The Theory of the Potential MacMillan

theoretical mechanics THETHEORY OFTHEPOTENTIAL ByWilliam Duncan MacMillan Dover Publications, Inc., New York, New York. ‘This new Dover edition first published in1958 isan unabridged and unaltered republication ofthe first edition. LibraryofCongressCatalogCardNumber:58-59865 Manufactured inthe United States ofAmerica Dover Publications, Ine. 180 Varick Street New York 14,N.¥. PREFACE ‘The purpose ofthepresent volume istogive thereader acon- neeted account ofacertain field ofmechanies which isvery useful from thepoint ofview ofaphysicist andvery beautiful from the point ofview ofamathematician. ‘Thebookisnotintendedasa treatise, butitishoped that thebird’s-eye view which ishere presented willsorve asanintroduction tothis very attractive field and stimulate somewhat itscultivation. ‘Thestudy ofthetheory ofthepotential may properly besaicl tohave been initiated byLaplace in1782 inthediseovery that thepotential function ofevery finite body satisfies acertain partial differential equation ofthesecond order which isnow known asLaplace’s equation, andthetheory ofthepotential is largely astudy oftheproperties ofthefunctions which satisfy it. Such functions were called harmonic functions byLord Kelvin about themiddle ofthelast century, and itisbythis name that they areusually known, Spherical harmonies were invented byLaplace inthepaper above mentioned. These harmonic functions are associated with thesphere andarepeculiarly adapted totheexpansion of thepotentials ofbodies which differ butlittle from spheres, although thepotential ofevery body isexpansible interms of spherical harmonics. ‘The zonal harmonies, which had been developed previously byLegendre, aremerely «particular type of ‘themore general spherical harmonies ofLaplace. ‘The next noteworthy advaneo inthetheory was made by George Green inavery remarkable paper which was printed privately in1828. Green was almost entirely aself-taught mathematician, Hedidnot reecive hisdegree ofBachelor of Arts until 1837, atwhich time hewasforty-four years ofage ‘Notwithstanding these handicaps thebrilliant originality ofthis paper marks itasoneofthemathematical elassies—a factwhich should bestimulating tothemore fortunate students ofthe present day. Inthispaper istobefound afundamental theorem ofmathematical analysis which isknown asGreen's theorem. vi PREFACE He also formulated the problem ofelectrical induction which later became celebrated under thename ofDirichlet’s problem, Itwas inthis paper that theterm potential function was used for the first time. Owing tothe manner ofitspublication Green’s paper was almost unknown for more than adecade. In the meantime many ofhistheorems had been rediscovered byGauss, Charles, Sturm and others. Inparticular, Gauss’ paperof1841covered somewhat the same ground asGreen's paper of1828, but the methods ofGauss were s0different from those ofGreen that bothpapersarefundamental inthisfield.Although, apparently,Gauss was not familiar with Green's paper, he,too, used the term potential function—without, however, any claim of invention. ‘The theory ofthe potential was very popular during the middle decades ofthenineteenth century. Avast literature was developed inwhich aretobefound thenames ofmany brilliant mathematicians. Interest inthesubject hasnotyetdied out, as isevidenced bythe long list oftitles ofmodern papors inthis field, The properties ofmany ofthefunctions which arise canbe made todepend upon thesolutions ofintegral equations, afact that isofgreat interest tothe mathematical students ofthis type offunctional equations, Aknowledge ofintegral equations onthepart ofthestudent, however, isnotassumed here. Itseems more practical toleave this aspect ofthetheory ofthe potential tothose who are devoted tothetheory ofintegral equations than toassume too wide aknowledge ofmathematical theory onthepart ofthe student. Itishoped that thepresentation ofthesubject here given will befound useful both tostudents ofmathematics and tostudents ofmathematical physics. W. D. MacMunan. ‘Tae Universrry oFCiicaso, ‘March, 1990. CONTENTS Paoe Puepace. . . v CHAPTER I ‘Tue Armraction or Frvite Bootes Srerion 1.The Law ofGravitation... . . 1 2.The Attraction ofSystems ofParticles sewed 8.TheComponents ofAttraction. ... :14. The Elementsof Mass... .2 5.Attraction ona Point... 2... 3 6.The Attraction ofaCircular Are onItsCenter. . 3 7,TheAttraction ofaStraightLineonaPoint... .. 48The Attraction ofaThinSheetonItsAxisofSymmetry... .5 9,The Attraction oftheFrustum ofaCone onItsApex. 7 10. Perspectivity wee 8 11,The Attraction ofanEllipsoidal Homecoid upon anInterior Point 10 12.The Attraction ofaSpherical Shell upon anExterior Particle 11 13,The Attraction ofaSolid Sphere upon anExterior Point... .14 14,TheMutualAttraction ofTwoStraightCollinear Rods...415, The Attraction ofaCircular Disk onIts Axis oe 15 16. The Attraction ofaBody ofRevolution onaPoint inItsAxis 16 17.Example—The OblateSpheroid ... : :718, The Attraction ofaUniform Rectangular Plate onaPoint inIts Own Plane... . 19 19,The Attraction betweert Two Rigid Bodies . oy Problems au CHAPTER IT ‘Twn Newrowtan Porentiat Function 20. The Potential Function Defined. : . oy 21.TheSignificance ofthePotential Function. .. se22. The Potential Function Exists. cee 28 23, The Existence ofDerivatives ofthe Potential . 27 24, Existonce ofDerivatives atExterior Points... ar 25, Existonce ofDerivatives atInterior Points... : 29 26,The Equation ofLaplace pee 27. Equipotential Surfaces, orLevel Surfaces... . : ae 28.TheLogarithmic Potential. .... 3529, The Potential ofaSpherical Shell 36 vii vill CONTENTS Seertow Pace ‘30.Potential of«Uniform Circular Disk along ItsAxis. 4 831,Potential of«Homogeneous Straight Rod. 2 32. ‘The PotentialofaHomogeneous Solid Ellipsoid forInterior Points 45 38,‘TheEquipotential Surfaces ee 0 34,The Components ofAttraction atanInterior Point, BL 85.The Attraction ofaSolid Homogeneous Ellipsoid upon anExterior Point—Ivory's Method... . 2 36,The Potential ofaHomogeneous Solid Elipsoid atExte-ForPoints ee eee eeeee 8B 87,Evaluation oftheElliptic Integrals. 58 388 MacLaurin’s Theorem : rr.) 30, The PotentialofSpheroidsatExteriorPoints... ..5.62 40.The Attraction ofaSpheroid attheSurface. Lol. 6841,TheAttraction IsaMaximum. . 26642,The Potential ofHomogeneous Elliptic Cylinders. oa) 43.The Potential ofaHomogeneous, Rectangular Parallelopiped 72 44,Tho ComponentsofForeefortheRightParallelopiped -.7 45.AGeneraliration RegardingDerivatives ofaPotential.....80 46.The Potential ofaBody ataDistant Point . st 47.The Terms ofHigher Degrees : 85 48,The Expansion fortheHomogeneous Ellipeoid. Ll os749.TheRightParallelopiped a 28850.TheInertialIntegrals. || Dill. se51. The Inertial Integrals Cannot AllVanish 0... 2. OL82.ABodyIsUniquelyDefinedbyItsInertialIntegrals. ||...9%Problems... vee : 4 CHAPTER UL Vector Freuos, Taroxems or GREEN AND Gavas 53. Definitions cee . 96 54. The Normal Derivative : ; oT 455. Relations between Certain Volume- and Surface-Integrals. 99 56. AVector Interpretation. . 2 10157.Generalized Orthogonal Coordinates Ss510258,Green’sTheorem... . : 10459.The Potential ofHomogeneous Bodies || : 106 ©.Example—A Non-Homogencous Spherical Shell 107 61,Existence ofHigher Derivatives ofPotential Functions, 109 62. Harmonic Functions... un 68. An Extonsion ofGreen's Theorem for Tia:monie Hunetions ut 64, ReductiontoTwoDimensions... . .nS 65.Analogy with Cauchy's Theory ofResidues 27166.TheSurfaceIntograloftheNormalDerivatives ofI/p.....120‘67.The Contour IntegraloftheNormalDerivativeoflogp.....122 68. ATheorem ofGauss . : 12369.Poisson'sEquation. im70. Poisson's Equation inTwo Dimensions 126 CONTENTS ix Srerioe PaceTI, An Extension ofGauss Theorem. . - 126 72. Green's Theorem Applied toTwo Potential Functions 128 78. Charneteristic Properties ofaPotential Function 130 74.Tho Average Value ofaPotential Function over aSphere 132 75. Maxima and Minima ofHarmonie Funetions 133 78. The Potential Energy ofaFinite Mass 136 77.The Potential Energy ofaHomogencous Sphere 1387B.TheHeatoftheSun... ee 13979.Relation between Certain Surface andLine Integrals .10 80. Stokes’ Theorem. a 43 81. Examples ofVector Curls as 82.The Vector and ItsCurl Are Orthogonal 145 83, Condition That aLine Integral Shall BeIndependent ofthe Path ‘ofIntegration 148 84,Condition That aSurface Integral Shall Depend upon theContour Only beens 49 Problems so... *152 CHAPTER IV ‘Tun Artuactions oFSunraces AND Lives 85. The Ovcasion forTheir Study 153 “Attenetions ofSurfacos 86. AUniform Disk . . 153 87, AnInfinite Homogeneous Universe. 155 88. Proper ancl Improper Intograls 157 189. Semi-convorgent Integrals. 16t 90. The Potential ataPoint ofthe Surface 165 91. The Potential IsContinuous across the Surface <5 166 92. The Normal ComponentoftheAttractionIsDiscontinuous aeross the Surface... a 168, 98,The Tangential Components ofthe Attraction Are Continuous 17194.Discontinuities intheDerivatives ofSurfacePotentials... ..17495.Example—A Non-homogencous Disk... 176196.Discontinuities intheSccondDerivatives ofSurfacePotentials .17897. Singular PointsoftheSurface 189)Attractions ofLines 98. AStraight Rod : 190 99. The Components ofAttraction. . 191 100. Attraction inthe Lino IsNot Well Defined 193 101. Asymptotic Expression forthePotential 194 102. The Potential ofaUniform Hoop 195 103. Evaluation ofthePotential According toGauss 2197 104. Asymptotic Expression forthePotential +200 CHAPTER V ‘Sunracr Distumonioxs of Marren 105. Transformation byReciprocal Radi ss 204 108. Application oftheTransformation toPotentials 207 x CONTENTS — Pace‘Or.ThePotential ofaUniform Distribution ofMatteronaSphere"200 108. ANon-uniform Spherical Distribution ee 200) 109. Inversion ofaHomogeneous Ellipsoidal Shell a 110, Centrobaric Bodies... Fee 212 111. The Center ofGravity ofCentrobarie Bodies 213 112. The Contral Eliipsoid ofInertia. . aie 113. ASystom ofDetached Masses Cannot BeCentrobarie =25 114. Theorems Relating toElectrie Images... 2216 1B. Level Layers, : 219 116. FamiliesofLevelLayers . 221 LIT.LevelLayeronanArbitrarilyGivenSurface 222 118,Robin’s Intogral Equation... 2... 27 119. Picard’s Solution ofRobin's Equation. 228 120, ExampleofaLovelLayer... ... ce 228h 121,LevelLayersonProlateSpheroids.|.- +.238 122,LevelLayersonEllipsoids. ..a +235, 128.ThePotential ofEllipsoidal Level Layers 239124,LayersofFiniteThieknoss. ... : ry)125. AFinite Shell Bounded byConfocal Spheroids. . .. |... 241 125, The Surface Density Necessury toProduce Given Potentinls 244 127, Green’s Problem... . _ . 246 128. Certain Physical Considerations... - 247129.TheExistenceofGreon’sFunction, .||| =248,180. Miscellancous Propertios ofGreen's Function 2.249131.TheGreenFunctionIsSymmetric. : 2.252182,TheNormal Derivative ofGreen's Function IsHarmonic. 253183.TheGreenFunctionfortheSphere,. 254134,TheNormal Derivative ontheSphere 237 135.Green's Equation fortheSphere. : ++260 136. AGeneralizationfortheSphere=.|_|Poe 2261 187.Green’s Equation foranyClosedSurface co.5268 188,AGeneral Theorem ofGreen's... : 269 Green's Problem fortheLogarithmie Potential 199. StatementoftheProblem, ss.289 140,Electric ImagesfortheLogarithmic Potcatial . Fes 20 Ml.ElectricImagesofControharie Bodies... am 142,TheExistence ofGreen's Function. are 143.Green's Function fortheCircle ner.) MA,ThePrinciple ofDirichlet andLordKelvin .TT 145.TheEquivalent Problem ofPoincaré... : 280 Problems... . : 282 CHAPTER VI ‘Two-taven Surraces ‘TheMethods ofNeumann andPoincaré 140.Various TypesofMass es 288 147.TheMagnetic Doublet 283 CONTENTS xi Sorrow Pace148.TheBarMagnet... .= ce 5285149. Magnetic Sheots—Two-layer Surfaces. ns 285 150. Closed, Uniform, Two-layer Surfaces es 286 151. Uniform Surfaces Not Closed . 2. 288 152, Surfaces with Variable Moments. +. 290 158. Discontinuities inthe First Derivatives 292 154. The Configuration Constant ofaClosed Surface . +s. 298155.TheSpreadofValuesofthePotentialonClosed, ConvexSurface295,156.Neumann's ProofofDirichlet’s Principle.... 296157, The Limiting Values ofthe Potentials W,on S... 299158.Harnack’s TheoremforHarmonie Functions... .2... .300159.CaseI—TheConstantCIsZero...2... 805 160. The Interior and Exterior Functions asPotentials ofthe Same SimpleLayer 2. 307 161. The Constant @IsNot Zero. . : 309 162,TheConstruction ofaSimple,LevelLayeronS.... ....SIL Poincaré’sMéthodeduBalayage 163,TheBalayageofaSphere..... .5BB 164. Existence ofaLevel Layer onaGiven Surface 314 165. Application ofHarack’s Theorem 318,168.Construction ofanInfiniteSystemofSphereswithinS.-...819167. ‘The Existence oftheRequired Harmonic Funetion. 2. 820 CHAPTER VII Spuemican Hanwoxtes 108.Definitions... ee 5 B25169. Examples ofSpherical Harmonics . 2s. 826 170. Homogeneous, Harmonic Polynomials. : +5826 171. Relation between Certain Harmonics. : 328, 172. The ExpansionofaPotential... 328. 173. Rotation about anImaginary Axis... . »330 174, Harmonies Which Depend upon rand: Alone... .831175.TheEquationofLaplaceforSurfaceHarmonics we5882176.ZonalHarmonics... 2... ce 5888177.ThePolynomials ofLegendre... . 335 178. The Expansion inTaylor's Series. 37179.TheExpansion inLagrange’s Series... . 2338180, Zonal Harmonies Given Explicitly.|.: 339 181.TheZerosoftheZonalHarmoniesAreAllResl +MO 182,CertainUsefulRelations nn and 183, The Zonal Harmonics AreOrthogonal Functions 343,184.AGeneralization ofthePreceding Formulas, so.845185. ARecursion Formula for Zonal Harmonies HG 188. The General Formula for Zonal Harmonies aT 187.TheGeneralExpressionforH - a) 188. Zonal Harmonics Expressed byCosines ofMultiples oftheArgument. . 350 xii CONTENTS: Seerior Pace189.Powersof»Expressed inTermsofZonalHarmonics .. 352190. ADefinite Integral RepresentationofZonalHarmonies... ..355 191.AnImportantPropertyofZonalHarmonics. ce.856 192. Expansion ofSin myinaSeriesofZonalHarmonics... ...837 193. The PotentialofaSolidofRevolution _ 360 194. The Oblate Spheroid 303 195. The Apparent Size ofaPlane Circular Disk 365, 196. The Potential ofaZonal Distribution ofMatter onaSpherical Surface. ee 366 107. Tessoral Harmonics... ss. po 368 108. Examples ofSolid Tesseral Harmonies : . 871 199. The Zeros ofthe Tesseral Harmonies 372 200, The Surface Integral oftheProduct ofTwo Spherical Harmonics ofDifferent Degrees... 373 201. ‘The Surface Integral oftheProduct ofTwo Spherical HarmoniesoftheSameDegree.. a: 874 202. The Expansion of1/pinaSeries ofTesseral Harmonies. - 378 203. The Expansion ofthe Potential of«Finite Body inaSeries of Tesseral Harmonics. . . . 380 204. The Expansion ofthePotential ofaFinite Body inaSeries of Inertial Integrals _ see +. 882 205. Laplace's Integral Equation. : 2.884 206. The Expansion ofanArbitrary Function inaSeries ofSpherical Harmonies . eee a 387 207. TheRepresentation of«Rational, Intogral Function . 390 208. Green's Problem fortheSphere . 2. 802209,‘ThePotentialof«SurfaceDistribution ofMatteronaSphere..393210, Differentiation with Respect toPoles 395 211, Derivation oftheTessoral Harmonies byPolar Differentiation 398 Problems . cae . 404 CHAPTER VIII Euurrsorat, Harwoxtes 212. Introduction... . 407 218. Definition oftheElliptic Coordinates. +. 407 214, Differential Relations... . 409215.TheEquationofLaplace.. : 410216. The Elliptic Functions ofWeierstrass 412 217. Spherical Harmonies inElliptic Coordinates 2.415 218. The Inverse Problem : 416219.TheFunctions ofLamé.| 418220, Determinetion oftheConstant N’ 419 221, Existence ofSolutions forClass I.. : 420 222, Existence ofSolutions forClass I|. 1. ee 223, Existence ofSolutions forClass IIT ee’) 224, Existence ofSolutions forClass IV. 425 225. The Products ofLamé 426 226. Liowville's Proof That AlloftheRoots AreReal a7 conrenTs xxii Seco Pao227. Particular Examples ofLamé's Functions . 429 228. ‘The Pattern asaFunction ofg: 435 229. Parametric Representations ofaSphere 436 230. ‘The Ellipsoidal Harmonies asaFunction of Xs... 438 231. The Surface Harmonic V:V3. . 439 282. The Spheroidal Surface Harmonic V+Vs, 44233.‘TheRootsoftheCharacteristic EquationConsidered asFunctionsoft . a 43 234. The Characteristic Equation Has NoMultiple Roots 45 235. The Functions ofLamé Are Linearly Independent 446 238. The Expansion ofAnArbitrary Function inTerms oftheEllip- soidal Harmonies : 4a 237. Surface Integrals... . : 449 238. The Coefficients ofan Expansion in‘Terms ofEllipsoidalHarmonies .eee =451 239, ‘The Roots ofLamé's Polynomials Are Real, Distinct, and Lie Detwoen afand et . . 452 240, Ellipsoidal Harmonies oftheSecond Kind. 454 2AL. The Potential ofanEllipsoidal Harmonie Surface Distribution ofMatter. . : 455 242, The Potential ofanEllipsoidal Homoeoid. ee 2487 243. Green's Problem onanEllipsoid. es 450 Extension ofthe Goneral Theory 244, Fundamental Functions, Ses 400) BmuooRarey .. bee ce 5468 Twoex. . .sr THE THEORY OF THE POTENTIAL CHAPTER I THE ATTRACTION OF FINITE BODIES 1.The Law ofGravitation—Newton’s law ofgravitation states that Every particle intheuniverse attracts every other particle with a force which isdirectly proportional totheproduct ofthetwomasses and inversely proportional tothesquare ofthedistance between them; thedirection oftheforce being inthelinejoining thetwo particle. Itwill beobserved that thelawapplies only toparticles, and not tobodies offinite size. 2.The Attraction ofSystems ofParticles.—Since forces are vectors, itisevident that the attraction ofmany particles, of mass ms upon asingle particle, ofmassms,isthevectorsumofthe|“>attractions oftheindividual particles. “>>(Fig.1).Inordertoeffectthissum, ‘> <=" itisconvenient toresolve theindi: *——" vidual attractions into their compo- Fi.1. nents along three mutually perpendicular axes. ‘The components ofthe resultant attraction, orthevector sum, along these three axes isthen thealgebraic sum oftheindividual components along these axes. 3.The Components ofAttraction—Let the coordinates of theparticle ofmass m;with respect toafixed setofrectangular axes be&,ni;f«;and letthecoordinates oftheparticle ofmass ‘mybex,y,z. The force acting onmywhich isduetotheattrac- tion oftheparticle m;isdirected towards m,and itsintensity is Reman, 1 2 THE THEORY OF THE POTENTIAL where n= VG= OF EE ‘The direction cosines ofthe line ofthe force are respectively ko my hte Hence the components ofthe foree along the three axes are mand *zmom, bmomE 5. Ifthere arenparticles m,,and ifX,Y, and Zarethecomponents oftheresultant attraction ofthe particles mupon theparticle ‘mo,then =- Sema(a=8), X=—Kk*my=aad =—KimeSUE) Y=—km>ra 0) Z= Kms Stmale=, Intheevent that thesystem ofparticles m,form acontinuous body ofdensity o(€, 1,£)these sums pass over into thedefinite integrals X=—Hefam,lp ye—HimefEan, (2),F ae~Hef2slim,|nr whereee r=VE- OF OG-FET, and dm =o dédndt. The value ofthe gravitational constant k*inc.g.s. units is 6.66X10-*. Itistheforcewithwhicheachoftwounitparticles attracts the other when the distance between them is1em. 4,The Elements ofMass.—The volume density atapoint isthe limit ofthe ratio _mass volume 4) THE ATTRACTION OFFINITE BODIES 3 atthat point, the density ofwater at4°C. being unity. The element ofmass is dm=odtdndt. Similarly for thin sheets orplates, which are regarded as surfaces, thesurface density atapoint isthelimit oftheratio mass ‘area atthat point, orthemass perunit ares ifthedensity isconstant. For asurface, theelement ofmass is dm=odtdn. Inthecase offine wires, which areregarded aslines, theline density stapoint inthelimit oftheratio mass Tength atthat point, orthemass perunit length inease thedensity is constant. For aline the element ofmass is dm =adé. 5.Attraction onaPoint.—The phrase “attraction ofabody onapoint” initself hasnomeaning, butsince theattraction is always proportional tothemass oftheparticle attracted itis convenient toassume that itsmass isunity. Hence, bydefini- tion, theattraction ofabodyonapointmeanstheattractionof thebody onaunit particle atthat point. Inthe simple examples which follow, thegeneral formulas ofSec. — 8arenotused,astheexamplesare<p Rsolvedbysimplermethods. Doubt- Z| less the student will find itinter- estingtoverifythesesolutionsby LYthe integration ofthe formulasEq.(3.2). oow6.The Attraction ofaCircular A Are onIts Center—Given afine Fu. 2 uniform wire ofmass «per unit length bent into theform ofacircular areABC (Fig. 2),which subtends anangle 2aatthecenter 0. Itisdesired tofind theattraction oftheareuponitscenter,theradiusof thecirclebeing.Itisevident from symmetry that the resulting attraction liesinthebisector oftheangle AOC, that is,along theline OB. 4 THE THEORY OF THE POTENTIAL Hence itisnecessary toconsider only thecomponents ofattrac- tion which areparallel toOB, thesum oftheperpendicular com- ponents being zero. Let the areAC bedivided into nparts each oflength As. Ifnisvery large, each part can beregarded asaparticleofmass ads. Let0betheangle which theparticle makes with theline OB. Then the component along OBofthe attraction ofthe particle atAsonthe point 0is Becos6, Imagine thechord AC tobeafine wire similar tothe are ABC. Let Acbetheprojection onthechord ofthelength 4s. Then Ac=As- cos 6, and themass oftheparticle atAcis ade = oAs 00s 0. Ifthis particle were moved tothepoint Bthen itsattraction onOalong OBwould be As cos 6 pardegeat, that is,itwould bethe same asthe component ofattraction oftheparticle atAsalong OB. But Asisany particle ofthe are, and Acistheeorresponding particle ofthechord. Hence ifevery particle ofthe chord AC were placed atthe point B their total attraction onthe point Owould bethesame asthe total attraction ofthe are ABC. The total mass ofthe chord is 2or sina,and therefore the total attraction ofthe are ABC onOis 7.The Attraction ofaStraight Line onaPoint—Let AB beauniform rodofdensity ,and letObeany point notinthe line ofAB. With Oas acenter draw acircle ofradius r,tangent tothe line AB (extended ifnecessary) atC. Draw the lines OAand OBcutting thecircle inthepoints DandErespectively, and then imagine the are ofthe circle DCE tobearod similar tothe rod AB. Let Asbe portion oftherod AB soshort that itsmass can beregarded asaparticle. Join theextremi- ties ofAetoO. These lines cut out aportion ofthe eizeular rod Ac. i) THE ATTRACTION OFFINITE BODIES 5 ‘The attractions ofthe particles atAsand Aconthepoint Oareequal; for, if@istheangle which theline OAs makes with thelineOC, Ac =As cos? 8, ‘The distance Ods isequal torsee6.Hence, theattraction of the particle 4sonOis ods_kaascost_odeFeecto~ tT which isthe same asthe attraction ofthe particle Acon0. Since each particle oftherodhasthesame attraction forthe point Oasthe corresponding particle ofthe circular are, both A o7 3 ig Fo. 3 asthe magnitude and dircetion, itfollows that thetotal attrac- tion ofthe rod isthe same asthe total attraction ofthe arc. Iftheanglesubtended bytherodatthepoint0is2a,theresultantattraction bisects this angle, and, bySec. 6,itsmagnitude is sina ae, a where ristheperpendicular distance from 0totherod. Iftherodincreases inlength, itslineremaining fixed, theangle increases and has the limit x/2 forarod which isinfinite inboth directions; and this limit isindependent ofthemanner in which the limit isapproached. Hence the attraction ofan infinitely long rodforaparticle, atadistance rfrom it,is, 2kte, a that is,itisinversely asthedistanee, and notinversely asthe square ofthe distance. 8,The Attraction ofaThin Sheet onItsAxis ofSymmetry— Consider theattraction ofathin, doubly symmetric, plane sheet upon apoint inthe axis ofsymmetry which isperpendicular 6 THE THEORY OFTHE POTENTIAL tothesheet. Byvirtue ofsymmetry, theresultant attraction lies inthe axis. Let0(Fig. 4)beany point oftheaxis OP,other than the point ofintersection ofthe axis with the sheet. With Oasa center, describe asphere ofradius 7,which istangent tothe sheet, andimagine that theplane sheet andthespherical sheet have thesame surface density c. 4 ‘Take an infinitesimal cone of solid angle Awwith itsapex atOwhich intersectsboththeplaneandthesphere, \ and which makes anangle @with the ‘ST axis OP. Let the area which this cone ofthe plane As,sothat Wa.4 ‘Ac=Ascos*6. ‘The component oftheattraction ofAson0which liesinthe axis OP is Be=Btcos8=iteSesoee Iftheparticle Acwere placed atP,itsattraction onOwould be BigAS=pigoest. Itsattraction, therefore, would beidentical withthecomponent oftheattraction ofAsalong OP. Therefore, theattraction oftheentireplanesheetonOisthesameastheattraction ofthe corresponding spherical sheet under thesupposition that the mass ofthespherical sheet isallconcentrated atthepoint P.Ifthesolid angle subtended atthepoint Obytheplane sheetis(theapparent sizeatOoftheplanesheet), theintensityoftheattraction ofthesheet on0is BeO?=Bee, @ The attraction depends only upon theapparent size atO andnotatallupon itsshape, except thatitmust bedoubly symmetric. Iftheplane sheetisnotdoublysymmetrie,theabove argument stillholds forthat component oftheattraction which isalong OP,butthecomponent oftheattraction which isparallel totheplane sheet, ingeneral, isnotzero. 8] THE ATTRACTION OFFINITE BODIES 7 The limiting value ofthecomponent which isperpendicular tothesheet is2k*ro, ifthesheet isextended indefinitely inall directions. 9.The Attraction oftheFrustum ofaCone onItsApex.— Iftheplane sheet oftheprevious section isregarded ashavingffinitethickness Aa,therelationbetweenitssurfacedensity«anditsvolumedensity@is «= Fa. Tmagine thefrustum ofahomogeneous cone, thebase ofwhich isdoubly symmetric, with the apex onthe axis perpendicular tothebase through thecenter ofsymmetry. Let thefrustum bedivided into nsheets, ofequal thickness Aa(nvery large). Iftheheight ofthefrustum ish,then A= nds. ByEq, (8.1) the attraction ofeach sheet onthe apex is Bow = adae Which has the same value for every sheet. Hence the total attraction ofthefrustum ontheapex is BGenda =Hohe, where «isthesolid angle attheapex. Inthe case ofaright circular cone, thegenerator ofwhich ‘makes anangle awith theaxis, thevalue ofwis tew=ffsinededs =2x1 —£08), ‘and the attraction is A= 2rk'GH(L —008a). Frustums ofequal height attract theapex equally. Consider any homogeneous cone, that is,one with aplane base ofany kind, and aninfinitesimal solid angle dowith itsapex attheapex ofthecone making anangle 6with theperpendicular ‘from theapex ofthecone tothebase. Ifthelength ofthis infinitesimal cone isr,thearea which itcuts outofthebase is da=rdw sec 6. Letdubeanelement ofmass ofthis infinitesimal cone which liesbetween two planes parallel tothebase atdistances pand p+ dpmeasured along the cone from the apex. Since the 8 THE THEORY OFTHE POTENTIAL thickness ofthiselement dis dpcos#anditsbase area isdu seo 0,itsmass is du=ap¥tada, andtheattraction ofthisinfinitesimal coneupon itsapex is A=fo=fierce=ordsJor Jo =00059=That, ‘The mass oftheinfinitesimal cone is a=Johda, sothat an Hence theattraction ofthehomogeneous infinitesimal cone upon itsapex isthree times asgreat asitwould beifallofits masswere distributed uniformly overitsbase. Since theheight ofevery suchinfinitesimal cone hasthesame value h,their masses areallproportional totheareas oftheir buses. "Since theproposition holds foreachofthem separately, itholds forallofthemtakentogether. ‘Thatis,theattrac- tionofahomogeneous coneuponitsapexis My(auf threetimesasgreat,andinthesamedirec-tion, asitwould beifthemass ofthe cone i B were distributed uniformly inaninfinitesi- mal layer over itsbase. 10.Perspectivity—Two lines, 1and Is, areinperspective with respect tothepointOiftheradiusvectorfromOisintercepted ébythetwolinesinaconstant ratio;thatis, Fro.6, if,inFig. 5, OB,_0A,OB, ~04, ~” forevery position ofthevector OB. Iftheratio ofperspeetivity fsp,theratio ofthelength ofthelines I,andlsalsoisp. Likewise, twosurfaces areinperspective with respect tothepointOiftheradiusvectorfromthepointOisintercepted bythetwosurfaces inaconstant ratio. Ifthe ratio ofperspectivity. is»,theratio oftheareas ofthetwosurfaces isp? 10] THE ATTRACTION OPFINITE BODIES 9 ‘Two solids areinperspective with respect tothepoint Oif their corresponding surfaces areinperspective with the same ratio ofperspoctivity. Thus, if,inFig, 6,thesurfaces C,and Cs areinperspective with theratio p,andifthesurfaces Byand By also areinperspective with the ratio p,then thesolid bounded bythesurfaces Byand C;isinperspective with thesolid which isbounded bythesurfaces Bsand Cs;andtheratio ofthevolumes ofthese solids isp2. Theorem I—Two lines which areinperspective with respect tothepoint Oand which have thesame linear density attract the point Owith forces which are ine versely proportional totheirratioof zh, perspectivity (or, inversely propor tionaltotheirdistances). “B, InFig.5lettian lsbeinpee =}spectivewiththepointOwiththerar ratio ofperspectivity p.Then —ffa, OB NL 0B,~* We Letdwbeaninfinitesimal plane rue angle which cuts out the ares . As:andAs, Letaandaybetheattractions ofthetwo particles ‘As;and As:onthepoint 0. Since thelinear density ofthetwo Tines areequal, themasses ofthese twoparticles areproportional tothetwo lengths As;and As;. Hence armOBEAsLd a2ms"Op) Beot Since thisconstant ratio holds forevery pair ofcorresponding. points ofthe two lines, itholds fortheir sum, and therefore holds for the attraction ofthe lines themselves. Inamanner quite similar the two following theorems are proved. Theorem II—Tvo surfaces which areinperspective and have thesame surface density attract thepoint ofperspectivity equally. Theorem III.—Two homogeneous solids ofthesame density which areinperspective attract their point ofperspectivity with forces which have thesame ratio astheratio ofperspectivity (or, which aredirectly proportional totheir distances). 10 THE THEORY OF THE POTENTIAL Tho Andromeda Nebula has about the same apparent size asthe sun but itis 6X10% times asfar away. Ifitwere spherical inshape with thesame density asthesun, itsattraction on the earth would be 6X10°° times that ofthe sun. Even though thedensity oftheNebula were thedensity ofthe sun divided by6X10%, thetwo attractions would beequal. 11.The Attraction ofanEllipsoidal Homoeoid upon anInterior Point.—A homogeneous shell bounded bytwo ellipsoids which aresimilar andsimilarly placed isealled byThompson and Tait anellipsoidal homoeoid. Forexample, thetwo surfaces oyeSthtaah and eye .athtasat bound such anellipsoidal shell. Itwill besupposed atfirstthat)isaninfinitesimal sothattheshellisverythin.Let Pbeany point within the shell. With Pasavertex deseribe aninfinitesimal solid angle which cuts theshell ata: a anda. Passa plane through Sy thislineandthecenter oftheN\ shell.‘Thecrosssectionisan\ ellipse (Fig.7),orrather, two similar concentric ellipses. P ‘The system ofchords parallel toaid:isbiscetedbythe ths conjugate diameter ofthis hee system,whichisalsotheeonju- gatediameter ofthesystem of chordsbibyoftheinner ellipse. Henee thechords bibsalsoare bisected bythisconjugate diameter, sothat ibs =agbs. Lotthemeasure ofthesolidanglebeaw.(Themeasure ofasolid angle isthearea which thesolidangle cutsoutofthe unitsphere which hastheapex ofthesolid angle asitscenter.) Since thevolume density oftheshellisconstant, themass ofthe particle which iscutoutoftheshell bythesolid angle ispro- portional toitsvolume. Since theangle and theshell are infinitesimal, this volume isanoblique cylinder, thevolume ofwhich isequal totheproduct ofthearea ofthebase and the 1) THE ATTRACTION OFFINITE BODIES wW height ofthecylinder; or,what comes tothesame thing, the product ofthearea of#perpendicular cross-section ofthecylinder and itsslant height. This last formula isthe one whieh is desired inthepresent case. Letm;bethemass oftheparticle ata:and msthemass ofthe particle ataz. Then, if¢isthedensity, m,=0+abi Pay-de, my=6aba:Pas’-Aa. ‘The attractions ofm,and msonthepoint Paretherefore Ay=BBY =BeGhAe, Pay Ay=BY, =Ke-GabiAo. Par Since aib; =Gaba, itfollows that Ai=Ae,and, since thetwo forees are oppositely directed, the resultant attraction ofthe two particles onPiszero. This istrue forevery infinitesimal ‘cone which hasitsapex atP. Itis,therefore, equally true for finite cones, and consequently fortheentire shell forwhich the value ofwis2r. Itfollows, therefore, that theresultant attrac- tion ofaninfinitely thin ellipsoidal homoeoid onaparticle any- where initsinterior iszero. Itwill bezero forany number ofsuch homoeoids, and is therefore zero for ashell offinite thickness. Inparticular, since aspherical shell isaspecial case ofanellipsoidal homoeoid, the attraction ofaspherical shell, which ishomogeneous in concentrie layers, onaparticle anywhere within itsinterior iszero. 12The Attraction ofaSpherical Shell onanExterior Parti-<le.—Itwillbeshowninthissoctionthatahomogeneous sphericalshell attracts anexterior particle just asthough allofthe mass ofthe shell were concentrated atitscenter. The method of proof used isdue toThompson and Tait. Let0inFig. 8betheconter ofthespherical shell ofradius aand thickness Aa(infinitesimal), and letPbetheattracted particle. Ontheline POtake thepoint A(the harmonic con- jugate) sothat PO. 04 =a". 2Natural Philosophy," Part HL,§471. 12 THE THEORY OF THE POTENTIAL ‘This canbedone graphically bydrawing thetwo tangents totheshellfromthepointPandthendrawing thechordwhichconnects thetwopoints oftangency. Theintersection ofthis chord withthelinePOisthepoint Awhich isrequired. For thechord CC isperpendicular tothelinePO,sothat thetri- angles OC,A and OC,P aresimilar, and therefore 0C;: OP::0A :0C,; from which follows atonce OP-OA =o. =~ G L- Po ofJALoy G Fro. 8. With thepoint Aasavertex construct anyinfinitesimal cone whose solid angle isAw. If«isthevolume density oftheshell, themasses ofthetwoparticles m;andmswhich arecutoutof ‘the shell bythecone atB,and B;are m=0+AoABy+Aa-see8, my= o-doAB,-Aa-see8, where 6isthevalue ofthebase angles inthetriangle B:OB2. ‘The attractions ofthese two particles upon the point Pare Ay=BeBw-Aasee9:(AB,BP, , ABAs a A=Weawaa-see8(8) Inthetriangles 0B,A andOPB,, theangle at0iscommon and thesides which include thisangle areproportional, since PO _OB, PO.O4 aORao OB, =04° or PO-04 =OB, =a* 13] THEATTRACTION OFFINITEBODIES 1B Hence, these two triangles aresimilar and the angle B:PO is equal to@ Likewise, thetriangles 0B:4 and OPB, aresimilar, and the angle BsPO isequal to6. Furthermore, from thesimilarity ofthetriangles, AB, _OB,|aB\P~OP~OP”ABs _OB:_a.BP~OP~OF ‘Onsubstituting these ratios inEq. (1),theexpressions forthe attractions become : A=Ke Awasec 9(4),oP, As=He:Ao:da-soe0(2),op, and, therefore, inmagnitude A,isequal toAs. The resultant oftheforces A:and Azliesalong thebisector oftheangle BsPBs, that isalong PO, and itsmagnitude isequal to 2k-Aw-2a A=ete the OP This formula holds forevery solid angle Awwhose vertex is atA,and therefore itholds fortheir sum, since the coefficient ofAwisconstant. Hence, onsetting BAe =2n, the attraction ofthe entire shell is A,=SeHeataaHM,w oP’ Since themass ofthespherical shell is4rva*Aa. But this isjust, what the formula would be, ifthe mass ofthe shell were all concentrated atitscenter. Ifthepoint Pisintheshell itself, only one nappe ofthecone intersects the shell, and forthis case oP ‘This result, however, depends upon the manner inwhich the limit isattained. Itisshown inSee. 86that the attraction ofa surface forone ofitsown particles isquite indeterminate. “4 THE THEORY OF THE POTENTIAL 13.The Attraction ofaSolid Sphere uponanExteriorPoint.— Ifm,isthemass ofahomogeneous shell and ifthepointPlies outside oftheshell atadistance rfrom itscenter, theattraction oftheshell onthe point isdirected towards the center ofthe shell, anditsintensity is ‘The resulting attraction ofanynumber ofsuch shells which areconcentric isdirected towards their common center, and its intensity issimply the sum oftheintensities fortheindividual shells, Hence, ifMisthetotal mass ofthesphere, itsattraction ‘upon anexterior point is 4aPMEs Itisnotnecessary that theshells shall allhave thesame volume density. Itissufficient that each shell separately shall behomogeneous. Itisevident that asolid sphere which is homogeneous, orhomogeneous inconcentric layers, attracts an exterior point just asthough itwere aparticle ofthesame mass. located atthecenter ofthesphere; andtwo such spheres will attract each other just asthough both were particles coneon- trated attheir conters. 14.The Mutual Attraction ofTwo Straight Collinear Rods.— ‘The preceeding examples have been solved byspecial methods oifABtf c Foo. 0. which appeal directly totheintuition. Some further examples ‘willbegiven which require themethod ofintegration. Consider first theattraction ofastraight rodonanexterior point inthe line ofthe rod. TetOAbearodoflength l:andoflinear densityex,Letde beanelement oflength ata distance zfrom 0. ‘The mass ofthe particle atdziso,dz. The attraction ofthis particle onapoint atadistance &from the point 0is wee,€- 2 14] THE ATTRACTION OFFINITE BODIES 15 andtheattraction oftheentirerodOAonthepoint£isthesum ofthe attractions ofallofitselements. That is jo(E—2)ONETg, ‘This istheattraction oftherodonaunit particle atthepoint £.Suppose csisthelinear density ofthered BC. Then the attraction oftherodOAonaparticle oxdé oftherod BClocated atthepoint £is 11) owe> -jae(De Letthedistance between therods beaand thelength ofthe second rod beJz. Then the total attraction ofthe first rod on the second isthe sum ofthe attractions onthe individual ele- ments. That is, meet1») (@+h@+h), owl (:Sy,7get=slogeh Fh)” and this isintensity ofthe mutual attraction ofthe two rods. 15.The Attraction ofaCircular Disk onItsAxis.—In Fig. 10 lettheradius ofthecircular disk berand letitssurface density be.Itisrequired tofind -theattraction ofthediskupon ‘apoint ofitsaxis, that is,the a] line which isperpendicular to the disk atitseonter 0. Let 0betheorigin ofasystem of polar coordinates p,6,andlet 2bethe distance from0along theaxis totheattracted point P. The mass oftheparticle ofthedisk whose polar coor- dinates arep,@is dm =opdpds, Pu. 10. and itsdistance from theattracted particle is Ra Vere Hence, itsattraction onthepoint Pis din_Kepdodd, RT pttat? 16 THE THEORY OF THE POTENTIAL and thecomponent ofthis attraction along theaxis is _Bepdpd? 2__Bospdpdd. ete RO pane ‘The total attraction ofthe disk istherefore +2ppdd *pdp_ Zea~veeffaio ~2ekeef >i Jo Gt-te98 oot +22)8 =+2bel2-4.) a+aleeeg+RA ° ‘The numerical value ofthe second term ofthe bracket is+1 according aszisnegative orpositive, forinextracting thesquare root of2?itisthepositive root which istaken. Since thisexpression changes sign i with zbut does not vanish with z, the attraction has afinite discon- tinuity,if¢»0,astheattracted ctpoint passes through thedisk. For adiskofinfiniteradiustheattrac- | tion isindependent ofthedistance zand has the eonstant numerical value 2rk'z. Inpassing from the negative side tothepositive side, that is,depositive, there isafinite jump inthe attraction equal to —4rk’s. If the earth were an indefinitely extended homogeneous plane, assome ofthe ancients believed, theacceleration ofgravity men 4gwould berigorously, instead of only approximately, constant. Itisevident from symmetry that there isnocomponent of attraction perpendicular totheaxis, 16.The Attraction ofaBody ofRevolution onaPoint inIts Axis.—Let thez-axis betheaxis ofrevolution. Ifthebody is homogeneous, aswillbesupposed, theresultant attraction isalong thez-axis, anditisnecessary tocompute only the#-component., InFig. 11,letAB beafigure ofrevolution. Bythethird formula ofEqs. (3.2) Z=kefSEsteenar: 16] THE ATTRACTION OFFINITE BODIES 17 but theresults ofSec. 15can beutilized, thus eliminating two ofthe three integrations. Consider athin cross-section of thickness dfatthedistance ¢from theorigin A. Ifisits volume density and o;isitssurface density, then v1=od. Ifzisthe coordinate ofthe attracted point, with z>B, Eq, (15.1) gives fortheattraction ofthedisk at¢ Qekte| —2 1a ly @=1 +r where ristheradius ofthedisk; and theattraction oftheentire body is Zeeve"LSa rer a) ulVe-oF +r 17, Example—The Oblate Spheroid.—Let the surface ofthe oblate spheroid bedefined bytheequation Pte oleet +Ge1 The radius ofthecross-section atthedistance ¢from theorigin isdefined bytheformula o mat—Si; sothat theexpression fortheattraction (Eq. (16.1) becomes + = pdZ=Ink'o| oe aeh ave-ptta— Ee Since (-2+¢=f)4% i Fa theintegral ofthis expression can beresolved into thesum ofthe twointegrals a a(% ceteSe tate |Goes seatmsve-pta—Se a, (Ht ws aiveita Oe 18 THE THEORY OF THE POTENTIAL ‘The first integral isthevalue of a 22ps,goale- pte =e between the limits +c and —c, After substituting the values ofthelimits andsubtracting, this value isfound tobe ~—_4 aoe ‘The second integral can bewritten rs a, : a2—ot) later(a*—ot+at)_(_act * @aepy4Goa aeatt) ‘The value oftheindefinite integral is 1Lo?=et+act,Bevan eee which fortheupper limit chasthevalue 1 a oh pce aa e+e4eee tant nyava ete Vue0) and for the lower limit go ete tee at ct egORO yg ae Me wae te Va—e@te) ‘The value oftheintegral istherefore @octe actertant OEE 4typSane VaeG—3 * Ve aet+e vena Vane atan BVEmataYORE, Hence, Zette|~e=gtOE, tonVY*| ee a 2 (a?-ca Anktoate|z=1Vai- Soe] +yercaw YS] Since themass ofthespheroid is 4 M=grate, theexpression fortheattraction canalso bewritten SieM| ety Ve @ 24TtyeraunSE} 17) THE ATTRACTION OFFINITE BODIES 19 From theexpansion tpegoleglsele seetantz =z get 5 wt itfollows that sy|143@>e)_3@—e |) z~vul eta 7 mw Ot Ifthisexpression iscompared with theexpression forthesphere itisseen that theattraction ofanoblate spheroid onapoint of itsaxis isless than theattraction ofasphere ofthesame mass at the same distance. 18.The Attraction ofaUniform Rectangular Plate onaPoint inItsOwn Plane.—Let thesides oftherectangular plate be2a Poy . ° Fro. 12. and2b,anditssurface density bec. Referred toasetofrectan- gular axes with origin atthecenter oftheplate and theaxes parallel tothesides oftheplate, theelement ofmass oftheplate isodfdy. Let the coordinates ofthe attracted point Pbez,y. Since there isnosymmetry inthe situation, itisnecessary to compute theX-and Y-components oftheattraction separately. The attraction ofthe element dmonthe point P,which is assumed tolieoutside ofthe plate, isdirected along the line joining dmtoP,and itsintensity is Kedédy (=e += The components ofthis element offorce along the z-and y- axes are _ _Bela ~atin ___ Holy =natin7» 33 (@- B+ 98 (@- 9+ —0F 20 THETHEORY OFTHEPOTENTIAL and thetotal X-and Y-components oftheattraction are x=~eeppe(z=Bandeey -oJ-ni@ =OF+ -08 Y=~k'ei|rie ~eJ~(@- + -oF Let the integrations forthe X-component betaken first, Since @-) a) 1 (@-o'+o-o8 “NG@- 9+ -0% itisevident that x=-#[*.—+—, - lie =a+ —E —_1___ |(+o +yo Now + =dn =logYADAV aE btya logVODAVETaFURby, fF(@—a+y—oe |UFOVea+GyFoF and +, dy togUAHAV +a) FyFoe= tog Ort roe I(etart@-9E —Y-D+VErar+y=H‘Therefore X=kelog (y=»)+Vea FU=Dy+0)+VeFa)?FWFHitz+VeHa+ UFO0)+eFayreyO and similarly, Y=kolog {e=a)+VGH FO=Die +0)+VieHa)?+WV+d)(ea) +VaFart y—O@—a) +Vie-a+Uy+H Itisunderstood, ofcourse, that inevery term inwhich the radical occurs thepositive square rootistobetaken. Although itisnotevident onthesurface, itisnotdifficult toverify that Xisanoddfunction ofxandanevenfunction ofy,andthat¥ isaneven function ofzandanoddfunction ofy.That is X(@) =-X(-a), XW) =+X(-w), Y@)=+¥(-2, YW) =-¥(-»). 18) THEATTRACTION OFFINITEBODIES a1 ‘These properties merely state that thefield offorce hasthesame properties ofsymmetry with respect tothez-andy-axes that the plate itself has. Itisinteresting tonote also that the attraction increases indefinitely ifthe attracted point approaches the edge ofthe plate anywhere. 19.The Attraction between Two Rigid Bodies.—Let the two bodies bedenoted byB;and Bs, Let o1bethe density ofthe body Byatthepoint £1,n,£1,andletozbethedensity ofthebody Byatthepoint f,m,$2. Ifthedensities arenotconstants, it, will beassumed that they arecontinuous functions ofthe coor dinates. The component ofattraction inthez-direction ofthe body B;onthepoint &,m2,£2,isbySee.3, SS. Hox(bi—&:)dbdndty, : —.,,4 }a.{(E: —&)*+(mn—me)?+i—$2)" where dm; =exdtidnidt isanelement ofmass ofBs. Ifdmz =oxdévdnadt isanelement ofmass ofthebody Bs,the z-eomponent Xofthemutual attraction ofthetwo bodies is evidently SSS,SSS,Kowwa(t—&s)dédndtdiednadts Oo 3 iB'as{(&: —&)*+(m—m2)?+(1—$2)" ‘There aresimilar expressions fortheother twocomponents ofthe attraction, Thus the determination ofthe three components of attraction oftwo solid bodies requires the evaluation of18 definite integrals. These integrals arenotsufficient, however, todetermine theline ofaction oftheresulting foree. Problems 1.Three particles ofmass m:,mz.and msareplaced atthevertices ofan, equilateral triangle ofwhich thesideis¢..Thecenter ofgravity ofthethree particles isatthepoint P.Show that theresultant attraction ofanytwo Oftheparticles upon thethird liesinainewhich passes through P.. Show‘thatthisisstilltrueifthelawofattractionvariesinverselyasthen"powerofthe distance. 2,Showthatforsphereswhicharehomogeneous theattraction ofthesphere forapoint upon itssurface is 4 jeer. 22 THE THEORY OF THE POTENTIAL 3.Asmall hole ofnegligible diameter isbored along aradius into a sphere which ishomogeneous inconcentric layers. Itisfound that the attraction ofthephere forapoint inthe hole isindependent ofthedistance ofthe point from the center ofthe sphere. What isthe law ofdensity of thesphere? 4.Show that the attraction ofahomogeneous cylinder oflength 21, radius r,and density oonapoint inthe axis ofthe eylinder atadistance2fromthecenterofthecylinderis Beko VOFaFH—VOREr—25} itthepoint isinside ofthecylinder, and aekte| VOFAP EH VORBEeOI) ifthe point isoutside the eylinder. ‘The negative sign before the term 21 istobetaken if2is postive and the lower sign ifzis negative. Thus the attraction allalong the axis ofthe cylinder isacontinuous function ofz. 5.Given astraight rod oflength Uand density ¢. Show that the attraction oftherodonapoint whose distances from theends oftherod are 7,and rs,Eq. (7.1), can bewritten A=eel, nr and therefore theequation ofthesurface, inbipolar coordinates, onwhich the intensity ofthe attraction iseverywhere the same is rire =constant, 6.If spherical segment iscut off sphere ofradius @and density «byaplaneatadistance2fromthecenter,theattractionofthesegmentonthe center point ofitebase is 2 2B ers _—Fpl—21)—(at—24); ‘andifthesegment isahemisphere theattraction is io, 1.Show that the attraction ofthe sbove spherical segment upon ite vertex, orits highest point, is 1p@=s Deals—aft3ve} 8.Astraight rodoflinear density isbent into the form ofarectangle ofwhich thesides are2aand 2b. Show that itaattraction onapoint inthe line which isperpendicular totheplane ofthe rectangle through itscenter is an.aoere Varopara tara! 8.The base ofahomogeneous isosceles triangular plate liesonthez-axis and thevertex ofthetriangle iesonthey-axis, The length ofitebase is 2aand thetangent ofitsbase angle éis«,Show that thecomponents ofthe 19) THE ATTRACTION OFFINITE BODIES 23 attraction oftheplateuponapoint2,y,outsideoftheplate,butlyinginitsplane,areX=WesinglogA-B, Y=Bycos#logC-D-E, where 4nthretay +Vialets)—iFlat2iP-a—2—ay+Viale+2)—lt+lo+2+autpeowete—ey+Vie=2)—teeta, oat+2=ay+Vala=2)—yl+[oat+2—ay}?ou[tetverarepo"a-24Ve-a tHDarastastay+Visa=a0=oetTeayta2tay+V[aa=at~yl+la—2Fay)? ant+2+ay+Vala+2)=yl+[oat—3=ay)? Showthattheattractionisinfiniteattheedgesoftheplate,andthatitis finite and determined along thelines which arethecontinuations ofthe edges ofthe plate. 10.Show that theattrction ofauniform rectangular plate onapoint in theline perpendicular totheplate through itacenter is z. a btele sin-t 8., Va" Jape Vere where 2aand2arethelengths oftheedgesoftheplateandzisthedistance ofthe point from the plate, 11, Matter isdistributed onthe curved part ofthe surface ofacone forwhichthelengthof«generatorisRinsuchawaythatthedensityisinversely proportional tothedistance from theapex. Show thatthe attraction ofthetoneforaparticleonitaxisatadistancefromtheapexandpfromitsbase ‘edge isdirected toward the apex and isproportional toR/(pz). For aninfiniteconetheattractioniinversolyasthedistancefromtheapex. CHAPTER IT ‘THE NEWTONIAN POTENTIAL FUNCTION 20, The Potential Function Defined.