ir{A'+^'),S"=1"+Ss"+4ir{A"+^")...{S0), where S=Xl+Ym+Zn,etc./imip[ymgX=Sl+8T-^S'T,etc. (31), A=al+l3m-hyn, etc.,implying OL=Al+A'r-{-A''r, etc.(32), and^=^l+7]m4-^n,etc.,implying f=^Z4-^T+^'7", etc.(33) ; and J,J',r',^,^',S"have stillthesamesignificanceasthat indicated in(8),§705,above. Nowby(9)wehave n-^=(.^-l)(/+g?), 47r^'=(t^'-l)(r+g)'), 47r^"=(^"-l)(/"+a") (34). Hence eliminating ^,^'^"from(30), =^(/+S)+47r^, S'=^\r-§0')+^'irA\ ^"=^"(r-t&")+47r^" (35). Putnow« /-f47r-=C, r+4^7r-,= C\r+47r^= C^..(36),I^K 'ST 'ST 'ST andlet :Cl+Cr+CT^F, Cm+am-hCV=G, Cn+an'+GV=H,\ implying ^>(37). G=:lF+mG^nH, C'=fF+mV-\-nH, C"=rF+m"G+n'H) By(35)wehave S=--a S'=^-^; ^"=C-0"... (38). •BT "ST "CT Hence ZcrtJT'zzr' "^ziT'ZJxro' tzTZcTZcr' 713.Now let Q=-L(|+5^+^)(40).OTT\ST'ST 'CT/ [Compare (13)of§707.] SubstitutingforS,S\S''their values by(31),wehave inQaquadraticfunction ofX,F,Z(corre- spondingintheelectro-magneticformulae tothefunction (Bi ofK,^,5Sinthepolar formulae). Now(39)becomes 3e=4.§-l',g9=4.§-(?,®=W§-i?...(41). 558 AMathematical Theory ofMagnetism. [xl. Eliminating X,^,5^bythecondition that^dx+|Bc??/+S^dz isacomplete differential, wehave d^dQ_d_dQ_ 1^/dH_dG\ d^_d^dQ_ 1_fdFdH\ ] dydZdzdY~4:Tr\dy dz/* dzdX dxdZ~4:ir\dz~ dx)'\,.^. dxdY dydZ~4-rr\dx dy)^ three linearpartialdifferentialequationsinX,Y,Z,equivalent totwoindependent equations, because -y-ofthe firstadded to -y-ofthesecond and-j-ofthethird constitutes anequationin which eachmember isidenticallyzero. Also,by(29), (5), (7), and(6),wehavedX^ dY^ c?Z_ dx dydz^ These four, (42)and(43,)equivalenttothreeindependent equa- tions, inwhich F,G,Harearbitrarily givenfunctions of{x,y^z), determine fully andunambiguouslytheunknown X,Y,Z throughallspace,aswillbeproved immediately bythepro- mised fresh demonstration. But first itmayberemarked that oneobvious wayofdealingwiththem leads usback toourformer^ analysis,thus :—Thethreeequations (42)simply expressthat isacompletedifferential. Hence theirmostgeneral integralis whereUsofardenotes anarbitraryfunction of(x^y,z).The firstmembers here aremerelyshortexpressionsforthelinear functions ofX,Y,Zwhichappearin(89)with>Sf,S',S"elimi- nated by(31). Solved forX,Y,Z,equations (44)give expres- sions which arethesame as(29)with^, tj,feliminated by (10),andae,H,5^by(7);andeliminating bythem Z,7,Z from(43)wehaveanequationfor17*identical with(11),which (§708)determines 17unambiguously throughallspace. 714. Second Proof ofDeterminateness andSingleness.—Let K,K',K"beanythreearbitrarily givenfunctions of{x,y,z)\ andputrrr i\{S-Ky {S'-kj {S'-kji,,,,,.. ^-i-J-J-.8^L-^ -^-'-^^'--^\d^dydz(45. [wherethesuffix isappendedtodistinguishfrom the(!Bof§§729 ...731 below.] 1 E .]Second Proof ofDeterminacy andSingleness. 559 Consider theproblemoffinding X,Y,Zsoastomake(!E,a minimum, subjectto(43). Denoting by\anindeterminate multiplier, accordingtotheordinary method ofthecalculus of variations, make unconditionallyaminimum. Theresultant equationsare dQ(S-K) _1d\dQjS-K) ^1dxdQ{S-K) _1d\^.dX ~^iTdx' dY ^irdy' dZ ^irdz^^' where foramoment Q{S—K)denotes thefunctionintegrated in(45).Ifweeliminate theunknownquantity \from these bydifferentiation, wehave three linearpartialdifferential equationsofthesecond order, equivalenttotwo,which with (43)determine theunknown functions X,F,Z.Considera- tionscorresponding perfectlytothose of§§206,709,710,show that theseequationscanbesatisfied throughallspace byreal finite functions X,F,Z,andthattheycannot besatisfiedby anyfunctionsdifferinginanypartofspacefrom one setof three functions whichsatisfythem.Wehave also, ofcourse, theoremsprecisely correspondiogtoTheorems I.and II.,(24) and(25)of§710. 715.Now letK=^G, K'=^'C\K"=7n"G"(48). This, asiseasilyseenfrom(37)and(40), gives dQ{S-K) _dQ_1^dQ{S-K)_dQ_l_-^ dQ{S-K) ^dQ1jj,,c.x. dX~dX 47r'dY~dY 47r'dZ dZ 47r^'' andtheequationsobtained byeliminating Xfrom(47)become identical with(42).Itisthusprovedthatequations (42)and (43)determine X,Y,Zunambiguously throughallspace. With theparticularvalues ofK,K',K"assumed in(48),we seeby(38)that(45)becomes e.=^J"1^^j^Jxdydz(^§b'+^'S^''+^'^S^"')...(50) ; andtherefore theproblemofmagneticinduction isreduced to makingthisconfigurational function aminimum, subjecttothe condition dXdY .dZ ^ /4o\roino ^ i -^+-^~+-1-=(43)of§713repeated. 71G. Goingback tothe firstproofofdeterminacy andsingle- ness,andparticularizingthevalues ofIBt, lEt', i^''of(26)bytaking u=-(47ra+F),b=-(47r^+G),to=-(4777 }-H)...(51) 560 AMathematicalTheory ofMagnetism. [xL. which invirtue of(5)satisfies(27),thesoleconditionobligatory onu,b;to,wemake thefunction^of(17)equalto 1/^^8" 8"'\o--+—7+— 52, easily provedfrom(8), (32), (36),and(38). Thuswehave andtheproblemofmagnetic induction isreduced tomaking thisconfigurationalfunction aminimum, subjecttothecon- dition thatXdw+^%+^dz isacomplete differential, S,S',S" being expressed byequations (38)and(8)interms of3C,^,^, theunknownquantities, and G,C\G"threearbitrarily given functions ofx,y,z. 717.Acurious relation between theconfigurationalfunctions (50)and(53)isproved thus:—Attendingto(7)andremember- ingthat^isaquadraticfunction of3C,^,^,put 2U^d^"*" dyd^'^dzd%) for itin(50)andperform integrations byparts.Wethus find /:/:/:-«</:/:/-^-^ (.4f-js-r/j)<-). orby(13)and(15) ^rrrdxdydz{'n!^'' +w'Si"'+'UJ"^"'')=l{IrdxdydzVp'{55), Nowtaking (53)substitute init,forS,S',S",their -values by(38).Wehaveimmediately ^=l|rrrdxdydz['m{^-'+C'~) +-m'{&'-'+C'"-)+'m"{^"-'+C'^)] +2|" [rdxdydz{'m^C+-m'&'C'+-af"^"C"\ (56). For (7,6",C",takingtheir values by(36),andattendingto (8),(28),and(32),wehave w^C+ ^-'^'(7'+ w"^"C" ='«;I^+ sr'I'^'+ sr"r5" +47r{5^ +5'^'+5'M") /dO dO dQ\=K^af+Srf-g+25s)+*^«" +S^+2v). Puttinor inthesecond member forK,IB,5^their values, I XL.] Polar andElectromagnetic Formuke. 561 dW, V•xu. fda dff^dy\, . --1— ,etc.,rememberingthatP="" I;/"+T"+^j>^^^ ^^' tegrating bypartsasusual, wefind the laststepbeing simply anintroduction ofthenotation of (15). Usingthis in(56), attendingto(55)and(50),and transposing, wefind <S,+^=^f" rrda:dydz('^G' +^'G''+7^"G"')...{oS). Compare §569(7), (8) ;§717 (55), (58) ;§731(99), (100). 718.Thetriple integral (58)denoted byEisofgreat import- ance, asbeingtheexpressionforthewhole kineticenergyinthe hydro-kinetic analogue (Chapter XI.below). Onaccount of thecorrespondence byopposites, which Iperceived someyears ago (§§733—739, below) between theforcesexperienced by solids held atrestinamoving liquid,andtheforcesexperienced bymagnetizedmatter inthecorrespondingcases ofthemagnetic analoofue, Iconclude that thediminution ofthevalue ofE produced bymotion ofanyportionofmatter, surrounded by spaceofuniform andisotropic permeabilityandnottraversed byelectric currents,isequaltotheworkrequiredtoeffect the motion. Beforeproceedingtoprovethispropositionitiscon- venient tonotice that the triple integral maybeputinto several other forms, eachhavingacharacteristicqualitysuitable foraclass ofapplications. 719. These transformations willbesimplified by,inthe first place, substitutingforelectric currents,ifthere areany,distribu- tions ofintrinsicmagnetization givingthesame contributions to, thevalues ofS,S',S";whichmaybedone inaninfinitevariety ofways,asweseebythefollowingconsiderations :— Foreveryclosed circuit substitute(§548)anopen mag- netic shellproducingthesamepotentialasthecircuitthrough- outspace, excepttheportion occupied bythemagnetized substance ofthe shell. The resultant force ofthe shell, T.E. V 36 I 562 AMathematicalTheory ofMagnetism. [xL. reckoned inthemagnetizedsubstanceaccordingtotheelectro- magnetic definition(§517, Postscript),willthroughout space be thesame asthat ofthe circuit. Thevalues ofB,S'S'\will beeverywhere unchangedifthewholemagnetizedsubstance thus introduced beplacedinspaceofzerosusceptibility (or unitpermeability), andbeitself ofzerosusceptibility. But thiscannot beifthere arecircuitscompletely imbedded in matter ofother than zerosusceptibility; if,forinstance, part ofthegiven systemconsists ofanelectric circuitthrough theapertureofasoftironring. Hence toavoid lossofgener- alitywemust suppose somepart,ifnotthewhole, ofthe intrinsic mao^netization, which wearenow introducinor tobe placedinportionsofspace havingintheoriginal data, sus- ceptibilitydifferent from zero. Themagnetizingforce inthese portionsofspacewillbealtered bythesubstitution ofmag- netization forelectric current, buttomake thewhole external effect thesame,wehaveonlytoaddinthem anintrinsic mag- netizationequaltotheinductive magnetizationlostbythe change. 