—In the formulas for theattraction ofafinite body onapoint, thethree components oftheattraction aregiven bymeans ofthree triple integrals, making nineintegrals altogether. Itisaremarkably interestingfactthatthesenineintegrals eanbereplacedbyonetripleintegraland three differentiations. Iet 2,y,2bethe coordinates ofthe attracted point and &n£bethecoordinates ofapoint oftheattracting body, of which dmisanelement ofmass. Then dm=odtdndt, where ¢isthedensity ofthebody atthepoints £,»,f,andis, therefore, ingeneral, afunction oftheletters £,»,¢.Let the function Vbedefined bythe definite integral. dm van|%,JA |a p=VEAP FG Wt EO thointegration tobeextended over theentire body. The constant k*depends upon the intensity ofthe force ataunit distance and the system ofunits employed. As itenters always asalinear factor, itwillbeconvenient todropit, and this will bedone hereafter, with theunderstanding that it istoberestored whenever necessary. The function Vthus defined isafunction ofthe coordinates 2,¥,£(not &1,£);that is,Visafunetion oftheposition ofthe attracted point. Itcantherefore bedifferentiated with respect tothese letters; that is ov 8 (dmGe7bpfiqr ete. (2) Iftheattracted point lies outside ofthe body, pdoes not vanish and therefore 1/p isalways finite within the region of pi 20) THE NEWTONIAN POTENTIAL FUNCTION 25 integration. Under these conditions’ itispermissible tointer- change theorder ofdifferentiation andintegration inHq.(2) and write 7 ayn a 2gv.San =~[Zen =X,Eq.) VL (2(\am =—(Yam =zSiaay fiStan=¥, oz 292 0. ln Ifthe function Vwere known, thecomponents ofattraction ofthebody onoutside points could beobtained bythedifferen- tiation ofVwith respect tothecoordinates 2,yand z;and V- isdefined byatriple integral. Itwillbeshown immediately that Vexists even iftheattracted point isinside theattracting body, and that even foraninside point thecomponents of attraction arethederivatives ofVwith respect to2,yand 2. 21.The Significance ofthePotential Function.—It iseasy toscethephysical significance ofthepotential function. Let theattracted point begiven adisplacement ofwhich thecom- ponents aredz,dy,dz. The element ofwork which isdone in effecting this displacement isaW=Xdr+Yay+Zaaviey4a, =Fede+Pedy+ae =av, and, therefore, thework done ineffecting anyfinite displacement against theattraction ofthebody is W=V(e1, yy21)—Ven Yn%)s where 2;ys2:istheoriginal position and2s,ya,sisthedisplaced position. Stated inwords, theamount ofwork done inmoving ‘particle from oneposition toanother against theattraction ofthegiven body isequal tothedifference inthevalues ofthe potential function forthese two positions. The potential function isascalar function ofposition. Its derivatives arethecomponents ofavector, namely theattraction ofthebody. IfAisthemagnitude ofthisvector, At=X?4 2428, +Hamxcx’s “Caleulus,” translation byCathestt,p.267. Also Gounsat- Heonicr, “Mathematical Analysis,” p.108. 2% THETHEORYOFTHEPOTENTIAL whatever bethe orientation ofthe coordinate system. Ifa system ofaxes ischosen sothat thez-axisisparalleltothevector A,then 7ai oz and the perpendicular components are zero. From this one concludes that thetotal attraction liesinthe line forwhich the derivative ofVisamaximum. 22,The Potential Function Exists.—It will be assumed that thedensity ofthe attracting body isgenerally acontinuous function, butthat there may beafinite number ofsurfaces across whichthedensity isdiscontinuous. Suchwouldbethecaseina sphere ifonehemisphere were made ofiron and theother hemis- phere were made oflead. Ifthenumber ofsurfaces ofdiscon- tinuity isfinite thebody can beresolved into afinite number of smaller bodies Bxforeach ofwhich thedensity iscontinuous, and then a @Im im ve (MA ; under these conditions each integral ¢dgdndtSSS exists iftheattracted point liesoutside ofthe body! forthe function @/piscontinuous over theregion ofintegration. Since each integral separately exists their sum exists, and therefore V exists. Furthermore, since theintegrand iseverywhere positive, ifristheminimum distance from theattracting body tothe attracted point, and Risthemaximum distance, Fig. 13, dm ‘dm ‘dim amy and, therefore, ifMisthetotal mass ofthebody, Myvoi ‘Astheattracted point recedes from the attracting body, both rand Rincrease indefinitely and therefore Vhasthelimit zero atinfinity inevery direction. Iftheattracted point isintheattracting body letthevariables bechanged topolar coordinates bythesubstitution *Gounsat, Hepaick, “Mathematical Analysis,” p.296, 22] THE NEWTONIAN POTENTIAL FUNCTION 7 =2+p.008¢0088,n= yt pcos sin &,faztpsing, adm=ep?cosedddp,J 0that theorigin ofthepolar system ofcoordinates isatthe attracted point. The integral which defines Vthen becomes V=SJffier 008ededddn, and bythesame argument asfortheexterior point itisseen that Vexists. Let cybethemaximum value of¢inB. With the attracted point asacenter and aradius Requal tothe distance ofthe most remote point ofthe attracting body, describe a sphere which will wholly enclose theattracting body. Imagine this sphere filled with matter ofthe constant density oy. ‘Then since alloftheelements ofthe integrals arepositive fimfe se Joe But Im Hhpmckfewoofffcosededddp ]s jz Jo Jo 5 =drop? Hence, forany interior point VS dreyR. 23,Existence ofDerivatives ofthe Potential—Owing to theimportanee ofthederivatives ofthepotential inthegeneral theory adireot proof will begiven notonly that thederivatives exist, but that these derivatives areequal tothecomponent ofthe attraction inthe direction inwhich thederivative istaken. Inmaking the proof, however, itisnecessary todistinguish between theinterior and theexterior points, and tomake the proof foreach class separately. 24, Existence ofDerivatives atExterior Points.—As first step certain upper limits forthederivatives of Bg ee i — a VEO Fy te 28 THETHEORY OFTHEPOTENTIAL will beindicated. Thus (1) _t=8 2(N<1, &()=-254,therefore, |xG<P #() 38e- 8?_1 a(4, dz\p, e e \ax"\p, & (0) iseOyge=F,|38(1)| 28, ae()>8G ee BCG)<i ete sy céaryz Let B,Fig. 13,bethe attracting PA body ofwhich dmisan element, Let drbethevolume ofdm,oitsdensity d and &,{the coordinates ofitscenter ofgravity. Let2,y,2bethecoordi- nates oftheattracted point p. The element dmexerts upon panattrac- tion ofwhich the z-component is —&=Ham, ie18. a and, therefore, fortheentire body, x==f252an= [2(2am Par ae and likewise, y=-[(%5%am= [2Sam, Par aau\p, Ze-[Glam =fa(;)em. Je FAG Bydefinition, thepotential is vef‘dm,Ise Itisdesired toprove that av a(noefra(3)e™ For convenience ofnotation, let 1 deen, sothat V=freley,2)dm, 24) (THE NEWTONIAN POTENTIAL FUNCTION 29 Bythedefinition ofaderivative ev odOF=timAI[ole+as,n.20m~fote.v.2idm]wan(12%4lag2% 7Jim,StBe+Jarhe+tas)fam Where@issomenumberlyingbetween 0and1;therefore, a(84m4timInef&, ,veSm+lim2[gue+Az)dm, Itisnot difficult tosee thet oo aJim,aSeZoole +oa2)dm= 0, for a 4 Boe tay <4, where pisthedistance from theelement dmtothepointps(2+ bz,y,2).SupposebistheclosestpointofBtop,andthatthedistance bp:=|.Then|isthesmallest possible valueofp, and 1 a 2az, 2m,das(Pole+edzidm<fPAEay=2s, 2»[i Jn F ‘AsAztends towards zero thepoint p:moves towards psnd theminimum distance ofp;tothebody Imay change. But ‘since pliesoutside ofB,there exists aminimum value ofJ, sayp>0,ifAzissufficiently small. Hence a a 2Mlim,atfae+eaa)dm<Tyas, Which vanishes with 42, Therefore, ave a/(rvefiayn =X from which theconclusion follows that thederivative exists and that itsvalue isthecomponent oftheattraction. 25,Existence ofDerivatives atInterior Points.—It wasshown inSee, 22that thepotential exists atinterior points. Itiseasy toshow inthesame manner that thecomponent oftheattraction 30 THE THEORY OFTHE POTENTIAL exists. Using thetransformation (Eq. (22.1)), theexpression forthe2-component ofattraction becomes zn k avx= - [25 fam= (2(7amxo[epfemeJoxlo} =SSfiecostv008oded0do, Inasmuch astheintegrand iscontinuous over theentire region ofintegration, ifiscontinuous, itisevident thatthislastinte- gral exists. Itisdesired toshow that thederivative ofVexists atinterior points, justasforexterior points, andthatitsvalue isX. Lettheattracted point pliewithin B.With pasacenter deseribe asphere ofradius rwhich lieswholly within B. ‘This spherical surface divides thebody Binto two bodies namely, thesphere $andtheremainder ofthebody B,. Let thecoordinates ofpbex,y,2,andletpbeaneighboring point with the coordinates z+Az,y,2.Let Vbethevalue ofthe potential atpandVthevalue ofthepotential atp.Then WotimY=V.Or “asso Ar Itwill beshown that ifXisthez-component ofthe attracting force atthepoint p,thevalue ofArcanbetaken sosmall that the value of V-V_y'[Wah-3]<6 where ¢isanysmall number given inadvanee. Let X,Xs, Xs, bethecomponents ofattraction, and V, Va,Va,thevalues ofthepotentials atp,duerespectively tothe body B,thesphere, andthebody By.Likewise, letX,Xs,Xs, V,Vs,Vaybethecorresponding values atthepoint 7.Then VeaVstVoy X= Xet+Xoy VeVstVa, X= Xs+Xa,j so that V-v Vs-Vs_y Vo,—Vn, [Yat -3]- [Pam]+[Pel"s) 25) THE NEWTONIAN POTENTIAL FUNCTION 31 SincepandFarebothexteriortoBy,itfollowsfromtheresults ofSec. 24that, whatever value rmay have, |4z|<rcanbe taken sosmall that [Voy—Vax 1 ‘There remains then forconsideration Vs-VsVe=¥e_ xy Let drbethevolume oftheelement ofmass dm, and leta betheupper limit ofthedensity within S.Then since Ben[252a,ls itfollows that Kl<afFobrosr, Ise sothat [Xd <dro ‘Anupper limit forthevalue of Vs—Vs B~ax” aalsocanbefound.Fromtheirdef/aSynitionsitfollowsthat aT ¢- ait 'Ve-Vo=[(5—))pa. Py } NowSJ} L_l_le-»ae pe Fi.14. and, asisseenfrom thetriangle inFig,14, p—A<[aal, Also, since : (-3)>0,on itfollows that 1 1 1 inte sothat a é 4Ve—Vel<|x)-oo)| =|. Perc feefet © 32 THE THEORY OF THE POTENTIAL thas already been observed that ffkder, js With ~asacenter describe asphere S;with aradius equal tor+ Ax|. This second sphere will then contain the first, and therefore [B<SGoeet <Ber. Js? Js," Ifollows atonce that [Vs—Volbse|<tree and_ Peat-x]<16reyr.ae For values ofrsufficiently small this expression isless than «/2, and therefore v-Vv_Fact -af<e ‘Therefore, thederivative ofVexists, and itsvalue isequal to X. The proposition isentirely general, and itholds whether theattracted point liesinside oroutside oftheattracting body. 26,The Equation ofLaplace.—In amanner quite analogous tothat ofSec. 2,itisarelatively simple matter toprove that thesecond derivatives ofVexist atallexternal points, and that eVLF BM, FV CBN ayGtJ,azo) —ay?~Jyavo) eV fH/1), aa?~ J, d2\p)™ But BI) 8aOF_1 ax*\p, * re HN) 8G _1 avo, ao @() _3@-9) 1 82*\p, ee and theirsum=2-3<9, ae 26] «THE NEWTONIAN POTENTIAL FUNCTION. 33, Hence,3/2), (1),2/1)JunSlee() +ai()*ae()]om =6 and therefore av av av “aat+ayetat = ® ‘apartial differential equation ofthesecond order which must besatisfied bythepotential ofevery finite body atallexterior points. The left member ofEq. (1)iscalled the Laplacian ofV. Ithasbeen designated bydifferent writers bythesymbols, AV, 4¢V,0V andVV. Inthepresent volume itwillbedenoted bythesymbol AV. This equation was first given byLaplace in1782 inpolar coordinates inthe form OV), 1 fg,QV), 1avroetain’ai(s7)+mroae ~% butseveral years later, 1787, hegave thecorresponding expression inrectangular coordinates. Since theform fortheexpression forp p= Ve-P +O-W Fe- isindependent oftheparticular rectangular system towhich itisreferred, itisnatural tosuspect that thesame istrue of Laplace’s equation. Thesuspicion iseasily justified byachange ofvariables, nam tartaytae, 11=vo+Biz+Bay+Bx,nay trat ny+7 where attaxttast=1, afi +ab +abs =0, B+ Bt+Bt=1, Bin +Bers +Bars =0, vitaettl, netnatnaa=0. Sincey y y ev_a av, aBemras +eas +Shaw itfollows that ora04,at ont azp+éy?tnaz? av av av +2B +Boonie +margege? 34 THE THEORY OF THE POTENTIAL av25gdoh Bemaa+Origa+atc av av ev +PashOxi0y1+260‘Oy1021+Pymae‘a BVoTyga0Va “ae GaTOagtoes av av eV tasesrag,+Gyan +MGRa, Ontaking the sum ofthese three expressions itisfound that av av av_av av, avast*ay?+oataneyetaes | ‘That is,Laplace's equation isinvariant under achange of rectangular axes, ‘The same thing istrue forLaplace's equation intwo variables. 21,Equipotential Surfaces, orLevel Surfaces.—If thepoten- tialfunction issetequal toaconstant VQ us) =6, a there isdefined asurface which isealled ane7uipotential surface, since thepotential has the same value atevery point onthis surface. Iftheconstant Cisregarded asaparameter, anentire family ofsuch surfaces isdefined. Since byitsdefinition Visa single valued function ofz,y,2,itcannot have two values atthe same point and, therefore, notwo surfaces ofthefamily intersect. These surfaces arealso called level surfaces forareason which willappear inthenext paragraph. LinesofForce.-If ,y,2,isapointonthesurfaceEq.(27.1), theequation ofthetangent plane atthepoint 2,y,2is av, eV, av,get—2+Sh—0+e-0=o, where &,1,¢arethe running coordinates ofthe plano. The direction cosines ofthenormal tothisplane are _lav. lav _lev, conanRs coseRI cosy =BU, inven ave ae-V(z) +(G+) - a7) THE NEWTONIAN POTENTIAL FUNCTION 35 But , ¥ y a a a wih Bh ete arethecomponents oftheattraction atthepoint x,y,2, and the direction cosines ofthe resultant attraction are xX Y, 2 TEPER.RoR OR R=eVXFVTR ‘These expressions areidentical with those ofthecorresponding direction cosines ofthenormal totheequipotential surface atthe point z,y,2.Hence thedirection ofthe attraction ofabody at any point isperpendicular tothe equipotential surface which passes through that point. Consider aline which isorthogonal toevery equipotential surface which itintersects. Ateach ofitspoints ithas the direction ofthe attractive foree, and forthis reason such aline iscalled aLine offorce. ‘The surface ofafree liquid atrestisalwaysnormaltotheforee whichisactinguponit.Henceitcoineides withanequipotentialsurface, and therefore equipotential surfaces arc level surfaces, Ifadrop ofwater could beplaced upon anequipotential surface and befreed from surface tension, itwould distribute itself over the entire surface. 28,TheLogarithmic Potential —InSec.7itwasfoundthataninfinitely long, homogencous, straight rod attracts anexterior point inadirection which isperpendicular tothe rod with an intensity which isequal to2e0/p where oyisthemass oftherod perunitlengthandpistheperpendicular distanceoftheattractedpoint. Imagine this rod parallel tothe z-axis and piercing the zy-plane inapoint whose coordinates are fn. Let the coordinates oftheattracted point bex,y. Then therodattracts thepoint 2,yjust asthough itwere aparticle ofmass¢=25 placed atthepoint &,7with alaw ofattraction #/p which isinversely proportional tothedistance,insteadofinversely propor-tional tothe square ofthedistance. ‘The components ofthis attraction are xa-0254 ya-045% ZH0.c * Abundle ofsuch rods, notnecessarily allalike, would attract justlikeanarea (thecross-section ofthebundle) with thesurface 36 THE THEORY OFTHE POTENTIAL density «and theinverse firstpower ofthedistance. That is, for such abundle z-t Y=am, x-fAdm,v--f- tam, p=VE +O-a For this law ofattraction the expression forthe potential becomes ve-f.logpdm,or @ os= dm vf.tog2 ifpreferred, where p»isanarbitrary constant. Itwillbeverified without difficulty that ev av x-% y-% @ ‘Thepotential inthiscase, forobvious reasons, iscalled the logarithmic potential. Ascan easily beverified, forexteriorpointsitsatisfies theequation ofLaplace intwovariables av. av itFn ‘The logarithmic potential exists atinterior points also, and Eqs. (1)and(2)arestillvalid. 29.The Potential ofaSpherical Shell—Let O,Fig. 15,be thecenter ofahomogeneous spherical shell ofradius a,Aaits TAN Fro, 18, thickness and itsdensity. Let Pbetheattracted point, and the line OPbedrawn, Let theradius OSmake anangle ¢with the line OP, and theplane OSP make anangle @with theplane of the paper. 29) THE NEWTONIAN POTENTIAL FUNCTION 37 Consider anelement ofmass dmoftheattracting surface atS. Expressed interms ofitscoordinates ofposition dm=catAa sinydyd?, and therefore o jo Jo 0 =2roaraaf2Me,bP sincepaViatarweTF isindependent of8. Ondifferentiating thisexpression forp,bearing inmind that and rareconstants, itisfound that sinede _dp p ‘ar’ and therefore V=20Sfdp. The limits oftheintegral depend upon whether the point P- isoutside theshell, orinside theshell. Ifthepoint Pisoutside theshell, rsVo=aretit [a =ita, o~ ro ‘thelastequality holding since themass oftheshell Mis ‘M =4roa*ha. Ifthepoint Pisinside theshell, oapet Weare a “hr (2) =Areata = @ From these expressions itfollows that thepotential ofahomo- geneous, infinitely thin, spherical shell isconstant within the shell, andthat itvaries inversely asthedistance from thecenter oftheshell ontheoutside. The limiting value ofthepotential ‘astheattracted point approaches theshell cither from theinside orfrom theoutside isM/a, and therefore thepotential iscon- tinuous across the shell. 38 THE THEORY OF THE POTENTIAL Iftheshel isoffinite thickness, bounded byasphere ofradius aontheoutside and asphere ofradius bontheinside, itis necessary tointegrate Eqs. (1) and (2)again with respect tothe radius; thus Vo=@Yatda 7 Je Aro, 3)=M=f2@-»-% and v=tof,ada =2no(a* —b*), which isnot conveniently expressible interms ofthe mass ofthe shell. For asolid sphere the radius oftheinterior surface iszero and Vo=M/r just asfortheshell. The limiting value ofthe potential inthehollow-interior oftheshell astheradius ofthe inner boundary diminishes is Vi=2roat =3M.' 2a Forapoint between thetwo spheres which bound theshell atadistance rfrom thecenter, pass aconcentric sphere through theattracted point. This sphere, which hasaradius r,divides ‘the shell into two shells forwhich thebounding spheres areb and rfortheinner shell, and rand afortheouter shell. The potential ofthe entire shell isthe sum ofthe potentials ofthe inner and outer shells, that is 4f-B : Vmfro=”+tee(at-#1) = gs 10_1),=soo(g0t—5a") This expression has the limiting value 2x0(a* —b*)onthe innerboundary andJro(a—08)ontheouterboundary. As these are the limiting values ofthe potential asthe attracted point approaches theinner surface from within thehollow, and theouter surface from theoutside oftheshell respectively, itis clear that the potential ofthe shell iscontinuous everywhere, although indifferent regions ofspace itisrepresented bydifferent analytic functions. 29) THENEWTONIAN POTENTIAL FUNCTION 39 Tneacli ofthese expressions theonly variable which occurs isr,‘thedistance fromthecenter. Hence, wherever theattracted point may betheattraction isalways directed towards this center. Itsintensity isgiven bythefirst derivative ofVwith Tespect tor,namely _ev.A=x Inthehollow oftheshell Visconstant; thefirst derivative iszero, and, therefore, theattraction iszero everywhere. Between the boundaries oftheshell 4bA=ap-) ‘This expression vanishes ontheinner boundary, and hasthe value 4(2-2) __M 3 a~ a ontheouter boundary. Outside ofthesphere A= M.--% which also has the value —M/a* onthe outer boundary ofthe shell. Hence the first derivative ofthe potential function is continuous everywhere and inparticular across the bounding surfaces ofthe shell. That is,the attraction ofthe shell isa continuous function ofthe position oftheattracted point. Itisnot the same, however, with the second derivative. ‘Within the hollow, the second derivative iszero everywhere. Between the boundaries ofthe shell ev 4 (2%we-r(+1). This isacontinuous function ofrwithin this region. Itdoes not vanish ontheinner boundary, however, but hasthevalue —4rc. Hence the second derivative has afinite discontinuity 4mo across the inner boundary. Onthe outer boundary the second derivative has the value 4 (28=aa+) On the outside ofthe shell the second derivative is av 48 ee,ort 3 r 40 THR THEORY OF THE POTENTIAL which iscontinuous everywhere onthe outside and takes the value 4 (2b ontheboundary. Thus incrossing theouter boundary from the inside the second derivative has thefinite discontinuity +410 (see Fig. 16). This study ofthepotential ofafinite spherical shell isvaluable because the integrals involved areeasily evaluated, and the Fro. 18 properties exhibited are characteristic ofpotentials and their derivatives ingeneral. The results are summarized inthe following table. Pommat or4Fosims, Howoanveous Seurnica Suet Ran|reo|bere |re v|eet=o|deo(r18s)|etav aye coEa o ace aio oy “(P getoF°|-3(+1))tee 30], THE NEWTONTAN POTENTIAL FUNCTION an 30,Potential of«Uniform Circular Disk along ItsAxis.— Let0,Fig.17,bethecenter ofthedisk,OPitsaxis,atheradius and¢thesurface density. Lettheangle @bemeasured from afixed lineinthedisk. ‘Then inpolar coordinates dm=erdrdd, and .Vine[PNRMmae[5 oyJoJo? a But, since pert, and isconstant intheinteg- 2ration, I| Ppdp =rar, F and therefore drTat dp. emde, Hence ro.18 vie=ne Vmaoeft, or V=droVatFt—Verh. @) Attention isagain called tothefactthatinevery casothe positive square rootistaken, sothatVisafuntion of24,asis evident from theform ofthepotential when expressed asan integral, Eq.(1). ‘The attraction ofthedisk for«point onitsaxis isgiven bytheformula a 1 1x2 =o -Ja) 3oven val ® ‘This isanodd function ofz,asevidently itshould be. The limiting value oftheattraction as2approaches zerofrom the positive sideis~2re; andasitaproaches zerofrom thenegative fide itis-+2rv. There is,therefore, adiscontinuity inthe attraction equal to—4re astheattracted point passes through thediskfrom thenegative sidetothepositive side, andthe magnitude ofthediscontinuity isindependent ofthesizeofthe disk. Itisnotindependent, however, ofitsdensity. Tftheradius ofthedisk ‘isvery great, thefirst term ofEq. (@)isnegligible. ‘Thesecond term isconstant oneither side 2 ‘THE THEORY OF THE POTENTIAL ‘ofthedisk, butopposite insignonthetwosides. Thus, if theearth were anindefinitely extended plane, asitwasthought tobebytheancients, with thesurface density ¢,then the acceleration ofgravity would be 9=2ra, andwould beindependent ofdistance from theearth’s surface. 31,The Potential ofaHomogeneous Straight Rod.—Let thelength ofrodbe21,Fig. 18,and letits 4?linear density bec. Letanelement oftherod Ms atadistance¢fromthecenterbedg.‘Then is themassofthiselementis «dn=odt, and ifpisthedistance from this element tothepoint P(z, 2)atwhich thevalue ofthe potential istobecomputed, then dm_pete y= (@-o(°% Let @betheangle formed bytherodandthelinep.Itisclear from thediagram that dysin6=pds, and therefore, yao (% jin 8 1tan56 @ =0log—j—- tan36 Since @always liesinthe first orsecond quadrant, thetrigo- nometrie identity tandg= jt41-1. @2?~Nianto tan holds. Now 2 2 tna= Fp tanh=2 ) Bymeans ofEqs. (2)and (3)thepotential can beexpressed explicitly interms ofzand 2;namely, V=olog22=Dt+VER— Fa =olog—E-DtVEWD te,-@+D04+VeFD +e 31] THENEWTONIAN POTENTIAL FUNCTION 43, or,ifp:and p:arethe lines drawn from the point Ptothe ends oftherod, = etl tps gigrtoe+2 Vee logTeo PEotos=2 The components ofattraction are now easily found. They are eV_offl1 11 X=ar71G:-ay-(Gatah av 1 1cari Gi~ny Imagine two particles each ofthesame mass asaunit length ofthe rod placed atthe two ends ofthe rod. Then the com- ponent oftheattraction oftherodatthepoint Pwhich isparallel totherodhasthecurious property that itisequal tothedif- ference ofthepotentials ofthetwo particles atthepoint P. Equipotential Surfaces—On anequipotential surface Vis constant. Let y Ceee<l @ ‘Then onanequipotential surface 1 L tan561=Ctan5%. @) ‘From theformula forthetangent ofahalf angle, 2tanbo tang=, 1—tan*ou itisfound that 2 1 1 ep tan gntan}tana 2 or,byvirtue ofEq.(5), 2 1 1 eC tgtr Ctande 2 and similarly 2 1 1ian*1 7tanyo tan36 44 THE THEORY OFTHE POTENTIAL Ifthefirstofthese equations ismultiplied byCandthensub- ‘tracted fromthesecond; andthenthesecond ismultiplied byCandthefirstissubtracted from it,there results thetwo equations 1 ce | a-ey, 1 tan,~ tang, ~~ 3 tmphe, ae tL =e) afan tam = 90 ioe \ / \SVbyy N\ Y/Y :O 7 N a NN, oo/i)\\ Fro. 10, Theproduet ofthesetwoequations eliminates thehalfangle,and gives 1__€Vc _1)_a-cy: tan6;~tan 6,)\fan@,~tand; ac” which reduces, bythesubstitutions ofEqs.(3),to (l=Oe|1—ytat_aForr tae pak ‘Thisequation, bythesubstitutions, ace a+o,_)Ton Gaghatts ©) becomes theequation forconfocal conics, Fig.(19), ze 2Peete a 31) THE NEWTONIAN POTENTIAL FUNCTION 45 Onreplacing the value ofCfrom Bq. (4), itisfound from Eq. (6)that e c=aml>o.sinh? Forpositive values of«these curves areellipses, and from its definition xmust bepositive. But theequation gee te c<P,poy Sah o<ese, ® represents the orthogonal confocal family ofhyperbolas’ which, byvirtue ofthe property oforthogonality, are the lines of force, Sec. 28. The parametric equations Ising Leos ¢*"Snhe’ *~tanbw” represents afamily ofellipses ifwisconstant, and thefamily of orthogonal hyperbolas if¢isconstant. - The equation oftheequipotential surfaces inthree dimensions is # tty Ppt ot which represents prolate spheroids. The orthogonal system of surfaces isrepresented by gee etePa eh and arehyperboloids oftwo sheets. The lines offorce arethe intersections ofthe hyperboloids with thefamily ofplanes which intersect inthez-axis, y= dy, where )isaparameter. 82The Potential ofaHomogeneous Solid Ellipsoid for Interior Points.—Let thesurface ofthegiven ellipsoid bedefined bytheequation Burge.BtRtaeh a 4“Statics and theDynamics ofaParticle,” p.356. 46 THE THEORY OFTHE POTENTIAL ‘ndettheinterior pointforwhich thepotential istobecomputed beP(x,y,2).Ontaking Pastheorigin ofasystem ofpolar coordinates »,y,8,withtheequations oftransformation E=2+pcosg0084,=ytpcosesin6, @f=ztpsing, din=op!cosededédp, thenthevalueofthepotential atthepoint Pis ‘dm +Hteon VacfPmef [Pfrcoseaeasa @ se “JazJoJo ‘Theupperlimitp,oftheintegration withrespecttopisafunction ofyandésincetheintegration isfromPtoapointonthesurface oftheellipsoid, 1fEqs.(2)aresubstituted inEa,(1)thesurfaceoftheellipsoid isfoundtoberepresented byanequation ofthesecond degree in This equation eanbewritten >Apt+Bo+0=0 ® where =208%vcost?,costysint|sin?y A=a te + =F0089.0088,yeosesind ,zsingBat —S ta ©) w2ae ye Cca+e+alL Thesolution ofEq.(4)which belongs tothepresent problem is a~-B+VBF-AC1BtVBRaC forAisnecessarily positive andforinterior points Cisnegative.Hence B*—ACispositive andgreater thanB*,Sincep,mustbepositive, thepositive signmustbetaken before thisradical. From thisexpression for1isobtained pit=2B=AC~2BV FAGfm BAC—BBVBRAC, #0that Eq,3becomes +Epeop—40—opBDACvores BBY=AC~2BVBFICoggig.(6) + 82)“ THE NEWTONIAN POTENTIAL FUNCTION a Consider first thepart ofthis integral which depends upon the radical +5pp VBFDAC R=ffBVBAC cosgga, -zJo ‘The elements ofthis surface integral can betaken inpairs the two elements being equal numerically butofopposite sign, At thepoints P,and Ps,let (P:) 0=6,¢= 5 and (Ps) 6=H+, 9=—e, sothat onaunit sphere P;and Parediametrically opposite points. Atthe points P;and Psthe quantities Aand Chave thesame values while Bmercly changes sign atthese two points. Hence, theintegral of BYBF= IC, 7 7 where diisanelement ofarea ontheunit sphere, taken over anyhemisphere isequalnumerically, butoppositein sign,tothesameintegral taken over theother hemisphere. That is,theintegral taken over theentire sphere iszero. Hence, R=0, and theintegral (Bq, (6)) reduces to 17th ptope —Vendofe[GAS omeet G)3 Ifthevalues ofA,B,and @from Eqs. (5)aresubstituted in Eq,(7),there results _[TH(?rfeostecosta 2?,costysin®® y? ve“fel.(esratt 5 @etf AP +3(2(xycostpsin8.088,ye.cosgsinysindsafe ar ve 4#20089singc0s6)cosdel?1p,(+3(**cosded,ea? aeoyffA 48 THE THEORY OF THE POTENTIAL Byproperly pairing theelements inthesecond integral itsvalue isseen tobezero. For example, intheintegral +2costpsin60088 itisseen that atthepoints P,and Ps,where (Pi) @=%, e=g, and (Pr) $= -h, P= ey ‘theintegrand hasvalues which areequal numerically butopposite insign. Hence theintegral iszero; andinasimilar way itcan beshown that theother two terms ofthesecond integral also are zero. Hence, +4p2foostecos?@ z?,costysin®# y*,sinty.2 cosvded@1,(*3eosped, xSeefet ed?(8) Now let 1,(+3008ededa, wvyf.fa3 Wisafunction ofthethreesemi-axes, a,b,and,butisindepend-entofthecoordinates z,y,andzoftheattracted points. If itiswritten loti cosededéw=vf-IBeatgwos,costesi, Sate’ ©) ivesores ete itisreadily verified that Eq. (8)eanbewritten _1aW,,, 1aW,, 1aW.,Vesa +5ete atOM or,again, onsubstituting thevalue ofC, 13aW 24 (law aw . ve(GrDp+(oe-Bet(e-Ee+W. (10) SinceWisafunction ofa,b,and¢alone(andtherefore itsderivatives also), thecvefficients ofz*,y?,and2*arefunctions of a,b, and conly. Itisevident atonce that Wisthevalue ofthe potential atthecenter oftheellipsoid. 32) THENEWTONIAN POTENTIAL FUNCTION 49 There still remains theproblem ofevaluating W. For this purpose, let cost,sinty _cost,sintyM="y +e ONpte ay Then Eq. (9)becomes 1tt ‘te a w=rfcoswefMore+Nave +9 3sootado =tofosoteaad eosede=no[2sede,fVMN Onreplacing thevaluesofMandNinthisintegralfromEqs.(11), itisfound that W=2roabet (*@——____esev____, Ta cteost g)(bisin® p+cfeos?9) anexpression which isnotsymmetric intheletters a,b,and c. ‘The symmetry can berestored however bythesubstitution , e sing = 5,OOVere where ¢isthenew variable ofintegration. The result ofthis substitution is @ W=roabe|“———_S______, 12) JVet +NE+8) . and ifthis form ofWisused inEq, (10), the expression forthe potential becomes - “1-8 we V=reat[(:@+sFreen) as XS See)Vereiers Forpurposes ofnumerical computation Wisreduced atLegen- dre’s normal form ofanelliptic integral ofthefirst kind bythe substitution sing=S18RY, org =acot*w —c¥cosect u,Vane . 50 THE THEORY OF THE POTENTIAL provided a>b>e and becomes w=—2eeabe_ 0S_da Va eo V1—RFsin?@ where wath 83.The Equipotential Surfaces.—The equation forthepoten- tial Vataninternal point ofahomogeneous ellipsoid can be written intheform, from Eq. (32.13), W-Vit ivy roabewatpty @ where 1.f“a, a So(a?+)V(a+80+8)CC?+8) aLf~ ds ,@ BO), SF )VETIE FTIE +s) i.f__—__ . Po Jo(et+8)(a+8)?+a)(ce?+5) IfVisaconstant, theequipotential, orlevel, surface also isan cllipsoid with axes which areproportional toa,8,and 7.By forming the differences between these expressions (Eq. (2)), it isreadily proved that Loiel esas and therefore a> B> x Inamanner quite similar itisalso proved that fo or a<&<of ‘These inequalities show that thelevel surface, which isan ellipsoid co-axial with the given ellipsoid, has itslongest axis coinciding with the longest axis ofthe given ellipsoid and its shortest axis coinciding with the shortest axis ofthe given elhpsoid, sothat the two ellipsoids aresimilarly oriented, but since #8 er aso<e 33] THE NEWTONIAN POTENTIAL FUNCTION 51 itfollows that thelevel surface isnotsimilar tothegiven ellipsoid, butismore nearly spherical. Hence thesurface ofthe given ellipsoid isnotitselfalevelsurface. Ifsurface tensions canbe neglected water placed upon the surface ofahomogeneous ellipsoid and subject tonoother force than the attraction of ‘theellipsoid will flow towards theends oftheshortest axis of theellipsoid. 34.The Components ofAttraction atanInterior Point— Since av y y@ a xe ya 2% itfollows atonce from Eq. (32.13) that: ° Jo(P+ )Va+)O+Fs) -2roabeds y= —y {°——__,_2maabeds 1of,Prove PHerH, OY Zf" 2reabedsa poCH INV ETHIE FTNETS) The resultant ofthevectors X,Yand Z,is,ofcourse, avector which isperpendicular tothelevel surface through the point 2,y,2Itrepresents the total attraction ofthe ellipsoid at that point. Consider theattraction ofasecond ellipsoid which issimilar and similarly placed upon the same point z,y,2.The axes ‘ofthis second ellipsoid areda,Xd,and Xe. Then ifX;isthe -component ofitsattraction onz,y,2 { 2roMabeds X,=-2({ --—__—_ a Jo(Ma? +s)V(ata? +8)(N20? +8)(AC? +8), or,ifthevariable ofintegration bechanged bythesubstitution i=, ‘itbecomesf.2QreabedrX:=-2| ——— 2S 428OFIVE ENE FINE ES) and, similarly, forYand Z.Hence theattraction ontheinside ofahomogeneous, ellipsoidal shell which isbounded bytwo similar and similarly placed ellipsoids iszero, aproposition which was proved inadifferent manner inSec. 11, The poten- 52 THE THEORY OF THE POTENTIAL tial, however, isnot zoro inthe interior. If >1isthe ratio ofsimilitude, thevalue ofthepotential inthe interior hasthe constant value or-DW. where Wisdefined inEq. (82.12). 35, The Attraction ofaHomogeneous Solid Ellipsoid upon an Exterior Particle—Ivory’s Method.—If thepoint forwhich the potential iscomputed P(z, y,2)isexterior totheellipsoid the limits oftheintegral which corresponds toEq. (32.3) arecom- plicated anditisnotpractical tocarry through theintegration, inthe manner ofSec. 32. Abeautiful method ofobtaining thecomponents ofattraction onanexterior point was given byIvory in1809, The method isbased upon acomparison oftheintegrals foranexterior point with the integrals fora corresponding interior point. The integrals for the interior points being known, thecorresponding integrals fortheexterior point eanbederived. Let thesurface oftheattracting ellipsoid E,bodefined by theequation etethan @) and lettheattracted point bedenoted byP2with coordinates Zy-¥n #2 Through thepoint P;pass anellipsoid Ezconfocal with E:, The surface ofthe ellipsoid Ezisdefined bythe equation wt yyt patos,gattea (&) but since itisconfocal with B,theaxes ofthe two cllipsoids arerelated bytheequations afaatte bi=btte ofaotte where «isthealgebraically largest root ofthe equation a re artet beet otte sh @) (The three roots ofthisequation arereal.)* Let %,J,%beany point within oronthe surface ofBy and let21,JZ beany point within oronthe surface ofBs. Statics andtheDynamics ofaParticle,” p.355. 35] ‘THE NEWTONIAN POTENTIAL FUNCTION 83 Aone toone correspondence between thepoints ofEyand Es isestablished bytheequations ra TO ® LetP,bethepoint onE;which corresponds toPsonEs. Then P,isinterior toHs,and Bzcan beregarded asahomogeneous solid with thesame density asE,. Let X:, Ys,Zsbethe com- ponents ofattraction ofE,onPs,and X,, Y:,Z;bethecom- ponents ofattraction ofZsonP,,” Then iefff,Bsdgudis,eB XefffBapages, where: A= V@aa +Gv) +Ga aa VGH aTG BPFG) and similar expressions for the other components. These integrals can also bewritten a(z a=fffian(G) esos 2(2)a orfffa) aedguda,. Integration ofthese expressions with respect tothe 2-variable sums uptheX-components oftheattraction along anelementary cylinder ofcross section dyde parallel tothe a-axis between the ‘two points where this cylinder intersects the surface ofthe ellipsoid towhich itbelongs. Hence X=She -ava.i my xinff(h-A)aran. where py:and pyarethetwo lines joining thepoint P;tothe two points where the elementary eylinder intersects the surface ofEs,and pxand ps:arethetwo lines which join Pstothe extremities ofthe elementary cylinder which terminates in ot THE THEORY OF THE POTENTIAL the surface ofEi. Itwill beassumed that theso two cylinders arecorresponding cylinders inthesense ofiq. (3),and therefore thetermini ofthese eylinders Rarecorresponding pointson _thesurfacesoftheellipsoids. Pesgh Lemma—IfRyand8,are COLEN anytwopointsonthesurfaceET} ofEyandReandSsaretheen corresponding pointsonthe a surface ofEs,thenthedistances RS: andRS; areequal. uobo. Inorder toprove this Iemma let the coordinates of Ryand RzbeLy,ms,mand ly,ms,nzrespectively, and letthe coordinates ofS;and SzbeXi, and Xs,us,v2. Then BiSe=(le—Ya)?+(om—a)?+(mi=98), and BEB =(le—1)? +(rms=ws)? +(ma= But since hati, =2, ete, thedifference between these two quantities is et yt (at_y) 4, (ot wa (et :KS!—KS=(31)bet(i-1)mot(-ne +(1~a8)as+(~3)ae+6i) ag BE gmat ome) Ont mat oo(istet=)*Cstoate =Wet =O Henee, thedistance 22,5; isequal tothedistance #35). Since alloftheconditions ofthelemma aresatisfied, itfollows that inEqs. (4) pu =pe pie =pas and, therefore, atcorresponding points theintegrands ofthese twointegrals areequal. Letthevariables g,and21inX:be changed bythesubstitution y= bY tgsays=Fae, de,=Say 35] THENEWTONIAN POTENTIAL FUNCTION 55 ‘The limits oftheintegral will bealtered and will become the sameasthelimitsfortheintegralinX,;anditisevidentthat Xe FX, and similarly, y,=S%y,= 22, abyA=abe" Henoe, ifthecomponents ofattraction ofEsontheinterior point Pi,that isXs,Ys,Zi,areknown, then theattraction ofEs onthe exterior point P;can becomputed. Using the values of these components asgiven inSee. 34with theproper subscripts, itisseen that X2=—2roasex:f~ —7Jo(ast+83)V(as?+82)(bs+83)ca"+83) ° dey Yq=—2rvaibscys ("————__,seyJ,GP+a)Va+be+lerbe) .ds, Zy=—2reabiexts ———. *onefertVa +lhe+aleFm) Inthefactors preceding the integral signs ast =az, ban =by esti =exes under theintegral signs thesemi-axes ofH;can beeliminated bythesubstitutions ataatts babi +e ateatte and then the substitution tens gives thedesired forms oftheintegrals. Dropping allofthe subscripts, which arenolonger necessary, thecomponents oftheattraction ofthegivenellipsoiduponanexteriorpointare = ds X=—2reabe: DarUrlWWrpcerteny/<erany wierd =| EVIE TIETS - ds ¥=—2re a _oo|SWEETIE TIE fO- éswareterers 56 THE THEORY OF THE POTENTIAL where a,b,care thesomi-axes ofthegiven ellipsoid andz,y,2ar thecoordinates oftheexterior point. ‘The lower limit «isthe algebraically largest root oftheequation . ae+ titearate tagen ® 36.The PotentialofaHomogeneous SolidEllipsoidatExterior Points.—The forms oftheexpressions forthecomponents of attraction ofahomogeneous solid ellipsoid upon anexterior point, Eqs. (35.5), areidentical with thecorresponding expressions foraninterior point, (Eqs. (34.1)) with theexception that foran exterior point thelower limit oftheintegrals isx,andforaninterior point itiszero. This similarity ofform leads tothesuspicion that thesame similarity holds forthepotentials also, and that = “G-s -v ¥=weed:f(:etePte=) ds XopeesVere Ne+A y isthe potential ofthecllipsoid atanexterior point. Inorder toprove that this surmize iscorrect, itisnecessary toshow that av _av av Bex, Pay, Taz. Ifthiscondition issatisfied, Vcandifferfromthepotential only byanadditive constant, Now ave ° ds> Ponder | ox f@HIVE TIOFINE+8) ox -(-ey 2)reales , Fe FF FF) VTE HEOE +) thefirst torm ofwhich isequal toX,Eqs. (35.5), andthesecond term vanishes byEq. (35.6). Hence thecondition ove az x issatisfied, and similarly with theother derivatives. 36] «THE NEWTONIAN POTENTIAL FUNCTION 87 Inorder toshow that the additive constant iszero, itis sufficient toshow that Vvanishes atinfinity, aproperty which ispossessed byevery finite body (Sec. 22). Just asinSee, 35,lettheellipsoid which isdefined byagiven xbedenoted byHs. ‘Thegreatest distance from theorigin to thesurface Eyisr=+/a?+x,assuming that, a>b>o, and the shortest distance isr=Vc? x.Hence forany point onEy et+ksersat+n, which shows that ifrtends towards infinity soalso does x,and conversely. Now i: i 5<[8-9 VEN TNE HS) JeGgVe and f. ds < Je(a?+8)V(a?+5)(* +s)(?+8) . ds JfOF )V@+NO+s\(+8) <[aawetemess< [1-2 .CH+IVE TIP IEH) Je§3f Hence (2) Qat+yt+e2 v 2, 2t+yit+2 -(Gi48 5")<seme<+(A+8 mo et a i a or,since PrPresate, 1/8,2a'Vv 1(8,2a? ils+39)<reabe<+vu+ac): Consequently as«tends towards infinity Vtends towards zero, which proves that Vasdefined inBq, (1)isthe potential oftheellipsoid atexterior points. Forx=0theellipsoid Z,which passes through theattracted point 2,y,zcoincides with thesurface oftheattracting ellipsoid. ‘Thus thelimit ofthepotential astheattracted point approaches the surface ofthe attracting ellipsoid isthe same whether the approach isfrom theinside (Eq. (32.13)) orfrom the outside 58 THETHEORY OFTHEPOTENTIAL (Eq, (1)), and thepotential iscontinuous across thesurface. Itistherefore continuous throughout allspace. . Itshould beobserved that the factor reabe which appears in Bg. (1)issimply themass Moftheellipsoid. ‘37.Evaluation oftheElliptic Integrals.—By thetransforma- tion rd wa eGo a>b>o where wisanew variable, theintegral lias" dsvel vane ("_—4§_— av JVETIEFIE+8) becomes_ v=al aw, 0 Va =wT =But) where waGahyae or,if w=sing, theintegral takes theform ad de=|se =Pee 1) ,\laee: (eo) ° where =,(28. ‘Thus theintegral »=F(w, k)isLegendre’s elliptic integral ofthefirst kind. The upper limit ofthelastform w,iscalledtheamplitude of»,andsinaiscalledsinamv,whichisusuallywritten sn». Similarly cosamy=env=VI—ante, deltaamv=dav=Vi—Manto. From these definitions itisevident that ersmo =55> and ais 2 as VEIT NE +) Va 37] THENEWTONIAN POTENTIAL FUNCTION 39 Bymeans ofthese relations itisnotdifficult toshow that ff——— -—8fe VEGI Gyo wares =tSfate, eH )VETIFINEFS)Cg—erado, f— de —of JVFIVEF IOPIES —(gr—eyoHP”° as 2prasntyes =F(t, JC+OVE FIOMEFS ge—aahoat where _ sae Fy ano,=o=wnt fe ‘The substitution ofthese expressions inEq, (35.1) gives Broabe (Eas—2)—atenty—ye—Sevel, (a=c8)=2tsnto—y*Gnty—onto[ae The last three ofthe above integrals introduce Legendre’s elliptic integral ofthesecond kind, namely, Bleab)=[VI Faintode. @ Since dsnvb= +onvdny, den?=—snvdnv, ddav .“d= —k'snveno, iteanbeproved simply bydifferentiation that f“sntodo=jalve—Hed], 0 vontva, Bla) AsnvcantyJodntv”? =a=) 1 dn, "ean?vy1snv.dnv, fonto”=roeae-He} 60 THE THEORY OF THE POTENTIAL Since =e =F), samat theexpression forVbecomes 2raabe 2 YTre, veVeeal[ -eoetes i8) 2 @at Noe @ +[pte wey teleRe@ 2-H) vere +le=ame|aEHETOOHO, Tthasalready been observed inSec, 35that thederivative ofthisexpression with respect toxvanishes, and therefore in forming thoderivatives ov av ov X=a’Y=a’aePra itisnotnecessary toregard «asafunction of2,y,and2.It issufficient todifferentiate onlyinsofaras«,y,and occur explicitly inV.‘Thecomponents ofattraction are,therefore, veryeasily derived anditwillnotbenecessary towrite them. 38,MacLautin’s Theorem—Let E,andH,betwoconfocal ellipsoids withsemi-axes a;,bs,c;andas,bs,c2respectively, each ofwhichishomogeneous, butnotnecessarily ofthesamedensity. Letz,y,2beapoint exterior tobothofthem. Let aetal bintapea2 and ey wttata tare a} represent thesameellipsoid, confocal toEyandHy,throughthepoint x,y,2.Then theequations aeatts batt, cteatts (1) definethevalueof«bymeansofwhichHycanberef Ey,and,therefore, canbereferredtoten 38] “THE NEWTONIAN POTENTIAL FUNCTION 61 Consider the two integrals Lefiz|Wat +an)+an)? 81) and heverter——) eV(aa? ea)a?Foa)i(ea?F8)* where i,j,and &areany positive integers. Ifaz*,bs*,ca?inthesecond ofthese twointegrals arereplaced bytheir values from Eq. (1)there results he=f."se;1VGFFerOFFeFaeteta) and ifthe substitution Seb 8 ismade thisintegral becomes - do h=ooo ee t=|Tar eae areee Alloftheintegrals which appear intheexpressionsforthepoten- tial Eq. (36.1) and forthecomponents ofattraction Eq. (35.5) are included under these forms. Hence ifV;and V:are the potentials ofZ;and EH,atthepoint 2,y,2which isexternal to bothellipsoids andifM,andM;aretheirmasses, itisevident at once that Va. Ve, m~ My and similarly Ho, Mey, A ts,Mi” Mi Mi" Mi My” My ‘This proves avery beautiful theorem due originally toMac- Laurin, namely MacLaurin’s Theorem.—Two homogeneous, confocal ellipsoids attractaparticle whichisexterior tobothofthemwithforceswhich have thesame direction and which inmagnitude are proportional tothemasses oftheattracting ellipsoids, MacLaurin proved this proposition forthe attraction ofellip-soidsofrevolution forpointsontheiraxesofrevolution. Legen- dreextended theproof toany points external totheellipsoids of revolution, The complete theorem asgiven above was first established byLaplace, although hismethod isnot thesame as that which isgiven here. 62 THE THEORY OF THE POTENTIAL Itfollows also that ahomogeneous ellipsoidal shell, finite or infinitesimal inthickness, bounded bytwoconfocal ellipsoids, attracts anexterior particle justasthough itwere asolid ellipsoid ofthesame mass andhomogeneous throughout. Likewise, an ellipsoid which ishomogeneous inconfocal layers attracts an exterior particle justasthough itwere ofthesame mass and homogeneous throughout. Thus theearth increases indensity from thesurface towards thecenter, butifthedistribution of matter within the earthissuchthatitishomogeneous inconfocal layers, theattraction oftheearth foranexterior particle isjust ‘thesame asthough itwere homogeneous throughout, provided the total mass isthe same inthe two cases. 39,The Potential ofSpheroids atExterior Points.—The integrals which areinvolved intheexpressions forthepotential ‘andthecomponents ofattraction ofthegeneral ellipsoid areall elliptic ofthefirstorsecond kind. They allreduce tointegrals ofanelementary character, however, iftwooftheaxes ofthe ellipsoid areequal;thatis,iftheellipsoidiseitheranoblateora prolate spheroid, "If,forexample, a=b>e, the ellipsoid isanoblate spheroid, and theexpression forthe potential is verote[oe ~roate(e?+y") 2@+s)Vei+s-as ods xfPOveseatsf—#__., w J.+ Vet 3Fevers K@+aer+o ‘The evaluation ofthese integrals gives 2roate a+yt=22)ee vmPrete (yt iy raat a@Sea) Nate reaeVe Fastty? xoate Oe +G-e Gt. @-eyapy where «satisfies theequation are tagenh 8) Although «isafunction ofz,y,andz,itisnotnecessaryto regard itassuch informing thefirst partial derivatives ofV 39] THE NEWTONIAN POTENTIAL FUNCTION 63 withrespecttoz,y,and;foritwasshowninSec.36thatevenforthe general ellipsoid the derivative ofthe potential with respect tox,y,or2,insofarasthese variables are involved implicitly through x,vanishes. Itisuseless, therefore, toform it,and - _av av, ov, X=ie Ywy Z=oe’ are obtained from Eq, (2)just asthough xwereaconstant.If, however, the substitution By # a@+K +n from Eq, (8)should bemade inthe second line ofEq. (2), although theexpression forVwould besimplified slightly, « could nolonger beregarded asaconstant informing the first derivatives, Iftheellipsoid ofrevolution isprolate, sothat ¢>a,Eq. (2) becomes imaginary inform though notinreality. Ifitisborne inmind that, Ve—Gaive—@, i=V-T, and that —isin~! i@=sinh 6, itisreadily seen from Haq,(2)that theexpression forthepotential ofaprolate spheroid ¢>ais te) 2Fae =oat2EH2)2 gins[PO Vvreat1+ea veces —mateo Feet yt,rootet yy@-@ ate @—a@Vepe Where«isdefinedbyEq,(3)justasbefore.Inobtaining thecomponents ofattraction from Eq.(4)bydifferentiation xcanbe regarded asaconstant. Itwill beremembered that forinterior points «iszero. ‘The expression forthepotential ofanoblate spheroid, Eq.(2) above, was derived directly from thedefinite integralsinEq.(1). Itcould have been derived with equal ease bylimiting processes directly from Eq.(37.3), byletting b*approach a2. ‘Thepotential oftheprolate spheroid isobtained byletting 6®approach c*. 40.The Attraction ofaSpheroid onthe Surface—If the derivatives ofthe potential ofanoblate spheroid Eq. (39.2) 64 ‘THETHEORYOFTHEPOTENTIAL ‘areformed andthen xissotequal tozero, theresulting expressions arethecomponents oftheattraction ofthespheroid forpoints onitssurface, They are 3M A X=-———[-e ++VI=@sinez, aaetaa *t . i 3M 7 ¥=-——S——[-e+ +VI-esin“ely, areV1=eet yp 3M — Zea- —_ -0-V1=@sinele; wel—ate °ie. whereMI=Sreatisthemassofthespherofdand¢=vent istheeccentricity ofameridian section, The intensity oftheattraction isafunction ofthelatitude butnotofthelongitude, since thespheroid isafigure ofrevolu- tion. For apoint inthezz-plane zeacsB, y=0, z=aVi—esind. whereFistheeccentric anglecorresponding tothepointx,y,2If¢isreplaced by¢=sin¢theintensity oftheattraction F= VxF+Yis 3M " s P=sargintdie—sin€008«)?+[4(¢c08¢—sinee 1 (¢—sin¢cos¢)*}sin? HP. Ifasphere ofradius Rhasthesame mass and density asthe spheroid, itsradius must satisfy therelationship 4eat =treR, M=5roa!cos¢=froR, and therefore 1 R= acoste, ‘Theintensity oftheattraction onthesurface ofthisephere is M M G=po a?cos! sothat H 3,cost F=3GOSE[(e—sine008d*+(Mecos¢—sine)*— (c=sin€0s¢)"Jsin® EY, (1) 40) THENEWTONIAN POTENTIAL FUNCTION 65 Onthe equator ofthespheroid, Hiszero, and 2 3cos? Fan.=3Gors(e—sin¢086, =~ 5+--) Forsmall values of¢theattraction ontheequatorofthespheroid isless than attraction onthesphere, and itisnot difficult to verify that itisloss forallvalues of€<#/2. Atthepoles ofthespheroid =x/2and 2 3g00594 Fre=3625(sin ¢—60080), no.+i+cee) sothat theattraction atthepole isgreater than the attractiononthesphereif«issmall.‘Thecoefficient of@,however, hasamaximum which isdefined bytheequation =DoTaint £=9Faint Po the solution ofwhich is c= 445, 0 =6958, For this value of€ Fru=1.0220, =$86approx., and forlarger values of¢thecoefficient ofGdiminishes and has the limit zero. Since the attraction onthe spheroid attho pole isgreater than the attraction onthesphere forsmall values of«while atthecquator itisless, there issome latitude forwhich itis justequal totheattraction onthesurface ofthesphere, This latitude isfound bysotting Fequal toGinEq. (1)and then solving forsin?E, The result is 4sin' , 5GSP=(=sin€008 intB=OE Sint©=Wine—€008)*=(e—ain¢086)? @)1, 119, .~3tapet co- 66 ‘THE THEORY OF THE POTENTIAL ‘The limiting value ofthisexpression for«=0is sinE=}orB=35°15'52”, and for«=0theeocentrio angle isthesame asthe latitude. Ifthefirst term ofthenumerator oftheright member of Eq. (2)isequal tothefirst term ofthedenominator, then sin Z=land #=90°, This condition is pt=sine—coos cost € the solution ofwhich is = 59°4/10", Ifcisgreater than this value there isnolatitude forwhich the attraction onthespheroid isasgreat astheattraction onthe sphere, Itiseverywhere less. This limiting spheroid and thecorresponding sphere isshown inFig. 21. Aa- 48 Fo. a, Pro. 22 41.TheAttraction isaMaximum.—It might beimagiagined ‘thatofallhomogeneous figures ofrevolution theattraction ofthe body onthepoint where theaxispierees thesurface isamaximum forasphere, buttheresults ofthepreceeding section showthat thisisnotso.Itwillbeofinterest therefore toinquire forwhat Figure ofrevolution ofgiven volume anddensity isthealtraction ‘upon thepointwhere theazispierces thesurface amaximum, ‘LetFig.22beahomogeneous solidofrevolution with»base theradius ofwhich isAB=a,andaheight AO=h.LetCD beathincross-section ofradiusrstadistance ¢fromtheapex©,where theaxispierees thesurface, Then, byEq,(16-1), 41) THENEWTONIAN POTENTIAL FUNCTION 67 thenumerical value oftheattraction ofthesolid upon thepoint ° F=one('{1- 3a =Ore['[1-2—las. J[vi+aI ‘Thesign isreversed since themagnitude ofFonly isofimpor- tance. The problem istofind thecurve ODB which makes F amaximum foragiven volume. That is,thecondition fires =const. also must besatisfied. Onmaking Aandrvary,itisfoundthattheconditions which oust besatisfied are arbre AaL (-yan)" =0, @ Saat *>vere and fp2rirds +ah=0. @) Let Eq. (1)bemultiplied bytheundetermined multiplier 2)and then subtracted from Eq, (2). ‘There results 7ants h 2—20 \rirds+(a?—anf1-52 —-\)an=0. o (2+8?) Vath If\ischosen sothat thecoefficient ofshvanishes, there remains 5 2 f(-inde =0; oN tp ot and since this must vanish forevery ér,itfollows that 1, (8) (t+ 8h also a=—* =@titava Fi (s) 1-~+— Ve tie Equation (8)istheequation ofthecurve sought inrectangular coordinates, Inpolar coordinates with Oasthepole and 0A asthepolar axis 8pcos8, str?=pt, andtheequation is p=eos. ) 68 THE THEORY OP THE POTENTIAL AtthepointBEig.(8)gives 3 (6)Mh =(i?+a%)F IfMiscliminated betwoon Eqs. (6)and(4)and theresulting expression isthen rationalized, itisfound that (i?+08)(8N* +4a*)a? =0, and therefore aiszero, Equation (4)then gives deh ° ro ‘The radius ofthesphere ofequal volume Risgiven bythe formula "as=Sane, or 5sokoaintogo=82, Son from which itfollows that h=RYS=171---R ‘Ameridian section ofthesolid ofrevolution ofmaximum sur- face attraction andthesphere ofequal volume isgiveninFig,23. ‘The intensity oftheattraction upon thepoint 0is P=2x0('(1- 2 —)a0 atea) aig=2eai(2-of,costsnsi) 4 =Soh, 41) THE NEWTONIAN POTENTIAL FUNCTION 69 ‘The attraction upon thesphere ofequal volume is 4G=rok. Heneo,P_LBh_ By ..39 @75R7 gv=1.0260 -=ggAPPTOX., and tho attraction ofthe solid ofrevolution ofmaximum attrae- tion upon the point 0exceeds the attraction ofthe sphere of equal volume byone part in38approximately. Itwas seon inSee. 40that themaximum attraction atthe pole ofanoblate spheroid exceeds the attraction ofasphere ofequal volume by about one part in45. Itisinteresting toobserve that this surface furnishes also the solution ofanother problem, namely; aparticle isattracted towards afixed point byaforee which varies inverscly asthe square ofthe distance. Itisrequired tofind the locus ofthe points furwhich thez-component oftheattraction isconstant. ‘Tho scomponent ofattraction is -— Gy +2a whore K?isthe force ofattraction at@unit distance. Oncom- paring thisexprossion with Eq. (8)itisscon that thesolution is thesurface drawn inFig, 23. 42,The Potential ofaHomogeneous Elliptic Cylinder.— Instead ofcomputing thepotential ofelliptic eylinders directly from thedefinite intograls thedesired result will beobtained by letting thelongest axisoftheellipsoid approach infinity. The cllipsoid, then, approaches anelliptic cylinder forwhich the equationisWyte 2>of Benet be Equation (35.6) becomes Woy Foyepatagenh which istheequation ofaconfocal cylinder. 70 THR THEORY OF THE POTENTIAL ‘The first factor intheexpression forthepotential ofan ellipsoidEq.(38.3), _Seoahe Vane hasthelimiting value 2rabe. ‘The amplitude oftheelliptic integrals Fand iscxwhere DgApe=e, sine=VorE Since «remains finite as@—»c,thelimit ofwsisx/2. ‘The Jimit ofthemodulus, petchme asa?tends towards infinity, isevidently +1. Hence, thelimit ofElly B)is,Eq. (882), 25)=fvTainede=+4; Jo and thelimit ofP(egh) is x i deri) = (Be = 40.()-ivrSaee- tis evident that theterm inthepotential which isexplicitly independent of24,?,and2"increases indefinitely asthelength oftheellipsoid increases. Itwill beshown immediately that theremaining terms have afinite limit, and, therefore, thelimit ofVisinfinite. This isnotsurprising forthemass ofthelimit ingcylinder isinfinite. This term corresponds totheconstant term ofthepotential, foritdrops outwhen Visdifferentiated with respeet toz,y,or2insofar asthese variables ‘occur explicitly. ‘The coefficient ofthe term in2°is 1aaTEeeB)—FloB)] 1 (-2), =poe eB B)-Fes B). 43) THBNEWTONIAN POTENTIAL FUNCTION 7 ‘Thelimit of(1—#)Eyas k*approaches +1,isevidently zero, Asforthelimit of(1—RF, “de “dp o F(w,k)=(a Se SE (om8)fyates <fivi-BVi-F 80that (LBP 2)<eVB, which has the limit zero ask*—+ 1. The coefficient ofthe term in2*therefore vanishes, Likewise, theterm 2 eos #) hasthelimit zero; and, since thelimit ofHis+1,thelimit ofthe remaining terms can bewritten down atonce. Atrue potential does notexist, butthevariable part is Qrode ffforte_ a(y—JeFx). Vow=pe{v(WiFe)+(VeF‘Dh Now (EE_ VEERA~VOTE P+. ViFE =VEE=(VER Vi +Vb a+ Vee) ee “REEVE FOEFH) and likewise 1-Vet=-#4,are StetVETO TE) sothat Vow.=—2rabe a(iFREVOFICTS) 2+apVETOER) from which thefactor 6?—c*hasdisappeared. ‘The components ofattraction are Y=, eotey__,PHAVOeT OC +H 2=——_et__. tet VETVE +e) R THE THEORY OF THE POTENTIAL Ifthepoint x,y,¢isintheinterior oronthesurface ofthe cylinder, «iszeroandtheexpressions forthecomponents of attraction become =bro, =~4robt.Yow wba, B= troy Ifb=thecylinder iscircular, andtheexpressions forthe ‘components ofattraction become —Practy, Proc,Yerape 2 eEK or,since inthis case, ct+x=y*+24, forexterior particles, =250i, gmDna2» Yo tottyLmProta ‘These expressions are the derivatives ofthe logarithmic potential V=-2rect logVik +2, which satisfies thepartial differential equation, See. 28, av ay ee a7 ‘The functions which satisfy thisequation areclosely related tothe functions ofacomplex variable* wayti, tava Itwill beobserved that thelogarithmic potential becomes infinite atinfinity. 43.The Potential ofaHomogeneous, Rectangular Par- allelopiped.—Let theorigin ofarectangular coordinate system be taken with theorigin atthecenter oftheparallelopiped and the axesparallel toitsedges. Lettheedges oftheparallelopiped be 2a,2b,and2cand, forconvenience ofnotation, letthedensity# bbetaken equal tounity. Lot thecoordinates ofthe attracted point be2,y,zandtherunning coordinates oftheelement of mass oftheparallelopiped be&,.Then vef2=[opefees, sp JoJo Jo. or,after multiplying numerator anddenominator by = (E- 2)+Ow +r-2, *Preamp, “Traite d’Analyze,” Vol. TL 43) THENEWTONIAN POTENTIAL FUNCTION 73 there results ae[fepe[enma cnn, aedon Jae i (=2)tt=a,&=2)4(iDNasinat. From this lastform itisreadily scen that Fees ety (Es2)|8(n= reLEIS) +) a(t=9+a('5)fst.Oy Since each term ofthe integrand isanexact derivative, itis possible tointegrate cach ofthem onec. Before doing so, however, itisdesirable todefine anotation astothelimiting values ofp.Let po=Vat FOWTC pu=V@= 2+= OHO pw=Vata +OF +098, aio=Vata FOR a+ =Oy ete, the subscript 1being associated with thepositive sign and the subscript 2with the negative sign, With this notation theexpression forVbecomes after integrat- ingonce 2fofleets-258]anin a +spies adSeat weJaal pre Powe +ff“ets-shee, @)oe oe ee ‘The numerators inthese expressions areallconstants with respect tothevariables ofintegration, andthesixintegrals areall 74 THE THEORY OF THE POTENTIAL ofthe same type. Itwill besufficient tointegrate one ofthem and then derive theothers from it, The first integral is ca)ff era)[fete renw! ne JayPio ae Jus Poe (eFa)+G—2*_(etay*poe Pionence®) mete)(*(72(f=2)4.8(120)_@ta* ‘ ete[™ola) a) Poeeve =Wiebe ene Tytb_ynb cre(OEE tere PS metaf(r. a) Since, ingeneral,~~ Smfptt =—loe2) p Vnitr ’ thesum ofthefour simple integrals intheabove expression is puns+(y+3) G+OE+0)log GB) z log BE +2)A““oeleca+yD) ante+0) FEFOU+2)leeeo) eto i 8)togEELS), - CFO02) 6te) Now pla =(Ha) + +) +e+e}, Pha =(2+a)? +(y— b)+(2+0)%, 0that. ‘ThoreforePha pha =by. pus+(y+5) _Mbon +pt—pins+40% pin+Y= 8) bps +phar —ein —40 =(oun+25)"=ptm_puss+pin+2, pin —Gun =2) pinpiss =205 and thesum ofthefour simple integrals canalso bewritten. Pur+pier+2b (+.a)(e+6)logSron+P - _ puss+pia+2i rearree 43) THENEWTONIAN POTENTIAL FUNCTION 75 pun+pus+2c +@+Oy+b)logpin+pus—26 Pin +i +2c—+OY~0)logome Consider theface oftheparallelopiped which isperpendicular ‘tothez-axis and atthedistance z=—a from the origin. This face contains four edges, two ofwhich areparallel tothey-axis and two parallel tothe z-axis. Through the attracted point P(z, y,2)pass aplane perpendicular tothey-axis. Any line L which isparallel tothey-axis willpierce this plane atacertain point O. The z-and z-coordinates ofthe point Pwith respect tothepoint Owill becalled forbrevity the coordinates ofthe point Pwith respect tothelineL;and asimilar convention will beadopted ifLisparallel toeither oftheother twoaxes, Any one ofthe four terms which occurs inEq. (5)isthe potential of ‘one ofthefour edges considered asastraight line ofunit density (Gee. 31)multiplied bytheproduct ofthecoordinates ofPwith respect tothat edge. Each oneofthesixintegrals inEq. (2)gives rise tofour of these edge potentials, making 24inall; but asthere areonly 12edges itisevident that each edge potential occurs twice, sothat inthe final result the terms inIq. (5)are multiplied bythe factor 2, ‘There remains thedouble integral (Eq. (4)) -eraf"f"nat jn Jn Pico which istheintegral of1/p* taken over the face oftheparallelo- piped =—a, Consider the infinitesimal cone which has the rectangle dndf asitsbase and thepoint P(xyz) asitsapex. Let dAbethearea cutoutofthesphere which has Pasitscenter and istangent totheface z=—a, Fig. 4,bythis infinitesimal cone. Sincetheradiusofthissphereisx+aitisevident that EAMnat=aA, and, therefore, theintegral af Pande: 76 ‘THE THEORY OFTHE POTENTIAL issimply theprojection oftheface s=—aupon thesphere which hasitscenter atPandistangent totheplane oftheface z= —a. (Fig. 24.) Inorder toevaluate it,itwillbeobserved that ace)_taht o=2) an\ pro Poo sothat 7" 7"- feo*s3] ; »LleFareGao pi| a cece = (—&9 bay, >SFaCHhmad“__@+ayt vty, ft erat aa owe Zl i Fra Now, since HM tant,Sat pmen where p=VETEFR, thefully integrated expression forDis = oftanEEOY+B_yyry$E)G=0) D=(«+a)’[tanteFa)an tan@apn Y=HeEt+e w=De=2] =tant ot (=0). aCo a Since there are four tan~"’s for each face and there are six faces, there are24such expressions inthecomplete potential. 43) THE NEWTONIAN POTENTIAL FUNCTION 7 If2,y,2,parethe coordinates ofapoint P,,and if wy=tant? =p ‘itiseasy tointerpret w,upon thesphere, which has theorigin asitscenter and passes through the point P:. Let Fig. 25 besuch asphere. Then, tag=4, sine=3 from which itfollows that tanw;=tan¢sinInthespherical triangle P,Q.Y,theangleatQ,isarightangle, sothat tan (90° —P,) =tan gsinw, and consequently wi=90° —Pi. 2 AHaFH] rN y Fro, 25 Itis,therefore, thespherical excess ofthe spherical quadri- lateral XQ,P,S,, and thearea ofthis quadrilateral is aren XQ.PiS: =pro. ‘The area ofthe spherical quadrilateral P:P2PsP, evidently is area PiPsPsPs =p*(w1 ++ws—tos—10). ‘The second integral ofEq. (2)can beobtained from thefirst byfirst changing the sign ofainthe first integral and then changing thesign ofalloftheterms, Acircular permutation oftheletters 2,y,2and a,b,cinthe first two integrals will ‘then give alloftheothers. Inthis way thecomplete potential 8 THETHEORY OFTHEPOTENTIAL ofthehomogeneous rightparallelopiped ofunit.density isfoundtobe: Vim+o)iy+8)logSrtomZe ~+Ny~BlogBetttBe FHa\e+o)logMtont HN ~0logPutt2+2 +~ay~2)logtn+Je ~~a)Y+b)logSeteaeBe +~ale~0)logMatoe+38 ~=ae+)logtetPe+ FOFDE+e)logBastoan+Be ~WHDG~6logntou+20 +O~De~co)logBetPen+Be ~G—Dle +)ogSatBan+Be 43) THENEWTONIAN POTENTIAL FUNCTION 79 Ly—ppltansEEMEFO yr@~ale+0) $e[tanENERO—aeCO Ft =Dom 88 =Do 1 Dean CHOY —1)_4@tay+b) HetortCEOERD —uaeeet a@-DY+) piealy- »| Fea FO EE am Le of tantSEVO+YD _tay@FW—B) +369[ta€=Opa =Opie 1@=a)y=)_pr@=aly+4}. i5 iin Tos 44,The Components ofForce fortheRight Parallelopiped.— ‘The components oftheforce due totheattraction ofaright parallelopiped X,Y,Zare, ofcourse, thepartial derivatives ofthepotential with respect tothecoordinates x,y,andz,On ‘theface ofitthis differentiation issuch adiscouraging task that a Tesort totheintegration formulas ave a(z _— a(2 xoteefffeQjane~~fffaC)asnee .pt), o ete, seemspreferable. Asamatter offact,however, itisnot necessary todifferentiate with respect tothecoordinates insofarasthese coordinates appear under thelogand tan~' symbols. Itis sufficient todifferentiate asthough these functions were constants, and arecognition ofthisfact makes thedifferentiation avery simple matter. Inorder toprove thisitwillbeobserved that Eq.(1)issimply 4(2V)/de insofarasxappears explicitly inEq.(43.2). The termsin2Vwhicharisefromthefirstintegral carryeitherz+a orx—@asfactors, andinordertoproduce Eq.(1)bydifferentia tion only these terms should beused. The terms inEq. (5) carry (x+a) linearly andthey areduplicated byterms inthe second andthird integrals, Hence, when thefactor 2isremoved ‘thesame terms appear inVnotduplicated. Similar remarks hold fortheterms which carry z—aasafactor. Allofthese 80 THE THEORY OF THE POTENTIAL terms contain logarithms. The terms which carry are-tangents donotreappear intheterms ofthesecond and third integrals, but asecond factor (z++a)appears, which, when divided by2, gives thefactor (¢+a)#/2. The derivative ofthis factor is just thesame asthederivative oftheproduct (x+a)(x ++a) when only one ofthe factors isdifferentiated. Hence, the correct value of8V/dx isobtained bydifferentiating Vwithrespecttoxonlyin80faras2occursinthefactorswhicharethe coefficients ofthelogerithms andaro-tangents. 45.AGeneralization Regarding Derivatives ofaPotential— Itwillberemembered that asimilar phenomenon appeared in tthepotential ofthe ellipsoid where x,therootofacubic equation, could beregarded asconstant intheprocess ofdifferentiation, ‘Thephenomenon isnotpeculiar tothese twopotentials, however, butappears ingeneral forhomogeneous bodies. Byitsdefinition mel{fet and’ ovwefff[o=meeme ib -a(a2|=ae=Placa._ a(f-2) |a(n—v) ,a(r—2Silas )+S) +(ES?)feos or <1 oy avweff[ES 2]offest oaInd+ >Jarae t-7f Lettheletters 2,y,and2éneofarastheyoccurexplicifly inEq.(1)bemarkedbyadash,%,7,2,sothattheycanbeidentifiedintheintegration process. ‘Then av) 1} a(2V) _ 1).or=~ae @ a@ We 45) ‘THE NEWTONIAN POTENTIAL FUNCTION 81 ‘Thecomponents offorce X,Y,andZaregiven bytheformulas wv a(t x=Eoff[2A)aeaer a(t>~<fJaG)seer ®) Pray, ple’ and similarly y=weff[zferas, Znweffifa Oy phy, ohn” where the distinction indicated bythe dashes does not occur. ‘Acomparison ofEqs. (3)and (4)with Eqs. (2)shows that x=22M, y-2@¥) 7.22N). oz oy a Since y a) Xe whereverzoccurs,markedandnotmarked,andsince lg x0, itfollows that thederivative ofV;with respect to2insofaras xismarked isexactly equal tothederivative ofVwith respect toinsofaras.zis notmarked; and, similarly, with respecttothe letters yand 2, 46,The Potential ofaBody ataDistant Point—Only in relatively simple casos cantheexpression forthepotential ofa body beobtained in»closed (finite) form onaccount ofthe <ifficulties incarrying outtherequired integrations, andrecourse must behadtoexpansionsinseries.Indeed,forpointswhichare atagreat distance relative tothesize ofthebody thefirst few terms oftheexpansion aresufficient formany purposes, and asthey aresimpler than theclosed form they aretobepreferred even when theclosed form ofthepotential isknown, InFig. 26,letBbeany body offinite dimensions. Let0 beany point within orwithout thebody which istaken asthe origin ofacoordinate system. Let@bethecenter ofgravity ofB,and Pany point distant from B, Let odr=dmbean 82 THE THEORY OF THE POTENTIAL clement ofmass ofB,atadistance pfromPandrfromO;and finally letOP=R.‘Then thepotential ofBat Pis vefs|, Iftheangle between randRisdenoted bya,then pt=REIRcosa+74, and 1ae a ?)-}11~2fcosa+jp) N P -f¥my \ 7 | ] t i / Mo. 26 ‘The binomial theorem gives theformula. _ Vouledy,1-3°5 4, Gaybate petty hat, and bytaking B=Qeosa~ Fees and then rearranging inpowers ofr/R, itisfound that 1 1 ir.7BtFROta+5AGcos?a~1) +426costa~3co8a) FETABScosta—30costa+3)-+---. (D ‘The coefficients ofthisexpression evidently arepolynomials ineosaandareknown asLegendre's polynomials, thecoefficient ofr*/R**# being denoted byP,(cos a). 46] THE NEWTONIAN POTENTIAL FUNCTION 83 ‘This expansion canbederived inanother way which isuseful inproving the convergence ofthe series. Ifr/R isdenoted bytheletter h,thefactors ofo*arep=REL—2hcosa+ht) =R= h(t he'), i=VHT; and Vd —pete) =hete)“3 pk: ‘Theexpansions for(1—he'e)~3 and(1—he~‘)~3inpowers ofjiareabsolutely convergent provided Whe) <1 and [hew'ol <1 respectively, and since le =Jer] =1, forallrealvalues ofa,both conditions aresatisfied if We and the expansions areabsolutely and uniformly convergent if Wal<ho <1. With 0asacenter and aradius atrifle greater than the distance ofthemost remote point ofBfrom O(the trifle can beas small asisdesired), describe asphere S. ‘The bodyBwilllic wholly within §,and theexpansion Hq. (1)isabsolutely and uniformly convergent forevery point Pwhich lies outside of S. Itcan, therefore, beintegrated term by term and the resulting series isconvergent and represents V. ‘Therefore ami 1 : ve{[#+afrcosadm+gisfo"cos?a—r#)dm tamforcosta—37cosadm+++.(2) TfMisthetotalmassofBandgistheprojectionofOGon R,itis evident atonce, that dm_M noose, _Mog, [e-% andf08am=Md.Ifcos?ainthethirdintegralofEq.(2)isreplacedby1—sin*a, itissoen that thethird integral ean bewritten Biren -omfsin?adm. & THE THEORY OF THE POTENTIAL ButJyr%dm isthemoment ofinertia ofthebody Bwith respect, totheorigin O,andfyr?sin?adm isthemoment ofinertiaof thebody with respect tothelineOP=R. Ifthese moments ofinertia aredenoted byI>andI»respectively, thefirst three terms oftheseries (2)canbewritten Mi Mg, %Mo-3le,vat ee ee ® ‘When theorigin istaken atthecenter ofgravity, giszero and theexpression is M,2lo—3IyVeRt aot : @ Ifthepoint Pisvery remote, thesecond and higher terms arevery small andthepotential isreduced essentially toits first term, which isthepotential ofasphere ofthesame mass ‘M. Itisevident from Eq.(3)thatforevery finite body limRV=M, (5) ms alimiting value which isoften useful. Expansion fortheLogarithmic Potential—For alogarithmic potential 1 v=[togt.five>odws, theintegral being taken over agiven area forwhich oisthe density anddwanelement ofsurface, sothatdm=oda. ‘Using thesame notation asabove andFig.26, 11 my =}5mll Be)2—hey, and 1toed=tog}—Log(=ht)=Bog(1he, Since bg a2dFHS sey itisfound very simply that 138(1—hel)—Fog(1=hes)= ls 1eosa+pk*cos2a+sheosSat++- 46) THE NEWTONIAN POTENTIAL FUNCTION 85 Henee, since r heR log2=logf+f008«+5Fr082a+3-7cos3a+ee, and vef.log+dm+ff,eosa:dm+sha2cos2amaR RJa BRD, “ 1,Mg,Io-2, Vi=Mogg+Getapts. © Thus Vcan bewritten V=Mog 4,RvR thelimiting value ofM,being Mg. 47.The Terms ofHigher Degrees.—The second and third integrals ofEq. (46.2) arereadily interpreted asintegrals of thecenter ofgravity and moments ofinertia respectively; thecenter ofgravity being theintegrals which involve thecoordinates oftheele- 7 y ‘ments ofmasslinearly, andthemoments Pans) ofinertia those which involve the coordi- natesofmassinthesecond degree,‘These BA| integralsarefunctionsofthebodyand| f thesystemofrectangular coordinates |77which arechosen, buttheydonot4 x depend uponthepointP.‘Theterms of mnahigher degrees depend upon integrals - which involve the coordinates ofthe elements ofmass inthe same way butinhigher degrees, andthese integrals, also, areinde- pendent ofthecoordinates 2,y,2ofthepoint P. Let Fig. 27represent acoordinate system with the origin atthepoint 0.Letdmbeanclement ofmass ofthebody B atadistance rfrom Owith thecoordinates &n,{.Let d,u,» bethedirection cosines ofthelineR,sothat 2 u vatdae web ond a ‘The angle aistheangle between thelines Randr,sothat rcosa istheprojection ofthedistance rupon theline ofR. Hence 1008a=NE+un+of 86 THE THEORY OFTHE POTENTIAL ‘Thesecond integral ofEq.(46.2) canbewritten a =Aftam+4%|nim2, rfcosean=2fe+fafipin+iframo E y z=HJdm+Bfpan+ween. ‘The third integral is Joe cost«—ran=ghfIMEan+ot)—(+98 +P)jdm ~Ot-Y-9 (yy, Cet w=, =Shfedm+OR ram\ (att 20f1 + ORE ean sy 3yz —3sBPend—Spa}tide—FeeJSeam. ‘The fourth integral canbewritten afr costa—3r?+rcosadm= Larf508tant)38tat+DOEtan+Vem 2a—Bay?—B22?(",=Baty+2y'—Bye(, aedRare ea —Satz—By%z+224(" 1arty—3y*—8yz" (4CRIESMenangBE "fend =Bent+12's—328", +heatpteBetftam ® =80! —Say"+1222" +e ftam 4pBate=Byte—8fran+ =323+1224? —3x2ae Sm —Saty —8y*+12y2* cyt +opr Jsittin+15|kntam. Thecoefficient off,e%r/t*dm intheabove expressions isa perfectly definite function ofz,y,and zaltogether independent 47) THENEWTONIAN POTENTIAL FUNCTION 87 ofthebody itself, andinEq. (204.4) there isgiven aformula which isvalid forany integral values ofi,j,and k, Ifahomogeneous body hasthe2y-, yz-,and r-planes asthree planes ofsymmetry, theintegralfj'ttemvanishesifeither,or kisodd. Forsuch bodies alloftheintegralsinEqs,(2)and(4) vanish andtheexpansion ofthepotential isvery much simplified, Itwillbelefttothestudent asanexercise toprove that the terms ofthefifth integral which donotvanish forsuch triply symmetric bodies areasfollows: Let A=fam, D=fentém, Befiy'dm, B=fy'sam, C=fim, P=fredm, Then 1 to 1s?a =afro costa—30cos+3)dm= aye{G4+8B+30—24D+OF—24Fxt +(—244 —24B +6C+162D—18H—18F)z*y* +(BA+8B+30—24D—248+16F)y4 +(+6A —24B —24C —18D +162E —18F)y*z* +(84+3B+8C+6D—24E—24F)z* } +(-244+6B—240—18D—182+162F)2*2* J. 48,The Expansion forthe Homogeneous Ellipsoid—In the case ofahomogeneous ellipsoid referred toitsown axes a8 coordinates axes, itiseasily found that _ Lyyas, =Lage, dm=1M’,fram=ga, fran gilts, fran=Met, Jam=2Ma', tam=2s =3Met, fe‘dm=apes,fe‘dmgps,fein=gglle, tam=LMatb?, setdm=A fevdm=3Maret, fenfdm=geMbr, attam=Amc’, fedm=JMctat, 88 THE THEORY OFTHE POTENTIAL theterms with odd exponents vanishing, The terms inEq. (47.3) become qopal(2at ~Y=efit+(mot4Db—oy?+ (—a? —b+2c*)z4]. ‘This expression issimplified somewhat byintroducing the eccentricities ofthemeridian sections oftheellipsoid bythe substitutions beatles), ct=aX(1 —04%), when itbecomes gopalen’ +esta"+(est—Bory? +(ox?—2es*)24]; ‘and,similarly,withthetermsinz,y,andzofthefourthorder.Uptoandincluding theterms ofthefourth order,the expansion is M, MatVag Meet test+(ot—Dey"+(ert—Des" Mat+sgppilOat +Bestest+90424 +(~T2at~1Beites?+18es!)2"y*|(py t(2hex! —Deter? +Dex8)y! +(—Test +162e:%e3* ~72es4)y%e? t+(Beit ~24ertes* +Q4es4)2¢ +(+18ex* ~18e,%e, —72er)2%rt} 4... 49.TheRight Parallelepiped.—For therightparallelepiped, theedgesofwhichare2a,2b,and2c,theintegrals correspondingtoEqs. (48.1) have thevalues 1 lane Linesgle, gat, Ee, lias le lageplat, SMBs, gic’, Luan: hbgov, gate, beta, 0thattheexpansion ofthepotential is M,MVimBtGilat — ~erat+(a?+2b—ony? M +(—at~b+2e*)et] +appel(Bat +8++Bet—240%?+Gd%ct—24erat)xt 49] THENEWTONIAN POTENTIAL FUNCTION 89 +(Bat +8b4+3c!—24a%? —24b%* +Bota)y! +Gat +B0¢+Bat +a%D? —24b%t —DActat)=t +(—24at —24b* +6ct+162a*b* —18b%e* —18c%a*)x%y* +(-+6a" —240! —24c¢ —18a? +162b%" —18cta")y*s? +(=Mat +6b—24te4 —18a%* —185% +162e%a*)z*2*} For the cube a=b=, and the terms ofthe second order vanish. The terms ofthe fourth order reduce to ATMOat+yhot+Baty?+By'et+Betztl, which shows that atlarge distances thepotential ofthecube is less than that ofasphere ofthe same mass. 60.The Inertial Integrals—It isevident from the preceeding discussion that theexpansion ofthepotential inpowers of1/R isafixedformsofarastheletters z,y,andzareconcerned. The coefficients ofthis fixed form aregroups ofintegrals ofthetype Senttam, which belong tothegeometry ofthebody and nothing else. As ‘thecenter ofgravity integrals and theintegrals ofthevarious moments ofinertia belong tothis class, theentire class will be called theinertial integrals. ‘Since thegeneral term oftheexpansion ofthepotential can bewritten down interms ofthese integrals (see Sec. 204), the general term oftheexpansion canbewritten down explicitly providing thegeneral inertial integral eanbewritten explicitly. ‘This eanbedone forthethree homogeneous solids which have three planes ofsymmetry, namely: Fortheparallelepiped bounded bythesixplanes ea, gab, tect, avbug . srqtrdtdndt = ;J[femeeae -Mesenere nereD fortheellipsoid bounded bythesurface yee.S+htaoh « ore =gM2P=Ua—Ul2r—MIgaopn, ffPervdidnt=eT aee where [2s-I]=1-3°5---- Qs), 90 THETHEORYOFTHEPOTENTIAL except when s=0,when thevalue isunity; fortheoctahedron which isbounded bytheeight planes rye+a+6+ct1 _ (2p)129)"2r)1 «fffersaednar =omeeae?BpBy aC, These integrals areexamples ofageneral classwhich eanbe evaluated forhomogencous bodies, Consider theintegral Servetdedyds ‘takenoverabody which isbounded bythesurface 2 (uy (2)8+Oy + Ifthesubstitution ‘2\e ye 27(J-8 (ies (2-5, {ismade, then tede dydy deataanne eo also PaO, ad, feet; s0that I=Jffieryteecaae- abicr Poa$atoyEff PEPa, whichisthewell-known Dirichlet integral,! theintegral intherightmember beingtakenoverthetetrahedron whichisboundedbythefour planes $0, 9=0, F=0, Etntped Hence P\r(9\p( pene(DG)r(5)By(2ar) rE+ptea +Duntouur, “Werke,” p,375,391.AltoGounsan-Hpnicx, “Mathe-matical Analysis,” Vol.,p.308. 50) THE NEWTONIAN POTENTIAL FUNCTION 91 For thevolume itself, p=g=r=l, therefore, 1 1 1 Vol.=oy71,1,141)riG+5t5+1) Hence thegeneral inertial integral ofthebody, ifitsmass isM,is +N(LEN(tN(14tat Jvein«aoeGCI) 2), Nr()r())i(Eet ate iat r)rG GS Fe ) 61,The Inertial Integrals Cannot AllVanish.—It hasbeen pointed outthat ifabody hasthree perpendicular planes of symmetry alloftheinertial integrals with oddexponents vanish. Even though negative (but notzero) masses areadmitted, not alloftheinertial integrals canvanish. Inorder toprove this itwill beassumed that the body iscomposed ofafinite number ofdiscrete particles instead ofacontinuous mass. Theultimate nature ofbodies, ofcourse, isnotknown but theideal ofavery large number ofdiserete particles ismore nearly inharmony with our notions astothestructure ofmatter than the ideal of continuity, andforthepresent purpose issimpler. Theproof will bomade inthree steps. (A)Theinertial integrals ofasystem ofdiscrete particles placed ona straight linearenotallzero. Forasetofdiscrete particles onastraight linetheinertial integrals arefinite sums, namely Lae paOL erry ee a Even though negative masses areadmitted, these sums cannot allvanish. Consider thefirst nsuch sums setequal tozero: mia?+man?+++ma?=0, for p=0,1,2,--+,a-1L (a) Itisassumed that the particles arealldistinct, that is,that notwo ofthe2’sareequal, and that none ofthemasses iszero. ‘Equations (1)however arelinear andhomogeneous inthemasses. 92 THE THEORY OFTHE POTENTIAL ‘Therefore, either themasses areallzero, orthedeterminant vanishes. Thatis Moo. seed ik ty ae =0; le? aah ssag? ee ‘This well known determinant has the value Awte-2), As, that istheproduct ofthe differences ofthe 2's. Itcannot ‘vanish, since allofthe2'ssredistinct byhypothesis. Hence, Eqs. (1)cannot allbetrue, and even thefirst noftheinertial integrals cannot allvanish. (B)Theinertial integrals ofasystem ofdiscrete particles lying ina plane arenotallzero. Forasotofdiscrete particles, ninnumber, lying inaplane the inertial integrals are Lmarve 79=0,1,% +++, & Foranyfixed integer p,consider thesums (mers+(marys ++++(mare)uet=0,q=0,-++,(n—1). (2) ‘There isnothing toprevent the’sfrombeing equal ingroups. ‘Therefore itwillbesupposed that WAY eeye=m, Voy)=Yayet=+=Yay=Tay Yaysth=Vanatt==Un=ey andthat the9'sarealldistinet. Let min? bo mate, =wy MayPaysr +t mgtay =bay Mayring FoFtty?=hw ‘Then Eqs. (2)become mot tam +s tune=0.g=0,-+-yn— 1. Thedeterminant ofthefirstsequations ofthissetlinear homo- geneous istheproduct ofthedifferences ofthen’swhich cannot 51] THE NEWTONIAN POTENTIAL FUNCTION 93, vanish since the 7'sare alldistinct. Therefore, allofthe y's must vanish, ‘This cannot betrue either, since wy=may? +maz? bo ++maha cannot vanish forevery p,byA. Hence Eqs. (2)cannot hold for every p,and proposition Bisproved. (C)The inertial integrals ofasystem ofdiscrete particles ina closed volume are not allzero. ‘The inertial integrals are Ymavrver —7,47=0,1,2,+++2 at Foranypairoffixedintegerspandq,consider the7simul-taneous equations (mizryssext +(mamaPyatyes’ +++++(mara?ynt) 2n"= 0, r=0,l,--+,n—1 (3) Ifequalities exist among the2's,let eae seaty Mayet =Raye = Bay =Say Fayaath Paget SS a eh and let many! +mataPyst +++++mayeayVay =hey Maybe? pYeysit “Fo +*+Mag” ZasYag =Hay MayDanyeV Tb MaenYat=Hae ‘Then Eqs. (3)become ee ee ceo Sincethedeterminant ofthefirstoftheseequations isnotzeroallofthey'sarezero.Butjn,4,...cannotvanishforeverypair ofintegers, byB. Henoe, Hgs. (3)cannot hold forevery triple ofintegers p,q, 7,and not allofthe inertial integrals can vanish, Proposition Cistherefore established. Imthe case ofasingle particle located atthe origin allof theinitial integrals vanish except thefirst one, namely Emardyfe? which isequal tothemass ofthesingle particle, which byhypoth esis isnot sero. With this single exception, itispossible togofarther and state that there does notexist afinite sub-set of inertial integrals which aredifferent from zero, ifalloftherest oftheinertial integrals arezero. 94 ‘THETHEORYOFTHEPOTENTIALSincethesubset isfinite, theexponents p,9,7arebounded. let p< g<y 7S foreveryintegral inthesubset J,andconsider thesubset Zs, Ta=Dimawteyereet =0,By=OA, @) Frat Bywritingwe marty T= My Eqs. (4)becomes Yameyenr=0 4,87=0,1- +,2a Thisisacomplete setofinertial integrals forasystem of particles us.Arepetition oftheprevious argument shows that every 4must vanish, andsince themasses oftheparticles are notzero, atleast oneofthecoordinates iszero, and allofthe particles lieinthecoordinate planes. Arepetition oftheargu- ment fortheplanes shows thatthey must lieontheaxes, and finally, arepetition fortheaxes show that theonly possibility is single particle attheorigin. 62.ABody IsUniquely Defined byItsInertial Integrals.— Bymeans oftheresults intheprevious section, itispossible toshowthatabodyisuniquelydefinedbyitssetofinertialintegrals.Suppose abody B;anditseetofinertial integrals Jisgiven, and suppose Bsisasecond body which hasthesame setofinertial integralsI.ThenB;~Bsisabody,withpossiblypositiveand negativemasses,whichhasallofitsinertialintegrals zero.But suchabodycannotexistunlessallofitsparticlesareofmasszero.Hence thebody Bsisidentical with thebody B,andisnotdis- tinet from it, Acomplete setofinertial integrals cannot bewritten down at random fortheintegrals arenotindependent, butifacomplete setofintegrals ofnparticles isgiven itispossible, atleast theoret- ically, todetermine theirmasses andtheirlocations. ‘Theproof istoolong forinsertion here. Problems 4,Letthedensity ofastraight rodABbechosen oothatthe potential distance¢fromtheendisequaltoone.Iftheend4,thepointPandthe DotentialatParekeptfixedwhilethelengthoftherodinereagesindefinitely, 52] THE NEWTONIAN POTENTIAL FUNCTION 95 show thet thelimit ofthepotential isoneeverywhere, except ontherod ital. 2.Aparticle ofmass unity isplaced atexch ofthethree vertices A,B,C ofanequilateral triangle, thecenter ofthetriangle being atthepoint D, andtheradius ofthecircumscribing circle being equal toa.Show that there arethres equilibrium points onthe cirele which has Dasacenter and radius r=.2847a; andthat thevalue ofthepotential atthese three points is3.1234, 3.Show that theexpansion ofthepotential intheneighborhood ofthe point Dinthepreceding problem is yndytetyn 29s 4Show that the potential ofauniform circular disk atone ofitsown points pis V=4az(t,3) ‘where BisLegendre's complete elliptic integral ofthe second kind forthe rnodiulus b~r/o,abeing theradius ofthedisk andrthedistance ofpfrom the center. 5,Show that thevalue ofthe potential ofa uniform elliptial disk, which indefined bytheequations zeacey, y=bsing, st«point ofits edge ia ay a208¢ V=[cosotant + «losee"aratePSsin¢tanh ses —_], VittiotBee. and atthe eonter ie v=40K(«5) ‘where Kis Logondre's complete elliptio integral ofthe first kind for the modulus k=6,theevcentristy ofthe ellipse 6.Show that thopotential of«homogeneous right cirular eone referred toasetofaxes which haw itsorigin atthecenter ofgravity ofthecone and itseaxia coinciding with the axis ofthe cone is M4 3 ap(q21yp)Bty=et iarCeeeeM(Bat+hh22*—82(2*+y*) vwihorehistheheightofthecone,aistheradiofthebase,andAfisthetotal mass 1.Ifa, 6,and 7arethesemi-axes oftheinterior equipotential surface of thehomogeneous elipsoid whose semi-axes areo,8,and ¢,show that 11,1 2 atta CHAPTER IIL VECTOR FIELDS THEOREMS OF GREEN AND GAUSS 53,Definitions.—If ateach point ofaregion R,which may bocithera volume, anarea, oraline,avectorisuniquelydefined, thenRanditsassociated veotorsiscalledafieldofvectors.‘Suppose there aregiven three functions ofx,y,and 2,viz., F,G,andH,which aresingle-valued and continuous. These three functions canberegarded asthecomponents ofavector WateachpointMofspace,foratthepointM,x,y,andzaredefinite numbers andsoalso areF,G,and H. Henoo, F,G, andHcanberegarded asthe2-,y-,and z-components ofa perfectly definite vector. Intheregion Rinwhich these things aredefined, there isafeld ofvectors.” ‘Thefunctions F,G,andHarescalar functions, that is,they farenumbers which depend upon thecoordinates 2,y,#.If ‘they arethree quite independent functions, thevector Wissaidtobeatriplyscalarvector. Suppose o(z, y,2)and itsfirst derivatives arecontinuous inRandy(z,y,2)isanyother continuous function. Avector Wcanbedefined alsobytherelations ag=yee, a Pave Gavi nay%. ‘Theexpression Fdz+Gdy+Hdzadmits anintegratingfactor, and (ae+Gdy+Hae)=de isanexact differential. Inthisevent thevector Wisdefined bymeans oftwoscalar functions, andthevector issaidtobea doubly scalar vector. Ifthefunction yisequal to+1,and =%, =, ae,PoyGnaHas thevectorWisdefinedbymeansofasingle funetion g,anditis said tobeasingly scalar vector, 96 5a) VECTORFIELDS 97 54,The Normal Derivative.—Let there begiven asingle valued function y(z, y,2)which admits unique derivatives with respect to2,y,andzinacertain volume V;andletthere be given alsoacertain surface Sdefined bytheequation Sle, v,2)=0, which lieswholly within V.LetP,beany point on8;,x,» thedirection cosines ofthenormal to§atP;,directed outward if$isaclosed surface; and P:apoint onthenormal near Py. Ifg:andgsarethevalues of¢(z, y,2)atthepoints P;andPs (ae a \ \ \ a1 u| |? Le 1k Fis. 28, respectively, thederivative of(2,y,2)normal tothesurface atthepoint P;isthelimit of ane PP: ‘asthepoint Psapproaches P;along thenormal. ‘That is,the normal derivative of¢atthepoint P;istherate ofchange of thevalue of inthedirection ofthenormal. Ifthecoordinates ofP;andPsarex,y,z,and x+dx,y+dy, 2+derespectively, and ifthedistance P,P: isdn,then itis evident from Fig, 28that dr= dn, dy=ydn, de=win, @ Also i=de=ae+ay+2 Tim(2—1)=dp=odz+ay+3 98 THE THEORY OF THB POTENTIAL Or,after replacing the values ofdz,dy,and dzfrom Eq. (1), =(128+22+22)an; ae(82+g,+Nin: that is te=0284284oe, PANig+Hay+"92 @ ‘Thus, thenormal derivative dg/dn isavector which has the direction ofthenormal and themagnitude ofwhich isgiven by Eq.(2).Thederivatives oe, ae, ae, az’ aya also arevectors which are parallel tothe2-,y-,and zaxes.IfthesurfaceSissimply¢itselfsetequaltoaconstant,that is Sa, y,2)=ol, y,2)=C, then Lae Lae Lagd=Rg BR TRG @) where _ fe)» (20) (0) and, since X*+x?+»*=1,Eqs. (2)and (8)reduce to deae_deae_dete_de R= 37 SydnGe7dn(4)IfNisthenormalatP;andLisanyotherlinethroughP:forwhich thedirection cosines area,8,v,andiftherate ofchange ofgalong Lisdenoted byde/dl, then de_oe4ie,oe.Hmoe+Ose+ae: and since ete, de,de,oe.de,azManayMan’ a~"a itcan bewritten dg_de,Tl thety) =¥cos(EX), from which itisevident that therate ofchange ofafunction is ‘most rapid inthedirection which isnormsl tothesurface along which the function isconstant. 54) VECTOR FIELDS 99 Ifthevector dy/dn atthepoint P,onthesurface y=Cis taken asthediameter ofasphere, thederivative inanyother direction isthe length ofthe chord ofthesphere which passes through is Pyinthegiven direction (Fig, 29). 55. Relations between Certain Volume- and Surface-Integrals.— Suppose there isgiven aclosed sur- face Swhich may becomposed of one ormany parts. Anelement of theenclosed volume Vwill be - denoted bythesymbol dr,and an 4 *® clement ofthesurface $bythe /* symbol du. ‘The direction cosines Fro20 ofthe normal tothe surface atthe element dwwill bedenoted bya,8,and y. Let Fbeagiven function of2,y,and 2,and consider the integral over thevolume V an, oF F Saber fffetes oF~Jfoiefe Onintegrating first with respect toz,thefunction Fisobtained asthe indefinite integral. Inthis integration yand #are eon- stants sothat theintegral [J represents thesumoftheclements ofarectangular parallelepiped oferose-seetion dydz parallel totheaxis, Fig. 30. ‘This elemen- tary parallelepiped enters the volume Vatthe element of surface da;and emerges atdes, re-enters atdwsand emergesagainatdeandsoon,Sincethesurfaceiscloseditemergesasoften asitenters. LetF,bethevalueofFattheelementdaanda,Bs,71thedirection cosines ofthe normal directed outward; letF2bethe value ofFatdeo;and as,82,‘2thedirection cosines ofthenormal, and soon, Then thevalue ofthedefinite integral is feee= roe —Pat os, 100 THETHEORYOFTHEPOTENTIAL and SSfzecse=Jfie:=F)+e—F)++++Myin, Now thenormal atdw;directed outward from Vmakes an ‘acute angle with thepositive direction ofthe z-axis. The element ofarea dyde ispositive, soalso istheelement ofsurface dis;Hence dydz=+andw. y f,Ci»yi ro. a0 But attheelement dw: the normal directed outward from V makes anobtuse angle with the z-axis, Since dydz and du: are,both positive, itfollows that dyde =—aude, and soon, Consequently Sffezecivae=ffie.—PatPaPet+++Jdyde =[lpexten+aPadan+aston+aPélos+=) Hence ‘ oFfwee=fetes, i) 55) VECTORFIELDS 101 and theoriginal integral taken over avolume isreduced toan integral taken over asurface; that is,atriple integral isreduoed toadouble intogral. Inasimilar manner itisproved that ayf.afipode, @ aHfee=fprtae. @) Ontaking thesum ofEqs. (1), (2)and (8)there results the important formula ‘aR,a@,aHf(E+eBla=fer+0042. Ezamples.—If thefunction Fisaconstant, say unity, Bq. (1) reduces to 0=fade, which expresses theobvious fact that thealgebraic sum oftheprojections ofanyclosedsurfaceupontheyz-planeiszero,IfFistaken equal tozsimply, Bq.(1)becomes Sara facies that isthe volume itself isex- pressed asasurface integral. me 56,AVector Interpretation.— IF, G,and Haresingle valued functionsofz,y,and2,theycan «(| beregardedasthecomponents of Ni avector Wwhich isuniquely po defined ateach point ofspace. Suppose there isgiven aclosedsurfaceSintheregionRinwhich the vector Wis defined. Let do beanelement ofthissurface at ¥ras81thepointM,andletMNbethe a normalto$drawnoutward,ando,8,itsdirectioncosines.Let W,betheprojection ofWonthenormal. ‘ThenW,=al+0G+yH. @ and Eq. (55.4) becomes ak,aG,anf(z+a+atag=foreae (2) 102 THE THEORY OF THE POTENTIAL ‘LetWrepresent thevelocity ofafluid and dwasmall hole inS through which thefluid isflowing. Letdabetheprojection of dwona plane perpendicular toW. Then, Fig. 31, Wade =Wydw represents theamount offluid which flows across theelement dw perunit oftime. Hence Wade =(@F+8G+yH)des iscalled theclement offluxacross du,orsimply theflux; andthe integral [rete represents thetotal fluxacross thesurface. ‘The quantity oF. at. a Emit ayte ®) alsohasphysical significance. ByEq. (2) fia=[rade CO) forevery closed surface S.Letthevolume belimited toasingle element sothat Ear=[Wrdo. ‘Theright sideofthisequation istheamount offluid which leavestheelement ofvolume perunittime.HenceEisthe‘amount offuidwhichleavesthevolumeperunittimeperunitvol-‘ume,andiscalled thedivergence ofthevector W.Equation (4) ‘thenmerely saysthattheamount offluidwhich escapes from ‘thevolume isthesame astheamount offluidwhich crosses the ‘surface perunit time. ‘57.Generalized Orthogonal Coordinates.—The normal com- ponent andthedivergence ofthevector Whaveameaning which isquiteindependent ofthecoordinate systemintermsofwhich they areexpressed. Letqr,dnguethecoordinates inanytriplyorthogonal system;bywhichismeantthatthetangent planesofthesurfaces uatyG2=C2,ga=Cy,Where ¢:,cxandcsareconstants, attheir common point ofintersection aremutually perpendicular. Then the Coordinates ofapointparerelated bytheequations, F=fd 9,99) Y=Sau 940), 2=falas,a2,00) 2 57) VECTOR FIELDS 103 Ifthecoordinates q:,q2,9aregiven infinitesimal increments, the point pundergoes adisplacement ofwhich thecomponents are =(2+(2)+(2)an= dema((Ge)+(Gh)+Gp)ee=Beto [aay(UV,(8\'yp, sy\&)+(H+(Fyam=Rat} Baw we ay dss=AG)+(4)+(i)ig,=Radas. ‘The directions ofthese components, which aremutually atright angles, will becalled theq:-,gs-,and qs-directions atthepoint p. The areelement dsis ds=VRidqe? +Redge +Redgs'. The elements ofarea onthesurfaces g:=¢1, 72=¢2,= and qs= ¢s,respectively, are doy=RaRadaaigs, dos=RaRdgeday, day=RRedadas; and the element ofvolume is dr=RRRadgrdgadas Returning now tothevector WV,letthecomponents ofWat the point pintheg:-,gxand g;-directions beW;, Ws, and Ws;and, ifpisonthegiven surface, letthedirection cosines ofthe normal, directed outward, with respect totheg:-, gs-and gsdirections be»:,v2,and»s.‘Thenthenormalcomponent ofWis W, =Wi +Ws +Ws Consider now fare =fWards). F s ‘The cosine oftheangle between thenormal andtheq;direction isn, Hence tvdo =dey =ReRsdgrdgs and fostrde=[pstates. Consider also thevolume integral 1a _f2Sordironan =f2estananan, 104 ‘THE THEORY OF THE POTENTIAL Arepetition oftheargument inSec. 55shows that 1 aSak @maron =fnara, and similarly 1 aSade denmad =fiery 1 aStd erro =fpete. Ontaking thesum ofthese three expressions, there results 1 a aSorte cen +Zany+ a rma Ws) |dr= ‘deo. (6) Lenaws|ir= [orate © Ifitisborne inmind that thisexpression holds whatever the volume may be,acomparison with Eq. (2)shows that OF,G,oH 1aaa*aytaeBameen) +deny +awa) ‘andtheright member ofthisequation isthedivergence ofWexpressed intermsofanytriplyorthogonal systemofcoordinates. IfWisasingly scalar vector U,then Wi, ouwen Gag Ban, and (Bap. (6)) 1au 1aU 13U_Rag7 Riag=7Raq= ‘Therefore U0,eU eu0Se4OY=av ® _1[8(Rki)42(RakaU),0(Rieav)|=Blea Ri,an)*aaRea)+aaRea isthe expression fortheLaplacian ingeneralized orthogonal coordinates, This formula isdue toLamé.' 68, Green’s Theorem.—Since doubly scalar vectors form a sub-class oftriply scalar vectors, theresults ofSecs, 55and 56 +Lamé, Journal deEcole Polytechnique, Vol. 23,p.215, (1833). 58) VECTOR FIELDS: 105 hold also fordoubly scalar vectors. Suppose yand yareany ‘two functions ofz,y,and zwhich arecontinuous together with their first derivatives and which admit second derivatives. Let =28, Gayl, nay;Pav Gaw5h Hmv5; oy and forbrevity ofnotation, let =e4te4He, de=3a+ay+ae (2) ‘Then theexpression forEbecomes OF.0G|8HBasti tbe =yay+24884OYBe,avae, ae +35ae+ayoytaeaz? and Eq. (55.4) becomes avde,ayBe,oyae! Silvae+ Oxaz*ayay+aea2| =[u(ad?+92%4420 -Sues+b+188). or,sinceae,40,de_ae,OF+oy+Ve=on that is,thenormal derivative ofy,Eq.(54.2), (OFde5Hde,Hae), fee Svseirs[(EG+SeaBila=[yiea. This formula, which holds forany two functions that satisfy theabove condition, isknown asGreen’s theorem. If¢and yareinterchanged inthis formula and theresult is subtracted from Eq. (3), the second integral will disappear, since itissymmetrical ingand y.There results thevery important formula an fiver-eanar= [(2e- Aa, which isGreen’s theorem initssecond form. Ifyisaconstant, sayequal tounity, thevector Wissingly scalar and Eq. (4)reduces to ae, fi‘Apdr=fFey, 6) 106 queTHEORY OFTHEPOTENTIALReduction toTwo‘Dimensions.—Green's theorem stillholdsif‘volumes arereplaced byareasandsurfaces bycontours. Tripleintegrals become double integrals, anddoubleintegrals becomesingleintegrals. ‘Threedimensions arereduced totwo,andtwodimensions arereduced toone. Equation (3)becomes ate,ate (3aeove)-f2p,© [vlGseSpae+ Ji.32ayayay)=Jevan™©wheredu=drdyisanclement ofareaanddk=VieFae isanelement ofaclosed contour. Similarly, Eq.(4)becomes ate4ate)(%28)law={(2-2[e- aa, @ andEq.(5)becomes #64May[2 Ss s=~ Laem @ 59.ThePotential ofHomogeneous Bodies.—Let £,7,{be@ pointofahomogeneous bodyofdensity ¢,and2,y,2,anoutside point, andlet p=VE= TOTO ‘Then atz, oe_1_(E~2),a ae > e deamy, % 1_o~vae ot ra at=2 de_1_atar pe and therefore =Ho HesOy2 d=getatopp Oo) _InEq. (68.4)identify ywithoand»withp,which eanbedonesince piscontinuous within thebody, and¢ isconstant. Itthen becomes obpdr=[=of2 fesedr=26f=of28, or,after removing thefactor 2, of{2ul {% Sipefiant @ 59) VECTOR FIELDS 107 ‘The leftmember ofEq. (2)isthepotential ofthebody inits usual form. The right member isthe surface integral ofthe normal derivative ofpmultiplied by¢/2. Thus, thepotential ofevery homogeneous body eanbereduced to surface integral, which involves only double integrals instead oftriple integrals, ‘This theorem isdue toGauss, although Gauss’ proof follows a very different line ofthought. Ttisasimple matter toextend this theorem ofGauss tocases where oisafunction which satisfies the equation ofLaplace,namely, heaeos do=Fat aatape78 forinthis event Hq. (68.4) becomes oa,=1((,9—20 fparANGan82a ®) ifoisidentified withyandpwithg.Theleftmember ofEq. (8)isthepotential initsusual form, while theright member isa surface integral. 60.Example—A Non-homogeneous Spherical Shell—Con- sider thepotential ofaspherical shell ofwhich the radius of the outer surface isaand the inner surface isb,andthedensityofwhich is a o=%, where raVEFPFP Since¢satisfies theequation of \ Laplace, the potential atthe point z,y,2,provided thepoint 2,v)#does notliewithin theshell itself, noronthesurfaces, is Poa.32. oo[|lap a(v v=oflan~oa(2)Jew The surface oftheshell consists oftwo spheres, one ofradius a and the other ofradius b<a, The normal derivative onthe ‘sphere ofradius aisdirected away from thecenter, while over thesphere ofradius bitisdirected toward thecenter (Fig. 32). Inboth eases itisoutward with respect tothevolume between thetwosurfaces, that isthevolume oftheshell. 108 THE THEORY OF THE POTENTIAL Since theradius ofthesphere isalways normal tothesurface a 4%, 80, 2san~ ta an a7! thepositive signtobetakenontheoutersphereandthenegative sign ontheinner sphere. Now t= (- a+ —w+ -2? =r—2Bcosy+RY, where Reavttyte, and¢istheanglebetween thelinesrandR.Hence a_1—Reose 30_ov,ar > ar rt Be_,2¢_solr!—rRcose+p*) ar Par~ rp =Get 418—R,2r%p ‘The element ofsurface dwis de =rd where daisthecorresponding element ontheunit sphere, and therefore (Fig. 15,p.36) dw=r*sinydydé. ‘The complete integral over theouter sphere is iap ao ca Bet+72—RF whererhastheconstant valuea.Sincetheintegrand isindepend-entof6,thedouble integral reduces atoncetothesingle integral Botbat—Rt roofPEOPsnody ‘Thevariables p,a,R,and¢arerelatedonthespherebytheequationpt=a?—2aRcos¢+Rt; and, since aandRareconstant, pdp=aRsinpdg, or sinode_dpak 60] VECTORFIELDS 109 Hence, ontheouter sphere 1((a_ao E839?+a?~RK?Af(fe—oto=vofa Ei, or +Re 1f(aa\, "e+B352+a?—R* 3(@-1Nie-redey according asthepointz,y,2liesoutsideofthesphereofradiusaorinside ofthesphere ofradius. Inthefirst case thevalue of theintegralis 2ra0¢psRt), ‘and inthe second case Araoa. ‘The corresponding integrals taken over theinner sphere are —PAR +b), and —Arorb, Hence a, (90—aVerf(& ofa =are4),or4rox(a—b). The mass Mofthe shell is M=cof"drrdr =2rao(a? —0°). Hence M 2M vee oo veh according asthepoint 2,y,2isontheoutside oftheshell, orwithinthehollowenclosedbytheshell.Withinthehollow,thepotentialisconstant. 61.Existence ofHigher Derivatives ofPotential Functions.— ‘The potential function ofanybody is where e=VE- +a +C—O 110 THE THEORY OF THE POTENTIAL ‘The existence offirst derivatives was proved inSecs. 24and 25, and foreither internal orexternal points ave a(v afl>Sue =—SeaG)-« o Aninteresting proof oftheexistence ofderivatives ofallorders, duetoRiemann, canbemade bytheuseoftheformulas which were established inSec. 