720. Asanillustration wemayconsider thefamiliar caseof Ampere's electro-dynamicsolenoid(§505, foot-note),withasoft iron core;—what iscommonlycalled abarelectro -magnet. First, supposethere tobenosoftiron core.Wemaydoaway with thecurrent andsubstitute auniformly andlongitudinally magnetizedbarofsteel, with flatends, occupyingthewhole internalspaceofthecylinder.This will, ateveryexternal point, givethesame resultant force asthesolenoid; and its resultant force, accordingtotheelectro-magnetic definition, will throughoutitssubstance bethesame astheresultant force of thesolenoidthroughoutthecylindrical space betweenplanes cuttingitperpendicularly throughitsends. Inthesubstance ofthesteelmagnet,theresultant force, accordingtothepolar definition, will(§479) bemerelytheresultant ofthe force calculable frompositiveandnegative planesofimaginary mag- netic matter coincident with itstwoends; and this iswhat would bethemagnetizingforceduetotheintrinsic magnetiza- tion ofthesteel if(§697)weattribute magnetic susceptibility toitssubstance, withoutdeprivingitofitsintrinsicmagnetiza- tion. Itisofverysmall amountexcept veryneartheends of "l ]Intrimic Magnetizationsubstituted forCurrents, 563 thebar,andis,throiigliouttheinterior, oppositeindirection to theresultant force ofthesolenoid. Topassthenfrom thecase ofabarelectro-magnetwith coreofsoftironorother substance susceptibleofmagnetic induction, toanarrangement producing thesame external effects with intrinsicmagnetizationofthe core instead ofelectric currents round it;wemayfirstgiveto thecoretheintrinsicmagnetizationofthesteelmagnet wehave justbeenconsidering,andsuperimpose uponthissomuch more ofintrinsicmagnetizationasshallbringthewholemagnetiza- tionofthecoreuptotheresultant oftheinductivemagnetiza- tionwhich ithasfrom the electric currents, andtheuniform longitudinal magnetizationwhich weattributed tothe steel magnet. Thecorethusintrinsically magnetizedand stillretain- ingitsmagnetic susceptibility,willactthesameuponallother magnets, andexperiencethesame action from them, asthe givenelectro-magnet. Thesame resultmaybealsoattained withoutattributingintrinsicmagnetizationtothe core, inany case inwhich itiscompletelysurrounded bymatter ofzero susceptibility ;asisthecase with anordinarybarelectro- maf]rnet orhorse- shoe electro-mao^net, unless itsends becon- nected byanarmature ofsoftironorothersusceptiblesubstance (thesubstance oftheelectric conductor being supposedtobe ofzero-magnetic susceptibility).For inanysuch case the substance ofthemagneticshells maybeplaced altogether outside thecore oftheelectro-magnet, byhollowing them so thattheymaypassclear ofthecoreround either endofit;or some ofthem round oneendandsomeround theother soasto enclose thecoreamong them. Then bysupposingthesub- stance oftheshells tobeofzero inductivesusceptibility, we have asysteminwhich thecore isinductively magnetizedin virtue oftheintrinsicmagnetizationofthe shells, toprecisely thesamedegreeasitwasunder theinfluence oftheelectric currents. Theexternal resultant force isthesame asthat of theelectro-magnet, being composedofaconstituent duetothe shells which isthesame asthatdue totheelectric currents, andaconstituent duetothemagnetizationofthecore, identical inthetwocases. 721. Supposingthen electric currents doneawaywithbythe processof§719,wemaysimplytake thedata tobe;^at any 36—2 564 AMathematical Theory ofMagnetism. [xL. point {x,y,z),intrinsic magnetization (a,^,7),andinductive permeabilities -cr, -cr','gt"along principalinductive axes(Z,m,n), {l\m', n'),{r,m",n").Thus(35)becomes where 5,5',5"denote thecomponents along theprincipal inductive axes, oftheresultant ofjf, C5r,|^.Hence for —in(40)wemayput (5+^+47r— )>S»,andsofortheother •BT \ 'C7/ terms. Nowbytheelementaryformula fortransformation of rectangular components, wehave (3I+S)5f+(F+5')^'+(F+5")S"=(ir+3e)X +{(S+i)F+(|^ +2)Z...(60); andbecause(jf+T)dx(CBf+^) dy-\-{^^-\- S^)dzisacomplete differential and —.— f--^+-7-=0,wehavedx aydz frrdxdydz[(§+K)X+i(S+WY+m+^)Z]=(61). J—coJ—coJ —00 Thus(53)becomes This isoneofthetransformedexpressions promisedin§718. 722. Tofindtheothers, substitute forS,8',S"their values by(59); andthenremarking that,bythetransformation of rectangular components, {E+^)^+(F+^V+(F+S'V'=(JF+3e)a +(©+l)i3 +(f^+2)7-(63), wefind Remarkingthat(jf+^)dx+{^+^)dy+(p^+%)dzisa complete differential, put Then integrating bypartsin(64)asusual, wefind ^00 ^00/.COp,AJ J^'2 J^^\ (66); I (70);XL.] FormulcBforExhaustionofEnergy. 565 where[asin§702(2)] PKdx^dy^dz) 4^ir\dx^ dy^dz)^^^^• Next, usingin{QQ) thesecond oftheseexpressionsforp,and performingasetofintegrations byparts:thenputting andperforming another setofintegrations byparts, wefind thefollowing twoformulae forE\— /../_„/_.'^'"'^^'^^[-{JF+ 3^)df-(©+i)©-(1^+2)1^+167r2 {^^+^,+^')] ^7r\dx dydz) \dx dydz)"*^ Lastly, replacingin(70)pandabythe firstformula of(67)and thesecond of(71), integrating byparts, andusing (68),wefind which mighthave been haddirectlyfrom(64)bytakingthe termXa+^yS+^7alone, andproperly modifying theintegral ofit.Each ofthethreeexpressions (62), (64), [QQ),isremark- able asgivingEbytriple integration limited tospace occupied byintrinsically magnetizedmatter :(althoughtheintegrations aremarked asextending throughallspace,theevanescence ofa,/5,7,A,A',A",andp,wherever there isnointrinsic magnetization,limits thetriple integralstospace where there isintrinsicmagnetization). Ontheother hand, theexpres- sions(70)and(70)hisareremarkable asgivingEbytriple integration through space occupied bymatterpossessing mag- netization, whetherintrinsicallyorbyinduction; that isto say,throughthoseportionsofspace where there isintrinsic magnetization, andthoseportionswhere thepermeabilitydiffers fromunity.In(53)and(69)theintegrationextendsgenerally throughallspace. 723. Forcesexperienced bymatter under magnetic influence.— Weshall stillsuppose, without lossofgenerality (§719), the electric currents inthegiven systemtobedoneaway with,and 566 AMathematical Theory ofMagnetism, [xL. aproperdistribution ofinducedmagnetizationtobesubstituted forthem. Let j5beaportionofmatteraltogetbersurrounded byspaceofzerosusceptibilityorunitpermeability. The forceandcouple experienced byB,regardedasarigid body, isdeterminable byanapplicationof§500,when thewhole magnetization (intrinsicandinduced)ofevery partof-5,and theresultant force atevery pointofitsvolume duetomagnet- ization elsewhere, areknown;or,viceversa,when themagnet- ization ofallother matter andtheresultant force ofBatevery pointofit,areknown. 724. Ishall conclude byadaptingtoourpresent case, in whichpartofthemagnetizationvaries invirtue ofmagnetic induction, themethod of§502 forexpressingtheresultant of magneticforce onarigid body,interms ofvariations ofa function ofitsco-ordinates, which in§503wasworked out forthecase ofintrinsic(orrigid) magnetizationalone. First, foramoment lettheinducedmagnetization becomerigid,andlet allthegivenmatter becomeunsusceptibleofmagneticinduction. Supposethewhole magnetizedsubstance tobedivided into infinitelysmall barslyingeach inthedirection ofthemagnet- ization, whether intrinsic orinduced, orintrinsic andinduced; and letWdenote theamount ofworkwhich would beundone inseparatingtheserigidly magnetizedbars toinfinite distances fromoneanother. By(7)of§569wehave Pr=if"rrdxdijdzV(p+cr) (72). J-CO./—00^—CO 725.LetnowBdenote anyportionofthemagnetized matter completelysurroundedbyspace ofzerosusceptibility ;and let A,prefixedtoanyfunction of(x,y,z),ortoanyconfigurational function ofthesystem,denoteaugmentation produced byin- finitesimal motion ofB,themagnetizationofBremainingun- changed (§72).Theworkrequiredtoproducethismotion will beATT;andwehaveby(72) AW^iT rrdxdydz[VA(p +a)+(p+a)AVl..{7S). J-ccJ—aoJ —00 NowapplyPoisson'sequation d'Vd'VdW' / , ,^,. d^-^df^'dl^-^'^^P^^^(^^^' .]Forceexperienced byanyPart 667 idwefind,bytwostepsofintegration byparts, rrdxdydz{p+a)AV=^[(\da;d2jdzVA(p+(7)..(7o). J—coJ -co J—ooJ -ccj -co [ence instead of(73)wemaywrite ATf=[" ["rdwdydzVA(p +(T) (76). J-coJ—ooj—00 726. Consider now(partofthesecond member ofthisequa- m) /•OO /«Q0 /.OO I / IdxdydzVAcr,J—OoJ—ScJ —00 Itinit[§722(71)] -=-(^-t*s-o ^' andperform integrations bypartsasusual.Wefind (78). Thesecond member ofthisexpressed [§712(33)],interms of Komponentsalongtheprincipalaxes ofpermeability, becomes -Trrdxdydz(JA'^+J'A'^'WAy') (79),J-QOJ—COJ-CO 01rhere J,J',J"denote the 3I+S>> etc., of§721,beingthe ^,dVdVdV .,, ,,,, )mponentsoi—^—,---j-,~-y—along these axes.Wehave by(9)"^^"^y"^^ 727. Remembering (§725)thatAprefixedtoanyfunction of[x,y,z)denotes theaugmentation which thefunctionexperi- enceswhenBismoved inanymanner asarigidbodywith its magnetization unchanged,while(80) expressestheactually varyinginductivemagnetization, weseethat, throughoutthe lume ofB^ 4-77- 47r147r 47r^ .Ay'=^;A^"+^A/"...(81), whereA^denotesaugmentation produced bygivingtheactual motion tojB,andmovingallother magnetized matter asifi 568 AMathematical Theory ofMagnetism. [xl. rigidlyconnected withit,theaxes of(x,y,z)being held fixed. Hence(79)isequalto -JTfffdxdydz [(J^A^-f-T'^^'+J'^^Lv^")]- TJTTJ—CDJ—CDJ—CO ^f IIdzdydziiiii-l)JA^J+{vj'-l)J'A,J'+{iu"-l) J"A^J"] (82), where -crmust beregardedasequaltounity throughallspace exceptthatoccupied byB.Nowusingthenotation of§730 (93),wehave dxd\j dz andrectangulartransformationgives JA/+/A/+JA/=^A,^+^A,^+^A,^(84). Usingthese inthesecond term of(82)andperforming integra- tionsbyparts,wereduce thatterm to dx dydz By(94)and(74)thisbecomessimply /•OO /•CO /•CO I I dxdydza-A^V. (86), where amust beregardedaszerothroughallspace except that occupied byB. 728.Now from thedefinitions ofAandA,itfollows that AjO-=Ao-;and J f[dxdydzf{x,y,z)=^0 (87), where f{x, y,z)denotes any functiondependent onthe configurationofthemagnetizedmatter. Hence bytaking /(a;, y^z)=orVweseethat /•OOfoo/•ao /"OOi»oc/•so I /dxdydz(TA^V=-I dxdydzVAo:..(88). Substitutingthesecond member ofthisequationforthesecond term of(82),andgoingbackthrough (79) to(78):then transposingandhalving, wefind f(fdxdydzVA<T=-^j j jdxdydz{J^Aw +J'^Ais'' +J''Aw")...{89). I ^]Force experienced byanyPart. 5G9 lally,usingthisin(76),wefind ATF= /*ridxdydz [FAp-^(J^^w+J'^Lw'+/"SAcr")]...(90).Stt ^H729.Now toprove §718: let5denote variation duetoany ^^^otion of5asarigid body,themagnetizationofevery portion ofmattervarying (accordingtoitsactualsusceptibility) with thevarying magnetizingforce towhich itissubjected. The orkrequiredtoeffect themotion ofBybeing infinitesimal, illbethesame asif(accordingtothehypothesisof§725) theactualmagnetizationwereeverywhere rigid. Hence if (IB—cdenote thework undone inremoving Btoaninfinite distance from allother bodiespossessingeither intrinsicmag- netization ormagnetic susceptibilitydifferent from zero(that istosay,permeability differingfromunity), and caconstant sofarasthepresentvariation isconcerned[tobearbitrarily assignedlater(731)], wehave SeB=ATr.(91). 730. Takingthevariation of{m),§722,wehave J—00J-(XIJ-00 A^ A'^A"^ astheterm ofthetriple integral depending on 1 ^H tt "cr zn" 'SOT doesnotvary.Nowputting P=i-(^r+^'J''+^"J''') (93), wehave,by(15), ,ddP ddP ddP .,_,, (ddP ddP ddP\ dxdV^dy^dV'^dz^dV Jdx dydzJ Hence I^K^\da; dy dz I AsPISaquadraticfunction of-v-,-7- ,-,- ,theexpression under theintegral signhere isclearlyasymmetrical function ,dVdVdV .dBVdBVdBV , of,-,-^, -J-,and-y— ,—7— >-^— ;andwemay writedxdy^dz dx dy dz-^ itthus :—I 670 AMathematicalTheory ofMagnetism. [xl. dPdBV dPdSV dPdSV .dVdxndVdy'^.dV dzd-^ d-j-^d-y-dx dy dz _dV^dPdV^dPdVdP 1j,^. axay dz 731. Takingtlie firsttripleterm alone andperforminginte- grations byparts,wehave £/:/:*dxdTdydV dzdV dxdydz rrf" /ddP ddP ddP\rrr^7^rr.^ \ dx dydzI Hence(92)becomes SE=-rrrdxdydzVSp +^jrldxd7jdz{J^d'Sf +J'^d-n/+J"^-d7u"){97). Comparingthiswith(90),andremarking that, accordingto thedefinitions ofAand B(§§725, 729),wehaveAp=Bp, Act=B-GT,Act'=Sct',andAct"=B^'\weseethat -BIJ=AW.(98), whichproves §718. Invirtue of(98), (06),and(91)wemayput ^^rco rco rco ^^^^^^y^ ^^^^ J-CDJ—COJ-00 By§566weseethat thisimplies assigningtocof§729a value equaltothework which, aftei"Bhasbeenremoved toan infinite distance, must beundone todivide intoinfinitelythin barsevery partofthesystem* possessingintrinsicmagnetization andseparatethese bars toinfinite distances from oneanother; their directionshavingbeen sochosen thatwhen uninfluenced themagnetismofeach islongitudinal. Thusweseethatthe function (Sexpressed by(99)isthe*'meehanical value" ofthe given magnetic system, accordingtothedefinition of§567 extended toinclude materialsusceptibleofmagneticinduction alongwithiutrinsically magnetizedmatter. Itisessentially positive. Were there nomagnetic susceptibilityinanyofthe *Notomitting Bthough infinitely distant,ifithasintrinsic magnetization. .]Forceexperienced byanyPart, 571 laterial concerned, itwould beidentical with the(Bof 569, 570. By(Q6)wehave ompare §717(55), (58). Fortheparticularcase ofzero sus- iceptibility(orunitpermeability) throughoutthesystem, (Band ^Khave thesamesignifications asin§569above. 732.Theexpressions (62), (64), (66), (69), (70), (70)Us,for E,and(99)forQ^,depend ontheexclusion ofelectric currents bywhich(§721)wesimplifiedtheformula formagnetic induction; butas(§719)thissimplificationdidnotinvolve anylossofgenerality,itisinreality proved that thecon- figurationalfunctionJE,expressed bytheformula notinvolvingtheexclusion ofelectric currents, represents by itsvariations theforcesexperienced bydetachedportionsof anysystem composedofintrinsically magnetized polarmag- nets, electromagnets,andinductively magnetized matter; thus:—Theaugmentationofthis functionproduced byany motion ofarigid portionorportionsofsuch asystem, through space occupied bymatter ofzerosusceptibility,isequaltothe workgained bypermittingthemotion. [AdditionofdateMarch 5th,1884. The student isre- commended toexercise himself bygoing throughthewhole investigationof§§700—732, forthesimplecase ofequal permeabilityinalldirections. Itwillthenbeseen thatthe seemingdifficulties oftheinvestigationasgiven above, are merelymathematicalcomplexitiesessential totheexpressionof theformulae concerned, when thematter isaeolotropic. Inrespectto"mechanical values" ofmagneticandelectro- magnetic systems,theinvestigationforthecase ofisotropic matter istobefound inArticle LXI. ("OntheMechanical Values ofDistributions ofElectricity, Magnetism,andGal- vanism")ofVol.I.,ofmy"Mathematical andPhysical Papers." \V.T.] XLI. HYDROKINETIC ANALOGY FORTHEMAGNETIC INFLUENCE OFANIDEAL EXTREME DIAMAGNETIC. . OntheForces experienced hySolids immersed inaMoving Liquid, [FromtheProceedings oftheRoyal Society ofEdinburgh forFeb. 1870.] 733. Cyclicirrotational motion*, [Y.M.§60(z)]once esta- blished through anapertureorapertures,inamoveable solid immersed inaliquid,continues forever after with circulation orcirculations unchanged, [Y.M.§60(a)]however thesolidbe moved, orbent,andwhatever influences theremaybefrom other bodies. The solid, ifrigidand leftatrest,mustclearlycontinue atrestrelativelytothefluidsurroundingittoaninfinite dis- tance, providedthere benoother solid within aninfinite distance from it.But ifthere beanyother solid orsolids atrestwithin anyfinite distance from thefirst, there willbemutual forces between them, which, ifnotbalanced byproper applicationof force, willcausethem tomove. Thetheoryoftheequilibrium ofrigidbodies inthese circumstances mightbecalled Kinetico- statics; but itisinrealityabranch ofphysicalstaticssimply. Forweknow ofnocase oftrue statics inwhich some ifnot alloftheforces arenotduetomotion; whether, asinthecase ofthehydrostaticsofgases,thanks toClausius andMaxwell, weperfectlyunderstand thecharacter ofthemotion, or,asin the statics ofliquids and elastic solids, weonlyknow that *Ther^erences [V.M.§§]aretotheauthor's paper onVortex Motion, recently publishedintheTransactions oftheRoyal Society ofEdinburgh (1869), which contains definitions ofallthenewterms used inthepresentarticle. Proofs ofsuch ofthepropositions now enunciated asrequire proof areto befound inacontinuation ofthatpaper. [They arefound in§§759—763,below.] I IT.]Hydroklnetic Analogy forExtremeDiamagnetic. ^73 some kind ofmolecular motion isessentially concerned. The theorems which Inowproposetobringbefore theRoyal So- ciety regardingtheforcesexperiSiced bybodiesmutuallyin- fluencing oneanotherthroughthemediation ofamoving liquid, though theyarebuttheorems ofabstracthydrokinetics,areof some interest inphysicsasillustratingthegreat questionof the18thand19th centuries:—Isaction atadistance areality, orisgravitationtobeexplained,aswenow believemagnetic andelectric forces must be,byaction ofinterveningmatter? 734. I.(Proposition.) Consider firstasinglefixedbodywith ^neormoreapertures through it;asaparticular example,apiece ofstraighttubeopenateach end. Letthere beirrotational circulation ofthefluidthroughoneormore suchapertures.It isreadily proved [fromV.M.§63,Exam.(2.)]*thatthevelocity ofthefluid atanypointintheneighbourhood agreesinmagni- tudeand direction with theresultantelectro-magnetic force, atthecorresponding pointintheneighbourhoodofanelectro- magnet replacingthe solid, constructedaccordingtothe fol- lowing specification. The"core"onwhich the-conductor is wound, istobeofanymaterialhavingextremediamagnetic inductivecapacity]-, and istobeofthesame sizeandshape asthesolidimmersed inthe fluid. Theconductor istoform aninfinitelythinlayerorlayers,with onecircuitgoing round eachaperture. Thewholestrengthofcurrent ineach circuit reckoned inabsoluteelectro-magnetic measure, istobeequal tothecirculation ofthefluidthroughthataperturedivided by 47r.The resultantelectro-magneticforce atanypointwillbe numerically equaltotheresultant fluidvelocityatthecor- responding pointinthehydrokinetic system. 735. Thus, considering,forexample,theparticularcase ofa straighttubeopenateach end, letthediameter beinfinitely small incomparison with thelength. The"circulation" will exceed bybutaninfinitelysmall quantitytheproductofthe velocitywithin thetube intothelength.Intheneighbour- *OrfromHelmholtz's original integrationofthehydrokinetic equations. +Eeal diamagnetic substances are,according toFaraday's very expressive language, relatively tolines ofmagnetic force, worse conductors than air. Theideal substance ofextreme diamagnetic inductive capacityisasubstance which completelyshedsofflines ofmagnetic force, orwhich isperfectly imper- vious tomagnetic force [orofzero"permeability," (§629)]. 574r AMathematical Theory ofMagnetism. [XLi. hood ofeach end,atdistances from itgreatincomparisonwiththe diameter ofthetubeandshort incomparison with thelength, thestream lines willbestraightlinesradiating from theend. Thevelocity,outwards from oneendandinwards towards the other, willtherefore beinverselyasthesquareofthedistance from theend.Generallyatallconsiderable distances from the ends, thedistribution offluidvelocitywillbethesame asthat ofthemagneticforce intheneighbourhoodofaninfinitelythin barlongitudinally magnetized uniformlyfromendtoend. 736. Merelyasregardsthecomparison between fluidvelocity andresultantmagnetic forces, Euler's fancifultheoryofmagnet- ism(§573)isthuscuriouslyillustrated. Thiscomparison, which hasbeenlongknown aspartofthecorrelation between themathematical theories ofelectricity, magnetism,conduction ofheat,andhydrokinetics,ismerely kinematical, notdynamical. When wepass,aswepresently shall, toastrictly dynamical comparison relativelytothemutual force between twohard steelmagnets, weshall find thesame lawofmutual action between twotubes, withliquid flowing through each, butwith thisremarkable difference, thattheforces areoppositeinthetwo cases; unlikepoles attractingand likepoles repellinginthe magnetic system,while inthehydrokinetic analoguethere is attraction between likeendsandrepulsion between unlike ends. 737. II.