55. ByEq. (55.1), OFfee=farees provided Fanditsfirst derivatives arecontinuous within B and on its surface. IfFistaken tobe Pe ? inthis formula itbecomes av av ao aodr BV f20\ae==|% ooGr 2) azSiG Sse Sit ® Ifthepoint p(x,y,2)isoutside thebody, 1/piscontinuous within BandEq.(2)isvalid. Ifitisinside, letasmall sphere 2of radius ¢bedescribed around thepoint p.The function 1/pis continuous within thevolume B,bounded bythesurfaces $and 2. Then WY_im!—(22,as,fa0.*we aay, — [.% dedeacallJsSse+n.38 Ifthepartial derivatives of¢existtheyarefunctions of£m,f, andthey canberegarded asdensity functions. Hence, thelast integral faedrSi BOE p Bs isaNewtonian potential foracertain distribution ofmatter within B,and itisknown from Sec, 22that wnf=[3Jee~Jp For thesurface integral over thesphere 2,thefunction phas ‘the constant value « Let. des=da, 61] VECTORFIELDS m1 where diisaninfinitesimal solid angle. The maximum value of« is+1, since itisacosine. Let oobethe maximum value of |c|within 2.‘Then fetes=«ft<4Pr 2 which vanishes with «.Since thesurface integral over Sdoes not depend upon ¢,itfollows that av a0dr aorSiPSF, © even when thepoint p(z, y,2)lieswithin thebody. From Eq. (8)itisseen that 4V/azisthesum oftwo potentials, oneofwhich isavolume potential and theother isasurface potential, foritrepresents thepotential ofacertain distribution ofmatter uponthesurface ofthebody. Thefirsthasderivatives everywhere, while thesecond hasderivatives everywhere except ontheeurface itself, Therefore Vhassecond derivatives every- where except onthesurface. Ifo(E,,¢)hasderivatives ofall orders, itispossible toproceed step bystep and show that V likewise has derivatives ofallorders except onthe surface, 62.Harmonic Functions.—If thefunction (x, y,2)and its first derivatives aresingle valued and continuous within and on theboundaries ofacertain region R,ifsccond derivatives exist, andifthefunction ¢satisfies theequation ofLaplaceate,ae,ate de=FatgetaeoO then ¢(e, y,2)issaid tobeharmonic within theregion R. LetSbeanyclosed surface within R,andgand yany twohar- monic functions. Then byGreen’s theorem, Eq. (57.4), ae_av Si-eae=0. @ If¥isequal toaconstant, say+1, this equation reduces to eau=05 fSeay=0; ” ‘that is,thesurface integral ofthenormal derivative ofanhar- monic function over any closed surface iszero. 63. An Extension ofGreen’s Theorem for Harmonic Func- tions.—Let Sbeany surface enclosing avolume B. Let &9,¢ 112 THE THEORY OFTHE POTENTIAL bethecoordinates ofapointofB,andletz,y,2bethecoordinates ofapoint plocated anywhere. Then thefunction ee oe aVE= FW +O isharmonic inBifthepointplies z outside ofB,since itsatisfies allofthe conditions ofSec,62.Ifgisany other function which isharmonic inB leg_ afl _y Sli-“Q)Je=o Interior Point—If thepoint plies within Bthe conditions ofthe theorem are not satisfied and Eq. (1)isnot applicable. Itispossible, however, toFro.33. describe asmallsphere©ofradius¢with itscenter atp(Fig. 33)andapply Eq.(1)tothevolume B,which isbounded bythesurfaces $and 1%,buttheintegral must beextended over both surfaces, That is lag_a(1) lag_a(t -SEE-8Q) +[E8-aQ)|e-o thenormal derivative tobetaken outward with respect toBy ‘onboth surfaces. ‘Ifthenormal derivative inthesecondintegral betakenoutward with respectto2itmerelychangesthesignofthesecondintegral, ‘andtheequation becomes normal derivatives outward with respect toboth surfaces. Since ‘theleftmember ofEq.(2)isindependent of¢,theradius of2, theright number alsoisindependent of«. ‘Thefirstintegral oftheright member is Ldeg, 18%, 0Ssant1fSee=9, byEq.(62.2),sinceyisharmonic in2,Inthesecondintegral a2)=2().-1 an ae\p) ~~o® 63] VECTOR FIELDS 113, and, since p=¢onthesurface of2,thesecond integral reducesto a(t 1~firdi(3)oo =#8fy Let Mand mbethe maximum and minimum values ofyon3. ‘Then damSfete=fies<4M, where dioisaninfinitesimal solid angle, and de=da.Sinceyiscontinuous and¢isassmallasisdesired,mandMcan bemade todiffer from thevalue of9atthepoint pbyassmallaquantityasisdesired;and,sincetheintegralisindepend- entof¢,itsvalue isrigorously —fe(2a=tro2. ‘Therefore, Tq.(2)becomes 1(flae_ a(t],ff}we(2)fw=oley2). @) Statedinwords,Ea.(8)saysthatifafunction¢isharmonicinavolume which isbounded byaclosed surface S,andifthevalue of¢anditsnormal derivative atallpoints ofSaregiven, then the Value of¢atanyinterior point ofSisdefined byEq.(8). Corollary I-—Tj ¢andexareharmonic withinaclosedsurfaceS, asidefromthefactthatg:hasasinglepolewithinS,sothat eoahI,whereTTieharmonic, thenL[[26_ae oudaeJl-de—eeaw, Corollary IT.—If oxandgsareharmonic within aclosed surface, asidefrom.thefactthatg,hasasinglepoleatthepointpiandya hasasingle poleatthe point py,80that 1 1slam, w-teneckeiy eat ie then 1 (00_202,=ep)— Af.(ete e96Niex(p)—ol), where ex(ps) means thevalue ofthefunction ysatthepoint ps. it ‘THE THEORY OFTHEPOTENTIAL Exterior Point.—If thefunction ¢(é;9,$)isharmonic outside of§,andifitvanishes atinfinity insuch »waythat se, 8, 0, Peeed ra VERSE remainfiniteforr=®,&similar theorem holdsforvaluesof atpoints outside ofS. “Assuming thatthese conditions ongaresatisfied, letthepoint ple,y2)lieoutside of8.Describe asphere 2ofradius RaboutthepointpasacenterlargeenoughtoencloseS,andletthevolumewhich isinside of2andoutside ofSbedenoted byB.Since plies within B,Eq.(8)isapplicable and lée_ a/l lde_ a/(l ls decor)» fF5 eb)J+f[558—ean(6)os where thenormal derivatives areoutward with respect toB. It cannot bestated this time that theintegral 1f aei.en ‘vanishes, because itisnotknown thatthecontinuity condition on ¢issatisfied inside ofSwhich iswithin 2.Butforvery large values of 1 1 and3areoftheorderR ag al 13anda)areoftheorderJy and dwisofthe order R?. Hence,theintegralfi[ie-eb(!)Jisoftheorder1/R. andvanishes atinfinity. Itisevident, too, that this integral is independent ofR,sothat itisalways zero. Therefore, if isharmonic outside ofSand vanishes atinfinity intheorder of 1/r,ifitsvalue and thevalue ofitsnormal derivative isspecified ‘atevery point onS,andifthepoint p(z, y,2)liesoutside ofS, then thevalue ofgatthepoint pisdefined bytheintegral 1[ide_a(t coe |bee(!)bo ® thenormal derivatives being taken inward with respect toS. 64 vectorFreLDs 115 64. Reduction toTwo Dimensions.—It isevident from See. 58 that ifyandyaretwofunctions ofxandywhich arecontinuous, together withtheirfirstandsecond derivatives, inanareaAwhich isbounded byaclosed contour c,Green's theorem initssecond form holds; that is _ a) fvse-conde=[ve-Ban, O) where =e4He, =o4 ae=FeFe,ay=Sh ® If¢and ysatisfy also theconditions Ap=0, Ay=0. @) then je_a)Lo: Se-Ba=o5 ® and inparticular ifyisaconstant ae .Li 0; ©) that is,theline integral ofthenormal derivative ofanharmonic function around anyclosed plane contour iszero. Extension oftheTheorem.—Let thepoint p(z, y)liewithin A, and letp=/(E= =)?+(7=y)*. Describe asmall circle 7 ofradius ¢with thepoint pasacenter, anddenote thearea lying inside the contour ¢and outside the circle ybyAi. Let ybe any function which isharmonic (intwo dimensions) inside of ¢,andlet¥=logp.‘Then both ¢andysatisfy Bq. (3)inAy, and Eq. (4)becomes a cy fi[S080—ologofan-fzlog»—etoen)[ar0) thenormal derivatives being outward with respect tocand out- ward with respect to7. Onthecircle ythefunction pisconstant and equal to¢,while thenormal derivative oflogpisequal to1/e. Since oe log«f3800, byEq,(6),theright member ofEq.(6)reduces to 1-1fear, 116 ‘THE THEORY OFTHE POTENTIAL andsincedk=déony,wheredéisaninfinitesimal angle, this integral becomes 7 =[ote=~2rete0) Hence, if¢isharmonic inside ofAandp(e,y)lieswithin A, a a ee,y)=affem,(og»)~$2logja (7) Corollary I—If G=lge+H, where Hisharmonic insideofA,itisstilltruethat L a Wp ele,9)=Lf[% -a. 6) This follows readily from Eqs. (7)and (4). Ifthepoint pliesoutside ofAand y=Visalogarithmic potential which isharmonic outside ofA,sothat, atlarge dis- tances, Eq.(46.6), _— V=tog +2, where M;isafunctionof1/RthathasfinitelimitasRincreases, then ; -1ffya ven=Ff[ric060)-IogeZla.—@) Inorder toprove this, describe asmall cirele about thepoint pand alarge circle 2ofradius Rabout any convenient point asacenter, thecircle 2being large enough toinclude Aand p wholly within it. Since Vand logpareharmonic inside ofthe area which liesinside of2andoutside of+andc,Eq.(4)gives a ave a aveSPpee0)~toeBa+[73doe0)~tog08Jn .7) av J]P2.006 0)~1gAF]a.20) ‘Since Visharmonic inside ofy,thesecond ofthese integrals is equalto+2rV(z, y),justasbefore. Asforthethirdintegral,forvery large values ofR, 1_N Jogp=—logh-No.., a 1MKogyo dat... 4) vacroR FIELDS 17 Hence ' a av 1 Vp.(log6)—logor=—(MN, ~Ms)(1+l0ogR) +==, and thethird integral beeomes ‘|-f[ann~Mytek 4...|0 which isoftheorder of1/2 and, therefore, vanishes. Equation(10),thereforereduoestoIq.(0),whichestablishes thetheorem,Corollary I.—If Uand Varethelogarithmic potentials oft20 masses Mand Nwhich lieinaplane area which isbounded bya closed contour ¢,then avL(%-von=0. any Corollary III.—If Uand Vhave thesame definitions asin Corollary IL,ifpismeasured from apoint pwhich liesoutside of¢, andif G=tog}+V, then: ai U@)=-xI(%~oR). 65.Analogy with Cauchy’s Theory ofResidues.—Equation (64.7) bears astriking resemblance toCauchy's equation inthe theory ofthecomplex variable, 1pf) $0)=ef{Om where f(t) isholomorphic within thecontour c,and 2=x+iy isapoint inthecomplex plane lying within c.The connection between thetwo theorems istraced byPoincaré asfollows:! Consider anarea Abounded byaeontour ¢intheplane ofthecomplexvariable¢ =&+in.Letf(¢)boafunctionof¢whichisholomorphic inA. Let f(¢) beseparated into itsreal and imaginary parts SOG) =FAG n)+ile, 1). 1“Théorie duPotentiel Newtonien,” p.149, (1809). us THE THEORY OFTHE POTENTIAL ‘ThefunctionsFandF,oftherealvariables&and7satisfythe equations oF_as, oF _Fs, @aE dn’ 7 agOF,|oF, OP,,aPs oF, OF Ps Fs 9, 2)agtae 7% ap+on ® ‘The function log(4)=L(G,1)+iLalé,») . y:<7 Fie. also canbeseparated into itsreal andimaginary parts; and if $—2isexpressed inpolar coordinates, poe=pel, then L,=—logp, r= -8, where =VESFGS, pater IY eeVERW EG ota Tae Now, letEq,(64.7) beappliedtothefunctionsF,andFy,which areharmonie withinA,withtherecollection thatlogp=—Lx. ‘There results thetwo equations ae Fy_yal Fv)xf(HR-oe ©oy a2 ((udPs— palaPow=a.f(Uerea. ‘Now imagine achange ofcoordinate systems, Fig.34, f=6 tsina~proosa,tindbenasnanay o 65) VECTOR FIELDS 119 inwhich foneispoint ofthecontour ¢,and aischosen sothat theaxis isnormal tothecontour directed outward and the n-axis istangent toit,directed forward. ‘Then attheorigin OF, _Fs,ORs_OF = 8h Gn’ am On” bau2 © Byvirtue ofthechange ofvariables, Eqs. (1)become Fy, oFy FF) oy(+R) cosa+(FE—Fe)sine=, Os4APYing—(OF:_oF? =0; (Rta)sina(&m)008a=05 and therefore, since thedeterminant isequal tominus one, aFs aR, OF: as, 8 mn Om Ok Equations (5)then show that attheorigin aFy_oF, OF: _aks,on OX’ ON n? andthese relations hold allalong thecontour, independent ofthe coordinate system. Likewise ohy als, aly Ola,Cn rr) rr? Equations (3),therefore, canbewritten Fiev)=gyffHaars~Pudla,° @) Fey)=ffChar~Felts, thetwo integrals being taken inthepositive direction around e. ‘Thefunction Ly=—tan-*—isnotsinglevalued, but L,=—logpissinglevalued sinceitisreal.Thefunctions F,andF;aresingle valued byhypothesis. Hence fata) =0, Sadar, =0; and iar. =-fPidts, [ladFs =-[Prats 120 ‘THE THEORY OFTHE POTENTIAL 80that Eqs. (6)canbewritten 1 Pye)=~xfPotts+Pat, Fey=+efi(Pd, —Fed). Now ae log2=ntilaandr-aL,+idLa; also {Oar=~.+Nada+iat). Therefore, 1(Oy 2 ~ Fy:-1[ @, PidPodrer Pada)—5falla+Fidl) =Figy) +p. That is 1 pf)gafeest 10 which isCauchy's theorem. The formula isapplicable only if lies within the contour c. 66,The Surface Integral oftheNormal Derivative of1/p.— Let there begiven aclosed surface $and apoint 0. With the fs SF Fie. 88 point0asavertex,takeaconewiththeinfinitesimal solidangledes,which cutsoutofthesurface Stheelements diy,dss, osey Fig. 35).Letthenormal toSattheelement. dw,directed out-ward,makeanangley;withtheaxisoftheconedirected awayfrou: thepoint0.Theprojection ofdexupon aplane perpen- 66) VECTOR FIELDS 121 dicular totheaxis ofthecone isp'di, where p,isthedistance along the axis ofthe cone from the point Otothe element of surface dix. Since theangle between theplane which istangent to$atdayand theplane which isperpendicular totheaxis of ‘the cone isthesame astheangle between thenormal atdw;and ‘the axis ofthecone, ¢y,itfollows that pda=daycose, ords=SE208-8, Ifthepoint0liesoutside ofS,theinfinitesimal conecutsthe surface aneven number oftimes; theangles qarealternately obtuse andacute, andthecos¢:alternately negative andpositive. Since theelements ofS,da, areallpositive, itfollows that dex086 at isalternately negative andpositive, Fd3.Henee,if2xisthenum- beroftimes theinfinitesimal cone pierces S, SSdov008or9, aw Since this istrue forevery infinitesimal angle da,itistrue for thesum ofallsuch angles, and therefore f.£28das=0. Js?IfthepointOliesinsideofS,theinfinitesimal conepiercesthesurface anoddnumber oftimes, 2x+1,and s208vidatas,=F pe ‘and since this istrue forevery such infinitesimal solid angle, it followsthat:f.£08Fy=fds=de.3sP F Ifthepoint OliesinSitself, atanordinary point ofthesurface, itisevidentthatJoost=fe=2n,3 s sinee intheneighborhood ofOthesurface liesentirely ononeside ofthetangent plane. Ifz,y,2arethecoordinates ofthepoint Oand &,»,¢arethe coordinates ofapoint on$atwhich thedirection cosines ofthe normal area,8,7,then o=VE=2FOF OS 122 THE THEORY OF THE POTENTIAL wt a posty gry, ta4, cosy=afTF4tity f=a,8,,0,magetig+15 ‘Therefore, Eq. (54.2), cos¢=3, a cose_1a__9/1,“FTpona3) and cosede Ggde. lssn\p Jo Henoe ayn 0if0isoutsideof8, fai)=|~25ifOisthesurface8,® 3p, —4rifOisinside ofS, thenormal derivative being taken outward with respect toS. If0isonthesurfacethevalueis—2r,ifitisatanordinary point ofS.Ataconical point ofS N44 itsvalue isthe solid angle ofthe envelopingconetakennegatively. fe67.The Contour Integral ofthe Normal Derivative ofLogp.—Let therebegivenaplaneareaAbounded byaclosedcontourc.LetObeany IN point intheplane and d@any infi- nitesimal angle inthe plane with its vertex atO,which cuts across thecon- tour ¢(Fig. 36). IfOis outside ofcit willcutacross theboundary aneven numberoftimes,butifitisinsidethe On, number ofcrossings willbeodd. Let Pio.20 @,Ms, ...bethe elements ofthe contour intereepted bytheangledéat thedistances :,px... fromO,andletys,es,...betho angles which theoutward directed normals make withtheaxisof theangle dédirected away from 0.Then ay=Deeoses,G=43---, a ‘Theangles y,s,.,.arealternately obtuse andacute, If 67) VECTOR FIELDS 123 axandpyareregarded asalways positive, thevalues inEq.(1) are alternately positive and negative, although numerically equal. Hence Syucoser_{0if0soutsideofc ape 49if0isonthecontourorinsideofSincethisresultistruéforeachinfinitesimal angle4,itistrue fortheir sum, sothat 082,_{ifOisoutsideofc, J. 2nifOisinsideofe. 1fO's ontheboundary c,thevalue isatanordinary point, but, ifOisatacusp, thevalue oftheintegral istheangle between the two tangents atthe cusp. AsinSec. 66, =2, cos¢=32, ® and 1 5 cose1998yy, vp pan anBG Hence a r__ Since a=0,or—2, @) according asthepoint0isoutsideorinsideoftheboundary. 68.ATheorem ofGauss.—By means oftheintegrals which have justbeen established, itisaneasy matter toprove acertain theorem which isdue toGauss. Suppose there isgiven aclosed surface Sandapoint 0.At thepoint 0there isplaced aparticle ofmass m. Ifpisapoint of$atadistance pfrom 0,theattraction ofmonthepoint p is—m/p? andthecomponent ofthisattraction along theexterior normal is —meose _a(m). Pane ‘Thesurface integral ofthisnormal component, orthefluxof thegravitational force duetomacross Sisa(m Oifmisoutsideof§, =(>)do=”1 Sal) {amet insideof8.o Ifthere aremany such particles andM;isthetotal mass insideof 4S,and M,isthetotal mass outside ofS,then 8 (sam_ fx(3mae=—4nM,, Cy 124 THETHEORYOFTHEPOTENTIAL which isindependent ofM.. Ifthedistribution ofmass My-+ ‘Mzisacontinuous one, thefinite sum becomes adefinite integral, dm limy™=[@oy, man fF where Visthepotential ofthegiven distribution ofmass, and Eq, (2)becomes ayVay= 3 fiada=—42M @) Expressed infull, Gauss’ theorem states that ifthere is.any Aistribution ofmatter which may beparly within andpartly without «aclosed surface S,andifMzisthe total mass within S,thesurface‘integralofthenormalcomponent oftheattractiontowardtheexterior, orthesurface integral oftheexterior normal derivative ofthe potential duetothe entire mass M,+Mi,isequal to—4zM,. Asimilar theorem holds intwodimensions forthelogarithmic potential. IfM;and AM,have thesame significance asbefore, and Visthelogarithmic potential, fVan=—22M (4) 69,Poisson's Equation—Suppose V(¢, n,¢)isany function of&,», and ¢which iscontinuous inavolume Bwhich isbounded byaclosed surface S,and,likewise, itsfirstderivatives. ‘Thon by Green's theorem, Eq.(58.5), _7, f.[aver=San ) Theconditions imposed upon Varesatisfied ifVisapotential function ofanyfinito continuous distribution ofmatter. Suppose Vissuch apotential function. ‘Then byGaus theorem, Sec.68, f.Way4M, sin and therefore fare=4M, 2) A Suppose o(%,1,£)isthedensity function which represents the distribution ofmatter inthevolume B.‘Then Maf,adr,B 69] VECTOR PIELDS 125 and Eq, (2)can bewritten fer+deo)dr=0. 2 This equation holds whatever thevolume Bmay be,andtherefore theintegrand vanishes identically. That is, AV+4x0=0; @) * ‘VvVv‘Vva a an aetattape=Are Os @ which isPoisson’s equation. ‘An illuminating derivation ofPoisson’s equation ean be obtained asfollows. Let Obeany point ofthe body and letoy bethedensity atO.Letasmall sphere 2bedescribed about the point O. ‘The potential ofthebody can beseparated into the sum oftwopotentials, V=Voths, one ofwhich, Vo, isthe potential ofthe sphere atO,and the other V;isthepotential oftheremainder ofthebody. Letthe sphere 2betaken sosmall that the density within itcan be regarded asconstant. Ifitsradius isa,then bySec. 25 Yedeel~ett»)} and AVe =—dr00. SinceOliesoutsideofthevolumeforwhichV1isthepotential, itfollows from Laplace's equation that AV, =0,and therefore atthepoint 0, AV =—4ra, which isPoisson’s equation. Poisson's equation holds whatever thesurface Smay beand whatever thedistribution ofmatter may be,provided only that ¢isanintegrable function. Itcanthereforeberegardedasholdingthroughout allspace.ItincludesLaplace'sequation,for,outsideofthebody, thedensity ¢iszero andPoisson's equation reduces toLaplace's equation. Tfitiswritten inthe form 1/av ,eV, a" c=ae +a+oy (6) 126 THE THEORY OF THE POTENTIAL itisseen that Poisson's equation answers the question “What distribution ofmatter will produce agiven potential, assuming that thepotential isdefined atallpoints ofspace.” Inordinary solid bodies thesurface isasurface ofdiscontinuity inthedensity, since atthesurface thedensity changes abruptly from acertain finite value onthe inside tozero onthe outside. Equation (5)shows that atleast oneofthesecond derivatives of Vforsuch abody also isdiscontinuous atthesurface, and the sum ofthe discontinuities ofthe three second derivatives is —4r times thediscontinuity inthedensity atthesurface. 70,Poisson's Equation inTwo Dimensions.—If »and yare functions ofthetwo variables zandywhich, together with their firstderivatives, arecontinuous inaplaneareaAwhichisbounded byaclosed contour e,byGreen's theorem, Eq.(64.1) - =((v%2- a; faeeanae=f(v58~ofan: and ify=1, ae f fSeay, Theconditionsongaresatisfiedbythelogarithmic potential Vofany continuous distribution ofmatter over anarea. Hence fAVde=fVa, 4 on and byGauss! theorem, Eq.(68.4) fBVay=—Dede =ofade, Hence‘ f.(AV+2re)du=0, and therefore AV =—2ne. 71,AnExtension ofGauss’ Theorem.—Gauss’ theorem can bededuced from Green's theorem, butitisjustassimple to give ageneralization ofit, Let(2,y,#)beanyfunction which isharmonic within a volume Bwhich isbounded byaclosed surface S,andlet 1 VECTOR FIELDS 17 V(e,y,2)beaNewtonian potentialwithinS,thatisthepotentialofabody B:bounded byasurface S:,Fig. 37. ‘The body Bymay liepartly within $and partly without, orwholly within, or wholly without. Then byGreen’s theorem, Eq. (68.4), - 7av_yar fi(PAV=Vae)drSat¥'38), @ the normal derivatives taken outward. Since ¢isharmonic inBand VisaNewtonian potential, Ae=0, and AV =—4re, inside ofB. Hence, Eq, (1)bocomesa_aeS(t-Vee)=~4feodr=~tefcm,@) where ‘cdr=dm, Gauss! theorem follows atonce bytaking ¢equal tounity, sothat Eq. (2)isageneralization ofGauss theorem. Ifthe body Byconsists of asingle particle ofmass m,sothat z ve%, > and ifthis particle islocated inside ofS,Eq,(2)reducestoHq.(63.3)asof/B ,course itshould. Ifthebody Bis a setofdiscrete particles >vey Yon37. Eq, (2)becomesav_yaoScie-Ve=~Yom, ) where gsisthe value ofgattheparticle m,and the sum inthe right member isextended over alloftheparticles which lieinside ofS. Eqs. (2)and (3)canberegardedasgeneralizations ofEq. (68.3). Reduction toTwo Dimensions—A. similar theorem holds intwo dimensions. Ifg(c, y)isharmonic inanarea Awhich isbounded byaclosed contour c,and ifVisalogarithmie potential ofany continuous distribution ofmatter over anarea A,which may liewholly outside ofA,wholly inside, orpartly outside andpartly inside, thenav_yaoSee—58)=a2fod, ® 128 THETHEORYOFTHEPOTENTIAL Or,if 1 = Simo 3, V=Ymilog isthepotential foradiserete sotofparticles av_yae\a=- 6)(se-738)"Inmegs 6)where gisthevalue ofgatthepoint &,matwhich theparticle nvisToeated, andthesum2isextended overallofthepoints inside ofA.IftheareaAisdivided intosub-aroas Au,As,-.- pounded byafinite number ofcontours, Fig.38,andifthe density, although discontinuous across acontour, iscontinuous ineach sub-area Ai,Eq.(4)isstilltrue foriteanbeapplied to each sub-area separately. If thesum oftheresultsistaken, > itisfound that Eq. (4)reap- pears fortheentire area A, since thedivision lines appear © \s e (+> ; Foo, 38 vo #9, twice inthe sum, the contour integrals being taken inopposite directions inthetwo cases. Since ¢,Vandtheir derivatives are continuous across these dividing lines, thetwo integrals taken along them, but inopposite directions, arenumerically equal though opposite insign. Hence,inthesumtheintegralstaken along thedividing lines cancel out, leaving only theintegral along theoutside contour ¢. 72,Green’s Theorem Applied toTwo Potential Functions.— LetVbethepotential function ofabody B,which hasasurface S,andadensity a:.LetVs,Bs,Ssandosbethecorresponding symbols forasecond body. Let S:beany spherical surface which contains B,and Bywholly within itsinterior, and let Bsbethevolume which liesinside ofS;and outside ofB,,and therefore includes thevolume ofBs,sothat thevolume ofBy isapart ofthevolume Bi. 72) VECTOR FIELDS 129 Since Vi,Vsand their first derivatives arecontinuous within Bs,Green's theorem initsfirst form, Eq. (58.3), applied tothe volume Bs,gives ‘AV,0V2,8V,aV2,aViAV: f,raver +f B+eee eeavy avs, =fveggie+f.Vignes thenormal derivative isoutward with respect toBs,andtherefore onS;itisinward with respect toSi. ‘The second integral intheright member ofBq. (1)can be evaluated. Ifthe center ofS;iskept fixed and itsradius R isvery large, Vsisoftheorder M/R and av, _arMh on 7OR ~~ BEAPPIONS where M,andMzarethemassos ofB,andBsrespectively. Thus, neglecting thehigher terms oftheexpansion, since du=R'dS, ‘theintegral becomes. ois =- 1 4,Ma,Sire =Mahf4gA, which vanishes with R=«. Hence, ifBsrepresents allspace outside ofBy,and 2V;/ans represents the normal derivative ofVs taken inward with respect toS:,Eq. (1)becomes AV,OV2,AVaVe,AVaV2\, fave +f(pe me= avyfVint @ Applied tothevolume By,Green's theorem gives ViAV,,AViaV:|AVLAV>" firarers f(Be ee eye= avyf,VG where 4V;/dn, means thenormal derivative outward with respect toSi,and consequently isthenegative ofV,/dn;. Onadding Eqs. (2)and (3)itis found that AVOV aVaVe,AVVs frane +fcTeaeteet ee=9,@) 130 ‘THE THEORY OF THE POTENTIAL theintegration being extended over allspace; and this equation can also bewritten @V18V2 |V:AV2 |AV,AV:=< wiefventetfGeTataeaeee) ifitisunderstood that o;iszero outside ofS,. Onpermuting V,and Vsandthen subtracting, there results f(VidVs—VsAV)dr=0, 1} oe° f(:V2 —o:¥i)dr =0, theintegration being extended over allspace. But since o:=0 outside ofB;andoz=0outside ofBs,this eanbewritten just aswell foWadr=fonVidr. (1) By By 73.Characteristic Properties ofaPotential Function.—In the preceeding pages ithas been proved that the potential function Vofanyfinite distribution ofmatter inaclosed volume, oronalimited surface, hasthefollowing properties: 1.Viscontinuous throughout allspace. 2,The first derivatives ofVexist and arecontinuous every- where, except possibly onagiven surface $onwhich there may beasurface distribution ofmatter. Oncrossing this surface there maybediscontinuities inthederivatives, thatis,theytend towards definite limits onboth sides asthesurface isapproached along thenormal, butthetwolimits aredifferent. The tangen- tialcomponents arecontinuous.3.OntheexteriorofS,AV=0,4,Ontheinterior ofS,AVisarbitrary. 5.Vvanishes atinfinity, Conversely, ifthere isgiven afunction V(z, y,2)which satisfies these five conditions and forwhich AV within Sand the discontinuities inthenormal derivatives onSarespecified, then, there exists oneandonly onedistribution ofmatter forwhich V isthepotential. ‘Toprove this, let AV=—4ra(E, 0,2) 73) VECTORFIELDS 131 within S,andletthediscontinuities inthenormal derivative onS be—4ra(, »,£). Since thefunction Visgiven, byhypothesis, itcanbesupposed that thefunctions «and ¢also aregiven. Let p= VE zt GW +Oe and letthefunctions V;and V;bedefined bytheintegrals ‘~fisWefieFa 3° thefirst ofwhich isavolume integral and thesecond isasurface integral. ‘Tho function V;isthe potential ofadistribution of matter within $which hasthevolume density ¢,and Vsisthe potential ofadistribution ofmatter on$which has the surface density %,Hence thefunction V:-+ Vsisthe potential ofa distribution ofmatter which satisfies allfive ofthe given conditions, Itremains tobeshown that this istheonly possible distribu- tion which satisfies the five given conditions. Suppose there existed another distribution ofmatter, thepotential Vofwhich also satisfied allfive conditions. Let W=V-(W+¥). ‘Then, onadmitting negative masses, Walso isthepotential ofa distribution ofmatter which vanishes atinfinity, forwhich ATV iszero everywhere, and normal derivatives are continuous across 8. InSee. 72theletters Viand V2denoted any two potential functions, InHq. (72.4) let Wve. ‘Then aw\?,(aw\:.(aw) theintegration being extended over allspace. ‘The first integral ‘vanishes since Wiseverywhere finite and AW iseverywhere zero. ‘The second integral, therefore, also iszero, which compels aw_aW Wgdeay~a everywhere, Hence, Wisaconstant, and since itvanishes at infinity, itvanishes everywhere, sothat VeNi+hs, 132 ‘THE THEORY OFTHE POTENTIAL instead ofbeing different aswasassumed, Consequently there isnoother distribution ofmatter possible. 74,The Average Value ofaPotential over aSphere.— ‘According totheextension ofGreen's theorem, Eq. (63.3), if¢isharmonic within avolume which isbounded byaclosed surface S,andifpisthedistance from apoint 2,y,2which is within S,then 1 [fle_a(t), _EB-eb)=otno.@ IfthesurfaceSisasphereandthepointz,y,¢isatitscenter,p anditsnormal derivative areconstant onS;andEq.(1)becomes 1fey,42 = 2) Bae +rapt een. ® If,inaddition tothese assumptions, ¢=Visthepotential ofa body which iswholly exterior tothesphere, then byGauss’ theorem, Eq.(68.3), lav andthefirstintegral ofEq.(2)vanishes, since there isnomatter within S. There remains then 1ear=Ven). ® ‘This canbeexpressed inwords asfollows: Theorem.—The average valueofthepotential function overany spherical surface which doesnotcontain anyoftheattracting matterisequaltothevalueofthepotential function atthecenterofthesphere. ‘Thistheorem holds alsoifVismerely harmonic within S. Ifthematter lieswholly within thegiven sphere, letasecondsphereS;concentric withthefirstbedescribedandapplyGreen's‘theorem tothevolume between SandSsIfwand ¢aretwo funetions whichareharmonic withinthisregion de WV, (jae_avSeonSGFan} thenormalderivatives beingoutward withrespect tothecenter ‘onbothspheres. Ifpisthedistance fromthecenter ofthetwo spheres, istakentobe1/p,and¢istakentobethepotential 74) VECTOR FIELDS 133 Vofamass which lieswholly within S,then yand yareharmonic inthevolume between Sand Ss,and 1av a/v 1av a(nSar-Vand)|e-Salean>Ving)J IftheradiusRsofthesphereSisverylarge,then,onS;,Vand1/pareoftheorder 1/I¢s, 8V/ananda(1/p)/dn areoftheorder 1/22, and duisoftheorder Rs, Hence, theentire integral inthe right member isoftheorder 1/2: andtherefore vanishes. Hence lev 1SE+aa=0; and, since pisconstant onS, 1 __1pav Mafre=Ao+n ‘Therefore1 uaefre-%. @) which, expressed inwords, states Theorem—The averagovalueofthepotentialofanydistribution ofmatter over any sphere which includes allofthematter inits interior isthesame asthough allofthematter were concentrated ina particle atthecenter ofthesphere. Reduction toTwo Dimensions.—It isproved inalike manner that analogous theorems hold intwo dimensions foracircle and thelogarithmic potential. 75,Maxima andMinima ofHarmonic Functions.—If o(z, y,2) isharmonie inavolume Bwhich isbounded byaclosed surface 8, then, byEq. (62.2), ae,fsas=0. a Suppose ¢hasamaximum atthepoint p,bywhich itismeant that thefunetion g(, y,2)has agreater value atthe point p than atany point intheneighborhood ofp. Ifasmall sphere 2 isdescribed with pasitscenter, itiscvident that thenormal derivative of¢iseverywhere negative on2,since thefunetion yhas amaximum value atp. Therefore, the integral ogJe 134 THE THEORY OF THE POTENTIAL isnegative. But since »isharmonic within 2this integral must vanish byEq. (1). This contradiction shows that if isharmonic in@volume which isbounded byaclosedsurface,it cannot have amaximum inthat volume, and asimilar argument shows also that itcannot have aminimum either. Theorem I—The potential function cannot have amazimum oraminimum atany point inempty space. ‘This theorem follows atonce from thefact that thepotential function isharmonic inempty space. The theorem does not, however, prevent allofthefirst derivatives ofthepotential from vanishing inempty space; and, since thefirstderivatives arethe components ofattraction, itdoes notprevent theexistence of equilibrium points (eg., thecenter ofauniform anchor ring). Such apoint which isneither 2maximum nor aminimum is called aminimax, foritisamaximum with respect tosome directions andaminimun with respect toothers (like theseatofa saddle). When such points exist they are always points of unstable equilibrium, Ifafunction isharmonic within avolume Band constant everywhere onthebounding surface S,then ithas thesame constant value everywhere inB. Ifitwere notconstant through- outB,itwould certainly have amaximum oraminimum point somewhere within B,which isimpossible. Therefore, itis constant, Similarly ifafunction isharmonic everywhere outside of8andhasthesameconstantvalueeverywhere on§andatinfinity,itisconstant everywhere outside ofS.From thisitfollows, by allowing $toshrink uptoapoint, that afunction which is hamnonie everywhere, including infinity, isamere constant. Iftwofunctions V,and Vareharmonie within Band take the same values everywhere onthebounding surface S,they aro identical everywhere within B.For their difference V= V;~Vsisharmonic within Bandequal tozeroeverywhere on S._Therefore, itisequal tozeroeverywhere within B. ‘From these properties ofharmonic functions ingeneral, there follow the two theorems: Theorem II.—If apotential isconstant overaclosed surfacewhichcontainsnoneoftheattracting massithasthesameconstanttalue throughout itsinterior, and 75) VECTOR FIELDS 135 Theorem III.—If thepotential duetoanydistribution ofmass has@constant value throughout anyfinite volume B,ithasthe same value atevery point ofspace which canbereached byanycontinuous pathfromBwhichdoesnotpasethroughtheattractingmatter, Inorder toshow that thislasttheorem istrue letB,Fig.40, bearegion ofempty space inwhich thepotential isconstant and suppose further that thespace intheneighborhood ofB also isempty, but that thepotential outside ofBdoes not have thesame value that itdoes inside. Take apoint Cnear the boundary ofBanddescribe about itasmall sphere which lies mostly inside ofBbutpartly outside ofit. Since Vand allof itsderivatives arecontinuous inempty space, itwould bepossible totake Csonear the boundary and theradius ofthesphere sosmall thatovertheportionofthespherewhichwas @& outside ofBthevalue ofthepotential would, beeverywhere greater (oreverywhere less) than thevalue within B. This, however, is impossible, since theaverage value over the. 40. sphere isthevalue attheeenter C. Hence theboundary ofthe region ofconstant potential can beextended until the above argument fails, and itwill fail only when attracting matter is encountered. Itwill beobserved that ifmatter isdistributed over aclosed surface insuch away astohave aconstant potential within and ‘onthesurface there isnothing intheabove argument toprevent thesurface density from vanishing atisolated points oralong certain lines,butitcouldnotvanish overafinitearea, Theorem IV—The potential function cannot have @minimum ‘intheinterior ofattracting matter, but itcan have amazimum. Letpbeaninterior point ofthemass, and let2beasmall sphere with patitscenter. ByGauss’ theorem the integral fWay=—4rM; 2On isnecessarily negative. Inorder that pmight beaminimum 4V/an would have tobepositive everywhere onZand therefore theintegral would bepositive. Since the integral isalways negative, aminimum within theattracting mass isexcluded and 136 THE THEORY OF THE POTENTIAL ‘the potential function cannot have 2minimum anywhere, except atinfinity. ‘There isnothing toprevent theexistence ofamaximum, how- ever, intheinterior oftheattracting mass. The potential ata distance rfrom thecenter ofahomogeneous sphere ofradius a, See,29,hasthevaluetrout-tr)fronwhichitisseenthat thepotential ofahomogeneous spliere has amaximum atits center. Reduction toTwo Dimensions.—All ofthe above theorems have their analogies for the logarithmic potential and harmonic functions intwo dimensions. 76.The Potential Energy ofaFinite Mass.—Suppose there is givenndiscreteparticles m;,...,m,.‘Thepotential functionofthis system ofparticles is 1S &mm oeWwe-3>txraat where oy=VG aEY WT ‘Thefactor1/2isnecessary sinceinthedoublesumasitiswritten each element occurs twice, asiseasily verified bywriting out theterms ofthesum inarectangular array. If7’isthekinetic energy ofthesystem, T-W=E thetotalenergyofthesystem,and—1isthepotentialenergy.Gravitational potential energy isalways negative, sothat as theparticles come closer together thepotential energy decreases. ‘The function Wrepresents theamount ofwork which must. be done upon thesystem inorder toeffect aninfinite dispersion ofthe particles. Itwas called the exhaustion ofpotential energy byThompson and Tait, because itrepresents thelossin potential energy from astate ofinfinite dispersion. If We ym jes, 76) VECTOR FIELDS 137 isthepotential ofalloftheother particles ontheithparticle, theexpression for Wcan bewritten W=Sma+mVa+==»+maVs) 12=amV. Iftheparticles form acontinuous mass, this expression passes over into thedefinite integral 1 1 Wearen afPeen (a) whereVisthepotential ofthebodyupononeofitsownelements dm =odr. Ifthesystem 6fparticles isregarded asforming two distinct bodies, which will bedistinguished bythe subscripts 1and 2, sothat Vjisthepotential due tothefirst body atany point t,y,2ofspace and V»isthepotential due tothesecond body, the limit ofthe double sum becomes 1 1 1 1 wehfvantBfvein+3veins$fvate ‘The last two integrals aremerely two different limiting expres- sions forprecisely thesame terms inthefinite sum, andare,there- fore, equal. This fact amounts merely toaverification ofEq. (72.7). Their sum represents theexhaustion ofpotential energy, duetothefact that thetwo bodies arenotinfinitely farapart. ‘Sinee byPoisson’s theorem, Eq.(70.5), a=pay, Bq. (2)eanbewritten 1 w=-bfvav dr, theintegralbeingtakenoverallspace. Onsetting Vi=Vi=V inEq.(72.4), itisseen that theexpression forWcanbewritten also 1 av)’, (avy, (avy theintegral being taken over allspace. But since 2(av), (aV)?,(avy 138 [THE THEORY OFTHE POTENTIAL isthesquare oftheattractive foree duetothebody, there resultsfinally 1Weafr‘dr. 3) IfinEq,(2)thesumofthelasttwointegrals isdenoted by Wy itisseenimmediately from Eq,(72.5) that Wa=[Vents =fVoesir wb[mae aHI, ® “GrJjal dzOz*dydy"OzG2|*" theintegral being taken over allspace. ‘According toLord Kelvin and Tait! itwas upon aproper interpretation ofthe formulas relating totheexhaustion of potential energy oftwo bodies that Green founded thewhole structure ofhisgeneral theorems regarding attraction. TI.The Potential Energy ofaHomogeneous Sphere.—In theinterior ofahomogeneous sphere theattraction ofthewhole sphere onapoint isduetothemass ofaconcentric sphere, the surface ofwhich passes through thepoint, ‘That is -M, 4 PewMe hoor, Ontheoutside ofthesphere F, M= Hence, iftheradius ofthesphere isa, BW=fredr+fFear. Sincetheforeeisafunction ofralone,lettheelement ofvolume drbetaken asthespace between twoinfinitely close concentric spheres, ‘Then ds=Aartdr, and ire : w=bfPree+ifPevar =fewfrar+Lan[er90 2ja @=Byagigs 4MPhi are 23Me 5a *Treatise onNatural Philosophy,” PastI,p.8, 78) VECTORFIELDS 139 78,The Heat oftheSun.—According tothetheory published byHelmbolts, in1854, itistheenergy released intheprocess of contraction that isthesource oftheradiant energies ofthestars. Onthis hypothesis the total amount ofheat radiated bythe sun inthe past can becomputed. Assuming that the sun is ‘uniform indensity, thetotal amount ofwork done incontracting fromastateofinfinite dispersion is W=Blorgs, where k*=6.66 X10-* isthegravitation constant (Sec. 20), M=1.99 X10" grams, and a=6.96 X10cm. Hence W=2.28 X10 ergs, =5.44 X10 calories since Leslorieisequalto4.19X10"ergs.‘Thesunradiates2.95 10** calories peryear, orapproximately 1.5calories pergram per ‘year, Hence, atthepresent rate ofradiation, theenergy released inthe process ofcontraction would besufficient tolast for18,- 000,000 years, ‘This theory oftheorigin ofthesun's heat was thedominant oneduring thelatter half ofthenineteenth century and thefirst two decades ofthe twentieth century. Both onastronomical and geological grounds itisnow recognized asaltogether inade- quate.’ Itseems much more probable that the source ofthe sun’s heat istobefound intheelectrostatic potential ofthe electrons within the atom rather than intheir gravitational potentials, ‘The electrostatic potential energy oftwo electrons ofopposite sign is yae > here pisthedistance between theelectrons and ¢theelectric charge, isequal to4.74 X10~° electrostatic units. Agram ofmatter isequivalent to6.08 X10 such pairs ofelectrons. Hence, theelectrostatic potential energy ofagram ofmatter is 1.38X105 Vi=EBX orgs, Ifitisassumed that when the distance betwoen the two elec- trons isequal tothe radius ofthe positive electron, which, 1MacMnutax, On Stellar Evolution, Astrophysioal Journal, Vol. XLVI, p.8%, (1018). Some Mathematical Aspects ofCosmology. Seience, Va LX, Nos, 1696-1597, (1925) 140 THETHEORY OFTHEPOTENTIAL according toMillikan, isapproximately 10-*, thetwo electrical fields are superposed and neutralized, the property ofmass disappears and thepotential energy takes thekinetic form of radiation, itisseen that 1gram ofmatter isequivalent to 1.38 X10" ergs ofenergy, orapproximately3X10"calories,8 result which isapproximately equal tothat given bythemodem theory ofrelativity, namely 9X10ergs, without anyconsidera- tion ofmodels. ‘This hypothesis does not furnish any basis forestimatingthepresentageofthesun,butifonegramofmatterisequivalentto3X10* calories and thesunisexpending 1.5caloriespergram peryear, itisevident that thesuncontainsareserveofenergyin itspresent mass sufficient tolast 2X10" years atitspresentrateofradiation, or20,000billionyears.Itisnotpossible toestimatehowmuchadditional matterthesunwillgatherinfromspace inthat length oftime. 79,Relation between Certain Surface and Line Integrals.— Itwill besupposed that atevery point m(z, y,2)ofacertain region R,avector Kisdefined whose components P,Q,Rare : single valued, continuous functions ofz,y,2which admit continuous firstderivatives.Itwillbeas- imesumed also that inthis region R there exists portion ofasurface S ‘ which has two sides and which is vA bounded byaclosedlineL.OneSide ofthesurface willberegarded aspositive, the other asnegative. Leta,8,7bethedirection cosines z ofthenormal erected ontheposi- Fro.al. tive side ofthe surface atthe Itwill beshown that surface element ds, eP_ap) free=f(ee22Vio, wheretheintegral intherightmember istakenovertheportion ofthesurface $which isbounded byZ,andtheleftmember is ‘thelineintegraltakenaroundtheboundaryLintheeounter-clock- wisedirection asseen from thepositive sideofS. Itwillbeassumed, atfirst, that both Pand+preserve their signs throughout S.This restriction will beremoved later. 79) VECTOR FIELDS 141 Ateach point mofSdraw aline mm, parallel tothez-axis and equal inlength tothemagnitude ofPatm. The locus ofthe point m,thus defined isasurface S,bounded byaclosed contour L;(Fig. 41). The straight lines which join thepoints of Ltothecorresponding points ofL;form acylindrical surface C. Since S,C,andS;taken together form aclosed surface, theprojec- tion ofthis surface upon any plane iszero, This fact will be formulated fortheprojection upon thez#-plane. Letdsbeanelement ofthecontour Lcutoutbytwo infinitely close generators ofthe cylindrical surface. The distance between theprojections ofthese twogenerators upon theze-plane isdz. Hence the area ofthe projection ofthat portion ofthe cylindrical surface which isincluded between thetwo generators above mentioned isPdz and the area ofthe projection ofthe entire cylindrical surface upon the2z-plane is, pe. Consider now anelement dwofthe surface S. The tangent plane atthis element makes thesame angle with thexz-plane as thenormal atthis element makes with theyaxis, that is,tho angle whose cosine [email protected] the projection ofduupon the aeplane isAdw, and theentire projection ofSis feeF Using acorresponding notation fortheS,plane, itisevident that theprojection ofS;upon thezz-plane is feehs ‘The sum ofthese three projections iszero. ‘Therefore fee[te+f,Bide=0. @ PaoJfae+J Lettheelement dw;correspond totheelement dwinthesense that deand do;arecutout ofSandS;bythesameinfinitesimal cylinder parallel tothez-axis. Since this infinitesimal cylinder isclosed, itsprojection upon thezy-plane alsoiszero, andsince theprojection ofthecylindrical surface upon thezy-plane iszero byitself, there remains do +nde =0, 42 THE THEORY OF THE POTENTIAL and therefore do,=—7de, wm ‘The substitution ofthisvalue inEq.(1)gives Spies[2tae=0 ® Fian ‘Lettheequation ofthesurface Sbez=f(x, y),then (e152) =flay) — 2=0. @) The direction cosines ofthenormal tothis surface a,8,+are proportional to oe, ae, ae. as"ay’oa" orif _% _o Pmye 175, ‘theequations eb @pog -1 hold, Similarly onthesurface S;, a bma.5. 1, (5)aaat ® From Eqs. (4)and (6), itfollows that =-§ =A,gap em (6) Forthesurface Si(zs, y1,2),itisevident that my naw Ame+Py2) Therefore co oP oP.met tae cay oP AP.Fart at tae Ifthevalues ofgandq;aresubstituted from Eq.(6)into the second ofthese equations, there results Br.—vb ,aP_oPmTYay~Poe Which shows that Eq,(2)eanbewritten aPoP’ fri=f 2), ) and thisistheformula itwas desired toestablish. 79] VECTOR FIELDS 143 ‘Therestriction that Pand shall preserve their signs isnow easily removed. Ifthiscondition isnotsatisfied ofitself, Scan bedivided upbyauxiliary curves insuch away that ineach portion Pandydopreserve their signs unchanged. Equation (7)holds foreach ofthese portions separately. Thesum ofthe portions isS,andthesumoftheeontoursis L,since theauxiliary curves have been described twice, butinopposite directions, whilethecontour Zhasbeendescribed butonce. Hence, Eq.(7)holds ingeneral. Therestriction serves merely tomake the geometric interpretation clear. 80,Stokes’ Theorem.—TFrom Eq. (79.7) two analogous equations areobtained bypermuting theletters. ‘These three equations are oPaP’ feefe_Pa, 2Q_a fede=f(r89-222), @ ORAR fre-[(@-Nie, Ontaking thesum ofthese three equations, there results ieee+iy+Res)=aR_00),(3B_a),(20_a SG 2)0-2) GF) @ which isStokes’ theorem. ‘The functions P,Q,Rarethecom- ™ ponents ofavectorKwhichisdefinedatw each point ofS.Let-A beapoint ofthis contour, and letK,bethe component of Kwhich istangent tothecontour atA. kK,ispositive ifithasthedirection ofa positive motion ofthepoint Aalong the Fro,42. contour, otherwise negative. Letdsbe ‘apositive infinitesimal displacement ofthe point Aalong the contour with the components dz,dy,dz, Ifthe vector Kisregardedasaforce,itisevidentthat Pdz +Qdy +Rdz =Kids it THE THEORY OF THE POTENTIAL isthe clement ofwork done inthis small displacement, and fice+Qdy+Raz)~xe istheamount ofwork done incompleting anentire cireuit. Itean also besupposed, that there exists asecond field of vectors Wintheregion R,and that thecomponents ofWare =2R_20, =P_8, =22_aP. Wen ae “Ay ® ‘This vector W,which isevidently related toK,iscalled thecurl ofthe veetor K. Furthermore aW. +6W, +Ws =Wn isthecomponent ofWwhich isnormal tothesurface S. Equa- tion (2)can, therefore, bewritten [kas=[Wede, and eanbeexpressed inwords asfollows: ‘Stokes’Theorem—Theworktwhichisdonebyavectoractingonapointwhichdescribesaclosedcircuitisequaltothefluxofthecurl ofthisvector across anycontinuous surface which isbounded bythe circuit, ‘This form ofstatement shows thatthetheorem does notdepend upon any choice ofaxes. Itwillbeobserved that thedivergence (Sec. 56)ofthecurl ofany vector vanishes identically, since OW.,Wy,aW,Oetay tGeO From thisitfollows that thefluxofthecurlacross anyclosed surfaceiszerosincethedivergenceofthecurlvanishesthroughout thevolume, Eq. (66.4). 81.Examples ofVectorCurls.—Suppose Visaconstant vectorwith the components X,Y,Z,andKisthemoment ofV with respect tothepoint 0,yo,20.IfVisacting atthepoint ,y,2,thecomponents ofKare P= (y—yZ -(@-2)¥, Q=(@—2)X —(z- 2)Z, R= @—20)¥ -(y—yo. 81] VECTORPreLDs 4s ‘Thecomponents ofthecurlofKarereadily found tobe —2X, —2Y, —27. That isW=—2V. _Asasecond illustration, consider thevelocity ofanypointofa rigid body which ismoving inanymanner inspace. Imagine asetofrectangular axesrigidly attached tothebody withthe origin atthepoint 0,andletm(z,y,2)beanypoint ofthebody. Asiswell known, themotion ofthebody canberesolvedintoa pure translation, which isequal tothatofthepoint 0,anda pure rotation wabout some axiswhich passes through thepoint 0.Letthecomponents ofthetranslation with respect toaset ofaxes which arefixed inspace, butwhich instantaneously are parallel tothesetofaxeswhich arefixed inthebody, beXo,Yo, Zs,andletthecomponents oftherotation «withrespect tothese same axes bews,ayos Then thecomponents ofthevelocityKofthepointmwithrespecttothesetofaxeswhicharefixedinspace are P=Xot 2,—Yor Q= Yo4+ws—te R= Zot yor —ty ‘The components ofthetranslation, Xs,Yo,Zo,and oftherota- tion, os,&y,«,,areevidently independent ofthecoordinates ofthe point m. ‘Hence thecomponents ofthecurl Wofthevelocity veetor Kare2s, 28,and 2v,. That isthecurl ofthevelocity of thepoint mistwice theangular velocity ofthepoint mabout the instantaneous axis ofrotation. Itwasthisfact which gave rise tothe term curl which isdue toMaxwell. Clifford called itthe spin. 82. The Vector and ItsCurl Are Orthogonal.—The moment ofaveetor isperpendicular totheveetor itself, andthevelocity ofaparticle ofarigid body, insofarasthat velocity isdueto spin, isperpendicular totheinstantancous axisofrotation. If, however, thetranslation ofthebody istaken into account, the velocity ofaparticle anditscurlarenotmutually perpendicular ingenoral. ‘The orthogonality ofaveetor anditscurl hasanimportant interpretation inanalysis which canbestated asfollows: ‘Theorem.—A necessary andsufficient condition thattheexpression Paz +Qdy +Razadmits anintegrating factor isthatthevector K (P,Q,R)anditscurlWaremutually perpendicular toeachother. 146 THE THEORY OF THE POTENTIAL The analytic condition fortheorthogonality ofavector K, which hasthecomponentsP,Q,andR,anditscurlW,Eq.