(Proposition.)Consider twoormore fixed bodies, such astheonedescribed inProp.I.[§734]. Themutual actions of twoofthese bodies areequal, butinopposite direction, tothose between thecorresponding electro-magnets. Theparticular instance referred toabove shows ustheremarkable result, that throughfluidpressure wecanhave asystemofmutual action, inwhich like attracts likewith forcevarying inverselyasthe squareofthedistance. Thus, consideringtubes openateach end,with fluidflowing through them,iftheexitendsbeplaced intheneighbourhoodofoneanother, andtheenteringendsbe atinfinite distances, themutual forcesresultingwillbesimply attractionsaccordingtothis law.Thelengthsofthetubes on thissuppositionareinfinitely great,and therefore, asiseasily provedfrom theconservation ofenergy,thequantities flowing outperunit oftime arebutinfinitesimallyaffected bythe mutual influence. [When anychangeisallow^ed intherelative XLi.] Hydrokinetio Analogy forExtremeDiamagnetic. 575 positionsoftwotubes bywhich work isdone, adiminution of kinetic energyofthefluid isproducedwithin thetubes, andat thesame timeanaugmentationofitskineticenergyinthe externalspace. Theformer isequaltodouble thework done; thelatter isequaltothework done;andsothelossofkinetic energy from thewholeliquidissimply equaltothework done.] 738. III.(Proposition.) PropositionII.holds, even ifoneof thebodies considered bemerelyasolid,with orwithoutapertures; ifwithapertures, havingnocirculationthrough them. Insuch acase asthis,thecorresponding magnetic systemconsists ofa magnetorelectro-magnet, andamerely diamagnetic body,not itself amagnet, butdisturbingthedistribution ofmagneticforce around itbyitsdiamagneticinfluence. Thus, forexample,a sphericalsolid atrestinthefield ofmotion duetoafixedbody through aperturesinwhich there iscyclicirrotational motion, willexperience from fluidpressurearesultant forcethrough itscentre equal andoppositetothatexperienced byasphereof infinitediamagnetic capacity, similarlysituated intheneigh- bourhood ofthecorrespondingelectro-magnet. Therefore, ac- cordingtoFaraday'slaw forthe latter, andthecomparison asserted inProp.I.[§734],itwouldexperienceaforce from placesoflesstowardsplacesofgreaterfluidvelocity, irrespectively ofthedirection ofthestream lines initsneighbourhood ;a resulteasily deduced from theelementaryformula forfluid pressureinhydrokinetics. 739. Ihave longagoshown[§646above] thatanelongated diamagnetic bodyinauniform magneticfield tends, astends anelongated ferromagnetic body,toplaceitslength alongthe lines offorce. Hence along solid, pivotedonafixed axis throughitsmiddle inauniform stream ofliquid,tends toplace itslength pei*pendicularlyacross the direction ofmotion;a known result (Thomson andTait's Natural Philosophy, §335). Again,twoglobesheld inauniform stream with thelinejoin- ingtheir centresperpendiculartothestream, requireforce to preventthem frommutually approachingoneanother. Inthe magnetic analogue, twospheresofdiamagneticorferromagnetic inductivecapacity repeloneanother when held inalineat right anglestothelines offorce.Ahydrokineticresult similar 576 AMathematical Theory ofMagnetism. [XLi. tothisappliedtothecase oftwoequal globes,istobefound inThomson andTait's Natura\ Philosophy, §332. 740. IV.(Proposition.)Ifthebody considered inIII.,§738 [beaninfinitelysmallglobe*, and]beacted onbyforceapplied soasalwaystobalance theresultant ofthefluidpressure,cal- culated foritaccordingtoII.and III. forwhateverpositionit maycome toatanytime, and ifitbeinfluenced, besides, by anyothersystemofapplied forces, superimposed ontheformer, itwillmovejustasitwould move, under theinfluence ofthe lattersystemofforces alone, were thefluid atrest,exceptin sofarascompelledtomove bythebody's ownmotion through it.Aparticularcase ofthispropositionwas firstpublished many years ago,byProfessor James Thomson, onaccount of which hegave thename of"vortex offreemobility"tothe cyclicirrotational motionsymmetricalround astraightaxis. [Additional, Sept. 14,1872.—Thesamepropositionholds fora globeofanydimensions, inafield offluid motionconsisting ofcirculation orcirculations withinfinitelyfinerigidendless curve orcurves forcore,andnootherrigidbodyintheliquid. Demonstration toappearintheProceedings oftheRoyal Society ofEdinburghfor1871-2. f] Extracts from twoLetters toProfessorFrederick Guthrie. [From thePhilosophical Magazine forJune 1871.] Glasgow, Ncn).Uth, 1870. IHAVE to-dayreceived theProceedings oftheRoyal Society containing your paper"OnApproachcaused byVibration," which Ihave read withgreatinterest. Theexperiments you describe constituteverybeautiful illustrations oftheknown theorem forfluidpressureinabstracthydrokinetics,withwhich Ihave beenmuchoccupiedinmathematical investigations connected with vortex-motion. 741. Accordingtothistheorem, theaverage pressureatany pointofanincompressiblefrictionless fluidoriginallyatrest, *[The proposition asoriginally published without limitation isobviously false, although that itissoIhave onlyperceived to-day.—Sept. 2,1872.] +Proceedings ofthePoijal Society ofEdinburgh, Mai'ch 4,1872. XLi.] Hijdrokinetic Analogy forExtreme Diamagnetic. 577 butsetinmotion andkeptinmotion bysolids movingtoand fro,orwhirling round inanymanner, throughafinitespaceof it,isequaltoaconstant diminished bytheproductofthe densityintohalfthesquareofthevelocity.Thisimmediately explainsthe attractions demonstrated inyour experiments; forineach case theaverage squareofvelocityisgreater ontheside ofthecard nearest thetuning-forkthan onthe remote side. Henceobviouslythecardmust beattracted by thefork asyouhave found ittobe;but itisnotsoeasyat firstsighttoperceivethat thesquareoftheaverage velocity must begreateronthesurfaces ofthetuning-fork next tothe card than ontheremoteportionsofthevibratingsurface. Your theoretical observation, however, thattheattraction must bemutual, isbeyond doubt valid, aswemayconvince ourselves byimaginingthestand which bears thetuning-fork andthe card tobeperfectlyfree tomove throughthe fluid. Ifthe cardwere attracted towards thetuning-fork, andthere were notanequal andoppositeforce ontheremainder ofthewhole surface ofthetuning-fork andsupport,thewhole systemwould commence moving, andcontinue moving withanaccelerated velocityinthedirection oftheforceactingonthecard—an impossibleresult. Itmight, indeed, bearguedthat thisresult isnotimpossible,asitmightbesaid that thekinetic energy ofthevibrations couldgraduallytransform itself into kinetic energyofthesolidmassmoving throughthefluid,andofthe fluidescapingbefore andclosing upbehind the solid. But "common sense" almost suffices toputdown suchanargu- ment, andelementary mathematicaltheory, especiallythe theoryofmomentum inhydrokinetics explainedinmyarticle on"Vortex-motion,"* negativesit. 742.Thelawoftheattraction which youobserved agrees per- fectlywiththelawofmagneticattraction inacertain ideal case which maybefully specified bytheapplicationofaprinciple explainedinashort article[§733... 740]communicated tothe Eoyal SocietyofEdinburghinFebruarylast[1870],asanabstract ofanintended continuation ofmypaper on"Vortex-motion." Thus, ifwetake asanidealtuning-forktwoglobesordisks *Transactions oftheRoyal Society ofEdinburgh, read29th April, 1867. T.E. 37 578 AMathematicalTheory ofMagnetism. [xli. moving rapidlyto-and-fro inthelinejoiningtheir centres, the corresponding magnetwillbeabarwithpolesofthesamename asitstwoendsandadoubleopposite poleinitsmiddle.Again, theanalogueofyour paperdisk isanequal and similar dia- magneticofextreme diamagneticinductivecapacity [§734]. Themutual forcebetween themagnetic andthediamagnetic willbeequalandoppositetothecorresponding hydrokinetic force ateach instant. Toapplytheanalogy, wemustsuppose themagnettogradually varyfrommaximummagnetizationto zero,thenthroughanequal andopposite magnetization back throughzero totheprimitive magnetization,andsoonperiodi- cally. The resultant offluidpressure onthedisk isnotat each instantequalandoppositetothemagneticforce atthe corresponding instant, buttheaverageresultant ofthe fluid pressureisequaltotheaverageresultant ofthemagneticforce. Inasmuch astheforce onthediamagneticisgenerally repul- sionfrom themagnet,however themagnet beheld, and is unaltered inamount bythereversal ofthemagnetization,it follows that theaverageresultant ofthe fluidpressureisan attraction onthewhole towards thetuning-fork,intowhatever positionthetuning-forkbeturnedrelativelytoit. . . . Nov. 23,1870. 743. ...There are,nodoubt, curiouslycloseanalogies between some ofthecircumstances ofmotion incontisaious fluids ofdifferent densities, andthedistribution ofmagnetic force inafieldoccupied bysubstances ofdifferent inductive capacities. Thus, ifinagreat space occupied byfrictionless incompressible liquiddenser insomeportionsthan inothers, a solid besuddenlysetinmotion, the lines ofthe fluidmotion firstgenerated agree perfectly [compare §§751... 763below] with thepermanentlines ofmagneticforce inacorrespond- ingly heterogeneous medium under theinfluence ofabar- magnet,tobesubstituted forthemoveable solidandplaced, with itsmagneticaxis intheline ofthe solid's motion. Asto amounts, thefluidvelocity multipliedintothedensityissimply equaltotheresultantmagneticforce ateachpoint,ifthe particulardefinition[the"electromagneticdefinition"(§517, Postscript)]oftheresultantmagneticforce inamedium of XLI.] Hydrokinetic Analogy forExtreme Diamagnetic. 579 heterogeneousinductivecapacity, giveninthefoot-note to [§516above] §48ofmypaperonthe"Mathematical Theory ofMagnetism,*"beadopted. Buthere theanalogy ends; therigidityinvirtue ofwhich asolid moveable inafluid mediumdifferingfrom itinmagneticinductivecapacity keeps itsform, does notexist[contrast §751below]inthehydro- kinetic analogue.. .. Report ofanAddress ontheAttractions andRepulsions due to Vibration, observed byGuthrie andSchellbach. [From theNorth British Daily Mail forDec. 15,1870;andProceedings of thePhilosophical Society ofGlasgoioforDec. 14,1870.] 744. Thespeaker began bystatingthatinteresting papers hadrecently appearedintheProceedings oftheRoyal Society andthePhilosophical Magazine, byProfessor Guthrie, inwhich some verycurioushydrokinetic phenomena were described. From hints andsuggestionsinhispaper,itseems that Prof. Guthrie connected inhisownmind these phenomena with^ possibilitiesofexplaining some ofthemore recondite actions innature; andhe(thespeaker)believed thatwhatgavethe greatcharm totheseinvestigationsforProf. Guthrie himself, andnodoubt also formanyofthosewhoheard hisexpositions andsawhisexperiments, was,thattheresultsbelongtoaclass ofphenomenatowhich wemay hopefullylook fordiscover- ingthemechanism ofmagnetic force, andpossiblyalsothe mechanism bywhich theforces ofelectricity and ofgravity aretransmitted. Thespeaker, however, didnotlayanystress atpresent uponthepossibilityofapplyingthese resultsdirectly toexplain magnetism. Hebelieved, onthecontrary,thatthe true kinetictheoryofmagnetism (andtheultimatetheoryof magnetismisundoubtedly kinetic) [compare §290and§546, foot-noteabove]involvesquiteadifferent class ofmotions from those towhich thebeautiful phenomena discoveredbyProf. Guthrie aredue.Herather wished topointouttheclose con- nexion thatexisted between thelawsofsome ofthese actions and thelaws ofmagnetism, which, whileinvolving some remark- *Philosophical Transactiom, June 21,1849. Published inPart I.for1851. [§§504—523 above.] 87—2 580 AMathematical Theory ofMagnetism, [XLI. able coincidences, involves certain contrasts decisiveagainst any hypothesis,such astheingenious one[§573above]ofEuler, explaining magnetism byfluid motiondirectly comparable with thatwhich forms thesubjectofthepresent communica- tion. 745.One ofthemost brilliantstepsmade inphilosophical expositionofwhich anyinstance existed inthehistoryofscience, wasthat[§634foot-note, and§G43above]inwhich Faraday stated, inthree orfour words, intenselyfullofmeaning,the lawofthemagneticattraction orrepulsion experienced by inductively magnetizedbodies. Hepointedoutthat asmall globeorcube ofsoft irontended inacertain direction when freetomove inthemagnetic field; while small detachedfrag- ments ofinductively magnetizedsubstances ofthekindwhich hecalleddiamagnetic,tended inthecontrary direction;and that theprecise specificationofthedirection inwhich the diamagnetictended "was fromplacesofstrongertoplacesof weaker force." 746. Bymeans ofdiagrams,thespeakerthenshowed the action ofmagnets uponsmallpiecesofsoftironinvariousposi- tions, intheseveral cases inwhich themagneticforce isdue toabar-magnet,ahorse-shoe magnet, andtwo bar-magnets placedsidebysidewith their similarpolesinthesame direc- tion.Adiagrammaticillustration of"the lines ofmagnetic force," inthecaseofabar-magnet,wasalsogiven.Inthecase ofthehorse-shoe magnet,itwaspointedoutthatthesmallglobe ofsoftironwould have apositionofstableequilibriuminthe linejoiningthepoles,iffree tomove inthehorizontal line bisectingthat lineatright angles ;this stableposition being thepointofgreatestforce. Theattractionexperienced would betowards thispoint ;sothat iftheglobe"wereplaced inside thispoint—that istosay,nearer thebend ofthemagnet— itwould seem toberepelledonthewhole bythemass ofsteel while moving towards theplaceofstrongestforce. Inthecase oftwobar-magnets placedsidebyside[§645above] with their similarpolesinthesame direction, itwaspointedoutthat, for eachpairofsimilarpoles,there isazero, orplaceofnoforce, mid-way between thetwobars,andnearlyinthelinejoining theends.Aglobeofsoftironmoveable midway between the XL!.] IlydroJcinetic Analogy forExtremeDiamagnetic.581 twobars isrepelled,asitwer^.^, from eacli ofthepointsofzero force, and finds apositionofmaximum force, which isoneof stableequilibrium,oneither side ofeither ofthezeros. Fara- day'slaw[§634, foot-noteabove] showed thatthesoftironwas attracted fromplacesofweaker toplacesofstrongerforce, quite irrespectivelyofthedirections ofthelines offorce and thussummed upagreat varietyofverycurious andpuzzling phenomenainonesentence. 747. Thisexpressionisperfectly applicabletosmall bodies atrestinanirrotationally movingfluid;with thesubstitution of"stream lines," instead ofFaraday's''lines ofmagnetic force," and"greaterorsmaller fluidvelocity,"instead of"strongeror weaker magneticforce." 748. Mathematicians were content toinvestigatethegeneral expressionoftheresultant forceexperienced byaglobeofsoft iron inallsuch cases;butFaraday,without mathematics, divined theresult ofthemathematicalinvestigation [§§638, 639,and§§671... 681above] ;and,what hasprovedofinfinite value tothemathematicians themselves, hehasgiventhem an articulatelanguageinwhich toexpresstheir results. Indeed, thewholelanguageofthemagneticfieldand "lines offorce" isFaraday's.Itmust besaid forthemathematicians thatthey greedily accepted it,andhave ever since been most zealous in usinf]^ ittothebest advantas^e. 749. Supposeatubesunk inaperfect fluid, andthefluid bysomemeans settoenter theoneendandflow outbythe other, theparticlesofitwould follow the lines ofmagnetic force. Themagneticfield offorce intheneighbourhoodofa bar-magnet corresponded exactlywith thestraighttubetaking water inatoneendanddischargingitattheother. Iftwo such tubes werepresentedwith likeends toeach other, they attracted, butwith unlike ends, they repelled,—thusacting differently fromtwomagnets placedinsimilar relativepositions. But,exceptinbeing precisely oppositeindirection, theresul- tant action between thesupposedtubes andthatbetween two bar-magnetsfollowsrigorouslythesame law,both astomagni- tude and astoline ofaction. This conclusion, andsome others, containingtheexplanationofmost oftheexperiments now tobeshown totheSocietv, hadbeenworked outmathe- 582 AMathematical Theory ofMagnetism. [xlt. matically bythespeaker,andcommunicated byliim tothe Royal SocietyofEdinburgh*. 750. Ithadbeenfound byFaradaythatthelines ofmagnetic forcewere diverted outwards from itselfbyadiamagnetic body placedinthe field. Ifabodyexisted ofextremediamag- netic inductivecapacity,thelines ofmagneticforcewouldpass altogetherroundit,andnone ofthemthroughit.This ispre- ciselythephenomenon,with reference tostream lines, which is metwith inthehydrokinetic analogue. Thespeaker thendrew attention tosome smallegg-shellswhich weresuspendedsoas tomovefreely,each inahorizontal circle. Byslightly waving thehand infront oftheegg-shells theywere attracted, andthe samephenomenonwasproduced byholdingintheirneighbour- hood avibrating tuning-fork.Thiscorrespondedtothebeha- viour ofadiamagneticinthemagnetic field, onlythat the direction ofthemotion wasopposite. Bymeans ofavery delicate anemometer itwasshown that thephenomenawere independentofcurrents ofair.Thespeakershowed that inwhateverposition,with oneexception,theforkwas held, theattraction wasproduced. Themagnetic analoguetothis forkwould beanon-magneticframe substituted forthetuning- fork,andbearing twosmall magnetslaid across theends, with similarpoles pointingtowards each other. Inthiscasethere would beazeropointinthemiddle, between thenearpoles. Thesame istrue ofthefluidvelocityinthecase ofthetuning- fork. Itwouldrepelthesuspended egg-shellsfrom thezeropoint; buttheexperimentwasoneoftoogreat delicacyforalecture- room. Some very interesting experiments uponflames had beenmade byMrTatlock, hisassistant, which thespeaker hadmuchpleasureinshowingtotheSociety. Avibrating forkwassupported horizontally,and theflame ofacandle broughtnearthevibratingends. Allthatpartoftheflame on alevel with theforkwasrepelled,andbentdown intheoppo- sitedirection, asifbyacurrent ofair.Onthevibration being stopped,theflame atonceassumed itsuprightform.Atall flame, obtained fromordinarycoalgas,wasnextbroughtinto *Proceedings, Royal Society, Edinburgh, February 1870 [§§733—740, above.] :li.] Hydrokinetic Analogy forExtremeDiamagnetic.583 proximitytothevibrating fork,when themiddlepartofthe imewasdrawn outtowards thefork, theupperandlower irtsbeing repelled.Inconcluding,thespeaker remarked, lat itwould beverywrongifhewere tosaythat these :perimentsonthehydrokinetic analoguecontained adirect opening upofthequestionofthemechanism ofmagnetic )rces. Theydidnotgoanywaytowardsexplaining magnetic )rces;but itwasimpossibletolookuponthem without feel- igthatthey suggested thepossibilityofsome very simple lynamical explanation. XLII. General Hydrohinetic Analogy forInduced Magnetism. February 1872. [Corn-pare §743ahove.l 751. Imagine aninfinitely fine-grained poroussolidper- meated byafrictionlessincompressible liquid. The con- stitution ofthesupposed porousmaterial will, forbrevity, bedesignatedasmolecular, andalthough wemight suppose ittodepend onperforationsinalldirections, andevery- whereopeningintooneanother allthroughacontinuous rigid solid, itwillgenerallybemore convenient toimagineit asmadeupoftwoclasses ofconstituents; —(1)small detached rigid particlesormolecules, eachsomehow heldabsolutelyat rest, unless wefind itconvenient toapplyforce toitandmove it:(2)closedinfinitelyfinecurves ofsolid matter. Itwillbe convenient tosuppose each molecule tobearing (thatisto sayasolid with atleast oneperforation through it) ;oratall events tosupposeaconsiderableproportionofthemolecules through anyfiniteportionofspacetobeannular. Thissup- position givesthefoundation(§§573... 