(80.3), is ‘aR_aQ‘aP_aRaQ_ar)- (222)+(&-a)+(2a)=* Ifthe differential expression Pdr +Qdy +Rdz admits an integrating factor ¥(z, y,2),sothat ¥(Pdz +Qdy +Rdz) =dU, (2) andtherefore vu au, au, élwat Wao R=Ze @) the function ymust satisfy certain differential equations. If thefirst ofEqs. (3)isdifferentiated with respect toyand the second with respect toz,thetworight members areequal, and therefore thetwo left members areequal. Inthis way, and by cyclical permutations oftheletters, itisfound that ymust satisfy thethree equations ay_gabap_a@ Pi-=oP-B) oY_pa_(20_a), GeRay~(33oy ® ob pth__(aR_oPRee—Pa(2a) Ifthe first ofEqs. (4)ismultiplied byR,the second by P,andthethird byQ,andthethree equations arethen added, Eq. (1)results. Hence, the orthogonality ofthe vector K and itscurl Wisanecessary condition fortheexistence ofanintegrating factor. Inordertoshowthatitisalsosufficient itis necessary toshow that ifEq.(1)holds, Eqs. (4)admit asolution. Without diminishing the generality ofthe proof itcan be assumed that Risequal to—l, ifitisnot zero identically; forEq.(2)canequally well bewritten (-9)(FytSay-a)=a, andthen byobvious changes inthenotation Rwould beequal to—1. Forthis value ofR,Eq.(1)becomes aP_paQ_aQ_oP,ee~Pas*ae~By’ © 82) VECTOR FIELDS 147 andEqs. (4)reduces tothetwoindependent equations Ov, pw, aP _ Setae t45,=9 oyQh4y2Q_ay+%e+¥ae= By changing the variables and taking dy/y =—dy, these equations aresomewhat simplified and become ae ae_oPae+Pon=Ge’ , ge+gfe-20. ° az*[9~ae" This isasystem oftwo partial differential equations with three independent variables, and since, byvirtue ofEq. (5), 44g8)(ae4pie)_(2,p2)(de,gae’ &+o8)(2+P2)é+P;(3+oe) thesystem iscomplete. Furthermore, since a a)(aP)\_(2,pa\(ad Gj+03\() -&+P3\(%) thesystem isconsistant. The solution ofEqs. (6),therefore, consists ofaparticular integral plus thegeneral integral ofthe homogeneous equations." Intheparticular case inwhich aP_aQg- Bao, ) Eq. (5)shows that 2g_aP, ©az~ay and thecurl Wvanishes identically. Equations (6)admit the obvious solution =const., and the differential expression Par+Qdy+Rdz =dUisexact. Conversely, ifitisexact, » isaconstant, Eqs. (7)and (8)aresatisfied, andthecurlvanishes. Hence the additional theorem: Theorem.—A necessary andsufficient condition thatthedifferen-tialexpression Pdz+Qdy+Razbeexactisthatthecurlofthe‘tector Kshall vanish identically. 'Gounsas, E,“Legons sur Vintégration deséquations aux dérivées parteles dupremier ordre,” p.68. 148 THE THEORY OF THE POTENTIAL 83.Condition That aLine Integral Shall beIndependent ofthe Path ofIntegration.—LetAandBbeanytwopointsintheregion Bin whieh the vector Kand itsfirst derivatives are continuous. Join thetwo points byany curve Q;whieh lieswholly within and letI;betheintegral 2 2 T=fiPar+Qay+Cae)=[i'Kas taken along thecurve C;. Join thetwo points Aand Bbya second curve C2,and letI;bethesame integral from AtoBtaken along thecurve C:(Fig. 43). Ifthevalueoftheintegralisindependentof lethepath, Ji=J,,and T=I-In= (Pde+Qay+Ras)=0, whereListhecircuitfromAtoBalongC;and Ais backfromBtoAalongC:.If,further,thearea1.43 bounded bythiscircuit lieswholly within R, sothat Kand itsfirst derivatives areeverywhere continuous, Stokes’ theorem (Bq, (80.2) gives - aR_aQ’ aPal Q_al - 1[[sGr- a)+058) GE) m=* Byhypothesis, AandBareanytwopoints inRandtherefore L isanyclosed cireuit. Hence J=0onevery surface $which is bounded byaclosed cireuit which lieswholly within R.Evi- dently then thecoefficients ofa,8,andyinJmust each bezero, foriftheyworenot,andif,say,32~3F50,aplanesurfaceS forwhich «=6=0,y=+1could betaken and acircuit L sufficiently smallthat2°—xwouldeverywhere havethe same sign, and theintegral Jwould notbezero, Asthis would contradict thehypothesis, itfollows that. oR_8Q_, aP_aR_, 3Q_ oPa Bam BenaTO everywhere within R,and therefore thecurl ofthe vector K vanishes identically. This ismerely another wayofsaying that Paz +Qdy +Raz =dU isanexact differential (Seo. 82) a4] VECTOR FIELDS 149 84.Condition That aSurface Integral Shall Depend upon the Contour Only.—A question which isanalogous tothat ofthe preceeding section isthefollowing: Under what conditions isa surface integral dependent upon thebounding contour only and notatalldependent upon theparticular surface which passes through the contour? Suppose there isgiven @vector Wwith thecomponents F,G, and H,which together with their first derivatives arecontinuous within acertain domain R.LetLbeany closed contour lying wholly within R,and S$any continuous surface which passes through andisbounded byL. If«,8,7arethedirection cosines ofthenormal toS,theintegral T=[lor+064aide=fiveteF 3 isthe flux ofWacross S. The question is:What conditionmustbesatisfiedbyWinorderthatthefluxacross$shalldepend uponLbutshallbethesameforeverySwhichisboundedbyL. Let S$;and S;betwo such surfaces across which the flux is thesame. Ifapositive direction along the contour isdefined, thepositive sides ofthese surfaces also aredefined and thenormals areassumed tobedirected from thepositive side ofthesurfaces. Let I,betheflux across S,and I;betheflux across S;. Sothat. I,=1;. The surfaces S,and S;bound acertain volume. Let. thenormals onone ofthesurfaces bereversed, thus reversing thesignofthefluxacrossthatsurface,sothatthenormalsareevery- where directed outward from the enclosed volume. The condi- tion I,=I;then becomes nefoP+00-+vEde=fWide=0,(1)suse vse and this condition holds byhypothesis forevery contour L which lieswithin R. This isthe same assaying that Eq. (1) holds forevery closed surface inR. ByEq. (56.2) theintegral I;isthesame astheintegral ar,a@,aHSee+349 2)=0. @ Since Eq. (2)holds forevery bounded volume which lieswholly inRitisevident that OF,aG,anazty+ae=9 150 THE THEORY OF THE POTENTIAL everywhere inR.That istosay, thedivergence ofthevector Wvanishes identically. Hence thetheorem follows: ‘Theorem.—A necessary andsufficient condition thattheintegral fer+0+ainae @F taken over anybounded portion ofasurface shall depend only upon thebounding contour isthatthedivergence ofthevector W shall vanish identically. Iftheintegral Eq.(3)depends only onthecontour Litshould bepossible toreduce thesurface integral toalineintegral. In order toshow how thisisdone, thefollowing theorem isuseful: Theorem.—If thedivergence ofagiven vector W(F, G,H) iszero, there exist infinitely many vectors K(P, Q,R)forwhich W ‘isthecurl. That istosay, ifthethree functions F,G,andHaregiven, and ifaF,0G,oHete tes ) there exist infinitely many triples offunctions P,Q,and Rsuch that OR_aQ\_ (aP_oR\ _ 2Q_ aP\_G-B)-"% (E-z)-% (F-|)-* © Itwillbeshownfirstthatthereexistsolutions ofEqs.(5)forwhich R= 0.Assuming that Riszero identically, Eqs. (5) become 0a P_ 2Q_ oP_Rap, Fue Ban. © ‘The second ofEqs. (6)gives Psfoe.v2)de, andthefirst gives“ =~ [Rev de +500, wherefis,atthemoment, anarbitrary function ofzandy.The third ofEqs. (6)now requires that af_(OF4\),_ >[Gree 84] VECTOR FIELDS 151 or,inview ofEq. (4), a, pialxef“eB which isthe same as aLHe ye, and therefore fo[inevsae, Eqs. (5)therefore, aresatisfied bythefunctions Pimflocinede a=f'Hen2) —fPenet, Rimo. Now letP,Q,and Rbeany triple offunctions which satisfies Eqs. (5),and let Pr=P-P, Q=Q-Q, R=R-R. ‘The substitution ofthese expressions inEqs. (5)gives I,90g sR_4Ie_8P_g, ay ee Oe az Oy therefore, ifU(z, y,2)isanarbitrary funetion ofz,y,and 2, au, au, _ou,an an ye Hence, any solution ofEqs. (5)can bewritten. au _au _au PHPitan Qaata R-h+a> whereUissomefunction ofz,y,andz,andthisestablishes thetheorem. ‘Theintegralfi(aF +0G+1H)dw 3 can now bewritten aRaQ\,4(aP_ak aQ_aPSlGr-&)+(eFe)+9G5~a) and, byStokes’ theorem, this isequal totheline integral fire+Qdy+Rds) p taken around thebounding contour L. 152 THE THEORY OFTHE POTENTIAL Itis easy toscethat thisintegral isindependent ofU,since aU au auSee+e =)=0. Problems 1.Ur=VEGFaismeasured fromapointO,whichmaybeether insideoroutside ofaclosed surface $,thevolume which isenclosed byS ingiven bythe formula 1f2 wohAfa 2.Show thatthe exhaustion ofpotential energy of«homogencous oblate spheroid forwhich aistheequatorial radius and theeccentricity of« eriian section, is SyyMEin WaSyAse, 8.IfMis themaximum value oftheharmonie function Vontheclosed surface 8,andmisitsminimum, show that within S msVSM. 4.Starting with Laplace's equations inretangular coordinates, derive thecorresponding equation forpolar coordinates CHAPTER IV ‘THE ATTRACTIONS OF SURFACES AND LINES 85. The Occasion forTheir Study.—In thedomain ofelectric ityitisfound that anelectrically charged conduetor inelectrical equilibrium acts asthough the surface were covered with an infinitely thin layer ofelectricity which attracts orrepels accord- ingtothelawoftheinversesquares, andthisleadstoaconsidera- tion oftheproperties ofsuch surfaces. Bythe term electrical density ofsuch asurface ismeant theamount ofelectricity per unit area. Inthesense ofvolumes the density isinfinite since the thickness istaken tobe zero. ‘But quite apart from theoccurrence ofsuch surfaces innature, the concept isvery useful even asamathematical fiction, as was seen inthe discussion ofSec. 61. Itisevident from the definition that the potential function exists atallpoints which arenotinthegiven surface, and the derivatives ofthepotential functions atallsuch points repre- sent the components ofthe attracting orrepelling force. It isfurther evident that thepotential function and itsderivatives arecontinuous atallpoints which donotlieinthesurface. Itis proposed tostudy, inthepresent chapter, thevalues ofthepoten- tialand itsderivatives intheneighborhood ofthesurface itself. Itwill befound that thepotential oftheattracted point iseon-tinuousalongalinethatpiercesthesurface,butthatthenormal component ofthe attraction has anabrupt discontinuity of 4no astheattracted point passes through the surface atanordi- nary point, ¢being thedensity ofthesurface atthepoint where itispierced. ATTRACTION OF SURFACES 86.AUniformDisk—InSec.30itwasfoundthatthepotential ofauniform disk ofradius aand density ¢atany point ponthe axis ofthedisk (which will betaken asthez-axis ofacoordinate system), andthezcomponent oftheattraction atthepoint p aregiven bytheformulas EF VA - #2) V=divare~V#, aZ2refvere al 154 THE THEORY OF THE POTENTIAL The potential iscontinuous along thenormal, but Zisdiscon- tinuous asthepoint ppasses through thedisk. Itwillbeinteresting toexamine thecomponent ofattraction which liesintheplane ofthedisk. The natural reply tosuch anenquiry isthat this component iszero from symmetry, but theenquiry isworthy ofacloser examination, InFig. 44,let C2with itscenter atO:beauniform disk ofdensity ¢. With any other point 0, sacenter, drawacircleC,tangentexternally toCs, ‘Then draw any small circle C;with 0;asacenter, and Fro. alastcircleCstangent externally toC;withtheradiusrs,such that theproportion none none holds, where r;istheradius ofthecircle C;, The circles C2 andCyareinperspective with respect tothepoint Os;for,if anystraight linebedrawn through thepoint O,andthedistances from 0,tothepoints ofintersections ofthislinewith thecircles Czand C,arep;andpy,then itistruethat otg,pe}* Since thecircles C;andC;areconcentric, itisevident that they alsoareinperspective with thesame ratio ofperspectivity «. HfCsandC,bekeptfixed butaisdiminished, thecircles C;and Czdiminish insizeandfora=0reduce tothepoint O,itself. ‘Theperspectivity relationship holds, however small @maybe. ‘Now imagine thedisk, which isbounded bytheCircle C2,is removed from thediskC;.Theattraction oftheremaining 88) THE ATTRACTIONS OFSURFACES AND LINES 155 portion ofthedisk Csupon thepoint Os,which will bedenoted byAcrey iswelldefined, since 0,isnotapoint oftheattracting surface. Iftheattraction oftheentire disk Csupon thepoint 01,which isapoint ofCs,istohave ameaning, then, this attrac- tion willbethelimit ofA,, a8adiminishes. That is Ac,=limAney Now Aer =Ara ~Aeey and Agra =0, byvirtue ofsymmetry. Hence Agneg =—Acgey Since thecrescent-moon shaped figures C4—Cand Cs—Cy areinperspective with respect tothepoint O;,their attractions onthe point 01areequal (Theorem II,Sec. 10). ‘Therefore Age =Accor But theattraction ofC,—Csisindependent ofwand remains constant as«diminishes. Hence JimAcpey =Acy=Acres Theattraction ofCy—Cson01isnotzero,foracircleCsofthesame radius asC,can bedrawn which divides the area C,— C3 into two portions, one ofwhich issymmetrical with respect to ‘the point 0;and theother lies entirely onone side ofaline through 0:. Since the radius ofCisentirely arbitrary, and the direction ofitscenter also, itfollows that A... isarbitrary both asto magnitude and direction. Consequently, the atiraction ofa uniform disk upon itscenter does nothave adefinite sense. 87.AnInfinite Homogeneous Universe.—An anslogous situa- tion arises with respect tothe resultant attraction ofaninfinite homogeneous universe upon one ofitsown particles. InFig. 45,lotPbeany particle, and Oany point chosen arbitrarily. About 0asacenter describe asphere ofradius OP. Ifthissphere isfiled with matter ofdensity ¢,thegravitationalforceactingatthepointPtowardthepoint0is(See.13), 4oe F=jrePO. 156 THE THEORY OF THE POTENTIAL Now letalarger sphere ofradius 0Qbedescribed about the point Oand theshell PQbefilled with matter ofdensity «.The resultant attraction ofthisshelluponthepointPiszero(Sec.11). ‘Hence theresultant attraction ofthesphere Qupon thepointP isproportional toPO,however large 0Qmay be. Passing tothe limit, the attraction ofaninfinite homogeneous universe upon thepoint Pis 4P=jPO. ButPOisarbitrary, both astomagnitude anddirection. Pio. 45 Neumann concluded from this that thepostulate that the universe isinfinite and essentially homogeneous isinconflict with thelawofgravitation, butitisclear that this conclusion rests upon stillanother postulate, namely, that every physical situation isuniquely defined, orthatnatureisneverambiguous—8postulate that, doubtless, will readily begranted. Quite likely, thelawofgravitation isonly aremarkably close approxi- mation and that itfails atsufficiently great distances. Thesame remark canbemade with respect toadistribution ofelectricity upon thesurface ofaconductor. Inadomain which isoftheorder ofmagnitude oftheelectrons, itisnotlegitimate toregard thedistribution asasurface distribution. Theapproxi- mation fails inadomain sufficiently small. 88] THB ATTRACTIONS OFSURFACES AND LINES 187 88.Proper and Improper Integrals.—If f(x) isafunction of which iscontinuous inthe interval aS$xS6,the integral . [ise () hasaperfectly definite sense, and iscalled aproper integral. If,however, f(z) isinfinite atone ormore points oftheinterval, theintegral considered asthelimit ofasum does notimmediately have asense. Suppose, forexample, f(a) isinfinite, but that elsewhere intheinterval f(z) iscontinuous. If¢isany small positive quantity, theintegral . ff,foe @ hasasense, however small «may be. Ifthelimit ofEq. (2) asetends towards zero isfinite, this limiting value isdefined tobe themeaning ofEq. (1), and theintegral issaid tobeimproper. Inanalogy with thetheory ofinfinite series itisconvenient to saythat theintegral Eq. (1)isconvergent ifthelimit ofEq. (2) exists, and that itisdivergent ifthe limit ofEq. (2)does not exist, aswill bethe ease when the value ofEq. (2)increases indefinitely as¢decreases, oreven oscillates indefinitely between finite limits. ‘Simple Integrals—Suppose there exists anumber £such that ifaszset M eal<ge where Mandaaretwofixed positive numbers. Then . " .frou=fiseae +fseeds, and. . “ou .Piserae <[ae +fis. ‘Thelastintegral isaproper integral andhasafinitevalueL. Flenoe re) cae weifroad=f{rae<a{Ga —|pndate Ja+«|fi) l-a@ l-a@, Ifa<1,itisevident that » | —a)ime limiffar<MELD 41, end Jug 2 Toe and thegiven integral isconvergent. 158 ‘THETHEORY OFTHEPOTENTIAL Hf,ontheother hand, inthesame interval, aSz$& N woras where Nand 6aretwo fixed positive numbers, the integral |‘ydzc| *|ide .Nsax, and therefore (8 oralsyfGrae e#7, If6>1, itisevident that nolimit exists and the integral is divergent.DoubleIntegrals,—Supposef(z,»)isacontinuousfunctionof zandy inandontheboundary ofacertain area S,The integral JSsesendy L Ss hasadefinite senseandisaproper integral.Butifatsomepoint0of @the area f(z, y)becomes infinite theintegral hasnosense dizeetly; itisnecessary togive itone. InFig. 48,letS,thearea of Fro.46, integration, bebounded bythe closed curve L,and letObeapoint atwhich f(z, y)becomes infinite. Forsimplicity itwillbeassumed that there isbut one such point. Around thepoint Odeseribe asmall closed curve C,and letthesymboliff,indicatethedoubleintegraloveria the area which lies between the curves Zand C, ‘Then the integral SS. mestn ® - isaproper integral, and ithasacertain value V. IfVhas a definite finite limit when thecurve Cshrinks down upon thepoint 0,independent ofthe shape ofthe curves through which C shrinks, then thelimit ofVisdefined tobethevalue ofthe inte- ral Eq, (3),and theintegral isconvergent. If,however, no limit exists theintegral isdivergent. 88] THEATTRACTIONS OFSURFACES ANDLINES 159 ‘Suppose, atfirst,thatitispossible todrawafixedcircleCiofradius r;withthepointOasacenter (Fig.47)inside ofwhich fic,»)iseverywhere ofthesamesign,saypositive; andthat‘thereexisttwopositive numbers Mandasuchthateverywhere inside ofCi u se <Bs whereristhedistance ofthepointz,y,from0.Drawasecond circleC;withOasacenter andradius rs<7. Then, ifthe integral overShasanysense, itis fiposmoc =fffosadaad F .-< tiimfff vided. L G € cers ‘Thefirstintegral intherightmember hasadefinite value. As forthesecond SfS(z,y)dady<SJ.Maras, awed Je. e ‘2r SS-cfeneni<Fit[r- ne} Je. Ifa<2,thisexpression hasafinitelimiting value. Anyothercontour C’canbeenclosed between twocircles Cy’andC2!(notdrawn inthefigure) withcenters at0.Then Sf<Shoes SSo-or er©SJone<JJere‘andsincethetwoextreme integrals havethesamelimitthe central onehasthesamelimitalso. Itfollows, therefore, that ifa<2,theintogral Eq.(4)isconvergent. 160 THETHEORYOFTHEPOTENTIAL Inasimilar manner, itisproved that iff(z,y)ispositive overywhere within C,,ifthere exist twopositive numbers M and8suchthat M ie >M and if8>2,then theintegral Eq.(4)isdivergent. Ifthere does notexist acircle C,with Oasacenter inside ofwhich f(z,y)haseverywhere thesame sign, itmay still be true that there exists acircle C;andtwopositive numbers M and a(a<2)such that everywhere inside ofC; 1 M.fewl<Te Ifthese conditions aresatisfied, theintegral Sfdocenicay isconvergent, and theintegral JSucternacay<fflpswlee isabsolutely convergent. The preceeding argument shows that if anintegral isabsolutely convergent, thelimiting value isentirely independent oftheforms ofthecurves Cbymeans ofwhich the limit isapproached, and theorder oftheintegrations can be interchanged ifdesired. ‘These results canbeextended readily tomultiple integrals of higher order, thevalue of«increasing byunity foreach increase intheorder ofthemultiplicity. Thus, fortriple integrals, the integral isconvergent ifa<3,and0on. Asanapplication ofthese ideas consider thevalue ofthepoten- tial ofaplane area S,forwhich thedensity function is, at& point Oofthe area itself. Let Obetaken astheorigin ofa system ofcoordinates. Then vetfSandy. Ifiscontinuous throughout S,asmall circle can bedrawn about theorigin inside ofwhich themaximum value of|o| <M. The value ofais1,and, since thisislessthan 2,theintegral isconver- gent. The potential, therefore, has adefinite value ateach point ofthearea S. 88) THEATTRACTIONS OFSURFACES ANDLINES 161 Consider acomponent ofattraction atapoint inside the attracting volume. Ifthepoint under consideration istaken as theorigin, x=~fffgecava. If,inasmall sphere ofradius pabout theorigin, themaximum value ofjo|isM,then, since |r|<rand <iIr®co theintegral isconvergent, for«=2,which isless than three. ‘The component ofattraction hasadefinite value. 89.Semi-convergent Integrals.—A series ofnumbers may converge without being absolutely convergent. Forexample 1,1 1,1 1 -p+g-atg-atc @) converges, and itsvalue isthelog2. But the series breeoreeare aresltgtgtagtgtgt diverges. Such aseries (a) issemi-convergent, and the limit depends upon the order inwhich the terms aretaken. Example ofaSimpleIntegral-—In «similarmanner, anintegral may beconvergent without being absolutely convergent. For example, consider theintegral +1ade hae [hae r=fvcdin=m[ 4-£9} SxaytimJoeJ3F where «and ¢aretwo small positive quantities which bound offtheorigin, and+/z? isalways positive. Itiseasily verified that T=limGi-2) and itsvalue depends upon the manner inwhich the interval ee shrinks tozero. Suppose \isany given constant, and =a. OS TFN then LL, “6 forallvalues of«. Therefore I=, which isarbitrary. 162 THE THEORY OF THE POTENTIAL ‘The integral ofthemodulus, however, nared_figl diverges. Double Integral.—Consider thedouble integral oP yanvoza8 aay,Ses where O0<t<a, O<n<b Since (tant Y)@Vat,aa %)“wry the value ofthis integral is tb tant tant214, tan” atanztan”a+tanz which isperfectly definite ifthevalues oftheangle arelimited to thefirst quadrant. ot ea Fro 45 Inorder toevaluate theintegral forboth lower limits equal to zero, lettheorigin bebounded offbyasmall rectangle ofsides & and, Fig. 48. The integral over theremainderoftherectangle, ofwhich thesides areaandb,is tan-? ©—tan-t 7. a G ‘The limit ofthisexpression asthepoint £»approaches the origin, depends upon thedirectionofapproach,forthisexpression representstheanglebetween thediagonal oftherectangle aband thediagonal oftherectangle tm.Theintegral converges, butit issemi-convergent 89] THE ATTRACTIONS OFSURFACES AND LINES 163 Itwill beobserved that l= 21 ery Se sothat a=2. Triple Integral—It isdesired tocompute theattraction ofan infinite, homogeneous universe on agiven point. Let the axes bechosen sothat thegiven point isattheorigin. With the point (—£, 0,0)asacenter describe asphere ofradius +>& ‘The center ofthis sphere is arbitrary, and therefore theseriesofboundingspheres(r Aincreasing) isarbitrary. Ifthe LYorigin ismoved tothecenter !ofthesphereandifthedensity /\ istakenequaltounity, the Z| integral is I=SSS Saedyde, ve Ifaplane ispassed through ‘the attracted point perpen-diculartothe2-axisitwillsep-aratethesphereintotwoparts, Fro.40.Fig.49. Inthesegment ofthesphere totheright ofthis plane x~Fispositive; inthesegment totheleft itisnegative. The integral willbetaken over thetwosegments separately, theone totheright, forwhich x—£>0,being taken first. Since z-t__@ (’) “ph az\p)? where Pao +e +e, theintegration with respect tozgives (Bq. (35.4) nef{LL-Ales inwhich RaVFFR R= NWP OY FAO te Te, arethetwo values ofpwhere theelementary column forwhich and zareconstants pierces thesurface ofthesegment. 164 THE THEORY OF THE POTENTIAL Forthefrstintegral, ven pie asisfound bysetting y= Rocosy, =Rosiny. ‘The second integral alsoiseasily evaluated bytaking y=qeSEER ay, =/p-GEESEr= E> any, inwhich Rand yaretheindependent variables. The ratio between thedifferential products dydz anddRdy isthejacobian' ofyand zwith respect toRandy. That is ay&y _aRap . dude=oF3ara aRa, or yds=ial+—RRaRAY. Hence7 yds_eff, thee ffEnon), j,+e Roary =FplVP=Be+28)—@— and T={(2%—[[aves * Ro R 2 fos 2 Heo oft ‘The integral over theother segment isobtained from Tyby changing £into—&andthen reversing thesignoftheentire expression. That isor s hezeFEEOHei}. ‘Thesumofthese twointegrals is T=ht+h= —4t, whichisarbitrary, since£isarbitrary, anditisindependent ofr.*Goursar-Hzonic, “Mathematical Analysis,” p.260 89] THE ATTRACTIONS OFSURFACES AND LINES 165 ‘From the manner inwhich theintegration has been carried out, itisseen that le=ffftptecauaeah-h os 3ototrvPSeHB ae"OEP =ea Forlarge values ofrthevalue isapproximately Tu=2rr, which increases asrincreases. ‘The integral ofthe modulus, ‘therefore, isdivergent, and theintegral Jissemi-convergent. 90.The Potential ataPoint ofthe Surface—In Sec. 88, itwasproved that thepotential ofaplane surface hasadefinite value ateach point ofthesurface itself. Itisdesired toextend this result tocurved surfaces. Let0beanordinary point onthesurface S,that isapoint at which the surface hasadefinitetangentplane.Drawthenormal ‘and thetangent plane atO,and then cutthesurface $into two portions byacylinder whose axis isthe normal and whose radius is6. Let 8;bethe small curved disk which iscut out of 'Sandwhich contains thepoint 0,and letS;betheremainder of thesurface, The potential ofS;onthe point Oisperfectly definite, since Oisnotapoint ofS. There remains forconsidera- tion only thedisk S,,the potential ofwhich is i=f.Sd,ow where dwisanelement ofthesurface S,and ¢,which isacontinu- ousfunction ofitsposition, is ny —LetP;bethepoint ofthe — Zsurfaceatwhichduislocated CES(Fig.50),let¢beangle which all the normal atP;makes with ‘thenormal at0,Pbethepro- Tea.00. jection ofP;onthetangent plane, and rdrd@ betheprojection oftheelement dwsupon thetangent plane. Then dy=rseegdrdé,OP=r, OP:=p, p=rsecy, 166 THE THEORY OF THE POTENTIAL if¥isthe angle between pandr. Hence Vsean bewritten +parsee¢ vin[Pree gaan a) Since atthelimit, ¢=¥=0,themodulus ofthe ratio secy/secyhasamaximum valueMonS;,andsoalsohas «amaxi- mum 2,if6issufficiently small.HenceV,isaproper aintegral and itsvalue isless than 2xM Zé. ‘Equation (1)still holds ata conical point ofrevolution of thesurface providedtheplane, instead ofbeing tangent, is perpendicular tothe axis of revolution attheconical point, ¢istheangle which thenor- mal makes with the axis of revolution, and9istheangleho.51 which pmakes with thenormalplane(Fig.51).Suppose theaxis ofrevolution istaken asthez-axis, and thenormal plane istaken asthezy-plane. Suppose further that thegenerating curve issufficiently represented bytheequation ema, —-1<850, where Then seog =VIFale +5%, sey =VIF at, Since 6isnegative, thelimit oftheratio is feecwlimBeg aite<. Therefore, if«iscontinuous on§;,thevalue ofV;isperfectly welldefined, nomatter howsharp theconical point may be. Ifs=0,thecone isanordinary right circular cone. 91,The Potential isContinuous across the Surface.—Let Sbethegiven surface, andOanordinary point ofS.Draw thenormal andthetangent plane at0,andanylineLwhich pierces thesurface at0.LetO;beanypoint onLnear 0. It 91] THE ATTRACTIONS OFSURFACES AND LINES 167 will beshown that the value ofthe potential ofSat0,varies continuously asthepoint 0;passes through thepoint 0. ‘Asinthe preceding section, letthe surface Sbedivided into two parts byacylinder ofradius 6,theaxis ofthecylinder coin- ciding with thenormal at0. Let S;bethedisk eutout ofS bytheeylinder, and letS;bethe remainderofthesurface.‘The value ofthepotential at0,due tothesurface S.iscontinuous intheneighborhood ofO,sinee Odoes not lieinS;. ‘That isto say, if00; =1,ifUoand U;arethepotentials ofS,atOand 0, respectively and if«isany positive number given inad- 4 ‘vance, thelength/canbetaken A 0smallthat PROSlu.-ul<k ‘i [Ue0<36 i however small 6may be. LetFig.52represent S;,and Fro,82. letVoand V;bethevalues ofthepotential ofS,atthepoints 0 and 0,respectively. ‘Then Yenfede, Niefode, ls. 0 sp Ifr,6,g,and ¥have the same significance asinSec. 90, these integrals can bewritten 1P00y fesoe¢ vom[ieegeom, vm[ILE Sa, and Vo<2rM28. Inthetriangle Q00,, lettheangles atOand 0,bedenoted by and a, Then, from the law ofsines, e sina, pi sina Ifyisthe angle which Lmakes with the normal, the limiting value oftheangle aasthepoint Qapproaches thepoint 0lies between 3—+and5+7,andthelimiting valueofsinais not zero. Let the maximum value ofthe modulus onSibe a [sin tae 168 THE THEORY OF THE POTENTIAL ‘Thiswillexistif5isnottoolarge,howeversmallImaybe,and Vi<2eMN26. Consequently 5canbetaken sosmall that \Ye~Vi)<2eM(N+125<ie | From thisitfollowsthatifWandWarethevaluesofthepoten- tialatOandO;respectively, duetotheentire surface S [We —Wil <|Uo— Us+|Vo-Vil<e, which proves that thepotential iscontinuous across thesurface. ‘Theargument holds, also, fortheconical points considered inSec.90,provided thelineZdoesnotcoincide with axisofthe cone. 92,TheNormal Component oftheAttraction isDiscontinuous across theSurface.—If thepoint 0ofthesurface Sistaken as theorigin ofasystem ofrectangular coordinates with thetangen- tialplane asthef-plane. theequation ofthesurface canbe written f=dak +den? +--+, @ assuming that thesurface isanalytic. Or,if E=rcos8, 7=rsind, theequation is =r%(ax0 cos? @+aossin?8)+r(-- +), theimportant point being that reo isfinite, Since thepoint 0isnotapart ofthesurface S:,thenormal component oftheattraction ofS;iscontinuous atO,sothat it isnecessary toconsider only thesurface Si. Atthe point 0\(z, y,2)thecomponent oftheattraction is Zef$2gay, Is. Pe Itwillbeshownthat,if«%0atO,thelimitofthisexpressionfor2=0isnot zero, assuming that the point 0,moves along thelineL,Fig. 52,and that thelimit for2positive isdifferent from the limit forznegative. 92] THEATTRACTIONS OFSURFACES ANDLINES 169 ‘The expression forZcan bewritten asthe difference ofthe ‘two integrals z=[fetefiSod.Jeo ase Consider thefirst integral Z;,which canbewritten. amfiomsss pe .=f ("(2) eee. Ifthemaximum values ofthemoduli of{/r* and p*/p,* onS;are Aand R,then |Z,|<2AMZRB, ifthe notation ofSec. 90ispreserved. Asthis expression vanishes with 4,thevalue ofZ,iscontinuous across thesurface. ‘There remains theintegral haf=eds. s.r ‘Letoobethevalue of¢atthepoint O,and consider theintegral 2=[2(B)e-« a=fAe-adofee ods ‘The ratio ntl =2'+o-y+e ot E- ta +O foragiven value off,isamaximum oraminimum forallvaluesofgand9,if§=zand7=y,according as g@ o> G-aA< +1 For, ifthesame quantityisaddedtothenumeratoranddenomina- torofafraction the value ofthe fraction isincreased ifitsvalue islessthan unity anddecreased ifitisgreater than unity. Hence, wherever thepoint £,7,¢may beonSi,thevalue oftheratio 71°/px* liesbetween +1and 2*/(¢ —2)*. Ifthedirection cosines ofthelineLarea,8,and y,thevalues ofthecoordinates ofthepoint O;are a-l, y=, z=; also,g=rcos6,a=rsin6, $=r2(a2ocos?@+cozsin?@)+-++. 170 THE THEORY OF THE POTENTIAL If§=candy =y,thenr?=Px?,wherex?=a?+6?=1—7%, and eo Dy! _, C= 2? Dy—P(e 008" OF azosin® )+== which hasthelimiting value +1for 7=0.Hence, theratio 2/p converges uniformly to+1asthevalue ofzdiminishes. For2sufficiently small then, essentially, 2 Za=feFile—olde. IfMisthemaximum value ofthederivative of«with respect tor onS,,then itistrue that Je=o <Mr, and te Phatdy ZugfMEae=uefwfne,sir Jo Jo where rf=at+7?—Derc08(0—6)+Pt} @ =P=Der60s(8—8)+74, and6,istheangle between theaxis andtheprojection ofL ontheplane. Since itmakes nodifference intheresultanditsimplifies thenotation, @—[email protected] with respect torthen gives |(2c?cos?@—1)8—Ixcos@ ZsSMly[~{Oe"cos’0—Db—eons?_- “8nf{a=costA)—Diabcos0+8 008 6 FTN ecod VE=DdcosOFF+6—Ikcos9 ae insCor|ee hil‘Thetermsintheintegrand whichdependupon@arefiniteforallvalues of@and innumerical value are less than (22+16+1e x My=Octbeth 44, O=Ovi mea T= VE=PldFO+5—lk begT+. Therefore Za<2MMyyl —2eMyl log1, 92]‘THEATTRACTIONS OFSURFACES AND‘LINES: 171 which vanishes with1.Itfollows thenthattheintegral Zs jscontinuous across thesurface.‘There remains, finally, forconsideration theintegral hewenoemaf,afae, art Jo9Jos Onreplacing thevalueof2=£1VT=eandperforming theintegration withrespect to7itisfoundthat r— {*| dccos 6—1 ty=taVi=e ("| ee sev’ ila=©cos?6)FF—xlcos6+1+irerales ‘anditisthelimitofthisintegral for1=0thatisdesired. Forfnadvaluesofxand5,however, thisisaproper integral, andtherefore thelimitof‘theintegral istheintegral ofthelimitoftheintegrand; thatistheorderoftheprocesses canbechanged. Hence — (Lt «cob limZe=toVI-afToecos?6?rd=tavI=fToews =£27 thepositiveornegativesigntobetakenaccording astheapproach,peeposfovfoee ismadefromathenegative orfromthepositive side."thediseontinuity inthenormal component oftheattractionastheattracted pointpasses through thesurface isthedifferenceLetmven thesetwolimiting values, or4zo,whereovisthevalue ofthedensity atthepoint ofpassage. 3,TheTangential Components oftheAttraction AreCon-tinuous-—The z-component oftheattraction ofthesurface Ss,vinichiatheonlypartofthesurface thatneedbeconsidered, is x-fine -fg(0)ede, 1s,PY js,TY \Pu, ‘AsinSeo,92,theratior/ox?converges uniformly to+1asthepointO,approsches thepointOalongthelineL.Hence,forgiven4,thereexistsan¢,‘whichvanishes withJ,suchthat fFFode+of£>7edu, Js,0% Pac 172 THE THEORY OFTHE POTENTIAL provided thislastintegral isfinite, andtherefore limX=limfEOFodes, BO aT Js. ‘The integralifSSFoderepresentstheattractionofaplane diskupon thepoint 0,,thedensity upon theplane diskbeing the same atthepoint &,7asupon thesurface S;atthepoint &,9,£. ‘Thediskis,therefore, non-homogeneous. Itisassumed however,that thedensity iscontinuous andthat there exists apositive number Msuch that, onS; |o—oo<Mr. Consider first thedifferenee between theattractions ofthe non-homogeneous diskandthehomogeneous diskofdensity os (thedensity atthepoint 0),thetwodisks having thesame radius 6. Let gn hefimte~ade oth (tece=ffSS—ordras. Itwillbeshown thatthisintegral iscontinuous along theline Lintheneighborhood ofthepoint Since nts (Ea) +(y— yt e, itisevident that: gaHse Hence 2ry mi<fff.chlo=odrdrdjo Jor te bas<ufwfdr, a Jo Onsubstituting, Eq.(92.2), rit=P=2lercos8+53, 93] THE ATTRACTIONS OFSURFACES AND LINES 173 and then integrating with respect tor,itisfound that 2iX<uf{s+1kc08@log(1?—2lxbcos@+§*)— ° Uecos8—1) 5—kecos8 lk6log1+SG tan"!ae wos6198tMSail anIVses?* «cos6 tan!£008? _\ag, VI=©cos*al} ‘This isaproper integral, and remains soeven for 1=0.The limit oftheintegral forJ=Oistherefore thesame astheintegral ofthelimit oftheintegrand for 1=0.Therefore [Xi] <23, and X;iscontinuous. ‘There remains still for consideration the limit ofthe -com- ponent oftheattraction ofthehomogeneous disk onthepoint O asItends towards 2oro; that is X,=limeof.E240.tes. 7 InFig.53,letC,bethecircle ofradius 6andcenter atOwhich bounds the homogeneous disk. Let O;onthe line Lbethe 6 et Foo. 88 attracted point. Drop aperpendicular from O,totheplane of thedisk, intersecting thedisk atthepoint p.Letthepoint ¢ lieinthe line Opand the distance Ogbetwice the distance Op=x. Withgasacenter andaradius 6drawacircleC2. ‘The portion ofthecircle C,which iseutoutbythecircle Ceis symmetric with respect tothe point p. Therefore the z-com- ponent oftheattraction ofthis portion ofthecircle iszero, by symmetry. The area Aofthe remainder ofthedisk (shaded imthediagram) isequal tothediameter 28ofthecircle Ci multiplied bythedistance Og,or2lx. Hence thearea is A=4xél. 174 THB THEORY OF THE POTENTIAL With these preliminaries disposed of,itiseasily seen that Xe=limoof,£4, and " du 1X]simoofYrs en Ae70Kdd SING =0. ‘Hence, thelimit ofthez-component oftheattraction ofthe homogeneous diskupon thepoint Q;,asthepoint 0;approaches ‘thepoint O,iszero, andfortheoriginal surface Si,therefore, this component ofthe attraction iscontinuous. The same argument holds forthey-component which, also, iscontinuous. Itfollows, therefore, that thetangential component ofthe attraction oftheoriginal surface Supon the point 0:iscon- tinuous asthepoint O,passes through thesurface. ‘The results ofSec. 86show that atthepoint Oofthesurface the tangential component ofthe attraction does not have a definite meaning. 94. Discontinuities inthe Derivatives ofSurface Potentials.— ‘Suppose there isgiven asurface S,onwhich there isagiven distribution ofmatter ¢,forwhich thepotential isV(x, y,2). Suppose further that thestraight lineLpierces thesurface Sat thepoint 0atwhich thedensity iso,and that O;isapoint onLnear0.IfAandBarethetangentialcomponents ofthe attraction and Cisthenormal component, theX-, Y-,and Z-components oftheattraction are ave ri X=aA+«B+al, aveSm¥=pul+OB+00, ave Wo 72d+nB+0, where a,8,yarethedirection cosines ofthenormal toSatO, ‘and a1,8,71;ax,Bs,72arethedirection cosines forthe two tangents at0. ‘Asthepoint O;passes through thepoint Othecomponents A and Bvary inacontinuous manner, but thecomponent Chas a discontinuity of4roy, and therefore thederivatives ofthe poten- 94] THE ATTRACTIONS OFSURFACES AND LINES 175 tial also have discontinuities. The numerical values ofthese discontinuitiesare av av aveGgTroe Gy=Aree, “gy=Aroory, a) where, asalready stated, a,8,+arethedirection cosines ofthe normal atthepoint 0. IfthelineZmakesananglegwiththenormaland1isthedistance 00,, the discontinuity inthe directional derivative aV/alis evidently VvOF=teas008 Discontinuities intheSecond Derivatives ofaVolume Potential attheSurjace—Suppose Bisavolume filled with matter of density o(#, 1,$)and bounded byasurface S. The volume density atthesurface will bedenoted byo:(§, n,{). InSee. 61 itwas shown that the first derivatives ofthe potential Vcan beexpressed asthe sum oftwo potentials, one ofwhich isa volumepotential forwhichthedensityisa¢/a&andtheotheris asurface potential, thesurface density being o»multiplied by ‘adirection cosine. Thus, Eq. (61.3), aV_[acd_as, oz~Si°fPhd ® thedirection cosines ofthenormal being «,8,7. Ifitisassumed that thedensity ¢admits second derivatives, Eq. (2)can bedifferentiated again. ‘The first integral, being a volume potential, has first derivatives which are everywhere continuous. The second integral, however, isasurface potential. Ithasderivatives which arecontinuous everywhere except on ‘thesurface itself. For Eq. (2)thenumerical value ofthejump alongthenormalis4raze,andfordzpositiveis—4rac»(See.15).Hence thefinite jumps inthesecond derivatives ofV,from Eq. (2)and similar equations foraV/ay and aV/az, are av 2 Ov OV aeed—4raya?, ayaa~4rox8a, dea roaya, av av 1 Oe Seay787008, GeArend, Gay=—Aree, avav. av : dre=A, Fagg tear =Ano. 176 THE THEORY OF THE POTENTIAL ‘The discontinuity intheLaplacian is av av av Gattap tGes7em which isPoisson’s equation, although theproof here holds onlyforthediscontinuities atregular pointsofthesurface.Discontinuities intheLogarithmic Potentials ofAttracting Lines.—The results whichhavebeenobtained forattractingsurfaces andtheNewtonian potential arealsotrue forattracting lines and the logarithmic potential. The change involves merely areduction ofunity inthedimensions. The potential itself iscontinuous across theattracting line; thenormal derivative has anabrupt discontinuity of2re, if¢isthelinear density ofthelineatthepoint ofcrossing; and thetangential component iscontinuous. 95,Example—A Non-homogeneous Disk.—The potential ofauniform oblate spheroid, Eq.(39.2), atanexterior point is 2reate gym) Goa ve Se (1- ee VaniWaay)8Se roate(VEFKe4yr)28)1) +00“(Ee —e ¢ where«satisfies therelation ety atae +aae7h Ifthepolar axis ofthespheroid cistaken very small relative tothe equatorial radius and atthesame time thedensity ois increased insuch away astokeep theproduet c7=a»constant, thespheroid isvery nearly adisk with asurface density which is, proportional tothethickness ofthespheroid, namely - 4,o=doe1-5 (2) where pisthedistance from thecenter, and atthelimit, for¢ equal tozero, itisexactly so. If2°+y?isreplaced by72,the potential ofaplane disk with thesurface density (2)isthelimit ofEq.(1)forcc=o;andc=0,namely a2) rive_2 Vndeeea(125)snSst (28)3 2a Vani tarteya)® 95]|THEATTRACTIONS OFSURFACES ANDLINES 177 where «isdefined bytherelation r a weet oh () The derivatives ofVwith respect toristhecomponent ofthe attraction which isparallel tothe plane ofthedisk, and the derivative with respect tozisthe component which isnormal to it.Since differentiation with respect to«isnotnecessary (Sec. 39), itisreadily found that av Vi len: aarwo asvary) oteou(Hsixrt 2-1), °a oN Vere Ve Now imagine that theat- L tracted point approaches the disk along aline Lwhich pierces thedisk atthepoint zo, ‘yoatadistance pfrom thecen- ter, Fig. 54. For simplicity, letthez-axis alsopass through this point. Then the para- metric equations ofLare Fro. BA. e=ptl, y=, 2=hy; and rt=p+ pa +(a? +6). ‘Thederivative ofVwithrespect toJis av_aVor|oVae a” Orat azal aVpa+Ua?+8),av arr rn a Asltends towards zero, rtends towards p,and thecoefficient of @V/ar tends towards a. Hence enOV " av ovtyr=[oe+0} Asfor«,itisthepositive root ofEq.(4). Ifp<aandztends towards zero, «also tends towards zero insuch away that im 2= ae,Injen ttB ® 178 THE THEORY OFTHE POTENTIAL butifp>@andztends towards zero, «tends towards p*—a*. Imorder thatLmay pierce thedisk, itisnecessary thatpshould bbelessthan a,andtherefore thelimit of2/-/« isgiven byEq. ©. Since thelimit of«iszero, itisseen from Eq,(5)that tim2=e007;reo or ‘a andfrom Eq. (6),that im2% == Ea 7 limGz=F4rooy1 —Ge @ But,sincethesurfacedensity«isgivenbytheformula omtout-8 Eq. (7)can bewriten alim5,=F2r0, and therefore timOY=A867aeFDror, feo ol~ 7%q% + 277% ‘The discontinuity is4rey, which vanishes attheedge ofthe disk, since thedensity ¢vanishes there. Forasimilar reason iftheattracted point approaches the disk from theoutside, sayalong the2-axis, theattraction remains finite, The explicit formula is . vE=@ 154 4 X=roelVEEgema) since forpoints onthez-axis, x=2*~ a. The limit ofthis expression, forpositive values ofz,as2tends towardsais—x*a», which isremarkable inthat itisindependent oftheradius ofthe disk. 96. Discontinuities in the Second Derivatives of Surface Potentials.—It willbefound inChap. VIthat thediscontinuities inthefirst derivatives ofthepotentials oftwo-layer surfaces depend upon thediscontinuities ofthesecond derivatives ofthe potentials ofsimplelayers.Itisnecessary, therefore, toexaminethese discontinuities, The analysis which issetforth here follows rather closely theargument given byPoincaré inthe sixth chapter ofhis“Théorie duPotentiel Newtonien.” 96] THEATTRACTIONS OFSURFACES ANDLINES 179 ThePotential ofaPlane Surface.—Suppose thegiven surface 'Sisplane andthatitisbounded byaclosed contour Cc.The potential atanypoint P(z,y,2)isthen Vie,2)=figesan, whereoisthesurface density and p=VE=P FU TED Itisassumed that oand itsfirst derivatives arecontinuous on 'S,andthatthesecond derivatives arefinite. This potential is aneven function ofthe argument z;that is V(x, +2) =V(x, ¥,—2)- ‘The first derivative ofVwith respect tozisanodd function of #;that is aV(z,y, +2)__aV(z, y,—2). oz oz , and the second derivative isagain aneven function. This ‘means that ifP(+z) and P(—2) tend towards coincidence at somepointofS,thepotential Vanditssecondderivative a*V/az*,having thesame values always atthetwo points P(+2) and P(—2), tend toward thesame limit, ifalimit exists, and are continuous across S, The first derivative 9V/dz, however, has ‘opposite signs, although numerically equal, atthese two points. ‘The limits atthe surface are, therefore, different, ingeneral, and thefirst derivative hasadiscontinuity incrossing thesurface. According toSec. 92,this discontinuity isequal to4c, where oisthedensity atthepoint ofcrossing. ‘The tangential derivatives, however, are continuous. For example ove afl‘az=Srl) “aden afrSaad) ote which gives, onintegrating byparts, wW__fe adidnozJeatJ.3E Thefirstofthese integrals isthepotential ofamass distribution oflinear density oonthecontour C.Itisevidently continuous 180 THETHEORY OFTHEPOTENTIAL everywhere except, perhaps, onthecontour itself. ‘The second integral isthepotential ofamass distribution onSforwhich the surface density isd0/dt, andthis, too, iscontinuous across the surface, bySec.91. Since thesame argument holds for2V//éy, itisevident that thetangential derivative inany direction is continuous everywhere except, perhaps, onthecontour. Sincetheintegral ffisthepotential ofanattracting line, allofitsderivatives arecontinuous everywhere except onthe line Citself. There isnoneed toexamine itfurther for dis- continuities across S.But thederivative with respect to2of thesurface potential f2ded soe p hasadiscontinuity equal to—4rd0/a%, while itsderivatives with respect tozandyarecontinuous across S.Hence av as nazO98thediscontinuity —ta92across8, likewise, a av " ao aSpazB98thediscontinuity —tn9!acrossSj while the derivatives [a A aae aeay arecontinuous across S.Since 4*V/dz? and°V/ay? arecon- tinuous across S,and avV=0 atallpoints notonS,itisevident that av eV avweave (G+) also iscontinuous across S. Itisworthy ofnote thatthese discontinuities donotvanish evenwhen thedensity atthepoint ofcrossing iszerounless the derivatives ofthedensity alsoarezero. Certain Properties ofthePotential ofaGeneral Surface— Returningtothenotation ofSee.91,letObeanordinary point ofthegeneralsurfaceS.Letthetangent planeatObetaken asthefr-planesothatthenormalat0isthef-axis.LetL 96] THE ATTRACTIONS OFSURFACES AND LINES 181 beany straight line which picrees Satthepoint 0,and let0, ofwhich thecoordinates arez,y,2,beany point onLnear 0. Let the surface Sbedivided intotwoparts byacylinder of oe radius 5,theaxisofthecylin- Adercoinciding withthef-axis. y,aLetS:,Fig.52,bethedisk SYcutoutofSbythecylinder, Xiand let8;bethe remainder ofthesurface. Letthesur- areface element dwbelocated at fa,08 the point Q,and letQsbetheprojection ofQonthefy-plane, sothatQQis¢.Finally let O0.=1, =r, OQ=r, O=r AQan, sothat at=(e— d+ ym? +@—H% nia @— s+ y—at +2, Pa Btn =Bot Iftheprojection ofthesurface element dwupon the&-plane isdédy and if«,8,yarethedirection cosines ofthenormal at. da,the potential ofSatthepoint 0;is ={Za0=f2a 2) v=fidefoajaen ® 2being thearea into which the surface $projects upon the fr-plane. Letthisplane area becovered with matter ofdensity@/. The potential ofthis plane surface atthepoint Q;isthen =|2azar 3) v=foeatin ® ‘The difference between these two potentials will bedenoted by theletter W,sothat 11\c W-=U-Ve=f.(ea-3)ates, w Let DW beany derivative ofW, and therefore ofthe form DW=fee1)adn, 182 THE THEORY OPTHE POTENTIAL where ¢issome function of£and7.Suppose that over Si k wish whereIissomefixed,positivenumber. ItwillbeshownthatDWtends toward alimit asthepoint O,moves along theline Ltoward thepoint 0. LetWbeseparated intotwoparts, oneofwhich corresponds to theintegral overS;andtheother theintegral overS:. ‘Thus W=mi+Ms, and DW =DW. +DW. since, byhypothesis, Dwi<fbildn,aT itisfound readily byintegrating, that |DW)| <2eks. Itisevident also, since |p|<k/r, that DW hasadefinite value atthepoint 0,which, separated into twoparts, asbefore, can bewritten DW® =DW. +DW. ‘Onforming thedifference, there isobtained DW ~DW® =DW, -DW,® +DW; ~DW, and therefore [DW ~DW®| <[DW, —DW,®| +[DW —DW. Itwill beshown that lim|DW —DW®) =0; ‘that is,given apositive number ¢,assmall asdesired, the radius 6and the distance Jcan be taken sosmall that |DW -DW®| <«. Obviously #canbechosen sosmall that ‘DW<ieandalso, [DW,®| <je sothat: DW,~DW.%|<2 96) THEATTRACTIONS OFSURFACES ANDLINES 183 Furthermore, since DW: iscontinuous inthe neighborhood ofthepoint O,thedistance Jcanbechosen sosmall that [DW-DW, <de, Forsuch positions ofthepoint 0,itisevident that |DW —DW®| <« Hence thelimiting value ofDW isDW, and thefunction DW iscontinuous across the surface. Ifthefunction ¢satisfies theinequality A kck, for6sufficiently small, thefunction ¢issaidtobeoftheorder n.Thus thefunctions 1/p: and 1/r; areeach ofthe order 1. The ratio p:/r: isoftheorder zero; likewise z-f ya, e-ti a e areoftheorder zero. Itisseen from Eq. (92.1) that ¢isofthe order —2; and p;—r;also isofthe order —2, since lanl <f. TheFirst Derivatives ofWareContinuous across S.—The first derivatives ofWare ow L 1\oWm—feeo(3,-A)saten, ow L 1\o,BF=~fvn(Zs7h)aem ow e-5_ z\eSa Ale For the derivative with respect tox, g, 11v=fe-o(Z- 3) If4issmall, yisvery near unity; and ifthe density does not vanish, o/yisoftheorder zero. The remaining factors ofycan bewritten @- (1 1 1\=O +At d)n-oo. 184 THE THEORY OFTHE POTENTIAL Thefirstfactor ofthisexpression isoftheorder zero; thesecond factor isoftheorder +3; and thethird factor isoftheorder —2. Hence¢isoftheorder+1,anddW/dziscontinuous acrossS.The same analysis, without essential change, applies also to theother two derivatives. Hence allofthe first derivatives of Warecontinuous acrossS. , The Second Derivatives ofWataPoint Where the Density Vanishes.—The sixsecond derivatives ofWare aw-(2-4) -(B-A) “ast=fi[ae®(Bs#)G73)yt aw-p(L-4 -G-Ak aJlo- »-4)-G- A)kom aw B@—5)?_Bet4-AE f(s" -8-G-A) em aw 1 1\ |eEE=floc-00-0(4-3)[recon ew 2-5 2\\cfea”J[so9-H) ow we — (2a -4)a[ls~(58-8)foen Suppose nowthatovanishes atthepoint O,butthat itsfirst derivatives donotvanish. Then /7isoftheorder —1. Con- sider anyoneofthesixsecond derivatives, say*W/dz*. The function ¢canbewritten =of8@=OF Q111) .HeeOOOGatatpanttpstra =(n-(4+ 2,41), which, itisreadily verified, isoftheorder +1. Therefore #°W/é2*iscontinuous across §atthepoint O.Itisthesame. alsofortheother fivederivatives, ‘Now thepotential of$at0,is,Eq.(4) U=W+Y, where V,Eq.