583above)forthehydro- kinetic analoguetoapermanent polar magnet.Thus(§574) cyclicirrotational motion ["Yortex Motion," §59(/)and§ 60{z)*~\throughaninfinitesimal solidringconstitutes aperfect analogyforaninfinitelysmallportionofapermanent polar magnet. Again, when thekinematicanalogyforalinear closed current(§535above)isdesired, weshallsupposeaninfinitelyfine closed curve, which toavoid circumlocution Ishall callanit3^oid {Proceedings, Royal Society ofEdinburgh,Dec. 18,1871),ofsolid material tobeplaced, threading through amongthe inter- stices ofthemolecules andeverywhere infinitelynear the lineoftheelectric current, butnotinanycasepassing through theperforationofanannular molecule. Byusingatemporary membrane drawn across suchanityoid ("Vortex Motion," §62) *Transactions, Boyal SocicUj ofEdinlurgh, April, 1367andDec. 1869. PermeabilityinHydrohinetic Analogy.585 Togenerate cyclicirrotational motion, with nocirculation through anyotheraperturethan that oftheityoid itself, aper- fecthydrokinetic analoguetotheelectro-magneticeffect ofa fixed linear current ofconstantstrengthisobtained. Aninfi- nitenumber ofityoids placed infinitelynearoneanother, no- where incontact, buteverywhere leavingsufficient interstices fortheliquidtoflowamong them, givesthefoundation forthe hydrokinetic analoguetoasolidelectro-magnet (§535above). 752. Letanycylindricalorprismatic portionofthesupposed porous solid, terminated byplanes perpendiculartothecylin- drical surface orsides, befixed inatube ofimpermeable mate- rialfittingclose toitallround, butleavingitsends free. This porous plugwillconstitute anobstruction, butnotanabsolute barrier, againsttheflowofaliquid throughthetube. Imagine nowtwoperfectly fittingfrictionlesspistonstobeplacedon thetube atanydistance onthetwosides oftheplug,and let thewholespace bounded bythepistons,thetube,andtheim- permeableconstituents oftheporous solid, beoccupied by frictionlessincompressible liquid. Let theliquidbesetin motion byforceappliedtoeither orboth thepistons. The motion willbedeterminate inevery partofthefluidaccording tothecondition [ThomsonandTait's Natural Philosophy, %^V7 , Example (3)]thatthekinetic energyislessthan that ofany other motion oftheliquidconsistent with thegivenmotion of thepistons.Ifthelengthsoftheclearportionsoftubebetween thepistonsandthetwoends oftheobstructing plugbevery greatincomparisonwith thediameter ofthetube,itiseasily seenthathowever coarse orheterogeneousbetheporousmate- rial,themotion oftheliquidwillbesensiblyuniform and in parallellinesthroughallthedistantpartsofthetube. But if theporousmaterial beinfinitely fine-grainedandhomogeneous astotheaveragestructure ofallequaland similar finitepor- tions, themotion oftheliquidwillbeuniform and inparallel lines atallfinite distances oneach sideoftheplug. If,asan extreme case, theplugbeacontinuous solid, withaninfinite number ofinfinitelyfinecylindrical perforations paralleltoits length,thevelocityoftheliquid throughitwould beuniform, andwould betothevelocity throughtheclear portionsofthe tube, intheinverse ratio oftheareas traversed, that istosay, 586 AMathematical Theory ofMagnetism. [xlii. intheratio ofthesectional area oftheclear tube tothesumof thesectional areas oftheperforations. Themass ofthefluid intheperforationsatanyinstant, would betothemass inan equal lengthoftheclear tube, asthesectional area ofthetube tothesum ofthe sectional areas oftheperforations ;and therefore thekineticenergyofthewhole motion intheper- forations would betothekineticenergyinanequal lengthof theclear tube, intheinverse ratio oftheareas, that istosay, intheratio ofthewhole sectional area ofthetube tothesum ofthesectional areas oftheperforations. Hence, generallythe greatertheobstruction offered byaplug consistingofanykind ofporous material, thegreaterwillbetheratio ofthekinetic energyoftheliquid permeating through it,tothat oftheliquid moving freelyinanequal lengthofcleartube;and(borrowing theword"permeability"fromLeSage), wemaysaythatthe permeabilityoftheplugisinverselyasthekinetic energyof theliquid permeating through it,when thevelocityofthefluid intheclearpartsofthetube isgiven. 753. Ifwewereonlyoccupiedwithhydrokineticsitwould benatural tocallthepermeabilityoftheclearpartsofthetube unity.Thiswouldmakeunitythemeasure ofperfect permea- bility, andwouldgivealwaysaproperfraction forthemeasure ofthepermeabilityofaporoussolid. But inview ofthe magnetic analogyitismore convenient tocallthepermeability ofsomeparticular porousmaterialunity, and todefine the permeabilityofanyother material asthenumber bywhich we mustmultiplythekineticenergyofthe fluidpermeating throughaplugofit,tofindthekineticenergyinaplugof equal lengthofthestandard material fixed inthesame tube. And further, forthemagnetic analogy (compare §732above)it isconvenient toattribute tothesupposed liquidsuchadensity that 47rtimes thekineticenergyofliquid permeatingasolid of unitpermeability,reckonedperunitvolume ofthewholespace occupied byporoussolidandliquidshall beequaltohalfthe squareofthe"flux;" theword fluxbeingborrowed from Fourier's theoryoftheconduction ofheatandadapted tothe usewehave tomake ofitbythefollowingdefinition :— 754.Thecomponentfluxinanydirection isthewhole volume oftheliquid traversingaplane perpendiculartothisdirection ILTi,]PermeabilityinHydrokinetic Analogy.687 3runit ofareaperunit oftime. Inthecomplicatedmotion of leliquid throughtheinterstices oftheporous solid, thecom- onentvelocity perpendiculartoanyplanemaybeincontrary directions atdifferentpointsoftheplane ;butinreckoningthe fluxwemust take theexcess(positiveornegative)ofthe quantity crossinginthedirection calledpositiveabove that which crosses inthedirection callednegative. Byconsidering atetrahedralportionofspace (whetherclear oroccupied by porous solid) bounded bythreemutually rectangular planes andafourthplane cutting themall,weseeimmediatelythat thecompositionoffluxes follows theordinary lawofthecom- positionofvelocities orthecompositionofforces;anelemen- taryproposition duetoFourier. 755. LetX,F,Zdenote, foranypossiblemotion ofthe liquid,thecomponentsofflux atanypoint {x,y,z)referred to rectangularco-ordinates. X,Y,Zmust(§540above)fulfil the equation dXdYdZ_,. d^^d^'^l^-^^^^' ailed the"equationofcontinuity." 756. Ingeneral thepermeabilityofaporoussolidmaybe supposedtobedifferent indifferent directions. When itisso thestructure isofcourse tobecalledseolotropic (Thomson and Tait's NaturalPhilosophy, §676; quoted above, §604, foot- note).Still denoting byX,Y,Zthecomponentsoffluxin three directions atright anglestooneanother, denote byQthe kineticenergy perunit ofvolume, which must beaquadratic function ofX,Y,Z.Hence, bytheordinary analysisofquad- ratic functions, weseethat there arethree determinate direc- tions{I,m,w), {V,m\ ri),iV\m', n'),atright anglestoone another, tobecalled(accordingtoanalogyofordinary usage) theprincipalaxes ofpermeability,andthree determinate con- stants-cT,' ot',-cj"tobecalled theprincipal permeabilities,in terms ofwhich wehave thefollowing expressionforQ:— ^1aiX+mT+nZ)^ {VX+m'Y+n'Z)^.{l"X+m"Y+n"Z)^) ,_, OTTIzer w vj ) 757.Now letussupposethewhole ofspacetobeoccupied byarigid poroussolid ofinfinitely fine-grainedtexture with differentdegreesofpermeabilityand aeolotropic qualityin 588 AMathematical Theory ofMagnetism. [xLii. differentparts; and letafrictionlessincompressible liquid initiallyatrest fillalltheinterstices. InaportionMofthe poroussolid(torepresentthe"inducing magnet"inthemag- neticanalogue),letsome oftheconstituent molecules bean- nular, and lettheaperturesofsome oftheringsbetemporarily closed byinfinitelythin flexible and extensible membranes. (Itisamatter ofindifference whether there beotherringsor noteither inMorelsewhere.) Letimpulsive pressure be appliedtothese membranes, uniform oneach, butnotneces- sarilyofequalvalues forthedifferent membranes;and in- stantlyletallthemembranes bedissolved. Themotion ofthe fluid willbeeverywhereirrotational anddeterminate["Vortex Motion," §62and§62(c)*], and willbeofthe class called polycyclic ["Vortex Motion," §60(x)*].Thekinetic energyof thewhole fluid motionproducedwill[Thomson and Tait's NaturalPhilosoj)hy,% ^V7Example {^)'\belessthan that ofany other motion consistent with theincompressibilityofthefluid, havingthesame normalcomponent velocityateachpointofthe supposed membrane surfaces. Apartial applicationofthesame theorem shows that ifweleave outofaccount thefluidmotion withinanysurface>Si,completely enclosing Jf,andconsider the normalcomponent velocityasgivenateachpointofthis sur- face,thekineticenergyofthefluidmotionthroughtherest of spacewillbelessthan that ofanyother motion with thesame normalcomponent velocityateachpointof8. 758.Tofindtheanalytical expressionofthiscondition let JJjdxdydz denoteintegration throughallspace exceptthat enclosedbyS.ThenX,F,^must, subjecttoequation (1),be such functions of(x,y,z)astomakeJfJQdxdydzaminimum. Hence, \denotinganindeterminatemultiplier, wehave /dBX dBYdhZ< JIJBQdxdydz+X=(3).Vdx dydzj Applyingtheusualprocessofintegration bypartstotheterms involving X,wefind SfffdxdydzX(^+^+^\=j:fdS\{ldX +mdY+n5Z) * Transactions, Royal Society ofEdinhuryh, April 18G7andDec, 18G9. XLii.]Kinetic Energy aMinimum. 