(3),isthepotential ofaplaneareatangent toSatOs.Since allofthesecond derivatives ofWarecontinuous across §atO,thediscontinuities inthesecond derivatives 96] THE ATTRACTIONS OFSURFACES AND LINES 185 ofUarethesame asthediscontinuities inthesecond derivatives ofV. ByEq. (1)these discontinuities are a(e) 4 au -18(2) mn20. ony, ayes? the remaining derivatives ee aaz?” oy ae” Oxdy being continuous. Since yisequal to+1atOandhas@maxi- mum there, thediscontinuities can also bewritten a aU4x:‘aE in aaaz! a aU ©) a The Density atthePoint OisNot Zero—If the &and y-axes are chosen soastocoincide with the tangents ofthe lines of curvature atthe point O,the equation ofthe surface Sintheneighborhood ofthepoint0is $= auof* +anon? +terms ofhigher degree. (6) Let ip, Fegem Faw oh oree oer Itisevident then that atthepoint O pega s:=0. Itwill besufficient tostudy the potential ofthe disk ofradius Sabout thepoint O,forthepotential oftheremainder ofthesur- face andallofitsderivatives arecontinuous at0. The potential Uthencanbewritten v~[im.f«dtdn, Jeoe Jena ms 21being aplane circular area with 0asitscenter. Itisevident that au_ffa/l)¢,ae>fala) 5 186 THETHEORY OFTHEPOTENTIAL where at=(@—e+ —)t@—De No O/1\_ _z-€& az\p:) — oe and,bearinginmindthatfisafunctionof¢and7throughtherelation Eq. (6), O(1) 2y2-fyentan) aptae and therefore (1) (1) _ahBeles)~~aeloi)~dep)?” or, 2/1). 2/1) _4/1dala)~dele)~dele)?* Therefore WW. _£81544 —(2/1) 9,4azfi5) fiwhee) o) aIitdywhere a/1\o a/l\ « teamfale)iiten Jem~3B)pifeen Thefirstintegral J;canbeintegrated byparts, giving =-(2.24 (2/2), dean, 8 afiatSa} a ® or,ifdsisanelement oftheedgeofthediskand8,isthecosineoftheangle between deandther-axis, ee 3(2) didn, 9) wefort La) S ® ‘Thus J;isexpressed asthesumoftwopotentials, thefirst of which isthepotential ofanattracting line,coinciding with the edgeofthedisk, onwhich thelinear density is—o8;/y. This Potential andallofitsderivatives arecontinuous atthepoint O, since 0isnotonthelineC.Thesecond, f.2(2\de,a) 96] THE ATTRACTIONS OFSURFACES AND LINES 187 isthepotential ofa distribution ofmatter ofdensity a(eBe distributed over thesurface S;. ‘This potential, and itstangen- tial derivatives also, are continuous across S. Therefore J: and itstangential derivatives arecontinuous across Sand, from Eq.(7),thediscontinuities in4U/dzanditstangenital derivatives PU agg OY Oat ‘azay are the same asthe discontinuities inJ;and itstangential derivatives. Ifafunction F(z, y,2)isdefined bytherelation. Pe-fPigu, 20) 9 itisseen that ee ae*"92’Ge”zde=Gyyaa‘ThefunctionFisthepotentialofasurfaceonwhichthedensityisop. This density vanishes atthepoint O,since p;vanishes at O. Hence, bythe preceeding case, J,=4F/dz, iscontinuous across S,while ads_ °F sccontinuit a(on.),Gz7indehasthediscontinuity 12(y) and ad,_ oF sccontinu (op:a=PFhasthediscontinuity an2(eB!), Since p;iszero and yisequal to+1 with amaximum at0, these expressions eanbesimplified, sothat a¢)=4nofPt= seg(B)=seo=teers3(ap: opr =s5(B) =Anos,=dros,=0. Since Jyand Jsarecontinuous across S,itfollows that aU//ax iscontinuous across S,aresult, ofcourse, already known. Sincethediscontinuities of@Yand2arethesameasthedis- ox? éxdy 188 THETHEORY OFTHEPOTENTIAL continuities of2?and‘J?itfollowsthatthediscontinuity is 8°U/az* isArops, andthat°U/azdy iscontinuous. ;Byanargument entirely similar (or,bysymmetry ifone prefers) itisfound thatthediscontinuity in*U/ay? is4rea Since theLaplacian iszeroeverywhere notonS, au eu euet=—\aat +ay) andthediscontinuities in9°U/a2* aretherefore —4ro(p2 +2) Inorder tofind thediscontinuities in PU ggg 2UBade yee? itisfound from theequation au Bahth that aU _ads, as an0a~ Getie Since ads_OFae” a andsinceinEq.(10)thedensity vanishes atthepoint 0,it follows that34/42 iscontinuous across S.Itisseenfrom Ea. (8)thatJjisthesumoftwopotentials, thefirstisalinepotentialwhich, together withallofitsderivatives iscontinuous across: 'Satthepoint0;thesecond isasurface potential forwhich the density is Yaeky Hence thenormal derivative: (derivative with respect to2)of Jyhas the discontinuity —4ry2 (2)=—452%Ary,2(2)=—408Zatthepoint0, andthisistherefore thediscontinuity inx. ‘Asimilarargument showsthatthediscontinuity inayis =4r 7 96] THE ATTRACTIONS OFSURFACES AND LINES 189 LetRiand Rybetheprincipal radii ofcurvature at0. Then, ‘since the ¢and »axes coincide with the tangents tothe lines of curvature of$atO, mp, andwaa Collected together and expressed interms oftheprincipal radii ofcurvature, thedensity and itsderivatives, thediscontinuitiesin the six second derivatives a a ed Ga Gye” ay’ yd? arerespectively fro dre 11 ae ae,eS ~tee(Fe+Ry0,ang —42. ay 97.Singular Points oftheSurface.—In thediscussion ofthe attraction ofsurfaces intheneighborhood ofpoints ofthesurface itwas assumed that thepoint Ounder con- sideration wasaregular point ofthesurface Ae inthe sense that the surface had adefinite tangent plane atO,that over acircle of £ radius 4about 0asacenter the coordinate {ofthe surface was continuous and less + than Br?, where Bisafixed number, and that thedensity was continuous with lo]< a0-+Mr, where Malso isafixed number. That these restrictions were necessary can beseen byconsidering theattraction ofa ‘cone upon itsapex and theattraction ofa ‘homogeneous rectangle upon apoint ofits Fra,85. edge. ‘Attraction ofaCone upon ItsAper—In Fig. 55,letAbethe apexofaconeandBCbeitsbase,whichwillbeassumed tolieina plane, Pass other planes through thecone parallel tothebase which divide theperpendicular from theapex tothebase inthe ratios 1/2, 1/4,1/8, 1/16,-- +. There areinfinitely many of these planes. Thezonesoftheconebetweentheconsecutive planesareinperspective with respect tothepoint A. ByTheorem II, See. 10,these zones attract thepoint ofperspectivity equally. ‘The attraction ofeach isfinite (not zero) and since there are 190 THETHEORY OFTHEPOTENTIAL infinitely many ofthem theattraction ofthesurface ofthecone ‘upon itsapex isinfinite. Attraction ofaPlane upon anEdge.—In Fig. 56letPbea point onanedge ofthehomogeneous rectangle R. Draw 1 ‘ P seriesofsemicircles whicharery inperspective withrespectto WIthepoint P,theradii ofwhich areintheratios 1,1/2, 1/4 1/8,-++, Theareas betweenconsecutive semicircles are in perspective withrespect tothe pointP,andattract thepoint Fr.58. Pequally. Since theattrac- tionofeach isfinite andthere areinfinitely many ofthem, the total attraction oftherectangle onthepoint Pisinfinite. ATTRACTION OF LINES 98.AStraight Rod.—The attraction and potential ofa straight rod, considered asaline with the line density o,was treated briefly inSeo. 31,butitwillbeofp interest toexamine thesubject moreclosely. 4LetAB, Fig. 57,beastraight lineoflength 21anddensity owithitscenter atO,andlet P P(e, y,2)beanattracted point. Letdm=dn| L odtbeanelement ofmassofthelineata¢| Hdistance ffrom O,pthelength oftheline 9 fa—* joining dmtoP,andr=~/2*+y Then thepotential Vis ' Hg veac(——*__.JVF+ =a IfpristhevalueofpatthepointBand:the “value atthepoint A,sothat Fie.57. at=P+etD, ota t+@-D', a) itisfound bydirect integration that =olog@Zt! Veele at From Eqs. (1)itisfound that Ale=pst—pst, (2) 98]|THEATTRACTIONS OFSURFACES ANDLINES 191 and therefore pt— ottah) AP ott osta sothat Alps+px?+41°—ps? Vim0108fin,—ptpatAP~oTogOttor+Dlox—or+2D=2108GF61—Bos=0+BD) ®=ologtet 2 soleaa From this expression forV,itisseen that thelevel surfaces, V=const., aredefined bytherelation pit er=2a, where aisaconstant. ‘The level surfaces are, therefore, prolate spheroids with themajor axis 22and foci atthe ends ofthe rod. If¢isthe eccentricity ofameridian section ofthe level surface, itis evident that L= ae. ‘Therefore, theexpression forthepotential, Eq. (8),becomes V=ologit? =2tanh-*e, For points which areremote from the rod aislarge and there- fore ¢issmall. The equipotential surfaces are very nearly spheres. For points ontheroditeelf pit ps=2b=2a, sothat¢isequaltounityandVisinfinite. Atpointsintheneighborhood oftheline¢islessthanunityandVisverylarge. ‘Atlarge distances ¢and Vboth tend towards zero. 99.The Components ofAttraction—Let Rbethe com- ponent ofattraction inaplane perpendicular tothe rod and Zthecomponent parallel tothe rod. Then avde aVde Rmar 7=8oF where ga 2, p++,atm pte +ede 192 THE THEORY OF THE POTENTIAL Now av|Be geal te,ae" T= dan Gta Be! and Bor oer,arp or pa amet! ap_enta oz Pe sothat oe (rircee(ett23), R=area(t +2)2-aeaatpe From Eq.(98.2) itisfound that 1Z=lot—02)=Flv od, oto =2a, b=ae. Hence / pate, p=a—e, and =o er 2QoezReeee"ator ® Or,ifriseliminated bymeans oftheequation oftheellipse r 2 @d—e) +a=1, there results oe VP et 2oerz R=Fe ven =—et, ise Poe? Fp @ which contain onlytwovariables, ¢and2. Iftheattracted pointmovesalongalevelsurface forwhich‘thesemi-axes areaand>,RandZhavemaximum numericalvalues, namely 2ce 2Qee a a ,aise ™™aa and, since b=avi-€, itisseen that Zonas_Rane5 99) THEATTRACTIONS OFSURFACES ANDLINES 193 ‘Suppose thevalue of iskept fixed with |e]<J, and etends toward unity. Inthisevent, theattracted point moves towards theattracting rodalong aperpendicular line. The limits ofthe components ofattraction are limZ=~pe limB=Fe astheattracted point crosses theattracting linetheZ-component varies continuously, but the R-component has aninfinite dis- continuity. Itwill beremembered that thepotential atthe attracted point alsotends towards infinity asthepoint approaches theattracting line. 100, Attraction inthe Line IsNot Well Defined.—Consider the attraction ofahomogeneous line upon any oneofitsown points. LetL(Fig. 58)bethegiven line and Oanyoneofits 2a £05 iL Grok G a Fro, 58. points. Let2a,and2azbethedistances ofthepoint Ofrom the ends oftheline, andletC:andC;bethemiddle points sothat C0 =a:andC,0 =a. Letagapbecutintheline Labout the point0andlettheendsofthelinealsobecutoffinsuchawaythat thepoints C,and C;stillremain thecenter points oftheir respective portions, asindicated inthelineI, If2hand2ls arethelengths ofthese portions, theattraction oftheline L ‘uponthepoint0willbeunderstood tomeanthelimitoftheattrac- tion ofL;upon thepoint Oforlk=a;andly=as. ‘Using thesecond formula ofEqs. (99.1) 2oez 2=amea oF fortheattraction ofeach portion oftheline, itisseen that, since z=aandez =l, h 1 a=tebaarte) whereAistheresultant attraction.Now let\beanyrealquantity, positive ornegative, andletIz berelated to1,insuch away that b h a are + 194 THETHEORYOFTHEPOTENTIAL As|;tends towards a,soalso does I,tend towards as,and the limit ofL,isL. But the limit ofAis inA=2h, which isanything whatever, since disentirely arbitrary. It follows that the“attraction ofalineupon oneofitsown points” does nothave adefinite meaning. 101. Asymptotic Expression for the Potential—For points lying inthezy-plane, which bisects therod perpendicularly, theexpression forthepotential is de 14+VPFR Var [to apogtVEFR Sve pr 8 Since livitR_ fyir_yr .SMt=etary to} and a(r\_1(r\* u(r)_3(r\*.. vos[1-+3(7)-7)+|G)BG)+ itfollows that forvalues ofrless than I 2 a(r\*_3(r\? ¥=totoe2+2o(5)~()+}@ andforvery small values ofrtheexpression +t V=20logbs @) 1,»«p_18.g00dapproximation.Suppose now that theattracted point Pliesnear therod, notatthemiddle point butatadistance 2from themiddle point. Lettherodbedivided into three parts L,,Ls,andLyoflengths 2h,2l2,and 2ls L withthecenter ofL;atthedistance zfromthecenteroftherod, Fig. 59. Then l=%*h+ht+2=%+h—-«# ‘The values ofthepotentials ofLyandLsatthepoint 2asgivenbyEq,(98.3)are re. 50. 1 Vimelog'5#, Vs=ologt¥, Ol] THE ATTRACTIONS OFSURFACES AND LINES 195 and therefore W+Va=artogVE=TES), ‘Atthepoint Pthesum ofthese potentials isthevalue atzplusfpowerseriesinrwhichvanisheswithr,sincethepoint2isnot apoint ofL;nor ofLs. ByEq. (8)the potential ofLs atPis 2s Vi=2elog™ +PS, where PS. istobeunderstood asmeaning apower series inr which vanishes with r.Hence the potential oftheentire rod Lat thepoint Pis Wtht+he= V=2elog*VE=STD|ps(4) 102. The Potential ofaUniform Hoop.—In Fig. 60, let Hbeauniform hoop which willberegarded asacircumference ’ p fl 4" meee H Cn$ — Fre. 00 ofacircle ofradius awith constant linear density 7.LetP beany point inspace notinH. From Pdrop theperpendicular PQ=<totheplane ofthehoop. Draw thediameter ofthe circle BOA which, extended, passes through Q. Letmbeany point onthecircle, and draw Pm=p, PA=p, PB=pr. Evidently pyand p;arethe minimum and maximum values ofpasthepoint mruns around thecircle. Iftheangle mOA isrepresented by20,the areclement is ds=2adw, and theexpression forthepotential is V=20f“de @ op 196 THE THEORY OPTHE POTENTIAL Ifthelength 0@isrepresented byr,then pia (r—a)+a,mQ?=r+a?—2arcos20, pia(tba)tat, ptart-fa? tet—arcosQu, The expression forptcanalso bewritten p= (r?+a?+24)(cos* w+sin*w)—2ar(cos* w—sin*«) =[lr—a)?+24]costo +[lr+a)?+24]sin? =x?costo +pa?sin? w. IiM=2rag isthemass ofthehoop, theexpression forthe potential, Eq, (1),becomes 2M (i deVie ee ee 2) aJVoitcos?+patsin? a ——Se Fro, 61. ‘Thisexpression showsthatVissymmetric inp,andpz,for ifwisreplaced by2/2—yitbecomes aM73 ay veaf‘Voitsin?¥+px?cos?y © and therefore Vor, a2)=V(p2, 1) Along theaxisofthehoop p:=ps,andifp,istheir common value, itisseenatoncethatthevalue ofthepotential along this axis V.is ven Pe Thefunction V(r, os),Eq.(2),ishomogeneous ofdegree 1inpandps,Therefore, :Vishomogeneous ofdegree 102] THE ATTRACTIONS OFSURFACES AND LINES 197 zero anddepends only upon theratio p:/o:. Inaplane which passes through theaxisofthehoop, thecurve bo eo% o where ¢isconstant, isacircle, forEq. (4)istheequation ofa circle inbipolar coordinates; and this circle, Fig. 61,divides thelineBCAD harmonically, since byEq. (4) AC _AD _ BC~BD“ IfKisthecomplete elliptic integral ofthefirst kind forthe modulus =1c? <1, itisseen from Eq. (8)that slong this circle veexe‘aye . : (5) M| Ve (128)ne4(12325V 103, Evaluation ofthe Potential According toGauss.— Equation (102.5) shows that ifthevalue ofthepotential were known atthepoint C,Fig. 61,itwould also beknown atall points along thecircle which passes through C.Itis,therefore, sufficient toknow the value ofthe potential atallpoints inthe plane ofthehoop which lieinside of thehoopitself,andGaussdevised a 1Smnvery ingenious method bywhich this ~canbefound. //\ Letthecircle inFig.62represent the hoopandBOCA represent thesame * Ja diameter asinFigs. 60and61. Let theareelement atthepoint mofthe hoop bedsandpthelength oftheline joining Ctothepoint m.Itisevi- red dent from thediagram that inthe oe infinitesimal right triangle ofwhich dsisthehypothenuse dseos¢ =pd6,andtherefore ado, > coe Tet thedistance OC=>. Then from thetriangle OmC itis seen that sing _sin@ rr 198 THE THEORY OFTHE POTENTIAL Hencey,fedaod e loVat—b?sin?6 eft 6.JoVa?cos?0+(a?—bi)sin?6 =M-ja, Va —8). ButitisalreadyknownfromBq.(102.3)that,whenthepoint PofFig.61coincides with thepoint C, Ve=M-f(a+b,a —b). «@) Hence Sa+b,a—2) =f, Va). ©) Or,since aisthearithmetic mean ofa+b and a—b, and Ve?—iistheirgeometric mean, foranytwonumbers mand thefunction fhastheproperty that fm,n)=(mee,vin). Now let mnime), m=Vi, m=3m+m), m=Vt ‘Then , . Fim,n)=fom, m)=fy ms)= + ‘Thearithmetic meanofanytwopositive numbers isgreater ‘thantheirgeometric mean. Since(m—n)*>0,itfollows that m?+2Imn+n?>4mn, and therefore y gm+n)>Vinn. Suppose, fordefiniteness, m>n,‘Thenthearithmetic mean is, lessthanmandthegeometric meanisgreater thann.Hence m>m>m--+, ncem<m---,m>n, m>m, ™m>Mmsss, 103] THE ATTRACTIONS OPSURFACES AND LINES 199 Consider the difference mai nar. Since nisi >my itis evident that igs —Rags <Mags —My 1<5lm+m) —me 1 <g(m —mi), and therefore m=rs<lm=n). ‘Thus thesequence ofnumbers m,ms,ma,ms-+-converges to& limit, and the sequence n,m, ns,ms, -»+also converges toa limit, and these two limits are the same. Gauss, towhom this analysis isdue, called thiscommon limit thearithmetic-geometric mean ofthetwo quantities mand n. The sequence converges rapidly 2sseen from theexample m=1,n=1/4) forwhich m,=625 ma 5 my=5625 1=.55901700my=56075850 nis=.56075580 mg=56075715 mq=.56075715. Returning now toEq. (4),letgbethearithmetic-geometrie mean ofthenumbers a+banda—b. Then Vo=Myla +b, ab) =My, 9) Mt 6) ¢ ‘and thevalue ofthepotential atany point along thecircle which passes through Cis vy-Me—»), G) om Ifthevalues ofp;and ¢(Eq. (102.4)), am and naeenet man, aresubstituted inEq. (102.5) and the result isthen compared with Eq,(6),itisseen that ifvisthearithmetie-geometrie mean of LandVI—F, theng=(0+b)y,and at+b_i IVeg(1:3)p04(1388Veego. eet) bea) ea(=) 200 THETHEORYOFTHEPOTENTIAL For thenumerical example given above a+b=1, e-bacap ek, ‘thisseries converges veryslowly, andatleast three hundred terms would berequired togivethesame approximation asthat given bym,andn,. 104,Asymptotic Expression forthePotential.—Ii thepoint Capproaches thecircumference, a—btends towards zeroandso alsodoesgwhile thevalue ofthepotential tends towards infinity. Itwillbeofinterest toexamine thenature ofthis singularity. Letthecircumference beseparated intotwoparts, oneof length 2!withthepoint Aasacenter, andtheremainder ofthe cireumference oflength 2ra—2. _7\|Thepotentialofthemainpartof a thehoopisananalytic function ofos "theposition ofthepoint C.Ifthe Ei point C’moves along theradius to oc t+ thepointA,itsdistance fromASSA, beingdenoted bytheletterr,the we potential isexpansible asapower Sseriesinrreducingtoacertain Sy,value atthepoint Awhich willbeFro,08 denotedbythesymbolVa.Ifa istheangle which theareoflengthJsubtendsstthecenter,itisreadilyfoundfromEq.(102.2)that moeHe=~20logtan% anah @ = a -s-(2).+20logE+Ps6) LetV«bethepotential oftheareoflength 27atthe point C. Then (Fig. 63) wea2|ames a JoVa?—2a(a—r)cos8+(a—r)*uado ) ae ,Vie=r)sintSotr8 104] THE ATTRACTIONS OFSURFACES AND LINES 201 IfLisvery small incomparison with theradius a,thepotential oftheareoflength 21differs very little from thepotential ofa straight lineoflength 2which istangent atA. LetVibethe potential ofthestraight lineatthepoint C.Then, using the same coordinates, tant asec!ad@ Vi=20fenlovesec’“aia @)Hse", asoc? 00 =20 _2500 db Onchanging the variable ofintegration bythesubstitution, ad=, ‘these two integrals, Eqs. (2)and (3), take theform a a ,ioe—AsinXt errssectan Vi=20 —_ C3),ay?tant+r The potential ofthe straight line, however, isindependent ofthe radius ofthe circle, a,and bytaking a= +2 these integrals reduce totheform ay Y=Vi=2%(O,SvME which istheform given inBq.(101.1) forthestraight line. That these two integrals have the same limiting form was tobe expected, since thelimit ofthe areisthe straight line. Expanded inpowers of1/a ing rearg eyMM 4a(a1)sin?X= ort) —MM 202 THE THEORY OF THE POTENTIAL isaseries which isconvergent forallpositive values of«andall values of}. Hence 1 ee are oo vee te {ioeAsin Rr VEEN Geayy tee, ) converges aslong as lto(e—1)sine&—a4i <| Poe If|\)<2ra, the numerator ofthis fraction isnegative, and theseries isconvergent if M=dole~1)sitRc which icertainly satisfied ifr <a.Equation (5)can, therefore, beintegrated term byterm, and, using theresults ofSee. 101, Vim2olog+--+, © theterms notwritten vanishing with rand 1/a, This analysis has assumed that rand 7were fixed while « increases; but since, intheunits oflength, V.ishomogeneous ofdegreezero,theresultisthesameasthough randIdiminished, with r<I, andaiskept fixed. Finally, onadding Eqs. (1)and(6),itisfound that veNt Va=Qelog4?+20log2+eer @ 8a=2olog 4..-, ‘theterms notwritten vanishing with rand|.From thesymme- tryofthecircumference itisevident thatthepotential isentirely independent ofI,andtherefore theterms inIvanish identically. ‘The remaining terms vanish with ralone. IfEq.(7)iscompared with Eq.(102.5) andr/aisexpressed interms of,itisfound without difficulty thatasymptotically, 1 ‘1\*, 1-3\3, 1-3 -5\3,Mla (Mp4 (L8\ en pee714Q)¥+ (aye era) Pho78 7B . 41.465871 logis ———;8Vi- Fk 104] THE ATTRACTIONS OFSURFACES AND LINES 208 and theapproximation gives the first seven significant figures accurately for/I— k*=.001. The asymptotic expression forthecomplete elliptic integral ofthefirst kind isaccordingly 4 K()=log.4. ©)2eeFB Inhis “Ezercises decaleul integral,” t.T,p.68,Legendre gives theformulas ide 1 9 4ea fitertBatt ---|togA Uae [takesght+JeeLeg Fae,-[b+Ft aa i7 13 4 fT=Paintody=[Bet+Saat+Jog5. L13, +[!=pat-Bet } where ky? =1—k*. CHAPTER V SURFACE DISTRIBUTIONS OF MATTER 105. Transformation byReciprocal Radii—Suppose there isgiven asphere Sofradius awith itscenter attheorigin ofa rectangular system ofaxes. Ifthepoint P(x, y,2)lieswithin thesphere atadistance r=a?+y?+#fromtheorigin Rn of RLZ Fro, 64, (Fig. 64)andthepoint Q(¢,»,£)liesonthelineOPextended atadistance p=«/f+9°+#*fromtheorigin,andif rolabes a) each ofthepoints Pand @isthe transform oftheother by reciprocal radii with respect tothesphere S. This transforma- tion sets upaone toone correspondence between the points interior tothesphere and thepoints exterior toit,thepoints intheneighborhood oftheorigin intheinterior corresponding topoints intheneighborhood ofinfinity intheexterior. Points ‘onthesurface areunaltered, and therefore, the.sphere issaid tobetransformed into itself, 208 105) SURFACE DISTRIBUTIONS OFMATTER 205 Ifthedirection cosines oftheline OPQ areX,«,and »,itis evident that zen, —=ph, y=Th, 1=Pity a=ry, £=pv, and therefore, since pr=a?, e=%2,a=Syoa(2)a, a a? c=rod yro aro Provided qisnotzero,theplaneP,(Fig.64), let my+ne+9 =0, istransformed into thesphere S:, la, ma? na? sateregepmae,4MAog, Bree Tet tr which passes through theorigin. Conversely, thesphere S; through theorigin istransformed into theplane P,;but the sphere S,,Fig. 66, attyttattletmy+net¢=0, which does notpass through theorigin (q=0),istransformed into another sphere Sz, bt, mat, nat, ot saegee eg Me, eg og,ee es which does notpass through theorigin. IfR,andR;aretheradiiofthespheres S,andSs,andifL;and Lyarethedistances oftheir centers from theorigin, then the centers ofthetwo spheres lieonthesame straight line through the origin and aly aR:hate =trRF thesign inthelastexpression tobechosen sothat R;ispositive. IfP:andP;aretwo planes which aretransformed into the spheres S;and Ss,Fig. 64,thetangent planes tothespheres at their points ofintersection make thesame angle with each other asdotheplanesP,andP».Indeed, itisevident fromthediagram that theplane tangent toS,at0isparallel toP;,and theplane 206 THETHEORY OFTHEPOTENTIAL tangent to8:atOisparallel toPs.Hencethetangent planesmakethesame angleasP; andP;do. Butthemutual nelinationof‘thetwotangent planesoftwospheres isindependent oftheparticular pointofintersection chosen. Hence thetwo{tangent planesatQ,whichisthetransform ofP,alsoformthesame angle asdothetwoplanes atP.Atransformation issaidtobeconform ifeverypairoftwointersecting linesistransformed intoanother pairoftwointer-secting linesinsuchawaythattheangles ofintersection aroPreserved. Itisclearthatthetransformation byreciprocaladilpossesses thisproperty, Aninfinitesimal length ds=V/dz*+dy+distransformed Intoanother infinitesimal lengthdy=/di4-dh? df', Sinve “He-aa- S08) deaae (28a o(ats f= ~F(2ae +(1=28g—(a0, a(n) +F(t—22 S(2 dex298i. ant 21—ofa, S28aeaE)+Sa—oar, itisverified readily that ae=Vas,ae Ifdaisaninfinitesimal ares,itstransform daisasimilarinfinitesimal area,sincetheangles arepreserved. SimilaeHtstProportional tothesquares oftheirhomologous lines Hence da=Sada,? Ifdtisaninfinitesimal volume, itstransform drisasimilarinfinitesimal volume,andtherefore, oedt=as, ermula whichcanbeverteddreetlybyforming theJacobianofthetransformation. ‘Thatis ae,y,2) dedyde =12¥,2) ee ae, ean. 106} SURFACE DISTRIBUTIONS OFMATTER 207 108.Application oftheTransformation toPotentials. —InFig. 65,supposePsisthetransformofP;,andM;isthetransform ofMi.Thatis Be2 aoa noe M, a Let the distance M,P, bedenoted byR,and MP: byRs. Letparticles ofmass m;and m;beplaced atthe f points M,andM;respectively. ‘Then thepotentials ofm:atthepoint P;and ofm:atthepoint P,arerespectively, -™, =™,Wak MR Since the triangles M,OP, and Fro.6 P.OM; aresimilar, itfollows that pm Ry tp Re Suppose also that the masses m:and m:arerelated, sothat m Tg,™ aTe then Yi omRe oro, Vi mR. ap Hence Vi=", =29, ati and the coefficient p/a depends only upon the position ofP: and isindependent oftheposition ofMf. ADiscrete SetofPoints—If there are many points M,‘° atwhich are located masses m,‘%, the potential ofthis setat the point P,is mi N=2Rw ‘The transform ofthissetofpoints isM:‘? atwhich areplaced particles ofmass ms‘, and the potential ofthe transformed setatthepoint P,is my Vi=tio 208 THETHEORY oFTHEPOTENTIAL Ifthemassesmandmarerelated, sothat mO nog mio a=50! then,asbefore, me pmo andtherefore Vi=27,=Sy, an AContinuous VolumeDistribution—Ut eachparticle oftheshovesetisregardedasoccupyinganelementofvolaicedr riapedeityo:whichmay,ofcourse,beafunction of%%andthenumberoftheparticlesisincreasedindefinitely, thediscretestofparticles passesoverintoacontinuous marcheeCceupies @certainvolume, anditspotential atPie edn, YsfRh ‘Thepotential ofthemass,transformed byreciprocal radii,upon thepoint P,is ‘rdry YeJ“ke i dm: adr on_a 1 ding~Sere=Gs=H ® itisstilltrue that Va=2,=Sy, @ana BySee.105, ary_(rit dy \a}i andthisrelation combined withEq.(1)gives Hence,theratioofthedensitiesatcorresponding pointsis‘directly ProportionaltothefifthPowerofthedistance oftheoriginal Pointfromthecenterofinversion 0, TineatandSurfaceDistributions—In easethedistribution ofmatteriscontinuous butiseitherlinearoeontasurface,the 106) SURFACE DISTRIBUTIONS OFMATTER 209 argument proceeds justasforvolumes, andthesameconclusion is reached, viz, Va= OM; () but, ifthedistribution isover asurface, thedensity isdirectly proportional tothethird power ofr,/a, and incase thedistribu- tionisalongalinethedensity isdirectly proportional tothefirst. power ofr;/a. Owing tothefact that thetransformation byreciprocal radii isofgreat value inthetheory ofelectrical attraction, Lord Kelvin, towhom theuse ofthis method isdue, called themass M; the electric imageofM,inthesphereS.107. The Potential ofaUniform Distribution ofMatter on a ‘Sphere.—In Sec. 29itwasfound that ifaquantity ofmatterM isdistributed uniformly over thesurface ofasphere ofradius a ‘thepotential isconstant inside ofthesurface, anditsvalue is Vi;=M/a. Ifthedistribution ofmatter istransformed by Lord Kelvin’s principle (Sec. 106) with thecenter ofthesphere Oasthecenter ofinversion, thedistribution isunaltered since the sphere istransformed into itself. TfP;isanypoint within thespherical surface atadistance p1from O,andPyisitstransform Eq. (105.1) atadistance p: from O,then byEq.(106.2) thepotential atPsis vy,=fy,-9MM,apag~pe whichisthesameastheresultfoundinSec.29byothermethods. 108. ANon-uniform Spherical Distribution.—Suppose S,(Fig. 66)isasphere ofradius a,which does notpass through the center ofinversion 0. Ifthere isauniform distribution ofmatter onS;ofamount M,, itspotential atapoint P,exterior toit,is M,/R;. IfthesphereS;istheelectric imageofS;inthesphereS,andifP;andOsarethetransforms byreciprocal radii ofP: and Oy,then byKelvin’s principle thepotential ofS;atthe pointPsis ue. py Ms MoM,Ven ak aR nk The ratio a/r, isindependent ofthepoint P,orPs. Hence the sphere Szattracts particles exterior toitjustasthough amass M,=2M, nm were concentrated atthepoint Os. 210 THE THEORY OFTHE POTENTIAL If,however, thepointP,liesinside ofS,,where thepotential Vi;isconstant, Psliesinside ofS:and thepotential atPsis oy, 2Meo Ms VenGM a a ‘andtheattraction atthepoint Psisjustthesame asthough ® mass M,= 2M,ay were concentrated atthepoint 0.‘Thedensityonthesphere$2,however, isnotuniform. Accord-ingtoSec.106,itvaries inversely asthecube ofthedistance from thepoint ofinversion 0.Inthediagram thecenterofinversion liesoutsidethesphereS:.HenceifP;liesinsideof'S,,Palies inside ofSy;andifP;liesoutside of8,Psliesoutside of a Ts ¥10, 66, Ss.Butifthecenter ofinversion Oliesinside ofS:,then Ps willbeoutside ofSsifP,isinside ofS1,sndwillbeinside ofSs ifP;isoutside ofS;. Theresults arrived atabove, however, holdwhether0liesinsideofS;,oroutside.IfS,isauniform, solidsphere ofmass M,8:alsois asolidsphere, butitsdensity varies inversely asthefifth power ofthedistance from 0. IfP,liesoutside ofS;and P;outside ofS:,then aM Vinee andPyisattracted justasthough allofthemass ofS3,which is equal toMa/r:, were concentrated atthepoint Os,which isalso thecenter ofgravity ofSs, 109) SURFACE DISTRIBUTIONS OFMATTER 211 109. Inversion ofaHomogeneous Ellipsoidal Shell.—Suppose there isgiven ahomogeneous ellipsoidal shell E(Fig. 67) which isbounded bytwo similar, co-axial, ellipsoidal surfaces and a sphere ofinversion Swhichhasitscenter atOinthehollowofthe shell. The shell Zisthe electric image ofacertain other shell Iwhichisobtained byinverting Ewithrespectto8,andwhichis bounded bysurfaces which are ofthe fourth order. Inthis inversion, itwillbeobserved that theinner surface ofEZbecomes. theoutersurface ofJ,andthatanypointP,within Zistrans- formed into apoint P:which liesoutside ofI. < WER 7<< (WN Gf, Q \\\ InSec. 11itwas found that ahomogeneous shell such asE, attracts aparticle anywhere initsinterior equallyinalldirections. ‘Thepotentialof£initsinterior,V;,isthereforeconstant(Sec. 34). Ifp2isthe distance ofP2from the center ofinversion 0, andaistheradiusofthesphere S,thepotential ofJ,thedensity ofwhich varies inversely asthefifth power ofthedistance from O,atthepoint Ps,isbySec. 106, ely, Bs and therefore, theshell Jattracts any particle which isexterior toitjust asthough amass equal toaV, were concentrated atthe point 0. 22 THE THBORY OF THE POTENTIAL Since thepotential ofZatpoints exterior toitisacomplicated funetion oftheposition oftheattracted point (See. 36), the same istrue, also, ofthepotential ofFatpoints which areinterior tot. 110. Centrobaric Bodies.—Thataspherewhichishomogeneous inconcentric layers should attract exterior particles just as though allofitsmass were concentrated atitscenter does not seem surprising, onaccount ofitssymmetry; but itcertainly doesseem surprising that such anunsymmetrical distribution of matter asthat which isexhibited byIinFig. 67,also should possess thisproperty. The illustrations which have been given above areexamples ofaclass ofbodies which arecalled centro- baric, Iftheresultant attraction oftheearth, oranyother mass,uponarigidbodyisequivalent toasingleforcewhichalwayspassesthroughafixedpointrelatively tothebody,irrespective ofrientation and distance, thebody issaid tobeeentrobarie. IfabodyBiscentrobarie withrespecttoagivenbody A,LordKelvinhasshownthatitiseentrobarie withrespectto allbodies. Imagine arodattached rigidlytothebodyA,andthe rodpivoted atapoint Osufficiently farfrom Bthat thesphere which canbedescribed byAdoes notcontain B.Inevery Position which ispossible toAbythisconstraint, theresultant attraction passes through afixed point @ofB.LetAtake » (very great) different positions distributed asnearly uniformly 4spossible overthesphere, Ifann'*partofAwere leftineach position theresulting distribution ofAwould bevery nearly a distribution homogeneous inconcentric layers overthespherical shell,andtheresultant attraction ofthisshellwould passthrough G.Thegreater nis,themore nearly cansuch adistribution be madeuniform, andthelimitforninfinite isashellhomogeneous inconcentric layers, which attracts Bwithaforce which passes through G.Buttheshellattracts Baswould aparticle atthe point0,Hence, Biseentrobarie withrespect toaparticle, and therefore, eentrobarie withrespect toallbodies, Withrespect toaparticle thelinesofforcearealways directed through thepoint@.‘Theyare,therefore, straight linesandthelevelsurfacesarespheres. BythetheoremofSec.74,theaveragevalue ofthepotential ofanydistribution ofmatter overany spherical surface which contains alloftheattracting matter in itsinterior isjustthesame asthough allofthematter were 110) SURFACE DISTRIBUTIONS OFMATTER 213 concentrated atitscenter. Hence, ifthe total mass ofBis XY,itspotential atallpoints which lieoutside ofthesmallest sphere which has itscenter atGand wholly contains B,is ye™, > and this formule holds evidently forallpoints outside ofB. 111. The Center ofGravity ofCentrobaric Bodies.—If a body iscentrobaric with respect toapoint @which isfixedrelatively tothebody,thepointGisitscenterofmass.Inorder toprove this, imagine thebody placed inthegravitational field ofavery massive particle atagreat distance, such asthe gravitational field oftheearth. Inthiscase, thecenter ofgravity coincides with thecenter ofmass, and since both points arefixed relatively tothebody, they must always coincide. Ifabody iscentrobaric with respect topoints which lieoutside ofthebody, itscenter ofgravity (orcenter ofmass) must lie inside ofthebody inthesense that every path from thecenter of gravity toinfinity passes through attracting matter. For example, thecenter ofgravity ofananchor ring does not lie within thebody inthis sense, but thecenter ofgravity ofa uniform spherical shell does liewithin thebody. Foroutside points thepotential ofacentrobaric body isV=M/p. Ifit were possible toreach the center ofgravity without passing through attracting matter, allalong this path the potential would beM/p. The average value ofthe potential over a sphere about thecenter ofgravity, sosmall that nomatter lies within it,isM/r, where ristheradius ofthesphere, andbySec.74 this isthevalue atthecenter. But M/pis infinite atthecenter. Hence, thecenter ofgravity cannot bereached byanopen path from infinity, andbodies whose center ofgravity eanbereached byopen paths from infinity cannot becentrobaric. The example ofSec. 108shows that ashell ofmatter canbe centrobarie with respect topoints outside oftheshell, andalso centrobaric with adifferent center, with respect topoints lying within theshell. But ifashell iscentrobaric forpoints lying withinitsemptyinterior,thecenterofattraction mustlieoutsideofthis interior, since thepotential function iseverywhere finite. ‘Ananalytic function which represents apotential function ina certain domain Aofempty space willcontinue torepresent the potential function inevery region ofempty space which canbe 214 THE THEORY OF THE POTENTIAL reached from Abypaths which pass through empty space only. Indifferent regions, which areclosed byattracting matter, the potential function will berepresented bydifferent. analytic functions, andthepoints atwhich theanalytic functions become infinite must certainly lieoutside oftheregions inwhich they represent thepotential function, since thepotential function is everywhere continuous. 112. The Central Elipsoid ofInertia.—The moment ofinertia ofabody with respeet toanaxis is SLpdm, where pistheperpendicular distance oftheelement ofmass dm from thegiven axis. Thus, themoments ofinertia ofabody with respect tothe2,y-,and maxes are fer eim, fet azydm, fet+vam. Suppose thegiven body Biscentrobaric and that itscenterofgravityisattheorigin.LetVbeitspotentialfunction, which,analytically, isdifferent indifferent regions. Deseribe asphere Saround theorigin, large enough tocontain Bwholly within its interior, sndletUbeanyfunction which isharmonic within Sand Which vanishes attheorigin. Then byGreen's theorem, Eq. (57.4), =((v8%—yal for VaU)dr=SC vee, a theintegral ontheleftbeing taken over thevolume ofthesphere, and the integral onthe right taken over itssurface. Since V isapotential and Uisharmonie, AV=-4ne, aU =0, «representing thedensity inside ofS.Onthesurface S vy=¥4%, W_ iM > on ” both ofwhich areconstant. Hence, Ea.(1)becomes ffVoss=Ffmde+fUt, oJne+ ByEqs. (62.2) and (74.3) aU,fee=0,five=0, 112] SURFACE DISTRIBUTIONS OFMATTER 215 thelast integral holding since Uvanishes atthecenter ofthe sphere, byhypothesis. Itfollows, then, that Svor=Suen=0. (2) ‘The volume integral can bereduced from thesphere tothe body, since ¢vanishes outside ofthebody. Suppose now that U=@+y)-@+2) which satisfies both conditions, namely, that Uisharmonic and that itvanishes attheorigin. "Then Sie+vam=fe+2am; that is,themoment ofinertia with respect tothez-axis isequal tothemoment ofinertia with respect tothey-axis. Asnothing hasbeen said about theorientation ofthebody Bwith respect tothecoordinate system, thisconclusion holds forevery orienta- tion; therefore, themoment ofinertia isthesame forevery axis through theorigin. From this, itfollows that thecentral ellipsoid ofinertia isasphere forevery body which iscentrobaric with respect toexterior points. 113. ASystem ofDetached Masses Cannot BeCentrobaric.— Suppose abody consisted oftwo detached portions, such as ‘MM,and MzinFig.68,andsupposefurtherthatthisbodywerecentrobaric. According to s Sec.111,thecenterofgravity _mustlieinsideofoneofthe_ Ui) masses,whichwillherebe7/7) G7takentobeM;,andthepoten--7/7Vy wal tialofthebodyis, CVM+™ Ll” ve—> ‘Fro.68. where pismeasured from @. Describe aclosed surface $about themass Ms, and take thesurface integral ofthenormal derivative ofVover S.By Eq,(68.1)ay, 2(ae=fFae=(hi+wefan(2)a6, 216 THE THEORY OF THE POTENTIAL since Gliesoutside ofS.But byGauss’ theorem, Eq. (68.3), fone=—4eMs.én Itfollows, therefore, that M;iszero and that allofthe mass isinM,. Acentrobaric body, therefore, consists ofasingle mass which isbounded externally byasingle closed surface ‘Itmay beintheform ofashell which isbounded bytwo closed surfaces, but itcannot consist oftwo ormore masses which are wholly detached. 114. Theorems Relating toElectric Images.—The following theorems relating toelectric images will beofinterest. Theorem I.—TIf abody iscentrobaric forexterior particles, its lectric image inany sphere whose center liesoutside ofthebody alsoiscentrobaric forexterior particles. Suppose B;iscentrobaric and that M,isitsmass and 0,its center ofgravity. LetBzbeitselectric image inthesphere S,ofradius a,and center atOwhich isoutside ofBy. Let P,beanypoint exterior toB;,andPsitstransform byreciprocal z zB eo Fro. 00, radii with respect toS;also, letOsbethetransform of0). ‘Then from thediagram, Fig.69, nlape Re © The potential ofByatPris =,v=Ro 2) therefore, thepotential ofB;atP:is Vo=27,22 am,aprRi~7Re! @) 114) ‘SURFACE DISTRIBUTIONS OFMATTER 217 or Mana ny where Mr=2My n obviously, isthemass ofBs,since theformula, limRV =M, me holds foranymass, eentrobarie orotherwise. Equation (4) shows that Byalso iscentrobaric. Theorem II—If thebody B,isashellwhich iscentrobaric for exterior particles, itselectric image, inanysphere whose center Oliesinthehollow interior oftheshell, Fig. 70,isagain ashell which iscentrobaric forinterior particles. Letthecenter ofmass ofB,beattheinterior point 01,and letPibeanexterior point. IfO;andPsarethetransforms &VYZZ, : ABN <p iy Fie, 70. of0,andP;,thepoint Oswilllieoutside ofBs,andPswilllie inside, The same notation asbefore canbeused, andEqs. (1),@),and(8)holdunaltered. Particles intheinterior of Byareattracted toward theexterior point 0sjustasthough mass Mja/r, werelocated there andthebody Bsdidnotexist. ‘ThepointOs,obviously, isnotthecenter ofmassofthebodyBs. IfBy,ofFig.70,isaninfinitely thinshellofmassM,,thenBy alsowillbeaninfinitely thinshellofmassM2,where, Eq.(106.1), Ms=fam=of=areBs a,7 218 THB THEORY OF THE POTENTIAL inwhichristhedistancefromthepoint0totheelementdm;,and Vsisthepotential ofB,atthepoint 0. Points intheinterior of B;areattracted toward thepoint Osjust asthough amass M= Me ri were placed atthepoint 0., Since B,iscentrobarie with respeet tothe point O,forexterior particles, itsexterior potential is ‘M,/R. Letitsinterior potential beV.Then, atallinterior points, R,being measured from thepoint 0, M Rin >o Onthesurface B,thisexpression vanishes, since thepotential iscontinuous across By. Deseribe asmall sphere 2about the point O;. Iftheradius ofthis sphere issufficiently small, M,/R: —Viscertainly positive on2.Sinceitisharmonic in theregion between 2andB;andvanishes onB,,itispositive everywhere inthisregion. Therefore, atthepoint 0, ¥,<M 7 Also, onmultiplying through bytheconstant a, Vs<MS; ni sothat Ms<M; ‘thatis,themass M;onByislessthan themass Matthepoint 0s. Ifthepoint 0,which isthecenter ofinversion, approaches thepoint O,,thecenter ofmass ofB;,thepoint 02recedes toward infinity, andtherange ofthevariation ofthepotential within the hollow ofB;diminishes, sothatif0,isveryremote thepotentialwithin B;isvery nearly constant. Ifthecenter ofinversion isatO:thepoint Osdoesnotexist andthepotential within By isconstant. ‘Thisisreadily seenfrom Eqs. (1)and(8),for aM,_aM, Vanee andasthepoint ©approaches thepoint 0,thelength p: approaches thelength R,,sothat Vv,Me. Mh, mo a which isconstant. Hence thetheorem: 114) SURFACE DISTRIBUTIONS OFMATTER 219 Theorem III.—If aclosed shell ofmatter iscentrobaric with ‘itscenter ofmass Ginitshollow interior, itselectric image inany sphere which hasGasitscenter isashell ofmatter forwhich the interior potential isconstant. The converse ofthis theorem also follows readily, namely, Theorem IV.—If thepotential ofashell isconstant throughout its interior, itselectrical image inany sphere whose center lies intheinterior iscentrobaric forexterior particles, and the center ofmass oftheelectric image isatthecenter ofinversion. If thecenter ofinversion Oliesoutside theshell, theelectric image is centrobaric with respect tothepoint Oforparticles intheinterior oftheshell. 115. Level Layers.—Suppose there isgiven asystem ofmasses which hasthepotential V.Suppose V=Cisanequipotential surface that entirely surrounds alloftheattracting matter, or only part ofit.Imagine that matter isdistributed over this equipotential surface insuch awaythatateach point ofit lav.=Fant ) that is,thesurface density ateach point isequal totheinterior normal derivative ofVdivided by47. Since theinterior normal derivative ofVisthemagnitude oftheattracting force F,this can bewritten also o=BF. ) With thisdistribution ofmatter upon it,thelevel surface becomes ‘alevel layer. LetP(z,y,2)beapoint exterior tothelevellayer S.The potential atPduetothematter inthelevel layer is o 11aV, v==2fe. f.adranpan where pisthedistance measured from thepoint P.Thefunction 1/pisharmonic inside ofS.Therefore, Eq.(71.2)gives 1aV, a/v dmLV (y(t +ae(2 Sine~[rene fF where, inthelastintegral, Brepresents thevolume enclosed by Sanddmisanelement ofthemass enclosed byS,Since V 220 THE THEORY OFTHE POTENTIAL isconstant onS,and1/pisharmonic insideofS,thefirstintegral intheright member iszero, Eq.(66.1). Hence 1lav, dm vengepant fe ® This result canbestated inthefollowing theorem: Theorem—The potential ofalevel layer atany point outside ofitisthesameasthepotential ofthematterwhichisenclosed byit.From thisitfollows that, sofarastheattraction atoutside points isconcerned, thematter inside ofalevel layer canbe replaced bythelevel layer. Since theattraction of‘thelevel layer isthesame asthatoftheenclosed mass atremote points, it follows thatthetotal mass ofthelevel Iayer isequal tothetotal ‘mass which isenclosed byit. Inempty space thepotential function canhave neither a maximum noraminimum; and, since thepotential isconstant on S,itmust have thesame constant value everywhere inside ofS.Hencetheattraction ofalevellayeratinteriorpointsvanishes.Itisinteresting tonotethatifanequipotential surface which encloses theentire mass were covered with negative matter 80as toform anegative level layer, thesum ofthetwo potentials V+Viwould vanish everywhere outside ofS,andthelevel layer would actasascreen tothe attraction ofthe matter within it. ‘The sum ofthetwo masses also would bezero. In theinterior ofSthescreen would have noaction and theoriginal masses would continue toattract justasthough thescreen did not exist. Reduction toTwoDimensions.—Suppose there isgiven asystem ofmasses inaplane forwhich thelogarithmic potential isV. Suppose V=cisanequipotential contour that encloses all,or only part, oftheattracting matter. Suppose that matter is distributed overthisequipotential contour insuch awaythat at each point ofitthelinear density is lave. Brin, ~BF @) Withthisdistribution of it "contour becomes alevel.head ponittheequipotential Arepetition ofthepreceeding nt ithmiDentalgivestheanalogoustheorem n°"SeFonarithmieane ntl ofseltreadolanypointousideof iematter which isenclosed byit. 116) SURPACE DISTRIBUTIONS OFMATTER 221 Since thepotential canhave neither amaximum noramini- mum inempty space and since thepotential isconstant along the level thread, ithas the same constant value everywhere inside ofit. 116. Families ofLevel Layers.—Suppose qi,q243isatriply orthogonal system ofcoordinates and that the level surfaces in empty space are g:=const. Expressed interms ofthese coordinates V(q3) isafunction ofgsalone. Using thenotation ofEq.(56.5) avavdn Rsdqs The element ofarea onthis surface is doy =RiRedgidgs. Hence, theelement ofmass inthelevel layer onthe surface q@=const.is 1RiR.aV am Feagtd @) Since Visapotential function, AV=0inempty space, and therefore, byEq. (56.8), 8(RRs aV @(RR, aV a(RR. dVaeae)+a an)tae ay=o© Since Vdoes notcontain q,and qs,thefirst and second terms of Eq. (4)vanish bythemselves. ‘Therefore a (RR.an(2av)=90, () which shows that theelement ofmass inEq. (3)isindependent ofgs. From this itfollows that: Theorem.—If f(q., qx,@)=0isany closed curve onthelevel surface qa=a,and iff(q:, gs,b)=0isthecorresponding closed curve onthelevel surface qx=b,theamount ofmaiter enclosed by thecurve inthelevel layer qs=aisthesame astheamount of matter enclosed bythecorresponding curve inthelevel layer qs=b. The theorem istrue forthecorresponding elements ofthetwo areas; therefore, itistrue fortheentire areas. Itfollows alsofrom Eq.(5),byintegration, that RyR: aV “Beag,71s90 222 THETHEORY OFTHEPOTENTIAL Wherefisafunction of:andqsonly;andsincea¥/dgs is«function ofgsalone, say av av 6)3g,=9) @) itisevident that BR:_fl..q), ©Ry 9)” thatis,RuRs/Rs isfactorable, onefactorcontaining g,aloneandtheotherindependent ofgs.Equation (3)thenbecomes 1 am=74aadda. ) Conversely,suppose atriply orthogonal system ofsurfaces 'sgivenandthatRifs/R, isfactorable intheformofEq,(7). Afunetion Veanbedetermined fromEq.(6)merely ‘by#quadrature, theconstant ofintegration beingchosen sothatVvanishes atinfinity. Thefunetion Vsodetermined iyhar.monic,vanishes atinfinity, andisconstant onthesurface gs=const. Hence, Visthepotential ofadistribution ofmaiteronthissurface forwhich =v,°" ERag; Theorem—Anecessary andsuffcient condition thatonefamily, gs=const,ofatriplyorthogonal systemofsurfacesmayalee beafamily oflevelsurfaces isthatRiRa/R, isfuctorable onshoformofEq.(7),andthatthereexistsaconstant Ceuchthat Socasdeas+¢ vanithesotinfinity. iY.LevelLayeronanArbitrarily GivenSurface—If itfaveknownthateveryclosedsurfaceisanequipotential surfacefefomedistribution ofmatterthatlieswhollywithinwene{hesurface,theresultsofthepreceedingsectionswouldprove{hetthere existsadistribution ofanygivenquantityofmaton:we Seon’Elvenclosedsurface$forwhichthepotentiallsconstanton QainotherwordsforwhichSitselfsanequi-potentiatnataceQuitelikely,ifisproperlyrestricted, thereasinfinitely many{plumedistributions forwhichSisanequipetonena surface,although thereisnoprooftothiseffect;butitistnetharnet 117) SURPACE DISTRIBUTIONS OFMATTER 223 exists one, and only one, surface distribution onSofagiven quantity ofmatter Mforwhich Sitself isanequipotential surface, atheorem which isduetoGauss, although hisargument isnot sufficient toprove the proposition. Suppose themass Iisplaced upon Satrandom and suppose also that theparticles ofMrepel each other instead ofattract. This last hypothesis does not affect thepotential V,but itdoes change thesign ofthepotential energy ofthedistribution which, byEq. (76.1), is WepfYeas, o)s where Vand oarethevalues ofthepotential and thedensity atthe surface element dw. Let Rbethe maximum distance between any two points onS. Then ‘im. M Vefe2R and ae weap which shows that, whatever thedistribution may be,thepotential energy, which isnecessarily positive, has alimit below which itcannot sink. Itdoes not prove that Whasaminimum, for, conceivably, the lower limit might beapproached bymany distributions and beattained bynone. Foranyinfinitesimal variationofthedistribution (thatis, 1variation in¢)thechange inthepotential energy is ow=ifoov+Vb0)dus; Is and, since thetotal amount ofmatter isconstant, a=[iede=o. @3 ‘The change inthevalue ofVatany given point is v=fFae, @ 36 226 ‘THE THEORY OFTHE POTENTIAL Inorder tomake sure that there isnoother solution, let Vobetheaverage value ofVonSfortheassumed solution. The variation ofWcan also bewritten wefe—Vo)bodu, (5) andthismustvanish forevery setof6¢which satisfies Eq.(2). LetPbetheportion ofthesurface $onwhich V—Vois positive, andNtheportion onwhich itisnegative. Letthe variations of¢intheregion Pbenegative andin‘theregion XNbepositive,butofsuchvaluesthat[edeiszero.Thenevidently thevariation ofWinEq.(5)isnegative andnotzero, sincetheintegrand isdecreased everywhere. Hence V=const. istheonly solution. ‘There cannot betwodifferent distributionsofthesameamount ofmatter onSforwhich thepotential isconstant onS.Suppose there aretwo different distributions which have constant potentials onS.LetV;andV2betheexternal potentials ofthesetwodistributions, andV,andV3their values onS. ‘Then Vi =eV whereyissomeconstant. Consider thedifference VaeVi- as ‘Thefunction Visapotential function which vanishes onSs andatinfinity. Itis,therefore, zeroeverywhere (Sec.75),and Vie as. But,since thetotal quantity ofmatter isthesame inthetwo distributions, thelimit oftheratio V/V; atinfinity isunity. ‘Therefore wel, and Vx=Vi, ‘There cannotbetwodifferentconstantvalues,butconceivably there might betwodifferent distributions which have thesame constant potential onS,butthisalsoisimpossible. Since the density onthesurface is(Sec. 115) 1av, o-Ee and since V;= V;, the normal derivatives are everywhere ‘thesame and, therefore, thedensities inthetwo distributions are identical. 117] SURFACE DISTRIBUTIONS OFMATTER 227 ‘There cannotbemorethanonedistributionofagivenquantity ofmatter onSforwhich thepotential isconstant onS. Itshould beobsorved that inthis distribution ofmatter on S,every portion ofthe surface iscovered with matter, ifSis aclosed surface. Ifitwere notso,there would beapath from theinterior of$,where thepotential isconstant, totheexterior, where itisnotconstant, which didnotpass through attracting matter. ByTheorem IIT, Sec. 75,this isimpossible and the surface must everywhere becovered, with thepossible exception ofisolated points and lines atwhich thedensity might vanish. ‘The theorem that the distribution which makes the potential energy aminimum also makes the potential constant inthe interior and onthe surface isaparticular case ofasomewhat more general theorem which isdue toGauss. Reduction toTwo Dimensions.—The above argument, without any essential modification, indicates, also, that there ean exist butone distribution ofagiven quantity ofmatter onany plane contour Cforwhich the contour becomes alevel thread; that is,for which the logarithmic potential ofthe distribution isconstant everywhere onC,and therefore within C,ifCis closed. 118. Robin’s Integral Equation.—If V,and V;aretheexternal and theinternal potentials ofaclosed level layer, then 1aVv. .1AV dimgang ON asan=O This isthe familiar discontinuity inthe normal derivative ofthe potential ofasurface distribution ofmatter. Ifmisa point ofthe surface, the normal derivative ofthe potential atmdoes not have adefinite sense. ‘Suppose m;isaninterior point onthe normal through mand infinitely near m,and m,isanexterior point onthesame normal also infinitely near m. Ifp;and p.are measured from m,andm,respectively andgistheanglebetween thedirection ofpand thedirection oftheexterior normal, then (Sec. 66) veef2085du,Vio[2Sodu. one Ispe On. Isee Hence =tim2(2%, =lim1(2S, ontimf,[23ds,0tingfFed, 228 THE THEORY OFTHE POTENTIAL Ifpmismeasured from thepoint mitself, theintegral I=f$08Yad, J,Pmt which represents thenormal component oftheattraction of Son m,haseperfectly definite sense andcanbeevaluated. Letaplaneperpendicular tothenormal atm,which isassumed tobearegular point ofthesurface, bepassed through thepoint ‘m;. This plane divides thesurface Sinto twoparts, an‘infinitesi- maldiskswith itscenter atm,andtherestofthesurface z ‘Thenormal components oftheattraction of2atthepoints m andm,differ infinitely little, since thetwopoints mandmdifferinfinitely littleinposition, Thelimitofthenormalcomponent oftheattraction of¢onmsasmsapproaches mis 2redirected towards theexterior, since thelimit ofthesolid anglesubtended bysatmis2r(Sec.8).Hence, thelimit of ‘thenormal component oftheattraction of2onm,alsois2x0 directed toward theinterior, since thetotal attraction onm:is 010. Thelimit ofthenormal component oftheattraction ofsat thepoint miszero, since thelimit ofsisaplane diskinwhich ‘mies. Therefore thenormal component oftheattraction ofS ‘onmisthelimit ofthenormal component of2onm,that is ‘2rem Hence Qn = [Fede”JsPm? ’ or q@) om=LfLeds,atef2fede, which isRobin's integral equation Ifthetotal mass onSisM,then Mafieds, 2)s and thedistribution ofmatter onSiscompletely defined by Eqs. (1)and (2). 119,Picard’s Solution ofRobin's Equation.—Combining thegeneral line ofthought ofNeumann inhismethod ofthe arithmetic mean with hisownmethod ofsuccessive approxima 119) SURFACE DISTRIBUTIONS OFMATTER 229 tions, Picard has given the following solution ofRobin's equa- tion! forasurface which iseverywhere convex, L208», ole)=gePREECE Dee, where theointhe left member isthe density ofthe distribution atthepoint from which pismeasured, and intheright member isthedensity atthesurface element ds, Letfbeanyfunctionwhatever thatiscontinuous onS.A series offunctions, f,,fa,...isdefinedbytherelations 1 0s ¢fagfea 1 (os ¢,fa=pe[FP hde,fag[5% =2 [oyfonge[tad Itwill beshown that, aside from aconstant factor, f,tends toward thefunction «which represents the density ofthedis- tribution inalevel layer. ‘The first ofthese equations canbewritten 1 focosyLs de, ixf,op Since ¢does notvanish onS,letAbethemaximum value ofthe ratiof/oandBbetheminimum; sothat(A+B)/2isthemeanvalue. Letabethat portion ofthesurface forwhich theratioJ/cisgreaterthanthemean,and6theportionforwhichitisless than the mean. Then anfsAfL28Fdy4ALBodarcsLas, J. 2 J, and A+B fcoose o00s enf,2ATR(78240+BES Fae, hey fpet f* ‘Prcano, &.,““Traité D’Analyse,” Vol.I,p.203. 230 THETHEORY OFTHEPOTENTIAL ‘These equations canberearranged soastoread 2hSAfreese, A-B(ote, Ie 2 J,6 £008G4,4A=Bocose, reneBfPte+AZf- ‘Theintegral overtheentire surface is2nv. Fortheother‘integrals, let «008 ¢ 208@ rd, = [TOSeg, =(EF, 2fESda,Deyf“ 0that6,and@yareeachlessthano(theintegrandbeingevery- wherepositive). Withthisnotation, theabove inequalitiesbecome h A= Be¢84-“S veers, forany point onthe surface, Iftwodifferent Pointsaretakenandaredistinguished bythe‘subscripts 1and2,theseinequalities give fucygA~Bow fn A=Bbse, am24ied neBt+aafgg-AaBbe fayA~Bone oF2oeoO 2oe Hence &~Be@-44458“(Gs+22). ‘Thecoefficients of(A~-B)/2 inthetightmembers oftheseequations areeachlessthan2;therefore B-Bl<wa -B), whereissomepositivenumberlessthan1.Sincethisinequal- ityholdsforanytwoPointsonS,itfollowsthatifA,isthemaximum valueoffi/oandB,isitsminimum value (4:~B)<y(a—B). 119] SURFACE DISTRIBUTIONS oFMATTER ——%81 Inasimilar manner, itisshownthat (As —By) <uX(A —B), where A,and B,are the maximum and minimum values ofthe ratio f,/e. Consequently, iff,tends toward alimit atevery point ofS,theratio f./a tends toward aconstant value, since the difference between its maximum and minimum values tends toward zero. Now |[eneens p=pe|Se a, tao and vatfetes,BroJs pF the dashes onthe letters indicating the value ofthe function atthesurface element dw. Onmultiplying thesecond equation byf.-: andsubtracting from thefirst, there results apelf(fee-fcosey. fewtennghf(FeSePas and since |e:-£3<u-B), itfollows that Ife—foul <w-(A —B)o<u(A —B)oo, where ooisthe maximum value ofo. Bywriting abt h-M+Gr-M +9 $e feds itisseenthatf,canberegarded asasumoftermswhichdecrease like the terms ofageometric progression. Ithas, therefore, ‘adefinite limit, and ifCissome constant. limf,=Co. 120. Example ofaLevel Layer.—Suppose two particles, each ofunit mass, areplaced atthepoints 0;and0:(Fig. 71), the distance between the points being 2. Ifthe distances 232 THETHEORY OFTHEPOTENTIAL ofthepointPfrom0,and0;arep;andps,theexpression forthe potential atPis vat4h, car Fro, 71, andalong anequipotential surface Visconstant. Ifthevalue ofthisconstant is2a/ltheequation ofthesurface is 1120 @ata 7 TheforceFwhich isacting atPistheresultant oftwoforces Ve?and1/p:*directed toward 0,andO,respectively. Its magnitude asgiven bytheparallelogram lawis Lai ay wy Hence, alevellayercanbeconstructed bydistributing matteroverthelevelsurface, Eq.(1),insuchawaythatthedensityis 1/1 Vi 1 4th=z] (44+2)454+4)--43, 2) caeta) SP @ andthislevellayerattracts pointsoutside ofitjustasthearticles atOsand0;do.Fora=3/4,thelevellayer‘isasingle closedsurface ofrevolution whichcontains bothoftheattractingarticles, For@=1,thelevellayerhastheshapeofanhourslass,atthepointmidway ofwhichtheforceFvanishes andtherefore, thedensity ¢also.Fora=8/2,thesurface consistsoftwoovalseachofwhichcontains oneoftheparticles, andtherefore (Sec.115)eachovalofthelevellayer,forparticles 120) SURFACE DISTRIBUTIONS OFMATTER 233 outside ofit,isequivalent tothe particle which iscontained within it. Ifthe particle at0,isleft undisturbed, but the particle atOrisreplaced byalevel layer which isasingle oval, thesystem isequivalent tothetwo particles at01and O;for particles outside oftheoval. ‘The potential inside ofthe oval isconstant; therefore, theattraction oftheoval onaninterior point isequal and opposite tothe attraction ofthe particle at0s, ‘The level layer surrounding Os,taken byitself, iscentro- barie foroutside particles, and attracts inside particles just as though allofitsmass were negative (repellant) and were con- centrated atthepoint Os. 121, Level Layers onProlate Spheroids.—The equipotential surfaces ofahomogeneous straight rod are prolate spheroids (ec. 98). Ifp:and psare distances measured from the ends oftherod, theequation ofthese surfaces inbi-polar coordinates is pit es=2a, whereaisthepolarsemi-axis. £pLet Xand ybeparameters defined bythe relations ' pita, apa2H d. andlet@bethelongitude withrespecttoapolar h,axis, which coincides with the rod. The surface ¢ X= const. isan ellipsoid; thesurfacex=const. isone ofthe sheetsofatwo-sheeted hyperboloid of revolution;and@=const.isaplane. These, np three surfaces are confocal and intersect each other orthogonally. Ifthelength oftherodis21,theequations oftransformation forrectangular coordinates are 2=TVA TEA) 00s8, y=FVWAVED tin8, @ eo™, thecoordinates 2,4,#being restricted totheintervals del -lsuS4 056525. 234 THE THEORY OFTHE POTENTIAL ‘Thedirection isnormal totheellipsoid, they-direction isnormal tothehyperboloid, andthe6-direction isnormal totheplane in thesense oflongitude increasing. IfP(z, y,2)isapoint onanellipsoid forwhich }=a,the normal displacement ofthepoint pduetoaninfinitesimal change inthe coordinates is =(2)+(24(@Ja- Meta, @ a=V5)+GQ)+G)a-\eae™ © forthe value} =a. Similarly thedisplacements along ameridian andalong«circleoflatitude ontheellipsoid arerespectively,BETH TaN, oF .VG)+@)+Gya-v and an)?|(au)? (a2)? lyin \@)+@+)6=VEREHe, @) also forthe value =a. ‘The value ofthepotential ofahomogeneous straight rod oflength 21andmass unity atagiven point Pis _1, attVago? © whereaisthepolarsemi-axis oftheprolate spheroid whichpassesthrough P. The normal derivative ofVis aBVanan~8aon’ which, onaccount ofEqs. (1)and(4),becomes ave -1f-—— (5)GoACC) “ Hence, thedensity ofthelevel layer onthisspheroid is os <a . ron ae =P —1) ‘Theelementofareaonthespheroidistheareaoftherectangle ‘ofwhich thesides aregiven inEqs.(2)and(3). That is, ba=VGTG Tats andtheelement ofmass isedw, or 1 dm=wited. 121) SURFACE DISTRIBUTIONS OFMATTER 235 Ifdm, istheelement ofmass inacollar ofwidth du,theintegra- tion ofthis equation gives du am.=J © and the integration with respect toufrom —1to+1gives the total mass +1, asofcourse itshould. The simplicity ofEq. (8), however, shows asimple distribution ofthemass inlatitude. ‘The distribution isuniform with respect to4,and therefore, uniform with respect to2,also; that isthemass included between any two planes which intersect thespheroid and which areparallel totheequatorial plane isproportional tothedistance between the planes. Asavaries, the mass included between any two hyperboloids remains constant. Ifthe rod isregarded merely asanauxiliary concept the spheroidal level layers being theprincipal one, itisevident from Eq. (4)that there areinfinitely many spheroids, foragiven mass, forwhich Vhasafixed value, say unity. Ontaking I=ae, where¢istheeccentricity ofameridian section,Eq.(4), lte 2=log defines therelation between! and ¢.The potential energy ofa unit mass distributed asalevel layer oneach ofthese surfaces is the same. Nowork isrequired inpassing from one ofthese distributions toanother. Itisinteresting tonote that ofallof these spheroids, which vary inshape from thesphere with aradius unity toastraight line ofinfinite length, the sphere isthe one which encloses aminimum volume. 122, Level Layers onEllipsoids.—It was proved inSee. 11 ‘that ahomogeneous ellipsoidal homoeoid attracts aninterior particle equally inalldirections, sothat theresulting attraction iseverywhere zero. This means, ofcourse, that thepotentialis constant inside the homoeoid, and that ifthe shell isinfinitely thin thesurface isanequipotential surface. Ifthedistribution ofmatter isasurface distribution, thedensity isproportional to thethickness ofaninfinitely thin homoeoid. ‘That is Idan, where hiisafactor ofproportionality. 236 THE THEORY OFTHE POTENTIAL Iff(z,y,2)=0isthesurface, and DAW af\? afy?OCU! then, Eq. (54.4), to =kLap, anddn=pif Since dfisconstant overthesurface, where hsisaconstant factor ofproportionality. The equation anti 4YMYZ-ie 2>B S@v2)-gogthagt =v! 0,a>, (1) represents afamily ofconfocal conicoids, ifgisregarded asparameter. Thethreerootsofthisequation, whenregardedfasacubic ing,aretheelliptic coordinates ofthepoint 2,v,2; thus! gt=SHAN =w)(*—0),a*(a? —b*) 21—BD —9)"—a9), p=Oe : C) 2103 ®ab? ‘Theorder ofthemagnitude oftheroots is a>garb >g>0> as andq=4inEq.(1)isane"lipsoid. ft pe=ON=0)-14 Taeqa —bn 4eq\enw net=(ON 9)_ a Far=g)b) =ges 49q\ enet ap=Oa ea) 14) Ta qo? —aa 4a4lcnet thecomponents ofthedisplacement ofthepointx,y,2dueto ‘aninfinitesimal change intheelliptic coordinates areRidgi, Redg:, andRdg. These three components aremutually 1Statics andtheDynamics ofaParticle,” p.355 +Bbid., p.380, 122) SURFACE DISTRIBUTIONS OFMATTER 237 orthogonal, thefrst twolying intheplane tangent totheellipsoid, andthethird being normal totheellipsoid. Itfollows atonce from Bq. (1)that apy?,(a)?4(a)*—4a (2)+Gi)+) =i and therefore ‘Rt =4R3*. ‘The density inthelevel layer ontheellipsoid isthen c=iFee oe=o). NG Maw } or,since gsisconstant onthe ellipsoid, the constant factors canallbeincluded inasingle factor ofproportionality, and then h Vai=99)(g2=9) The element ofarea ontheellipsoid evidently is do=RRedardas, or (1=VG =OG—7) dy= Vs=99)02—19)gga We =ala =nle —we and theclement ofmass, dm =od, is Mgu—eaddgudns dm = ———_M_—dds____. @) IVE =1a —ae (= OT BT This expression fortheclement ofmass isindependent of92, and therefore, itholds foreach member ofthefamily ofconfocal ellipsoids. The constant Acan bedetermined bythecondition that the mass ofthe level layer isequal tounity. Hence, on integrating over anoctant oftheellipsoid, aef°ier= aortas he JuJoVie WG —Mar@=WOge =f.gu=dg,fge JeVG =aa: —Vaso Vt =aOF=ga fr— slr SS JnVia =a)qs—ba JoVa =a0 =aaa ‘Thesubstitutions \/a? —gq:=V/a* —8sinyinthefirst and third integrals, andVj =bsin¢inthesecond and fourth 238 THE THEORY OF THE POTENTIAL integrals, effect the reductions toLegendre’s normal forms, and itisreadily found that 19y7EK+EK~KR), where Kand EareLegendre’s complete elliptic integrals ofthe first and second kind, forthe modulus sabeate* and K,and Z;arethesame quantities forthecomplementary modulus k:*=1—k*.But,asLegendre proved, EK.+EK-—KK,=> whatever value k*may have, Hence, foralevel layer ofunit mass 1 1hep and o=. 4)i 4eV (ai=@(G2 =9) For gs=0,theellipsoid is@plane double elliptical sheet, Therefore, asingle plane ellipse will bealevel layer ofmass unity ifthedensity onitis 1 OVante ItisfoundfromEq.(2)that,forgs=0, By _amat pf=} gay Hence, thelines ofconstant density areellipses which aresimilar tothegiven ellipse. Thedensity attheedge (g:=0)isinfinite, andatthecenter, where g,=a?andgs=BF,itis ss°°=rab ‘The average density istwice thedensity atthecenter. Itwillbeshown inthenext section that theequipotential surfaces ofthiselliptical diskaretheconfocal ellipsoids. Equa- tion (3)shows thatifthedouble elliptical diskshould expand through theseries ofconfocal ellipsoids, retaining always the constant mass unity, thesame element ofmass would befound inthetube defined byqi,q1+dq,a1,a2+gz,euch constant. Thedensity would vary from onesurface toanother, butthe element ofmass would remain thesame. 123} SURFACE DISTRIBUTIONS OFMATTER 239 123.ThePotential ofEllipsoidal Level Layers.—The direct computation ofthepotential ofanellipsoidal level layer involves. verydifficult integrations which, probably, have never been carried out. Butthepotentials canbeobtained without diffi- culty from theexpressions forthepotential ofahomogeneous solid ellipsoid simply bydifferentiation. ‘Thepotential ofahomogeneous, solidellipsoid atanexterior point 2,y,2,isEq. (36.1) = “(-f- w v~vvatrf(@arePreri) —— HtVETS TNFS where a,8,7,arethesemi-axesoftheellipsoidandxisthepositive Toot oftheequation a a srarnt eee teeth Thepotential ofthesimilar ellipsoid whose axesare(+a, (1+), and(1+d)yatthesame point z,y,2is .a nr ae a W=soabyf(149-335 B+sFri) ~ a Vie $6 F565) where xzisthepositive rootoftheequation a ar .Btn tette TNE ‘Thepotential oftheshellwhichhasbeenaddedis,therefore, V=W-U; andfor\very small awvemee ‘Thelower limit oftheintegral, xz,is,ofcourse, afunction of A,butthisfactcanbeignored inthedifferentiation, since the function within theparenthesis vanishes fors=n. Ondif- ferentiating thenwithrespect todsofarasdoceursexplicitlyandthen setting \equal tozero, there results V=dreadaf”. oe STN ee +(BF NOE 8) 240 THETHEORYOFTHEPOTENTIAL ‘The mass oftheellipsoid is M=Froaby(l +5, and themass oftheinfinitely thin shell is aM =troasydn. Since themass ofthis shell isunity itfollows thet 1=readin, and therefore, thepotential oftheellipsoidal level layer ofunit mass is a“2), VEFOP FIG TS) Atthe surface x=0,and, since the potential iscontinuous across thesurface and constant intheinterior, the value ofthe potential atallinterior points is yelf . “2S Vie FORT IG TS For the elliptical disk y*=0.The potential isconstant over thesurface forwhich risconstant. ‘That is,each member ofthefamily ofconfocal ellipsoids isalevel surface. 124, Layers ofFinite Thickness.—If aninfinitesimal amount ofmatter isdistributed over alevel surface insuch away, however, that the surface density iseverywhere proportional to8V/dn, the potential due tothe matter sodistributed is constant within the surface, and itwill remain constant ifthe matter expands s0astofillthevolume between two infinitely close level surfaces, the amount ofmatter associated with each element ofsurface remaining proportional toaV/an just as before. The volume density inthisnew distribution, however, isproportional to(aV./an)*. Inorder toshow this, suppose theamount ofmatter distributed ismal and that itisdistributed inashell ofuniform thickness di, The amount ofmatter associated with each clement of surface is lev aeantes and this amount ofmatter isexpanded tofillthe element of 124] SURFACE DISTRIBUTIONS OFMATTER 241 volume dadn, dnbeing the distance between the two level surfaces. Ifoisthevolume density 1av odin=22Vdua ‘After removing the common factor dwand then multiplying through bydV/8n, this expression becomes 1/av\? adv=aa)‘al; and, since dVand dlareconstant over thesurface, this equation can bewritten avy:o=(ir)=P, a where/isconstantoverthesurface,andFistheintensityofthe force atthepoint under consideration. This distribution ofmatter leaves the level surfaces undisturbed. Infinitely many such layers canbebuilt upinto ashell offinite thickness forwhich the internal potential isconstant, and the external ‘equipotential surfaces are the same asfor the original mass. Ifthetotal mass ofthe shell isM,itsattraction onparticles exterior toitisjust the sume asthe attraction oftheoriginal mass. The factor ofproportionality hinIq. (1)may vary from layer tolayer inany manner whatever, continuously or discretely, but inany given layer the volume density ispro- portional tothe square ofthe resultant force atthat point. ‘The electric image ofany ofthese shells inany sphere whose center lies inside ofthe shell isacentrobarie body (Sec. 114) forexterior particles. Green's theory, therefore, shows how to construct aninfinite variety ofbodies which possess the interest- ing property ofattracting outside particles just asthough the body itself were aparticle. 125. AFinite Shell Bounded byConfocal Spheroids.—In See. 121 alevel layer was constructed onaprolate spheroid. AAsitis desired toextend these results tothe construction of shell offinite thickness bounded bytwo confocal prolate sphe- roids, the notation and formulas ofthat section will beuseful inthepresent one. 242 THETHEORY OFTHEPOTENTIAL Forahomogeneous rodoflength oflength 21itisconvenient totransform tothecoordinates 4,4,@which were defined in ‘Eq,(121.0), namely: _Loe=BYE=4)0086, yoJoe=YE=#4)sin8, z-™ Ifthemass oftherodisunity, itspotential atanypoint ‘onthesurface oftheprolate spheroid which isconfocal withthe ‘endsoftherodandforwhich)isthepolarsemi-axis is,Eq. (121.4), 1A+L Vmay ® ‘Theforceacting atanypoint Pofthissurface isgiven byEq. (121.5) Fs a an VRBO = ‘andtherefore, thevolume density atPis =(3Pek o=Nan)~OFHot-where hisconstant over thesurface ofthespheroid, butcan varyfromonespheroid toanother. Thatis,hcanbeafunction ‘of)butnotafunction ofuor6.Letthisfunction bechosen sothatintheplane oftheequator, where »=0,thedensity is constant andequal tooo.Then h=ood*(M? —F*),and otone @) Since thethree displacements defined inEqs. (121.1), (121.2) ‘and(121.3) aremutually orthogonal, theelement ofvolume isobtained bytaking their product. ‘That is ar=MFFinda; and theelement ofmass, dm=odr, is dm=Fadl. 125) SURFACE DISTRIBUTIONS OFMATTER 243 Henee, ifaisthepolar semi-axis oftheouter surface, and b that oftheinner one, the total mass Mis wo(2 (tta9ff Pvnauce a 4 @)=too,Medd=Groo(a?—b4), which isthesame asthat ofahomogeneous spherical shell ofwhich the bounding spheres have the radii aand band the density ofwhich isthe equa- torial density ofthe non- — homogeneous spheroidal shell. (ree) The surfaces ofconstant aN i} density aredefined (Eq. (2)) wN\ /} bytherelation x#/M* =acon- AC. a, stant. Since NN AY oN CA mitps=2, pr—pr= 2, PSs Saitisevidentthat,iftheratio|— /uisconstant, soalso isthe ratio p:/p: constant. Thesurfacesofconstantdensity\\,p71 are,therefore, spheres which \7.7S a have their centers ontheaxis. (~~ — ofthespheroid; furthermore ‘ye SON these spheres divide theline fof SN ofthehalf-rod externally and /; ainternally inthesameratio, Ne) asisindicated bythedotted ea ines inFig.73. pete Itisknown from thetheory ofSec. 115that this spheroidal shell possesses the property that itsinternal potential isconstant. Itisdesired toascertain the value ofthis constant, Ifan infinitesimal amount ofmatter dM isdistributed over alevelsurface inaccordance withEq.(2),itspotential atexternal points and onthe level surface itself isthe same asthough it were uniformly distributed over theline oflength 22which joins thetwo foci. Equation (1)then gives Lydtl. aV=5logaM; 244 THETHEORY OFTHEPOTENTIAL land thisisthevalue ofthepotential throughout theinterior ofashell ofinfinitesimal thickness which isbounded bytwo level surfaces, thedensity ofwhich isdefined byEq.(2). From Eq.(3),itisfound that dM =4ro<Xdn. Hence=42"?logtL, aV=227)" logSD, andfortheshell offinite thickness on fyoeMEY, veofpf2loga. ‘Theintegration ofthisexpression givesthevalue oftheinterior potential, viz.: _2oof ysgedt24poeat—2)+art. va2[xlog$A+FlogO~F)+e |Atpoints exterior totheshell thepotential isthesame as though allofthemass oftheshellwereuniformly distributed alongtherod;thatisMM, d+!V=3 le, andtheshell attracts anexterior particle justthesame asthe rod does. ‘Theclectric image ofthis shell inanysphere whose center liesintheinterior oftheshell, is,ofcourse, acentrobaric body. 126, The Surface Density Necessary toProduce Given Potentials.—It wasshown byGreen asoneoftheearliest appli- cations ofhistheory thatifaclosed surfaceSisgiven,ifVi(z,v2) isharmonic (See. 62)inside ofSand V.(z, y,2)isharmonic outside ofS,vanishing atinfinity intheorder of1/r,andif V;=V.onS,there exists oneandonly onedistribution of matter onSforwhich V;istheinternal potential andV,is theexternal potential. Inorder toprove this,Iet2,y,#bethecoordinates ofapoint pinside ofSand§,1,¢thecoordinates ofanypoint on8.If thedistance between these two points is paVER FOTO 126) SURFACE DISTRIRUTIONS oFMaTTER 245 thevalue ofV;atthepoint pisgiven bythesurface integral, Eq.(63.3), 1 10V; afl theexternal normal derivatives being taken. Outside ofSthe function 1/p isharmonic. The function V-also isharmonic and vanishes atinfinity intheorder of1/r. Hence, asisproved inSec. 63foranexterior point, 11a. a(t o~ffLoan~Pons) ® InEq.(1),thenormal derivative istaken outward with respect toS,and inEq. (2)itistaken inward. The inward normal derivative and the outward normal derivative of1/p differ only insign; onthesurface S,V:=V-;hence, ontaking the sum ofEqs. (1)and (2), itisfound that _1(favs, a.Jaw Vile,v2)=PaleFan,|= i) Ifthepoint p,atwhich thepotential isevaluated, liesoutside ofS,itisnecessary only tointerchange the subscripts eand i inEqs. (1)and (2),and therefore, also inEq. (3); that is _Life, a.Jaw, Vides2)=Ese+orof cS) theform oftheright member remaining unaltered. Now letadistribution ofmatter bemade onSinsuch away that thesurface density is 1fav. ,av, 5) o=rab+x} © which isdefinite and unique since the normal derivatives of V,and V,aredefinite and unique. The potential due tothis distribution ofmatter is vefof,an Ifthepoint atwhich Visevaluated liesinside of8,Eq. (3) shows that V=Vi,and ifitliesoutside ofS,Eq. (4)shows that V=V.. Hence, one and the same distribution ofmatter on Sproduces the potential V;inside ofSand the potential V. outside ofS. 246 ‘THETHEORY OFTHEPOTENTIAL 127. Green’s Problem.—The functions V.and V;are not as independent astheabove proposition might lead one toinfer. Indeed, ifV.,harmonic outside of§and vanishing atinfinity, isgiven arbitrarily, there exists, atmost, but one function V,which isharmonic inside ofSand equal toV.on S. Suppose there aretwo such functions, and that thesecond funetion is V;+W. Then Wisharmonic inside ofSand equal tozero onS. Itis,therefore, (Theorem I,Sec. 75)equal tozero every- where inside ofS,and there can bebut one function, ifany atall,that isharmonic inside ofSand equal toV,onS. Suppose theexterior potential isgiven, then Green's problem for the interior ofasurface can beformulated asfollows: Green’s Interior Problem.—Given aclosed surface S,does there exist afunction V(x, y,2),which isharmonic within Sand which takes @given continuows setofvalues onS? ‘The equation, Eq. (63.3), vend ELLE r2Q)foaw requires aknowledge notonly ofthevalues ofVibut also of the values of8V;/dn onS. But since, ifthe function V,exists atall, itisunique, V;and aV;/an arenot independent, and Green’s formula requires more information than the question premppons. Green himself observed that ifthere exists afunction 1G=A+4H, (2) inwhich Hisharmonic inside ofS,equal to—1/p onS,sothat Gvanishes onS,and admits afinite well defined normal derivative onS,thevalue ofV;atanyinterior point isgiven bytheequation 1 ae " Viz,y,2)=~bfrit, (3) anequation which isobtained from Corollary IinSee. 63by taking ¢=Viand ¢,=G,and remembering that G=OonS. ‘The function G,which isknown asGreen’s function, isassociated with the surface S,and isentirely independent ofthe function Ve ‘The problem, therefore, can bere-stated asfollows: Given a closed surface $andapoint pwithin italwhich pvanishes, does 127] SURFACE DISTRIBUTIONS OFMATTER 247 there exis afunction G= +HYinwhich His harmonic inside ofSandequalto—1/ponS? Evidently asimilar problem exists fortheregion exterior tothe surface S. Itcan bestated asfollows: Green's Exterior Problem.—Given aclosed surface S,does there exist afunction V.(z, y,2),which isharmonic outside ofS, vanishes atinfinity, and takes agiven continuous setofvalues ons? ‘The equivalent reduced problem isGivenaclosedsurface$andapointpousideofitatwhichp tanishes, doesthereexistafunction G=™4-H,inwhichHis harmonic ouside ofS,vanishes atinfinity andisequal to—1/p on Ss? 128. Certain Physical Considerations.—From certain physical considerations, Green was satisfied that the answer tothese questions isinthe affirmative. Suppose the surface $isa perfect conductor ofelectricity which ismaintained atzero potential under allcircumstances byawire, which also isa perfect conductor, connected with theearth. "Ifaunit particle ofpositive electricity isplaced atapoint pinside of$acertain charge ofelectricity isthereby induced upon S. The potential atanyother point inside ofS,duetoboth charges ofelectricity, is 1 G=ath where 1/pisthepotential due totheunit charge atp,and H isthepotential due tothesurface charge induced onS. Since His. potential due toasurface distribution on§,itis harmonic inside ofS;and since the surface Sisgrounded, itspotential isalways zero. Hence Gvanishes onS,and Histheharmonic function inquestion. Asecond example from thedomain ofphysics isthefollowing: ‘Suppose theindividual points ofthe surface ofabody are main- tained atconstant temperatures, although thetemperature may vary from point topoint inany continuous manner over the surface. Inthe course oftime, the interior ofthe body will reach astate ofthermal equilibrium, inwhich thetemperature 248 THE THEORY OF THE POTENTIAL atany given point remains constant. The function Twhich represents the temperature ofthe steady state isharmonic, for theequation 47’=0means that theheat received anddischarged ateach point isthesame (Sec. 56). ‘The temperature also takes prescribed values onthe surface. Therefore, 7’satisfies the conditions required inGreen’s problem 129.TheExistence ofGreen's Function —Givenanyclosedsurface $and apoint pcither within itorwithout it. Let p bethedistance measured from ptoany point. Ifpisinside ofS,itisrequiredtofindafunction Hwhichisharmonic insideofS,and equal to—1/p onS. Ifpisoutside ofS,itisrequired tofindafunctionwhichisharmonic outsideof8,vanishesatinfinity,and isequal to—1/p onS. Letasphere >ofradius abedescribed about thepoint pasa center, and letthesurface Sbetransformed into the surface S* bythemethod ofreciprocal radii (See. 105) with respect tothe sphere 2.Let aquantity ofmatter Qbedistributed upon the surface S*insuch away that thethepotential due tothedis- tribution isconstant within and onS*(See. 117). The electric image ofS*inthesphere Zcoincides with the given surface S,and theresulting distribution ofmatter onS iscentrobarie with respect tothepoint pforpoints outside ofS ifpisinside, and forpoints inside ofS,ifpisoutside (Theorem IV,See. 114). Ifpliesinside ofS,the quantity ofmatter Qcanbechosen sothat themass ofthedistribution on$isunity, andthepoten- tialofthis distribution atalloutside points is1/p. IfVsisthe interior potential ofthis distribution, V;isharmonic inside ofS andisequal to1/ponS,since thepotential function iscontinuous across S. Hence the function H=-V; satisfies alltherequirements ofGreen's interior problem. Ifpliesoutside ofS,thequantity ofmatterQcanbechosen sothat thepotential ofthedistribution on$is1/patallinterior points. IfV.isthe exterior potential ofthis distribution of matter onS,thefunetion V,isharmonie outside ofS,vanishes atinfinity andisequal to1/ponS.Hence H=-v. 129) SURFACE DISTRIBUTIONS OF MATTER 249 isafunction which satisfies allofthe requirements ofGreen's exterior problem. ‘The existence ofasolution ofGreen’s problem forany closedsurfaceSisthusmadetodependuponGauss’theorem thatthere exists one, and only one, distribution ofagiven quantity ofmatter upon agiven closed surface Sforwhich thepotential isconstant onS.Indeed, Gauss’ problem isequivalent toGreen’s problem. 130. Miscellaneous Properties ofGreen’s Function. (a)Green's interior function, which isalways relative toa closed surface Sand afixed point p,isthe potential ofacertain distribution ofmatter; namely, aparticle ofpositive matter of ‘unit mass located atthepoint pand acentrobaric distribution of negative matter relative tothepoint ponthesurface S,ofwhich the total mass is—1. The potential ofsuch adistribution is zero on§and everywhere outside ofS,forthenegative matter repelsanexterior particle withthesameintensity andinthesamo straight line asthepositive particle attracts. Hence, @iszero onSand everywhere outside ofS. (®)The exterior and interior potentials ofthesurface distribu- tion onShaving been determined, namely, vee AT, i thedensity onSwhich isnecessary toproduce these potentials is given byEq. (126.5); namely, 1favy ,av, onpape+v2]| w1@ *aeOne ) Inview ofthis result, Eq. (127.3) becomes: Vie,va)=~VG9Deleweb Nee, @) where z,y,zarethecoordinates ofaninterior point p,and &,9,¢ thecoordinates ofapoint onthe surface. Therefore, ifV;is anyfunction which isharmonic inside ofS,andifitsvalues are known onS,itsvalue atany point pinside can beobtained by integrating theproduct —oV; over thesurface. 250 THETURORYOFTHEPOTENTIAL Letp;with thecoordinates 1,ys,2:bethefixed point and p with thecoordinates x,y,zbethevariable point. Then Green's function relative tothe fixed point p;is G=}+Hlenmvaie ws where n= Vea TOW tm, and G,=0,ifthepoint pliesonS. Ifthepoint p;isinterior tothesurface, G,isGreen’s interior function relative tothe surface Sand the point p.;and ifp; isoutside ofthesurface, G,isGreen’s exterior function relative tothesurface Sand thepoint pi. (©)TheInterior Function isPositive Everywhere within S.— Since mG, =1+pls, and H;isfinite everywhere within S, lim,eG,=+1. Hence, onasmall sphere 2with itscenter atthepoint p,thefunetion G,ispositiveandverylarge.OnSitvanishes; andin theregion between 2andSitisharmonic. Itis,therefore, Sec. 75,positive everywhere inside ofS. (d)TheExteriorFunctionisPositiveEverywhere outsideofS— Ifitisborne inmind that, fortheexterior function, H;vanishes atinfinity, thesame argument asabove shows that theexterior function iseverywhere positive outside ofS. (e)InTheir Respective Domains theGreen Functions areEvery- where Less than 1/p.—Since H,=—1/p; onthesurface and is harmonic inthedomain ofitsvalidity, itisnegative everywhere within that domain (except atinfinity where it 3 vanishes). Hence G,<1/peverywhere, save at infinity where G;vanishes. Gf)The Same Point butDifferent Surfaces.— Suppose thefixed point: p;lieswithin thesurface S,,and that thesurface S;iswholly enclosed Fro.74. bythesurface S,(Fig. 74). Let G bethe Green function relative top;and S,,and@,'® betheGreen function relative top:andSs,sothat 6=L+H, wy253,49, G0 = EMM wyaim wd, 130] SURFACE DISTRIBUTIONS OFMATTER 251 Within and onthesurface S;,itisevident that Gi? —GO =Hy —HO OnSy,thefunction Gi>0,by(©),x0that2+#7,>0. Also Hy? >—1oS. ms HY=+7onSe Hence Hy® —Hy >0onSy. Since H,® —H,® isharmonic inside ofS;and positiveonthesurface,itispositiveeverywhere insideofS;,andtherefore Gx>Gxeverywhere withinS,. a ‘Thispropositionisstilltrue,ifthesurface ‘S;lieswholly outside ofS:(Fig, 75), and if thesymbols have thesame significance as before, sothat G, isthe exterior Green function for the surface S,relative tothe pointp;.Theargumentissimilar. Since G, >0atallpoint outside ofS.,and Fro.75. GY =0onS,, GE. —G6, =H —1, >00nSy, and, since itisharmonic, itispositive everywhere inside ofS;. Therefore G,® >G\" everywhere inside and onS;. (9)SurfacesonWhichGreen'sFunctionisConstant.—Let Sbe 1closed surface, and @theGreen function relative tothe interior point p. Then Gvanishes onS,ispositive everywhere within S, and becomes infinite atp. IfGyisapositive constant, thesur- face S;,onwhich G=Go,lieseverywhere within S,and ifGois very large the surface @=Godiffers butlittle from thesmall sphere 5 1‘5 o=a 2@? since Hiscontinuous inS(Fig. 76). For very small values ofGo,thesur- face differs but little from the Bio.78. surface S. LetGsbegiven. Then desoribe asmall sphere 2about the point p.Inthespace between ®and thesurface So,thefunction 252 THE THEORY OFTHE POTENTIAL Gisharmonic. Itsmaximum values areon2anditsminimum value ison S. Inthevolume between SandSo,@iseverywhere lessthanG». Ithasderivatives ofallorders, sinceitisapotential funetion inempty space. Itisevident, therefore, that atall points ofthesurface Sptheexternal normal derivative ofGis negative. Consider now thesurface integral ofthenormal derivative ae av oHSst SiG) *Sse ByGauss’ theorem, Eq.(68.1), thefirstintegral intheright member isequal to—4x; andsince Hisharmonic inside ofs thesecond integral iszero (Eq. (62.2)). Hence agfule=ote 181. The Green Function isSymmetric.—Consider theGreen functions relative toasurface Sand the two interior points ‘pxand ps.LetG(pi, p)betheGreen function relative tothe point p,,and G(p2, p)betheGreen function relative tothepoint p:(Fig. 77), thepoint p being thevariable point. Itwill be shown that Foo.7. _GPyP2)=CPPr). For simplicity ofnotation, thefunctions G(p:, p)andG(ps, p)willbedenoted byG,(p) andG,(p). From Corollary II,Sec. 63,itisseen that ifS,isthe surface onwhich the Green function has the constant value G;=« L G2 3G)éf(om-oF,ye=Gl)—Od, which isindependent of«,provided S;contains both p;and p: inits interior. Onthe surface S,the function G;has the constant value ¢: which can beassmall asdesired. Hence L G2, _a:(GrteSica =Efisoa 131) SURFACE DISTRIBUTIONS OFMATTER 253 ‘The function Gsisnot constant onS,,but itiseverywhere positive. If¢:isitsmaximum valuc, then, since 3G,/dn is everywhere negative, Lf gar ef ay| waefoo <-ef,an ‘Therefore 180;_gerramosonyae<ate, which vanishes for¢,=0,since also vanishes with «, Hence G(p2) —Gx(p,) =0 rigorously, since itisindependent of¢,and theGreen function Gps, ps)=Gx m) issymmetric. Itwill beobserved that this proof does not assume that the normal derivative ofHexists and iswell defined onS,aswould bethe case ifthe integration had been taken over the surface Sdirectly. 132. The Normal Derivative ofthe Green Function isHar- monic.—Assuming that derivatives ofthe first three orders exist andarewell defined onS,itiseasy toshow that thenormal derivative ofG(p:, p)onSisanharmonic function ofthepoint p,. Let 2,u,»bethe direction cosines ofthe exterior normal to Satthepoint p(z, y,2). Then (liq. (54.2)) aG_\aG,a0,0GonOe+Hay+ae’ ‘The partial derivatives inthe right members are functions ofthecoordinates ofboth pand p,;that is,they arefunctions of ©,VY;2%24Yy21. The direction cosines \,4,varefunctions of2,y;2,butnotfunctions of21,yi,21. Let a. a, a Aant? aye*ae” Then O_O, aedigg=MageTtHagyTMG a a a GAG) +mg(AiG) +75,(0:6). 254 THETHEORY OFTHEPOTENTIAL ‘The function Gsatisfies theequation ofLaplace intheletters 2,y,2byhypothesis, andsince itissymmetric inthecoordinates ofthepoints pandp,,itfollows that, AG =0, and therefore ag Aah =0. 183. The Green Function forthe Sphere.—Let S»,Fig. 78, with the radius a,and center at©bethe given sphere; and DerayeA7sseG ee Se oe S,syannBy Fro. 78. let0bethepoint inside ofS;with respect towhich theGreen function istobedetermined. Deseribe anysphere Sofradius aabout Oasacenter, and letS;ofradius a;and center O: betheinverse ofS;with respect toS.IfCO=rand 00,=r, thefollowing relations areobtained from Eq.(105.3): ar aayneg asa @ Ifthepoint 0sistheinverse ofOwith respeet tothesphere ‘Ss,and00;isdenoted byr2,then rae, a FR From thisequation and thefirstofEqs. (1)itisfound that noeae8 @ which shows that thepoint 0;isalso theinverse ofthepoint 0,with respect tothesphere S.This factpermits thelocation ofthepoint ;graphically bytheusual method. 133] SURFACE DISTRIBUTIONS OFMATTER 255 Now letS;becovered uniformly with alayer ofmass My, and letS;bethe electric image ofS;inthe sphere S. Let P,beany point outside ofS,,and P,itsinverse with respect toS;also letOP: =R:, OsP2 =Rs. Since thecenter ofSliesinside ofS,,thepoint P,lies inside ofS; The potential ofS;atP,is M: Vie=Re Therefore, byLord Kelvin’s principle, the potential ofS:at Pris (Sec. 108) aM VanRY thesubscripts ¢andidenoting external andinternal respectively. Also, thepotential ofS,attheinternal point P;is Vy=, a therefore thepotential ofS:attheexternal point P,is Vue 2M,a Inorder that themass onS;shall beequal tounity (Sec. 129), itisnecessary that thecoefficient of1/p, inV2.shall equal unity. Therefore M,== oe, (byEa.(1))5 and thedensity onS;is 1* Fraay ® This value ofM;makes 1 a Vemoo Va Ry Green's function, then, isthe difference between these two potentials, both taken atthepoint P:,namely alm,G-paTR o Itisevident from this expression that Green’s function forthe sphere can beregarded asthe potential oftwo particles: The first ofmass +1located atthepoint O,,and thesecond ofmass —a;/r located atthepoint O.. 256 ‘THE THEORY OF THE POTENTIAL Equation (4)remains unaltered even though the point Ps liesoutside ofSs. Itrepresents, therefore, either theinterior function ortheexterior funetion. Itisasimple matter toshow directly that Gvanishes ifthepoint Psliesonthesurface ofthe sphere Ss The Green Function isSymmetric—Referring toFig. 79, theGreen function relative tothesphere Sand thepoint 0, atthepoint Q,is 1_a e-1-5, | Fra. 7, and theGreen function relative tothepoint Q,atthepoint Oris laCn-r where Qoistheinverse ofQ,and Osistheinverse ofthepoint 01. Itisproposed toshow that these two expressions are equal The triangles CO,Q: and CQ,02 aresimilar, for mea, th=a, and theangle atCiscommon. Since two sides areproportional‘andtheincludedangleisthesame,allthreesidesareproportional.That is, tt a a ‘From this itfollows that rR=f, 133] SURFACE DISTRIBUTIONS OFMATTER 257 andthetwoexpressions forGareequal. Toputthesymmetry inevidence, itisasimple matter toshow that R= =VEE =Oae. 134, The Normal Derivative ontheSphere—Taking Green's funtion inthe form l_a o-)-& the normal derivative of@is Gil, a .inii008Fp+=F,cosWR @ for,according toEq. (66.1), 20 oe|ORL 20=cos, 2=coneR. & Fro. 80. Itiseasily proved directly however. Since p?=r?+a?—2arcosar, thereresults,onvaryingaandp,thedistancerandtheangle© G@remaining constant, a_aancean~aa>eosne and similarly forR. From thetriangle (Fig. 80)OP0,, there isobtained rat=a?+R*—2aRcosmR. @) Since @vanishes onthesurface, andrr;=a?always, R=%, nat. 258 THE THEORY OF THE POTENTIAL Onmaking these substitutions inEq. (2)and then multiplying ‘through byr3/a*, there isobtained a?=12+pt—2rpcoswR. Likewise 1=at+p*—2ap00ip isobtained directly from the triangle CPO, The difference between these two equations gives a?—12=(acosfp—rcos*R). Iithis equation isdivided through byap* and the coefficient 1/(ap*) isreplaced byitsequal a/(rR*), there results at—1?_costa_acosnitTar BE RE ® Acomparison ofEqs. (1)and (3)shows that, aG _rt-a? on=apt @) TheDistribution ofMatter ontheSphere.—By Eq. (130.1) the density ofthematter distributed over thesphere, taken posi- tively, is ~-1@_id-7oSEeén~deaps” that is,inversely proportional tothe cube ofthe distance from thepoint 0;. The minimum value ofpisa—r,andtherefore, the maximum value of¢is ~1a+r omia ‘This expression shows that themaximum density tends toward infinity asthe point ofinversion (the pole ofGreen's function) approaches the surface ofthe sphere. Diametrically opposite, the density tends towards zero, The matter shows astrong ten- deney togather about the point Hh ofinversion. The dotted lines in Fro,81. Figs. 81and82show thedistribu- tionforr/a=.Sand.9 respectively. IfRistheradius ofthesphere ofinversion and a-rsRsa+r, 134] “SURFACE DISTRIBUTIONS oFMATTER 259 itiseasily shown that theamount ofmatter which liesonthe spherical capwhich isinside thesphere ofinversion is attr _ orm= or ark 9 Pia. 82 ‘The limit ofthis expression asra is+1forevery R. At thelimit allofthematter onthesphere liesinside ofthesphere ofinversion, however small that sphere may be. The rest ofthesphere isbare. The Normal Derivative IsHarmonic.—Let the coordinates ofthepoint 0,be2,y,2,and thecoordinates ofthepoint P, which lies onthe surface ofthe sphere, be&,f,80that Fert oeaatLet aye trtene [@- P+ G-+@-H# Itwill beshown byadirect computation that Nisharmonic with respect tothe variables 2,1,2 The first differentiation with respect toxgives aN 2_8-OG ty tea), or ; and thesecond differentiation with respect to=gives ON 2_Wee) _Be—a) 15—HA—at) act pt o a 7 260 THETHEORYOFTHEPOTENTIAL Hence, also, BN21My—9)_B62a9)15—HGFad, ay? BF om * a ON 2_idle —1)_30%—a),15~Hot—0) pa eS # uy ‘The sum ofthese equations gives AN=Sip—(a—9+vy=9)+eleD)+r?—at If @-Feete,teddy te, =~ D+F EDS aresubstituted intheright member ofthis equation, itisfound that AN =0; which shows that the normal derivative ofGisanharmonic function ofthecoordinates ofthepoint 0. 135. Green's Equation forthe Sphere.—It was proved in Sec. 127forany closed surface Sthat ifVisharmonic inside ofSandifitsvalues areknown onS,itsvalue atany interior point p(z, y,2)isgiven bytheequation Venue) =pfSrcnto Forthesphere this equation becomes 1a?—7 Veewa=25[2S2V6 adae, @ where Pettyta pao tat +e—De Inspherical coordinates, apoint onthesurface is §=asinyg,cos6,n=asinsin6),£=008w; andthecoordinates oftheinterior point pare 2=rsin go008A, y=rsin gysinbo, 2=7COSyp. Inthis system, thenorth pole oflatitude liesonthez-axis and thezerooflongitude onthez-axis. Theargument ¢isthedistance from thenorth pole. 135] SURFACE DISTRIBUTIONS OFMATTER 261 Take anew system ofreetangular axes X,¥,Z,with thesame origin but with the Z-axis passing through the point p,the X-axis lying inthe zy-plane and the positive end ofthe Y-axis lying always inthe northern hemisphere. The equations of transformation are w= aX +mY +ard, y= BX +BY +82, eanXtn¥ tnd, . where the a,8,7,'s are the direction cosines ofthe new axes with respect tooldones. ‘The values ofthese direction cosines are! a,=—sino, az=—608¢C08Ho,a3=SinaCOSOy; Bi=+005 , Br=—cos aasin% 8s=sinaysin 9; n=0, v2=+sine 1s=008 yo. Ifgisthepolar distance and @isthelongitude with respect to theX,Y,Z-system, then forpoints onthesurface ofthesphere thepolar distances and the longitudes inthe two systems are related bythe following equations: sin¢,c08@;=a;sinycos#+azsinpsin9+a3cosy,sing:sin6,=B;sinycos9+Brsinysin0+B;cos¢, cosyi=71sin¢cos0+y28inpsin6+3cosg. Inthecoordinates ofthe new system deo=a?singdpae, pt=at+1—2ar cos¢, and, since pisindependent ofthelongitude, Green’s equation (Eq. (1)) can bewritten a(Tare ae Vena=p), ao sinede |VieAds, 2) theintegration with respect to6depending only upon thefuntion V(¢, 6). The integration ofthis equation bymeans ofspherical harmonies isgiven inSee. 208. 136. AGeneralization for the Sphere.—Green’s equation was derived upon theassumption that Visharmonic inside of Sand that itsvalues are given onS. Suppose the values of +Station and the Dynamics ofaParticle,” p.987. 262 THE THEORY OF THE POTENTIAL Varegivenonthesurface, continuous butotherwise arbitrary, without any statement astothenature ofVelsewhere. Green's equation, 1fart Wee,v2)=Pn 1S)de, a certainly defines some function ofz,y,zinthe interior ofS. Ttisthepurpose ofthepresent section toshow that thefunction ‘80defined isnecessarily harmonic inside ofS,and that thelimit ‘ofWasthepoint papproaches thesurface isthevalue ofV atthe point ofapproach. TheFunction WIsHarmonic.—Any functionFcanbeexpressed inrectangular orinpolar coordinates, where z=reosgcos8, y=reosysing, z=rsing. Consequently _oF, a, oF,_oF, OF, oF OF=a+ay!+Ftd=oror+ae!”+Fra(2) forany setofvariations. Ifthe variations are taken radially, ég=50=0,and the relation between the other variations are a Fyyu2fT Hence Eq. (2)becomes oF oF OF OF. "Sp~on+Vay+G5" @) Returning toEq. (1),take 1 EVGnn=o ® and therefore atrt Ww=Faeroe Now let U=fimso sothat UisaNewtonian potential forthedensity «.Then o,f al? or sor\o and, since pt=1?+a?—2arsing, 136) ‘SURFACE DISTRIBUTIONS OFMATTER 263 itfollows that a(t Zoe argh2r2()=—3(rt~arsino) 1 =-he- a+p. Therefore oo fs, atrf are=figeorfirodeo .=W-uU. Hence wavs 2tco _ av, a, av~ve Fy+) Bydifferentiating twice with respect to2,y,2there isobtained aw _ au au, au au aa?7PGa3+(2+Yayaa8+au) ew aU FU aU auap~Pap+(ssa +05+aan) ew _aU aueu|aw ae7aetcnaFvayaet+22). ‘The sum ofthese expressions is sau ole? xy? 2 AW=5aU+(2.+Yay+#2)au=0. HenceWisharmonic, since,evidently, itiscontinuous.W=VontheSurface oftheSphere.—It stillremains tobe shown that, ifthepoint papproaches the point Ponthesurface along any line whatever, thelimit ofW(p) isV(P). Suppose papproaches Palong aline which makes anangle with thenormal, and that wo)=(9fa. Imagine Pasthe origin ofasetofrectangular axes, z,y,z, with the axis directed toward the center ofthe sphere, and 264 THETHEORY OFTHEPOTENTIAL therefore coinciding with thelinePC,Fig. 83. The coordinates ofpinthis system arez,y,2and PA =z =pocosh. PSincezisaconstantintheprocess Ky‘ofintegration,theexpressionforW Lycan bewritten f @—r z W=oexf.ote Since a?—1?