589 where jjdSdenotesintegration overthewholeboundingsurface ofthespaceincluded inthetriple integral, andI,m,narethe direction-cosines ufthenormal. Fortheinfinitelydistant partsoftheboundarythedoubleintegral vanishes, asbyhypo- thesis there isnomotion there;and fortheboundaryofM (whichistheremainder oftheboundaryofthespaceincluded inthetriple integral)thedoubleintegral vanishes, because the condition thatthenormalcomponent velocityisgivenoverthe boundaryofM,requiresthat UX+mhY-nhZ=0. Hence asQinvolvesonlyX,F,Z,andnottheir differential coefficients, thevariationalequation (3)gives dQ^^dX dQ_dX dQ_d\dX~dx' dY~dy' dZ~dz^^^• Theseequations,with(1)and(2),§§755,756,and IX+mY+nZ^F(5), forevery pointoftheboundaryofM,whereNdenotes the given normalcomponent velocity,suffice todetermine X,F,Z forevery pointofspaceexternal toM.Comparing them with equations (43), (42),and(40)of§713above, weseethattheyare simplytheequationsofthemagnetic inductionthrough space external toM,due toanydistribution ofmagnetizationorof electric currents within if;if-sr, -cr', -cr"bethethreeprincipal magnetic permeabilities, and(I,m,n), {I',m',n),{V\m",n")the principalaxes atanypoint (a?,y,z) ;X,Y,Zthecomponents oftheresultant force atthesamepoint accordingtotheelectro- magneticdefinition;andNitsnormal componentatany pointofasurface Mjwhichcompletelyencloses theinducing magnet. 759.Considering nextthefluidmotion within thespaceM, and itselectro-magnetic analogue, weseefromequations (42) of§713 above, that d^dQ_d^dQ^ d^dQ__^dQ ^dQ_d_dQ dydZ dzdY' dzdX dxdZ' dxdY dydX^^^' wheretheyarenotzeroareequaltothecomponentintensities oftheelectric flow(§539above),at{x,y,z),inadeterminate distribution ofelectric currents, which, with themagnetism inducedbyitthroughout space, producesresultant electro- 590 AMathematical Theory ofMagnetism. [xlii. magneticforce(X,Y,Z)atanypoint {x,y,z).Suppose nowany motion tobegiven (§751 above)tosolidmaterial inspaceexternal toM^orany cyclicirrotational motion oftheliquidtobe generated bytheaidofmembranestemporarily stopping aper- tures ofsolids inthespaceexternal toM;this will alter the motion already existing bycompoundingwith itthemotion which thesupposedactions external toMwouldproduceof themselves intheliquidifgivenmotionless. Nowfrom(4)it follows thatthroughout M,thevalues ofthefunctions(6) arezero forthesecondsupposed componentofthemotion. Hence, throughout Mthefunctions(6)beinglinear functions ofthefluxcomponents,remainunchangedinthealtered motion oftheliquid.Itfollows that their valuesthrough anyportion ofspace, throughoutwhich themolecular constitution ofthe solid matter iscompletely given,aredeterminable from the cyclicconstants ofthefluidmotionthroughalltheringsinthis partofspace, independentlyofthemolecular constitution, orof circulations through aperturesinotherpartsofspace. From this, lastly, weseethat ifMbemoved inanymanner, transla- tionallyorrotationally,with allitsparts kept rigidly connected, andtheaxes ofco-ordinates moving alongwithit,and ifitbe broughttorestinanalteredposition,thevalues ofthefunctions (6)willbethesame astheywere before themotion. This motion ofifasarigid body implies,ofcourse, motions and changesofmolecular arrangementinthesolid matter ofsur- rounding spacewhich arealtogether arbitrary, subject onlyto thecondition ofmaking wayforM, 760.Theanalogy maybefurther extended toinclude there- sultant forceexperienced bytheinducing magnet,orbyany moveable solidportionofmatterexperiencingitsinductive in- fluence. Todothis,consider theeffect ofanyvariation ofthe solidmatter concerned inthehydrokinetic analogue. First, it must beremarked thattheeffect ofthechangeinthemolecular distribution ofthesolidmatter inthespaceMuponthemotion ofthefluid, cannot bedetermined frommereknowledgeofthe changewhich itproducesinthataverage qualityofthematerial which Ihave defined above(§752)asitspermeability.For without changingthepermeability wemaysoalter themolecu- lararrangementwithinMastochangetoanydegree weplease XLIT.] Analogy ofForce. 591 thefluxofthefluid inthisspace, andtherefore alsothefluid motion through spaceexternal toM.Conceive, forinstance, aninfinitesimal molecularchangetobeproduced which, withoutalteringthe"permeability"ofthegroup,shallvery much contract infinitesimalapertures throughwhich there is circulation. Thismaybedone either byalteringtheshapes ofinfinitesimal molecularrings,orbybringingother molecules towards theaperturesofringssoastoobstructpassage through them. The circulationthrougheachaperture remains ("Yortex Motion," §59*) constant, but itisclear thatthewhole kinetic energy maybediminished asmuch asweplease bythesup- posed process. 761.LetnowAdenote thesolid matter inanyportionof spacewhichmaybeeither thewhole ofMoraltogetherexternal toM,Letthepermeabilityoutside ofAbeuniformthrough some finitespaceallround it.Keeping Arigid throughout, alter itsposition infinitesimally ;keepthepermeabilityun- changedinthespace immediately contiguouswithit,byforces appliedtosurroundingmoleculesobligedtogivewaytoit duringitsmotion; andkeepallotherportionsofsolid matter inexternalspace rigidlyconnected with one another. The workdonebyforcesappliedtoAandthesurroundingmole- cules toproducetheir supposedmotions must beequaltothe X)^CO oo augmentation experienced bytheintegralI I 1Qdxdydz. This isthesame astheamount ofwork requiredtogivethe correspondingmotion totheportionofmattercorresponding toAinthemagnetic analogue;aconsequenceof§731above, with theconsideration thatboth inthehydrokinetic system IT., ,1 -i nddQ ddQ ,andthemagnetic analogue,thevalues oi -j--py—ir'J/>^^^'> are(§759above)notaltered bythesupposed changeof^'s position. 762.Thenecessarily complicatedcharacter ofthedynamical actionrequiredtoproducethesupposedmotion ofAand re- arrangementofthesurroundingmolecules disappears altogether inthecaseinwhich afinite shell ofspace contiguouswithAall Transactions, Royal Society ofEdinburgh, April 1867andDec. 1869. 592 AMathematical Theory ofMagnetism. [xLii. round isfreefrom solid molecules. Inthis casethe(general- ized)componentforcesrequiredtogiveanyinfinitesimal motion whatever toA(compare §502above),willbesimplythedifferen- tial co-efficients ofQwith reference tothecorresponding co-ordinates;andtheforcesrequiredtobalanceAinanyposi- tion, willbeequalandoppositetothese forces. Hence the force requiredtobalanceAinthiscase ofthehydrokinetic systemwillbeequal andoppositetotheforcerequiredtobal- ancearigidbody correspondingtoAinthemagnetic analogue. Inthelatter, theanaloguetothespace round A,clear ofsolids, buttraversed byliquid, may [notwithstandingthe different convention(§753above) moregenerally adopted] beair. This particularconventionbeing adoptedforaninstant, themagnetic analoguefor allportionsofspace occupied bythe"porous solid," described in§751above, orbycontinuous finite solid substance, willbediamagneticmaterial ofanypermeability fromunity (thatofair)tozero(thatofideal substance ofex- treme diamagnetic quality). TheanalogueofMmaybeeither arealordinary electro-magnet consistingofanelectric current, ordistribution ofcurrents throughsolid conductors ofdiamag- netic material;oranidealpolar-magnet (§697above)ofdia- magneticinductivequality. But itistoberemarked thatby choosingairforthemagnetic analogueofspaceunobstructed by solids inthehydrokinetic system, weexclude allferro-magnetic induction from theanalogy. 763. Usingnowthegeneral propositionof§761 ,andmaking theproper particular suppositions regardingthemoveablebody A,wenotonlyprove PropositionsII.and III.of§§737,738 above, butextend theirapplicationtorealbodies ofanydegrees ofdiamagneticinductivecapacityinstead oftheideal bodies of "extreme" diamagnetic quality (zero magnetic permeability) imaginedinthosepropositions. 'V | ) | | | |- t Petit = | te g 5Digitized byMicrosoft ® — Tof€ooe.Pcc(f^2S7. Pfa/0.3. ( iDigitizedbyMicrosah®| INDEX, Accumulator, uniform current, §408- 411 Action ofasmall plane closed circuit onanelement ofanother complete electro-magnet ormagnet, §546 ^olotropic, §604,foot-note Analogy, Hydrokinetic, §§573-583, 738-763 Atmospheric Electricity, early observers of,§267 method ofobserving, §§262- 266 newapparatus forobserving, §391 Notes on,§§392-399 Observations on,§§296-300 onthenecessityforinces- santrecording, andforsimultaneous observations indifferent localities to investigate, §295 Atoms, size of,§400 Attractions andrepulsions duetovi- bration observed byGuthrie and Schellbach, Eeport ofanaddress on the, §744 Attraction ofauniform spherical sur- faceonanexternal point, §87 •propositions inthetheory of,§§ 187-205 Capacity ofconductors, §§51-56 Cavendish, §34,foot-note ratio ofthecapacityofadiscto that ofasphereofthesame diame- ter, §235,foot-note Certain partialdifferential equations, theorems with reference tothesolu- tion of,§206 Coercive force, §§609,630 Collector, water dropping, §§262,266, 287 burning match, §§261,286 Condenser, sound produced bythedis- charge ofa,§302 Conducting andnon-conducting elec- trified bodies, ontheattractions of, §§144,148 sphere, determination ofdistri- bution ona,§77 T.E.Conducting surfaces external and in- ternal, §97 Conductors, insulated, §71 ofelectricity, §68 Conditions towhich thedistributions ofgalvanism insolidandsuperficial electromagnets issubject, investiga- tion of,§§539-546 Cone, area ofsegment cutfrom a spherical surface byasmall, §86 orthogonal andoblique sections ofasmall, §85 thesolid angle ofa,§81 Cones, definitions regarding, §80 Contact electricity, new proof of,§ 400 Coulomb's experiments, §25 Crystalline andnon-crystalline bodies, theory ofmagnetic induction in,§§ 604-624 Cyclic irrotational motion, §733 Density, electric, §330 Diamagnetics, repulsion of,§§643- '646 Diamagnetic particles, reciprocalaction of,§§695,696 Dielectric, §§36,447 Dip,lineof,§441 Distribution ofelectricity onacircular segment ofasphere, §§231-248 Distribution ofelectricity, mechanical value of,§§695,696 ofmagnetic matter necessary to represent the polarity ofagiven magnet, §§473,474 Distributions ofmagnetism, solenoidal andlamellar, §§504-523 ofmatter, mechanical value of, §§661-563 Electricity, atmospheric, §§249-301 ontheelementary laws ofstatical, §§25-50 conductors of,§68 non-conductors of,§67 ofacharged conductor rests en- tirely onitssurface, §68 twokinds of,§58 38 594 Index. Electric current, strength of,§532 accumulator, onauniform, §§408-411 equilibrium, §66 machines founded oninduction andconvection, §§416-425 Electrical density atanypoint ofa charged surface, §§69,93,138 forces, superposition of,§63 influence onaninternal spherical conducting surface, §§102-105 onaplane conducting sur- faceofinfinite extent, §§106-112 quantity, §61 Electrification oftheatmosphere, what isknown regarding the,§§253,296- 301 how experiments may be made forascertaining the, §§254- 262 Electrified bodies, lawofforce between, §64 surface, repulsion onanelement ofan,§88 spherical conductors, mutual at- traction orrepulsion between two, §§128-142 Electrometers andelectrostatical mea- surements, Keport on,§§341-390 classification of,§§343-385 Electrometer, definitionof,§341 absolute, §§307-309, 339, 358, 363 newabsolute, §§364-367 divided ring, §§263-270, 345-357 electroscopic, §305,foot-note long range, §§383,384 standard, §§379-382 portable, §§263, 277,368-378 Electromagnet, definitionof,§434 Electromagnets, §§524-554 linear, §536 superficial, §537 •soUd, §538 Electromotive force required topro- duce aspark inairbetween parallel metal plates atdifferent distances, measurementof,§§320-340 Electroscope, Bennet'sgold-leaf, §387 Bohnenberger's modification of, §388 Electrostatic force andvariations of electric potential, relations between §337 produced byaDaniell's bat- tery,measurement ofthe, §§305- 319 Electrophorus, reciprocal, §427 Elements, division ofsurfaces into, §79 Equilibrium, electric, §66 Ellipsoid, attraction ofahomogeneous,onapoint within orwithout it, §§21-24 Ellipsoid, uniform motion ofheat in an,§§11-20 Faeadat's researches, §27 onelectrostatic induction, §36etc. onspecific inductive capacity, §46,etc. Law, experimental illustrations of,§§654-664 deduced from thelaw of energy, §§674-687, 745-750 _ Ferromagnetic anddiamagnetic mag- netization, relations of,tothemag- netizing force, §§664-668 Ferromagnetics, attraction of,§§634- 642 Field ofmagnetic force, orfield of force, §605 Force atapoint due toamagnet, §605 analogy of,§§760-763 Forces experienced byinductively mag- netized ferromagnetic ordiamagnetic non-crystalline substances, remarks on,§§647-653 bymatter under magnetic influence, §§723-732 bysolids immersed ina moving liquid, §733, etc. "Frequency" electric, §294 Galvanometer, §341 mirror, §350 Gauss, §§187-481 Geometrical slide, §346 Green, essay ontheapplicationof mathematical analysis tothetheo- ries ofelectricity andmagnetism, §§25,156, 163, 167,481 potentialatapoint, §37,foot-note quotation from, onsome experi- ments byCoulomb, §234 Guthrie, Professor, extracts from let- ters to,§§741-743 Harris onthelawofelectric force, examination of,§26 Heat, uniform motion of,§§1-24 Heterostatic electrometers, §385 Holtz's electrical machine, §429 Idiostatic electrometers, §385 Images, electric, §§127,208-230 Imaginaryelectrical points, §116 magnetic matter, §§463-475 Induced magnetism inaplate, §§156- 162 Induction, magnetic, §§604,624 plate, §357 Index. 595 Inductive action, curved lines of,§39 capacityofasubstance, principal, §611 Inductively magnetized bodies inposi- tions ofequilibrium, onthestability of,§665 ferromagneticordiamag. netic non-crystalline substances, re- marks ontheforces experienced by, §§647-653 Insulated sphere subjectedtothein- fluence ofanelectrical point, §§89- 95 Inverse problemsofmagnetism, §§684- 601 Intensityofmagnetization, §§461, 462 Isothermal surface, §1 Isotropic, §604,foot-note Laplace, §481 Lamellar distribution ofmagnetism, characteristic of,§514 Laws ofstatical electricity, ontheele- mentary, §25 ofmagnetic forces, §§452-453 Lamp's Memoir onIsothermal Sur- faces, §20 Lettres deM.William Thomson, A.M., Liouville, extraits de,§§208-220 Lines ofelectric force, §§39,251,256 ofmagnetic force, §605 offorce, diagrams of,§§632,633 Liouville, surunpropridt^ delacouche electrique enequilibre klasurface d'uncorpconducteur, §163 noteonthesubjectofelectric images, §§221,230 Lightning, onsome remarkable effects of,observed inafarmhouse near Monimail, §301 Leyden phiaJ, capacity ofa,§§51,etc. Magnet, definition ofa,§434 Magnetic agency oftheearth ona magnet, §438 axis, §§440,494 centre, §494 field, §605 force atanypoint, total, §605 • thecharacteristic ofmag- netism, §§432,433 axioms of,§606 induction, determination ofthe conditions of,§610 general problem of,§§700- 732 laws of,§607 Magnetic induction, aprincipalaxis of,§611 inductions, superposition of,§ 607Magnetic moment, §§458-460 polarity, §§443-447 sheU, §§506-512 solenoid, §§505,507,509,611 strength, §§454-456 susceptibility, §610 permeability, §628 analogues of,§§625-631, 751-756 Magnetism, mathematical theory of, §§430, etc. Magnetization, directionof,§462 intensity of,§461 intrinsic, §698 Magnetized matter, mutual actions be- tween anygiven portions of,§§476- •501 Mathematical theoryofelectricity, actual progress inthe, §74 ofelectricity, objects ofthe, §73 Measurement byelectrometer, inter- pretation of,§336 Mechanical theory ofelectricity, de- monstration ofafundamental pro- position inthe, §§149-155 value ofadistribution ofelectri- cityonagroup ofinsulated con- ductors, §138 Mouse-mill replenisher, §426 Mutual action between twomagnets consists ofaforceandacouple, §§ 496-501 between twomagnets ex- pressed interms ofafunction of their relative position, §§502-505 Nicholson's revolving doubler, §429 Oersted, §524 Plane conducting surface, electrical in- fluence ona,§§106-112 Pliicker's hypothesis, §666 Polar magnet, §549 inductive suceptibilityofa, §§697-699 mechanical values of,§§ 564-572 Polarity, §443 Poles ofamagnet, §§443,549 Poisson, Memoirs of,onthemathema- ticaltheory ofelectricity, §25 theory ofmagnetic induction, §604 quotations from, regarding mag- necrystaUic action, with explana- tions, §§620,621 Potential atapoint, §37,foot-note atanypointintheneighbour- hood oforwithin anelectrified body, §129,foot-note «S 596 Index. Potential, electric, §335 ofamagnetic shell atanypoint, §512 ofaclosed galvar circuit ofany form, §§555-560 Potentials, equality and differenceof, §249, foot-note Potential-Equalizer, §§422-426 Proof plane, §§25,foot-note, 35,330 Quantities ofelectricity, measurement of,§828 Keplenisher, §§352,418-421,427-429 Eesultant electric force atapoint,de- finitionof,§65 duetoauniform sphe- rical shell, vanishes foranyinterior point, §78 atanypoint inanin- sulating fluid, §331 magnetic force atanypoint, §§ 479-515 Size ofAtoms, §400 Specificinductive capacity, §45,etc. Spherical conductors, geometricalin- vestigattons with reference tothe distribution ofelectricity on,§§75, etc. geometrical investigations regarding, §§113-127 conducting surface, electrical in- fluence onaninternal pointofa, §102 surfaces ofwhich thedensity va- ries inversely asthecube ofthe distance from agiven point, attrac- tion of,§90 Solenoidal distribution ofmagnetism, characteristic of,§513Statement oftheprinciples onwhich themathematical theory ofelec- tricityisfounded, §§57,etc. Stratum ofairbetween twoparallel or nearly parallel plane orcurved me- tallic surfaces maintained atdiffer- entpotentials, §338 Strength ofelectric current, §532 Superficial density ofmagnetic matter, §§471,472 "Surface oftheearth," definition of, §250;generally negatively electrified, §252 Telegraph wire insulated intheaxis ofacylindrical conducting sheath, electrostatic capacity of,§§54,etc. Terrestrial electrification, extremely rapidvariations of,§259 magnetism, onthe electric cur- rents bywhich thephenomena of, maybeproduced, §§602,603 Thal^n, magnetic susceptibihty ofiron, §630 Theearth, agreat magnet, §436 The earth's action onamagnet, sen- sibly acouple, §§439,442 Theory ofelectricity, oncertain defi- niteintegrals suggested byproblems inthe, §§166-185 ofmagnetic force, elementary demonstrations ofpropositionsin the, §669 Tyndall, Professor, correspondence with, §§694-696 Unit strength, §647,foot-note Varley's instrument forgenerating electricity, §428 Volta connection byflame, §§412-415 CAMBRIDGE :PRINTED BY0.J.CLAY, M.A.ANDSON,ATTHEUNIVERSITY PRESS. il§;dt>2 |;—Digitized byMicrosoft® re| Digitized byMicrosoft é BINDING SECT. AUG81983 PLEASE DONOTREMOVE CARDS ORSLIPSFROM THISPOCKET UNIVERSITY OFTORONTO LIBRARY Plqrsical & AppliedSci. S3 - oe ;: fs) SN eee