=apy cos k—po®, Fie.Si itisevident that in andim,prooa 72% ‘The integral f.2edsso isthe normal component oftheattraction ofasurface with thedensity (Eq. (4)) =,v.otra Itwas proved inSee. 92that thelimiting value ofthisattraction is2re. Hence thelimiting value ofWis =20-2"+.= limW(p)=20°2nGs=VP). Hence Green’s equation aa" (VEnn), Ve,y,2)=Ff.das @ definesafunctionwhichisharmonicintheinteriorandtakestheprescribed setofvalues onthesurface. ‘The above proof, which ispurely analytic incharacter, is given byPoincaré. Aproof which makes astronger appeal tothe intuition can beobtained from the results ofSee, 134. Green’s equation canalso bewritten, Eq.(130.2), Wane) =+£V(69,tedo, (3) 136] «SURFACE DISTRIBUTIONS OFMATTER 265 ifthenegative matter inthedistribution isreplaced bypositive matter. Let the integral over the whole sphere beseparated into thesumoftwointegrals, oneofwhich,toistakenoverthe spherical capdescribed inSec.132,andtheother,SListaken over theremainder ofthesphere. Ifthepole ofGreen's function (z,y,2)issufficiently close tothe surface, the radius ofthe sphere ofinversion can betaken sosmall that the value of V(E,1,2)isessentially constant overthecap.Hence SVG 4Neds=Veofiede, and SV GsmDodoSVowfede, where Voisthevalue ofVatthecenter ofthecap and Vinx is themodulus ofthe maximum value ofVonS. Since Jimfede=1andlimfiedo=0, itisevident that limW=limf,Vode =Vo. 187. Green's Equation for Any Surface—Greon’s formula, Eq, (127.3), Ved =P renngoe vesnd, where 2,y,2arethe coordinates ofthepole ofGand &n,£ arethecoordinates ofthe surface clement dw, was derived upon the assumption that Visharmonic inside ofSand that its values onSare known. Ifacontinuous, but otherwise arbitrary, set ofvalues is specified upon S,theright member ofEq. (1)defines acertain function ofx,y,2which can bedenoted byW(x, y,2). Since the integrand isfinite everywhere onS,itisevident that W(z, y,2)iscontinuous and single valued everywhere within S. Derivatives ofallorders exist, since they exist for@. Therefore 1 ‘ag aw=-Efre»a()au=0, 266 THETHEORY OFTHEPOTENTIAL since the normal derivative ofGisan harmonic function of z,y,%(Sec. 132), The function W,therefore, isharmonic inside ofS,since itissingle valued, has derivatives ofthefirst twoorders, and satiafiea theequation ofLaplace. lanCD Inthecase ofthesphere, itwasshown inthepreceeding section that as0,the pole ofG,approaches the surface, the value ofWtendstowardthevalueofVatthepointofapproach. It isdesired toshow that thesame property holds forany surface. Letp,thepoint approached, bearegular point ofthesurface a foe 'Sinthesense that ithasadefinite tangent plane and two prin- cipal redii ofcurvature. Itisthen possible todescribe two spheres S,and S;which are tangent tothe surface atp,one ofwhich hasaradius equal totheminimum radius ofcurvature and theother themaximum. Letathird sphere bedescribed about pasacenter with aradius R. IfRissufficiently small, 137] SURFACE DISTRIBUTIONS OFMATTER 267 the portion ofthe surface intercepted bythis sphere will lie wholly between S;and Ssiftheradii ofcurvature have the same sign, and outside ofboth ofthem ifthey have opposite signs. ‘Let Obethe pole ofGreen’s function. About 0asacenter describe asphere Z,theradius ofwhich Ris kept fixedasthepoint approaches thepoint p. Letthesurface S*bethetransform ‘of$byreciprocal radii with respect tothe sphere 2,Fig. 84 ‘The portion ofSwhich lies inside of2istransformed into theportion ofS*which lies outside of2;and the portion of 'Swhich liesoutside of istransformed into theportion ofS* st — Fro. 86, which liesinside of2. Asthe point 0approaches the point p,theportion ofS*that liesoutside of2expands and rapidly ‘approaches theform ofasphere, Fig. 85,while theinterior portion contracts. Itismuch asthough thesurface 8were afilm of soap solution, the portion lying within 2being blown into a large bubble while the remainder ofthesurface contracts toan insignificant irregularity upon it. ‘That thebubble approaches theform ofasphere asitincreases insize isseen from Fig. 86, ifthe two radii ofcurvature have the same sign. ‘The portion ofthe surface Swhich lies inside thesphere ofinversion 2also liesbetween thetwo spheresS,andS;whichistheregionthatistransformed intothevolumebetween thetwospheres S;*and S;*, ‘The point pistransformed into thepoint q;and since S;and S;aretangent toeach other atp,the spheres S* and S:", which are the transforms ofS: and S;,are mutually tangent atg. Asthe point ofinversion 268 THE THEORY OF THE POTENTIAL Oapproaches thepoint p,thepoint recedes insuch away that theequation Op-0g=Rt isalways satisfied. Ifthetwo radii ofcurvature ofSatphave different signs, theportion ofSwhich liesinside of2liesoutside ofboth S, and S;, Asisseen from Fig. 87,theregion which isoutside of 'S;and S;transforms into the volume which lies between S,* and $2", of Fao. 87. Ineither case theportion ofthesurface $which liesinside of©istransformed into aportion ofS*which lies between ‘S\*andS:*andwhich passes through thepoint g.‘The portion ofSwhich liesoutside of©istransformed into aportion of ‘S*which liesinside of2.Itisevident therefore that ifOpis very small thesurface S*isessentially avery large sphere with small irregularity near p.Consequently, inthedistribution ofalevellayeronS*,thedensityisverynearlyconstant, andinversely proportional tojg,Sec. 133, andthelimit ofthis density asOpdiminishes iszero. Consider now theelectric image ofS*. Itcoincides geo- metrically with 8,butnearly allofthemass liesonthatportion ofSwhich isinside of2.Letthisportion ofSbedenoted by Candtheremainder ofthesurface byD.‘Then Sede=1=[lode+rods. 137] SURFACE DISTRIBUTIONS OFMATTER 269 Since the limit ofoover Diszero itfollows that the limit ofthe lastintegral iszero, whatever Rmay be. Hence imfeds=1, forevery Rwhich issufficiently small. Since V(,,f)isfinite onS,itisevident also that SiVode $[Vineefrode, which hasthelimit zero. Consequently limW=lim{,Vede, forevery Rsufficiently small. But Rcan betaken sosmall that Visessentially constant over Cand equal tothe value which ithas atp. Hence lim[,Vodo =V(p){iodo=Vip), and lim W=V(p) 138. AGeneral Theorem ofGreen’s.—Given aclosed surface Sand acontinuous setofvalues V(E, ,¢)onS,Green’s equation (Eq. (137.1)) defines afunction V,(z, y,2)which isharmonic within Sand equal toV(g, 1,¢)onS,provided GisGreen's interior function. IfGisGreen’s exterior function, thefunction Vz, y,2)80defined isharmonic outside ofS,isequal toV(t,n,¢) onS,and vanishes atinfinity. Itfollows atonce from theanalysis ofSeo. 126that V;and Y,aretheinterior and exterior potentials ofadistribution of matter onSinwhieh, Eq.(126.5), _Tf, av],c=rab+al Hence, thegeneral theorem due toGreen and also toGauss: Theorem.—Given aclosed surface Sand acontinuous setof valuesV(E,n,t)onS,thereexisteoneandonlyonedistribution *ofmatter onSforwhich thepotential onSitself isequal toV. GREEN'S PROBLEM FOR THE LOGARITHMIC POTENTIAL 139, Statement oftheProblem.—It ispossible also tostate Green's problem foranattracting line and thelogarithmic potential. Fortheinterior problem itis 270 THE THEORY OF THE POTENTIAL Given aclosed plane contour C,itisrequired tofind afunction which isharmonie inside ofCand which takes @given continuous setofvalues onC. For theexterior problem itis Given aclosed plane contour C,itisrequired tofind afunction hich isharmonic inevery closed region outside ofC,and takes given continuous setofvalues onC. TEwillhave been observed that practically allofthetheorems relating tothe Newtonian potential have their counterpart forthelogarithmic potential. Itwould beexpected, therefore,thatthemethodwhichsolvesGreen’sproblemfortheNewtonianpotential isalsoadaptable tothelogarithmic potential. There isonestep inthemethod, however, where there isadifference, and that isin thedefinition ofelectrical images. 140. Electric Images for the Logarithmic Potential—In See. 108(using thenotation ofthat section) Mzissaid tobe ‘theelectric image ofM,ifeach clement ofvolume ofM,isthe transform byreciprocal radii ofthecorresponding element of ‘M,,andifforthecorresponding clements ofmass ts dm,=gam. For alogarithmic potential and aplane area, Mzwill becalled theelectric image ofM,ifcach clement ofarea (orline) ofMz isthetransform byreciprocal radii ofthecorresponding element ofMy, and ifthecorresponding elements ofmass are equal; that is dms =dm. Referring toFig,65,letSbethecicle ofinversion with radius aand center ofinversion atO.Let aparticle ofmass mbe Tocated atMyand asecond particle also ofmass mlocated atMs which isthetransform ofM;. LetP;beanyother point andP itstransform, Let Uybethe potential ofthe particle atIf atthepoint P;,and Uzthepotential oftheparticle atM:at ‘thepoint P;. Then N=miog Us=mlogB where Reis anarbitrary constant; also, asinSee. 106, s)Halk, nispona, Bata Be 140) SURFACE DISTRIBUTIONS OFMATTER 271 ‘The difference ofthetwo potentials gives Ry_ Tm Us=Us+mlogi!=Us+mlog Ro Ro, =U,~mlog ©?+mlogBe, or Us=Uy—Un+mlog® ® where Ujo isthe value ofU,atthe point O,and therefore inde- pendent oftheposition ofP;orP:. Suppose there isadiscrete setofparticles My, and acorre- sponding discrete setM:. There will beacorresponding set ofEq. (2). The sum ofthis set gives the relation between thepotentials ofthe two sets atP;and P;. Thus, ifVsis thepotential ofthefirst setatP;and V;isthepotential ofthe transformed set atP:, sothat Vi= 3, Vi=3Uy, M=Im, then Va=VinVin+atog. ® Iftheparticles form acontinuous aggregate, such asaline orarea, Eq. (3), evidently, isstill true. Ifo;and o;are the densities ofsuch aggregates, itiseasy tosee, onaccount ofthe relationship dm; =dms, that a.=S11 (linedensities), a ® a1=Fo:(arealdensities). 141. Electric Images ofCentrobaric Bodies.—It isnow easy toprove theorems analogous tothe four theorems ofSec. 114. ‘The bodies referred tointhese theorems are, ofcourse mass distributions onplane areas, and thetheorems hold only inthe plane. Theorem I.—If abody iscentrobaric forexterior points, its electric image inany circle whose center liesoutside ofthebody also iscentrobaric forexterior particles. 272 THETHEORY OFTHEPOTENTIAL LetMbethemass ofthegiven body. Since itiscentrobaric, itspotential is(Fig. 69) Vi=Mlog7 2 2, z z Is Lip Foo, 00 and thepotential ofitselectric image is Ro Ry Ro Va=Mog#!~Mlog™+Mog? Ron, =Mlog5 whieh, byBq. (1),becomes Va=AtogBe; that is,the electric image attracts exterior particles toward thepoint 0;just asthough allofitsmass were aparticle atO.. Theorem I1—If thebody Byisaring which iscentrobaric forexterior particles, itselectric image Bsinany circle whosecenterOliesintheemptyinteriorofthering(Pig.70)isaringwhich iscentrobaric forinterior particles. Since B,iscentrobarie itspotential forexterior particles is Ke Mtog thepoint 0;being thecenter ofattraction. Since P;liesoutside ofB,,thepoint P;liesinside ofB:. Hence thepotential of 141] SURPACE DISTRIBUTIONS OFMATTER 273 theelectric image ofB;,that isBs,atpoints interior toByis Ro_ Ry, Ve=Mlogy?—Vio+Mlog Rit =MleeRP,—Vio, =MogReVi =Mog@+(140BoVio) °£§DL> E SS 228,Liam Laas To. 70. Since thepoint Oisinterior toBi,thevalue ofitspotential at Oisnot known. Since itdoes not depend upon the position ofthepoint P;orPs,theexpression within theparenthesis issome constant, thevalue ofwhich isnotknown. Denoting,byQo#certainconstant, theexpression forVzcanbewritten = Q, Va=MlogRY which shows that particles interior toB:areattracted towards O,justasthough aparticle ofmass Mwere located there. ‘That is,B:iscentrobaric forinterior particles. ‘Asthepoint0tendstoward 0,,thelines2,andp,tendtoward equality, andthepoint Osrecedes indefinitely. Hence thelimit ofV2asOapproaches 0,is Re Re li2=Mik -—Vi= Mk =Vay limV; 08ipsVv lOot =MogBt—Vu,=const, Vinbeing thevalue ofV;atthepoint 0, Hence 274 ‘THE THEORY OF THE POTENTIAL Theorem IIIf aclosed ring ofmatter iscentrobaric with itscenter ofmass Ginitshollow interior, itselectric image in any circle which has Gasitscenter isaring ofmatter forwhich theinterior potential isconstant. ‘Theorem IV.—If thepotential ofaring isconstant initsinterior (orifitisalevel thread), iteelectrical image inany circle whose center lies intheinterior iscentrobarie for exterior particles, and thecenter ofmass oftheelectrical image isatthecenter ofinversion. Ifthecenter ofinversion lies outside thering theelectric image iscentrobarie with respect tothepoint Oforparticles inisinterior. Since Re Vi= Vi- Vet Mlog®, pe ifthepoint ofinversion isintheinterior, Vi-Vu=0, and therefore, forexterior particles, Ro Ve= Mi ’°Foe which states that exterior particles areattracted just asthough1particleofmassMwerelocatedatthepoint0. Ifthepoint ofinversion 0liesoutside ofthering, Vis constant but notequal toVis. For this case Vs=ilog22+const. and the electric image iscentrobaric for interior particles. They are attracted just asthough aparticle ofmass Mwere located atthe point 0. 142. The Existence ofGreen's Function—According to Corollary I,Sec. 64,ifpismeasured from apoint inside ofC; if G= log +4, whereHTisharmonic insideof©andequalto—log}onC; and ifUisany other funetion which isharmonie inside ofC,then 1 UG,»)=-2[rian 0) 142] SURFACE DISTRIBUTIONS OFMATTER 275 ‘The same equation holds, byCorollary Ill, Sec. 64,ifpis measured from apoint outside ofC;if G=log”+H,log +H, whereHisharmonic outside ofC,isequalto—log°onC, and acts like alogarithmic potential atinfinity, and ifUisa logarithmic potential outside ofC. Ineither case, GisGreen's function for the given contour. Inorder toestablish the existence ofGreen’s function, letp bethepoint from which pismeasured. Deseribe acircle with any convenient radius aabout pasacenter. Let C*bethe transform ofCbyreciprocal radii with respect tothis circle, and letaunit ofmatter bedistributed along C*insuch away ‘astomake italevel thread, which isalways possible forprop- erly restricted contours, bySec. 117. The electric image of this thread inthe circle ofinversion coincides with thegiven contour C,and theresulting distribution ofmatter onCiscen- trobaric withrespect tothecenterofinversion—centrobaric for outside particles ifpisinside ofC,and centrobarie forinterior particles ifpliesoutside ofC(Theorem IV,Sec. 141). Let Viand Vebethe interior and exterior potentials ofthis centrobaric distribution. Ifthepoint pliesinside ofC, Vi=log’?jon” ‘The interior potential V;isequal tolog po/p onC,since the potential iscontinuous across C,Sec. 95,and itisharmonic inside ofCsince itisapotential function inempty space. Hence, Green's interior function is G=log®- Ve.log® Ifthe point plies outside ofC,the interior potential ofthe distribution onCis Vimlog+K where Khas certain constant value. The exterior potential V,,isequal tothe interior potential onthe surface. Hence the function @=log—VitK 276 THE THEORY OF THE POTENTIAL vanishes onthecontour, and istheexterior Green funetion which was sought. 143. Green’s Function for the Circle.—Given acirclo Cs,Fig.88,witharadiusofa;andapoint0atadistance7;fromthecenter Aofthegiven circle. Describe acircle Cofradius with thepoint 0asaconter. Let thetransform ofthecircle Cswith respect tothecircle Cbethecircle C,. Ifthecenter of CiisatO,atadistance 7;from Oand itsradius isa,,then, by Eq. (105.3) af = aor A \ ;eKi cAWoTh — ey) : Fra, 88. Ifaunit mass isplaced upon C;insuch away astomake it 2level thread, the density onC,isconstant and isequal to negli ad@tont1Gray~2aay The electric image ofthis distribution onC;inthecircle Cisa distribution ofmatter onC;which iscentrobaric with respect tothepoint Oand forwhich thedensity is(Eq. (139.4), -%,,- 1.wonOR Fea, whore risthe distance from 0tothe element ofmass dms on thecircumference ofCz. From this itisscen that thedensity onC;varies inversely asthesquare ofthedistance from thepoint 0. ‘The potential ofthedistribution onC;atanexterior point Pi\s(See. 42) Ry Vs=logFe 143) SURFACE DISTRIBUTIONS OFMATTER 277 Hence, thepotential ofthedistribution onC,attheinterior point Ps,which isthetransform ofP,inthecircle C,is,Hq. (189.3), Va=ViViotlogB.1 Be ‘The potential ofC,atinterior points isconstant and equal tothe value atthe circumference; that is, Ro. Vio=loga Hence Ro_1Ro Ro Va=logjp!—log+log5 or Rea, Ve=logpits and, since moper Re ® this becomes=togBaVa=108Ry Green’s interior function, therefore, is =lowFB?—JogRott G=logbelogRr or, tor Ft—toe2!=low#—tow%. @=log5?—logSt=log5?—lowF @) ‘This expression vanishes ifthepoint P2liesonC1;for Ra _Ry byEq.(2). IfpzliesonCz,P;liesonC,andthen R,=a, ‘sothat onthe circumference Ra _a pet Equation(3)isalsoGreen’sexteriorfunctionifthepoint Olies outside ofCs. 144. The Principle ofDirichlet and Lord Kelvin.—Ihe firstefforts togive amathematical proof oftheexistence ofa 278 THETHEORY OFTHEPOTENTIAL solution ofGreen's problem were made byLejeune-Diriehlet' andbyLord Kelvin,* andtheassertion that asolution ofGreen's problem always exists hasbecome known asDirichlet's principle. Weierstrass pointed out that Dirichlet’s conclusion did not follow from hisergument, butinasmuch astheargumentissimple and illuminating itwillbegiven here. ‘The problem is:Given aclosed surface S.Does there exist afunction which isharmonic within Sand takes aprescribed, continuous setofvalues onS? Itwillbenoted frst that iftheproblem admits ofany solution itadmits but one. For, suppose Visasolution and V;also isasolution. Then V;—Viisharmonic within Sand vanishes atallpoints ofS.Hence V;—V:vanishes everywhere within S,andVsdoes notdiffer from Vs. ‘Suppose thegiven surface isdefined bytheequation I@ y,2) =0, and that the prescribed setofvalues onSisg(z, y,2). If A(z,y,2)isanycontinuous function ofitsarguments, thefunction gthh Q) which contains thearbitrary element h,also takes theprescribed set ofvalues onSand satisfies the condition ofcontinuity. ‘There are infinitely many such functions. Inthis infinite setoffunctions there isone, V,which makes thevolume integral avy?,(aV\?_,(aV\*Sl(@)+ Ga)+Ge)* aminimum. For the integrand isnowhere negative, and if wv Vgee~Gy~2 throughout thevolume, B,which isenclosed byS,thefunetion Vwould beaconstant and could not take aprescribed set ofvalues onS. Let Ubeany continuous function which vanishes onS. ‘Then V+U, +sVorlesungen Uber dieimumgekehrten Verhiltniss desQuadrats der Entfornung Wirkenden Krifte,” p.127 +Treatise onNatural Philosophy,” Vol. I,p.170, 144) ‘SURFACE DISTRIBUTIONS OFMATTER 279 where¢isanarbitrary constant, belongstotheclassoffunctions defined inEq. (1); and theintegral AV+AU)?,(AVEAU?,(AV+WN)G, oefC y+ 3y+(e ie isgreater than theintegral av. AV, (av since, byhypothesis, Vmakes theintegral aminimum. ‘The integral Jcanbeexpanded inpowersoft;viz., aVaU|aVaU,avaU" 2 ‘aUu\* (au\? (aU\® +efile) +(ar)*Ce) Suppose theintegral which isthecoefficient of2tintheabove expression were different from zero. Itwould then bepossible totake ¢sosmall numerically and ofsuch asign that the sum ofthelasttwo terms inEq. (2)would benegative and therefore J<1. Asthis contradicts the hypothesis that Jisaminimum ofthis integral, itfollows that forevery continuous Uwhich vanishes onS aVau,avau,ava" fi‘azdztaydytaehe=0. ® Now byGreen’s theorem initsfirst form, Eq.(57.3), aVaU,aVau,aVau av fave+Sf‘dzoz+Gybytona-Sean ‘Therightmember ofthisequation vanishes, sinceUiseverywherezero onS.The second integral iszero, byEq. (3); therefore, ‘the first integral also iszero. Aside from the conditions that Uiscontinuous andvanishes onS,thefunction Uisarbitrary. IfAVwere not zero everywhere within S,the function Ucould bechosen insuch away that UAV was everywhere positive orzero, and therefore, the integral not zero. Itfollows that AViszero everywhere, and the function Vwhich makes the integral Jaminimum isharmonic within S. AsVtakes the prescribed set ofvalues onS,itisthe function which was sought. 280 THETHEORYOFTHEPOTENTIAL ‘The criticiom ofWeierstrass was that this argument fails to distinguish between theexistence ofaminimum and theexistence ofalower limit. The fact that theintegral Jhas alower limit, does not justify the inference that there exists afunction V which makes Jaminimum. Suppose, forexample, from thetotality ofplanes o(z,y,2)=art+byt+catd=0, that oneissought which renders theintegral ‘ae\* ‘ae\* ‘ae\* o>flGy+Gy* Gye taken over aunit sphere attheorigin, aminimum. ‘The argu- ment ofDirichlet would lead totheinferenee that such aplane existed. Inthis case, however, Jiseasily evaluated, and Joho tute), ‘The lower limit ofJiszero, and there areinfinitely many planes forwhich Jisassmall asmay bedesired; but aplane forwhich J=0doesnotexist. Itistrue, however, that invery many cases theminimum doos exist, Green, Lord Kelvin andDiriehlet were entirely correct in their conclusions that asolution ofGreen’s problem exists for avery wide class ofclosed surfaces, notwithstanding thefact that their arguments areinsufficient. 145. The Equivalent Problem ofPoincaré.—A problem which isequivalent toGreen's problem hasbeen formulated byPoin- caré.' Itcan bestated asfollows: Given aclosed surface S,does there exist afunction V(z, y,2) which satisfies thefollowing conditions: 1.Vanditsfirst derivatives arecontinuous within S. 2.Thesecond derivatives ofVexist andarefinite. 3.Intheinterior ofS,theequation AV =—4r0 ‘issatisfied, where oisagiven finite andintegrable function. 4.Vvanishes onthesurface 8. +-Thdorie duPotential Newtonian,” p.167. 445] SURFACE DISTRIBUTIONS OFMATTER 281 Inorder toshow the equivalence letitbeassumed atfirst that asolution ofGreen’s problem exists. Let the function Ubedefined bytherelation U=f,off,e ‘the integral being taken over the volume enclosed byS. Since U,obviously, isaNewtonian potential, itsatisfies the first ‘three conditions ofPoincaré’s problem, but not the fourth. Since asolution ofGreen’s problem isalways possible, there exists afunction U,which satisfies the conditions. AU,=Owithins, FU=—U,onS. ‘The function VaU+U ‘then satisfies allfour ofthe above conditions, and asolution ofPoincaré’s problem also exists. Conversely, letitbeassumed that asolution ofPoincaré’s problem always exists; itwill beshown, asaconsequence, that asolution ofGreen's problem always exists. Let Ubeany function which iscontinuous, which has con- tinuous first and second derivatives within S,and which is equal toUsonS. ‘There are, ofcourse, infinitely many such functions. Let ¢bedefined bytherelation 1 o=4zhU- Let U;bethe function which solves Poincaré's problem for thisparticular ¢,sothat AU, =—4ro, and U,=0 on. ‘Then, evidently, V=eU+U, satisfies the conditions ofGreen’s problem, since AV =0, within S,and V=U,onS. Since theexistence ofasolution ofGreen’s problem implies theexistence ofasolution ofPoincaré’s problem, and conversely, thetwo problems areequivalent. 282 THETHEORY OFTHEPOTENTIAL Prbiens, 1.Stow by«lining proc fom theepere thatthe Gren fet miniseto ent port inde Hane oath, were piseared from thepoint and pi measured frm theope Bae peat theinte pane arte Don? ta tanesh rom theplane, thedtrbutn ot saat ontheple socnnsy tomake fener wh Parison onseo thepan aretzid tovanl thepoint rand on theotbersde tard the pot SLi pte ine ate Beigua, ptheoptic] imag ofpin theyre yb apal imag hs thetinny pyheago Bear Meets Std hertore an theimage ofpeathe mops Poste wutparla areplaced ap an andegtive waipare wreetsnt depte duet thfurpce vances 0hepeleSmohemplane:ttnheepanerevelmarae,TeareoseiiStgind aanang on oplealsrface wih ogi athecna tevaline ©vic iswery eat Fil hetution ofdeny Sand mae which buat bytenes neh teh pensive asi and thomsn thepostive ae an themay ath Metco heoper ich nates trae ena,‘SietatPotiomSanterncningiemethodofeatsslesioeg and erate poiiv tnd gai man fo thetng‘otheangleetshisanoe3a wip intate nya ofretetons, show low toconsrut Goec fnation Tro etenglar paral SDs thepotent a's enrol hythemed a sett 4.Sow that, fF 9) isthe equation ofLaplace, theuntin Gp (ME ain a sho sinha 9= EP CHAPTER VI TWO-LAYER SURFACES ‘THE METHODS OF NEUMANN AND POINCARE 146. Various Types ofMass.—Up tothe present point, it has been assumed that the foree exerted bythe given bodyuponaunitparticlehasbeenattractive. Iftheforceisrepellant,itisevident that itissufficient merely tochange thesign oftho potential Vorthe sign ofthe mass ofthe particle, provided ofcourse, that the force continues tovary inversely asthe square ofthe distance. Gravitational, elcetrie, and magnetic forces obey this law and the potentials forthese forces differ from oneanother only intheconstant factor ofproportionality which isassociated with the potential; this factor ofpropor- tionality depending upon the forees under consideration and the system ofunits which isemployed. Itispossible tospeak ofgravitational mass, orelectrical mass, ormagnetie mass asthe case may be, for, after all, itis theforces that aredealt with physically, and mass isaderived concept which isfound tobeconvenient; and this concept is found tobeconvenient forallthree types offorce. Inthecase ofgravitation, the forees are always forces ofattraction, and there isbutone kind ofgravitational mass. This mass isalways regarded aspositive and itsconcentration isdenoted bythe term density. Inthe cases ofelectricity and magnetism both attractive andrepellant forces occur, and therefore, ineach case itisnecessary toassume the existence oftwo kinds ofmass; the one positive and the other negative. Two positive masses ofelectricity repel each other, and s0also dotwonegative masses; but two masses ofunlike sign, that is,one positive and one nogative, attract each other. ‘The samo statements hold also formagnetic masses. The notion ofdensity, mass per unit, volume ormass per unit area, islikewise applicable. 447. The Magnetic Doublet—Imagine two magnetic masses, +uand —x, numerically equal but ofopposite signs, placed ‘ona 284 THE THEORY OF THE POTENTIAL atthepoints Nand Srespectively, atadistance Japart (Fig. 89). Letthere beplaced atany point Pinspace aparticle ofmagnetic mass equal to—1. Then theparticles atNandP attract each other,while the particles at§ ?andPrepel each other. If NP =p SP=p., Ny thepotentials atPduetothemagnetica masses +4and —yrespectively are s = +4 =-4.Fra,$9, et Vem my ‘The potential atPduetoboth masses issimply thesum ofthese two potentials. That is, =ve#—4. VitVs=V=pepe This expression canalso bewritten (2-3)Lit 1Pe) Pi bs, v=a =m where m=[1iscalled themagnetic moment. Letthedirection ofIbetakenfromStoward N.If|isinfinitely smalldlthe pair ofmagnetic masses -+uand —yissaid toform adoublet, oramagnetic clement. For such adoublet Lik = pdl-lirm2 = yO(2v=wdtimTwan) where pistheline joining thedoublet tothepoint P,and dl isthedistance between the components ofthe doublet measured from Stoward N. BySee. 54, a(t) _a(t)poo be, AG)=aG)orB- Iftheangle fpisdenoted by¢,theexpression forthepotential becomes Ve OO, mon e.ualEe= 147) TWO-LAYER SURPACES 285, Ifthedirection cosines ofthedoublet area,8,7,ifto,yo,20 arethecoordinates ofthedoublet, andifz,y,zarcthecoordinates ofthepoint P,then, (Sec. 54), afi afl afl a(t). ,2(2)9a(2 ~|2(2)+52) +7aG)} Since Vismerely the difference between two Newtonian potentials, itisevident thatValsoisharmonic inevery domain that does notcontain thedoublet itself. 148. The Bar Magnet.—Imagine astraight bar oflength 1 built upofmagnetic elements placed endtoend. Iftheeross- section ofthe bar isinfinitesimal, itisseen from Eq. (147.1) that thevalue ofthepotential ofthebaratany exterior point Pis +v=“f1Sal; or,since dl-cosy=—dp, "do (1d ‘That is,thebarmagnet acts atexterior points just asthough magnetic masses equal to+4and—mrespectively were placed ‘ontheends ofthebarandthere were nomagnetic masses else- where. Thenorth pole ofthemagnet isatthemass +and thesouth pole at—4. 149. Magnetic Sheets—Two-layer Surfaces.—It ispossible also toimagine asheet built upofmagnetic clements placed sidebysidewith thepositive masses ononesideofthesheet andthenegative masses ontheother, with theaxes ofthemagnetic elements everywhere normal tothesheet. Since thethickness ‘ofthesheet isinfinitesimal, itcan beregarded asasurface made upoftwolayers—a two-layer surface—as contrasted with thesingle-layer surface which hasbeen studied hitherto. Let debeanclement ofarea and oitsmagnetic moment: perunit area, orthedensity ofitsmagnetic moment. ‘Then edwisthemagnetic moment ofthesurfuce element dw. ‘The 286 THETHEORY OFTHEPOTENTIAL potential ofthiselement foranegative unit mass placed atan exterior point P(z, y,2)is £20845,=92(;)eane) and forapositive unit mass atthepoint P _e0089), 9/1prde=+55 ‘This expression isindependent ofthesense along thenormal, forifnchanges sign ¢also changes sign. Regarding themass atthepoint Pasnegative, thepotential oftheentire surfaceis au Ve-Seals)-a/va/vay+fleas) +(5)+vais)J@ theintegral being extended over theentire surface. Let ay=Vifie=Vs, foe=v. sp se se If,inthese integrals, ac,fo,and yoareregarded asordinary mass densities itisevident that V1,Vs,and V,arethepotentials, ofsimple layers onS,and Eq.(1)canbewritten Vi, V2, Vs ve[Bee @ Since the derivatives ofsimple layers have discontinuities incrossing thesurface (Sec. 92),it gS istobeexpected that discontinui- "ties exist inthepotential ofdouble layers incrossing thesurface. 4 160.Closed, Uniform, Two- layer Surfaces.—Iet dwbean clement ofatwo-layer surface, .and letdébethe solid angle 10.90. which issubtended bydwatagivenpointP,sothatdéistheareawhichthissolidanglecutsoutofaunitspheredescribed aboutPasacenter,Fig.90.Then, aswas seen inSec. 66, de=Fd, of 150) TWO-LAYER SURFACES 287 Thesolid angle déwillberegarded aspositive when thepositive side ofdwisturned toward Pand negative when the negative side ofdwisturned toward P. Ifoisthe magnetic moment oftheclement du,thepotential ofthesurface atthepoint Pis vmfoa(p) =[ete =fs. 1sonhe, ls s Ifthe surface $isclosed and ifeisconstant, that is,the magnetic moment isconstant over thesurface, then atan external point Pthepotential Viscro, since thesolid angle pierces thesurface aneven number oftimes, andtheelements Yi \ \W Y SA Tra, 1 dw,which itcuts out ofthesurface present positive and negative sides alternately tothepoint P. (Fig. 91.) Ifthepoint Pisinterior tothesurface thesolid angle pierces thesurface anodd number oftimes. Hence, thepotential has thevalue +4ro ataninterior point. Itis+-4re ifthepositive magnetic mass isontheinside ofthesurface, and —4re ifthe negative magnetie mass isontheinside ofthesurface. Finally, ifthepoint Pisinthesurface itself andifthesurface hasadefinite tangent plane atthis point, thevalue ofthe potential is+20 according asthepositive ornegative magnetic mass isontheinside ofthesurface. There aretwo possibilities inthiscase; oneinwhich thesurface liesentirely ononeside ‘ofthetangent plane, and theother inwhich itdoes not. If thesurface liesentirely onone side ofthetangent plane, the integral fdaisevidently 2rsincethesolidangled&pierees the ‘surface anodd number oftimes onone side ofthetangent plane 288, THETHEORY OFTHEPOTENTIAL and notatallontheother. Ifthesurface does notlieentirely ‘ononesido ofthetangent plane thesolid angle d@pierees the surface an odd number oftimes onone ssideofthetangentplaneandaneven CAnumber oftimes ontheother (Fig. 92). = ‘Thevalue oftheintegral isthesame in\) bothcases. 1) IfthepointPliesinthesurfaceata conicalpointoronanedge,thevalueof CVx thepotential atPistotimes thearea ? cutoutoftheunit sphere bytheenvelop- fio.02. ingcone orwedge. Hence the following theorem: Theorem.—The potential ofaclosed two-layer surface, onwhich themagnetic moment isuniformly distributed, isconstant andequal tozero atallpoints exterior tothesurface; itisconstant and equal to4n0 atallinterior points according aspositive ornegative ‘magnetic mass isontheinterior ofthesurface; itisconstant and equal to+2na atallpoints ofthesurface itself which have definite tangent planes. Asthepoint Pmoves along anormal atanordinary point ofthe surface, the value ofthe potential has discontinuity of+4n0, ifPpasses through thesurface from thenegative side tothepositiveside,and—4reifPpassesfromthepositivesidetothe negative side. Ifthe point Pisinthe surface itself, thevalue ofthepotential isthearithmetie mean ofthevalues onthe two opposite sides ofthesurface atthe point P. IfVoisthevalue ofVatapoint Psofthesurface, and ifVas andVy.arethelimits ofVasthepoint Papproaches thepoint Pofrom thenegative and positive sides respectively, then Von =Vet 2n0, Vo =Vo—Bre. 161. Uniform Surfaces Not Closed.—The discontinuity in the value ofthepotential, just mentioned, exists and hasthe same value even ifthe surface isnot aclosed one. Toshow this etthe given surface S$bebounded byaclosed contour C. ‘Through thecontour @pass another surface 2,such that $+2 forms aclosed surface, and let2becovered with auniform two-layer material ofthesame moment asS. IfVisthepoten- 151] TWO-LAYER SURFACES 289 tialduetoSandUisthepotential dueto%,thepotential of theentire closed surface is,evidently, V+U. BySec. 150, U+Vhasadiscontinuity of+4rrc ifthepoint Pcrosses this closed surface atany point, say through the surface S.Since thepoints ofSlieoutside of2,thepotential Udueto iscontinuous across S.Hence, thediscontinuity ofU+Visduetothesamediscontinuity inV,sinceUis continuous. Ifthesurface isuniform, butnotclosed, andisofsuch shape ‘that astraight line through Ppierces itinbut one point (Fig. 93), and if@isthe solid angle which the surface subtends at P(that is,theapparent size), then V=+09, ifthepositive side ofSisturned toward P,and V=-<9, ifthenegative side ofSisturned toward P. Ifone sheet ofthe surface, asseen from P,projects upon another sheet ofopposite sign the two sheets insuch projection contribute nothing tothe potential. Consider, for example, anopen hemisphere ofradius awith itspositive side outward resting onan2y-plane with thez-axis astheaxis ofsymmetry. ‘Atany point inthezy-plane, outside ofthecircle which forms thebase ofthehemisphere, thevalue ofthe potential iszero, since thepositive sheet ofthehemisphere projects exactly upon thenegative sheet. Atallpoints situated above the2y-plane, but outside ofthe hemisphere, the potential isequal to+o times theapparent sizeofthebase ofthehemisphere; andatall points situated below thezy-plane thepotential isequal to—o times theapparent size ofthebase. Along thez-axis 290 THETHEORY OFTHEPOTENTIAL ‘ V=Vi=—2n0( 1+———— ifs<+a, (+yaaa) and 2 , V=VV,= +2no( 1-——— ife> bh+(Vara) ife>+a. ThelimitingvalueofVifor2=ais~2re(+4.)anav2, thelimitingvalueofVsfor2=ais+2re(-vayThe discontinuity is limV;—limV3=4re. Thevalue ofVfor2=ais dinetin) =Fe ascanbeseen byadirect integration. 162. Surfaces with Variable Moments.—It was shown in Eq. (149.2) that thepotential ofany two-layer surface can be written a¥s,aVs,aVs ve-[%+ ay+eh where vimfas,Y=feeWeStesp Is se «,6,7beingthedirectioncosinesofthenormaland¢themoment,attheelement do. Hence Vs,Vs,and Vscan beregarded as potentials ofsimple layers forwhich the surface densities are ao,Be,and ye. ‘The potentials ofsimple layers arecontinuous across thesurface (Sec. 91), butthefirst derivatives ingeneral arediscontinuous, According toEq,(94.1), thediscontinuities are forUt, —trae-a =—Arost, avs =toa for$F, nb9=—teop ava forOn” —4rye-y =—4roy*. 152) TWO-LAYER SURFACES 201 Hence, thediscontinuity inVincrossing thesurface is 4ro(a? +6°+42)=dro, @ thesign, ofcourse, depending upon the direction ofcrossing. ‘The potential iscontinuous across any point atwhich ¢vanishes. Suppose that Pyisafixed point ofthe surface Sand that V»isthevalue ofthepotential atPy. IfV,isthevalue ofthe potential atavariable point Pnear Poonthe negative side ofthesurface and V,isthe value ofthepotential forapoint onthepositive side ofthesurface near P,and ifVxo and Vyo arethelimits ofV,and V,asthepoints Papproach Po,itis desired toshow that V»isthearithmetic mean ofVacand Vo, and therefore, Vao =Vo—2re0,Vyo=Vot+aevo\ @ ¢ybeing thedensity ofmoments atthepoint Po. ‘The potential atany point canbewritten vef.2008ea,4.f@=0)00894, sp Is ee * or,bysetting wm[eea0aee Isp? . Is eo , thepotential canbewritten V=Wt+u. Itwas shown inSecs. 150 and 151 that equations similar to Eqs. (2)hold forthepotential W,which isthepotentialofadouble layerofconstantmomentdensity. Itwillbesufficient, therefore, toshow that the function Uiscontinuous across the surface atthepoint Py. But this isevident from Eq. (1),forUisthe potential ofatwo-layer surface forwhich thedensity ofmoments (@—>)vanishes atPo. Itis,therefore, continuous across the surface atP,and Eqs. (2)isestablished. Itmight beremarked that inGreen’s formula, Hq.(63.4), =f20aa0—(1%. thefunction&isrepresented asthedifferenee betweenthepoten-tial of«two-layer surface, forwhich the moment density is 292 THE THEORY OF THE POTENTIAL 4,andthepotential ofasimple layer forwhich themass density is08/an. 163. Discontinuities intheFirst Derivatives.—The potential ofatwo-layer surface with variable moment density can be written (Vs, Va, aveve~(+4HH), where nefetes,Ve[Pao,Te[ean ee ee se 4,8, 7arethedirection cosines ofthenormal, and¢isthemoment density atthe element dw, The functions Vi, Vs, and Vs arethepotentials ofcertain simple layers onS. Itisevident that the first derivatives ofthe potentials of double layers arerelated tothesecond derivatives ofthepoten- tials ofcertain simple layers onthesame surface. Forexample, av_(8Vi ,V2, a°Vson(az?+azay+or). ‘The discontinuities inthesecond derivatives ofthe potentials ofsimple layers were studied inSec. 96, From the results ofthat section, itisfound that thediscontinuities of at" Ozay’ Oz02" under theassumption that thez,y-axes areparallel tothe tangents ofthelines ofcurvature atthepoint ofcrossing O ofthe surface, are ae tramp,0, —Ara or,since aiszero and+is+1atthepoint 0,thediscontinuities are a 0,oO —Ag av. 3o. ts. Hence,thediscontinuity in3%is+4r%2; andinasimilar man- neritisfoundthatthediscontinuity inzis+403" 163) TWO-LAYER SURFACES 293 For the derivative with respect to2, OV_(#Vs,Vs,ave a(&+at=), the discontinuities are a (aetry(82 - ani(*), 4+72(22), Ara(p2+4) Since, a=6=0, y=1, atthepoint O,these discontinuities areexpressed more simply by a—4rose ‘Aroa Aro(ps+92): The total discontinuity of8V/d2 is,therefore, da,ap ~4ee(pa+utht2), Now a ry en SeVit pitat VIF ptt at =2h, =om, nn® a=% Atthe point 0, ag = anaga =Bao, from which itisfound thatda,28 pitatceti alsoiszeroatthepoint 0.Hence, there isnodiscontinuity in 8V/dz incrossing thesurface atthepoint 0. ‘These results canbestated asfollows: Thenormal derivative ofthepotential ofatwolayersurface icontinuous across thesurface; thetangential derivatives, however, havethediscontinuities ca oo 154.The Configuration Constant ofaClosed Surface.— ‘Suppose agiven closed surface Sisdivided intotwoparts. 294 THE THEORY OFTHE POTENTIAL ‘This division may boassimple asthodivision oftheearth’s surface into anorthern andasouthern hemisphere, oritmay be fscomplicated asthedivision into theIand and water aress fftheearth's surface. Let these two parts bedenoted by@ and 8,sothat S=atp. ‘Assume further that the surface $iseverywhere convex, 80 ‘that any given straight line pierces itnot more than twice, and cos¢isnever negative. Lettheintegral ffosta0=fe a)ja Ja denote thototal solid angle subtended bythearea aatafixed point ofthesurface denoted bythe subscript 1.Likewise, theintegral S26 2 denotes the solid angle subtended bythe area #atanother fixed point ofthesurface denoted bythe subscript 2.Under thehypothesis ofconvexity, thesolid angle subtended bythe entire surface atany point whatever ofthe surface isless than oratmost equal to2r. Hence os[aos ofaos25, J. os and therefore, osalfa+fas] ) Inorder that thelower limit ofBq. (2)beattained, itisnecessary that every element dwofthe area ashould beedgewise as seen from the point 1,and 0,everyelementofthearea8 on should beseen edgewise from thepoint 2. Thus, ifthepointListheapexofarightcireular moo cone(area a)andthepoint 2 istheapex ofasecond right circular cone (area ),thebases ofthe two cones having the same diameter, thesurface formed byputting the two cones 164] TWO-LAYER SURFACES 295 together base tobase (Fig. 95) would beasurface forwhich thelower limit could beattained. ‘The eos¢inBq,(1)vanishes everywhere. Such surfaces were called double stars byNeu- mann, and are excluded from consideration. Single starred surfaces donot exist inthe class ofclosed surfaces. For agiven surface $and agiven manner ofdivision ofthe surface, there exists aconstant ¢>0,whatever thepair offixed points may be,such that 1 —<2 als forallpossible divisions ofthe surface; and forallpossible selections ofthe fixed points, ¢has aminimum which will be denoted by1—.Hence foragiven surface there exists a constant },which isindependent ofthe mode ofdivision ofthe surface, and independent ofthe choice ofthetwo points, such that 1 1 -osi-Z[f aofiaa]srcr ® Neumann called \the configuration constant ofthe surface, 155. The Spread ofValues ofthe Potential onaClosed, Convex Surface.—Let the potential ofadouble layer onS, evaluated forsome particular point ofS,bedenoted by2rV\. Then whfottran =Bf Vinhfoh%a0=Lfoa a) Let Mbethe maximum value of¢onS,and mitsminimum value. Draw thelines on$forwhich «=(M+m)/2, which ispossible since «isassumed tobecontinuous. Let the area forwhich «>(M+m)/2 bedenoted byaand thearea for which «<(M+m)/2 bedenoted by@. Since «aud cos ¢are everywhere positive, itisevident from Eq. (1)that Visif.ao+Gemdi,arJar 4 px and M+m "m . vieGem aosBfa. 296 THETHEORYOFTHEPOTENTIAL Byvirtue ofthe relation Stet fae=2, these equations canalso bewritten vsM- =f as,oes o) M-m mViem¢e f@ Ina likemanner, forasecond fixed point onS, visu emf do,alsa ® M-mf 4 Vizm+rof.a Equations (2)or(8)show that the maximum value ofVon$ isdefinitely less than Mand itsminimum definitely greater than ‘m,unless «isaconstant, inwhich ease V= M=m. On subtracting the second ofEqs. (2)from thefirst ofEqs. (8), there results 1ie e); v=Vis(=m(-af +fel): and therefore, byEq. (154.3), Vi- Vis (fm,0<d<1, ® wherever thetwo fixed points on$,atwhich thepotential was evaluated, may be. Hence, the maximum difference between thevalues ofthe potential atany two points onSdoes not exceed the maximum difference ofthe densities multiplied by 2,theconfiguration constant ofthesurface. ‘This important theorem isdue toNeumann. 156. Neumann's Proof ofDirichlet’s Principle—The first rigorous proof oftheexistence ofasolution ofGreen's problem was given byC.Neumann.’ His proof islimited toclosed surfaces which aregenerally convex, although they may have Neumann, Dr. Cy,“Untersuchungen aber das Logarthmische und Newion'sche Potential” 1877. Leipag 1656) TWO-LAYER SURFACES 207 corners andedges, butexcluding surfaces which hecalled double stars (Sec. 154). Itwill berecalled that inGreen's problem, itisrequired to find afunction that isharmonic inside ofagiven closed surface (interior problem), orharmonic ontheoutside andvanishing ‘atinfinity (exterior problem), and that takes aprescribed con- tinuous setofvalues onS. ‘Neumann’s method ofthearithmetic meanshows howtobuild adouble layer onSwhose external, orinternal, potential will have therequired properties. Suppose thefunction which is given onSisf(&»,¢).Form thepotential ofadouble layer on§whose moment density isf/2n, namely, 1 COS 1 —1a Pde=»[fae. Winge[Sneae=aefF © Ifpismeasured from apoint ontheoutside ofS,thefunction W,istheexternal potential; ifpismeasured from aninside point, W,istheinternal potential; ifpismeasured from apoint onS,thefunction W;isthevalue ofthepotential onthesurface, and thisfact willbedenoted bythesuperscript zero. Iffiseverywhere positive and thepositive magnetic mass isontheinside ofS,sothat daispositive, itisevident that W,ispositive. SinceW,‘°iscontinuous onS,W/2ncanbe taken asthedensity forasecond potential W.,thus: ei girsWakIWide; and, sequentially 1 Wi=2[werd,af, O) L - W.==|Wada,ar). ‘Suppose Misthemaximum value offonS,and mistheminimum value. IfS;isthespread ofW,® onS,that is,thedifference between itsmaximum and minimum values, then, byEq. (155.4), Sis (M—m), (2) 208 THE THEORY OF THE POTENTIAL where }istheconfiguration constant ofthesurface. Ingeneral, ifS,isthespread ofW," onS,then S,SS. SS 8) S(t me. Since disafixed constant less than unity foreach surface ofthe class considered, itisevident that the limit ofS,asn inoreases iszero. That is,thesuccessive double layers tend toward constant density onS. IfMyand m;arethe maximum and minimum values ofW,®, itfollows from Eq. (155.3) that M>Mi>Ma> ++>>Mere Mer oo) gymem <me< + <mer<meSve That is,thesequence ofmaxima isadecreasing sequence, while thesequence ofminima isanincreasing sequence. Since the spread tends toward 2oro, itfollows that thelimit ofthemaxima isthesame asthelimit oftheminima, and thefunctions W. tend toward adefinite constant value, or Wo =0. 6) ———_ pid Pid, G hy aa Fro. 96. Onthestraight line L(Fig. 96), letpoints bemarked atdis- tances mand Mfrom the point 0. The interval mM, which will bedenoted byS,isthespread of¢onS. That is msosM. atallpoints ofthe surface. Similarly, mark the points m:, my... 5May Mayo... Thon msWO SM, ms Wi SM, +--+, atallpointsofthesurface. Since7"andobothlieintheinterval mM, itfollows that, IW —0}5So, even though thevalue of1’: istaken atonepoint ofthesurface 168] TWO-LAYER SURFACES 299 and the value of¢istaken atanother. That is,this inequality holds forevery pair ofpoints onS. Inasimilar manner, IW -WSS, |W. —W.| <Ss, [Was —Wa SSoy Indeed,sinceWarliesbetween m,andM,foreverypointonS,andforevery p,itfollows that (Wars —Wal SSay andtherefore, byEq.(3), Wass —Wail SS. Oy ‘Thelimit ofWg,» forpinfinite is,Eq.(6), Wi =C. Therefore [C—WA =Sod, and C- Sars Wi SC +80. ) 1657, The Limiting Values ofthePotentials 7,onS.—The limiting values ofthepotentials W., asthepoint atwhich the potential isevaluated approaches thesurface from theinside orfrom theoutside (internal orexternal limits), willbedenoted byW.9 and W,. Ifthepositive magnetic mass isonthe inside ofS,and ifW, isthevalue ofW,atthepoint ofthe surface which isapproached, then itisseen from Eqs. (152.2) and (156.0) that WO-W Of WO=WO4+s; a and similarly, from Eqs. (156.1), WO =WOW, We =WO +47, 0 =WO —Wo) WA =WOE WO, WW Wo, We 2wo 4LO, From these equations and Eq. (156.6) itfollows atonce that [Wu] =[Wl —Wana! SSid%; @) 300 THETHEORY OFTHEPOTENTIAL and therefore Jim|W] =0. ‘Theexpressions fortheinternal potentials canbewritten 2C=Wasi =(C—Warr) +(C—Wo); and therefore 2C —Wy] S25", @ from which itfollows that Tim [F740 =2. Itisevident, therefore, that thefunctions W., W.®, and W.® have the limiting values, fornequal toinfinity, 0,C, and 2C,inagreement with thetheorem ofSee. 150foruniform two-layer surfaces. 158. Harnack’s Theorem for Harmonic Functions.—It is desirable toturn aside from themain lineofNeumann’s argument, foramoment, inorder toprove that thelimit ofthesum ofa sequence ofharmonie functions isitself harmonic. Suppose @) ® Fro, 97 Us, UxUs+++,18asequence offunctions of2,y,2which have thefollowing properties: (a)Each ofthefunetions U,ispositive everywhere within acertain domain B. (®) Each ofthe functions U,isharmonic within the domain B. 158] TWO-LAYER SURFACES 301 (c)There exists apositive number M,which isindependent ofnandoftheposition ofthepoint z,y,z,provided itremains within B,such that Uit+U2tUst -+++Un<M. The sum UsUitUrt+Ust -+++Uat +> evidently converges atevery point ofBandrepresents acertain function ofx,y,2.Itwillbeshown that Uisharmonic within B. LetPybeanypoint within B(Fig. 97). With Poasacenter describe asphere Swhich lieswholly intheinterior ofB,anda second sphere Sowhich liesinside ofS.Lettheradii ofthese ‘twospheres beaandayrespectively. Letdwbeanelement ofthesurface ofthesphere S,andP(z, y,2)beanypoint lying within thesphere So. Also let PP=1, Pia=p. Since U,isharmonic, byEq. (135.1), I = (e=2ves =[Speagr UGDae, @ where U.(z, y,2)isthevalue ofU,atthepoint P,andU.(E, n,$) jisthevalue ofU,atthesurface clement dw. If Afe1Ddo, itisevident that, yarfue, de; and therefore A. <4raM. Hence the series AtArtdst 9 tAgt oeey allofwhose terms arepositive, isconvergent. Now let eon Saar =908v2). 302 THE THEORY OFTHE POTENTIAL Ifthepoint Premains inside ofSo,itisclear that pcannot vanish, since itisnever lossthan a—as.‘Therefore, »andall ofitsderivatives arefinite inside ofthesphere Ss,andtherefore, have maximum values. Let lel<Go, ae ae! ae]|\<6. {3¢|<G,zl<G, |a%p| ae \d*¢|El<e, [Fs]<esGl<@n ete. Then Uafeet<GA, s aU. aeWefives<G.A., aU. He,guefest <Gide Hence, theseries UtUrtUst +++<GeQA, <4ra*MG, Uy, Uy, Us, .tet ait <1.DA,<4ra®MG,, BU;,Us,aU; .at+GetGgtt <A, <drat, areabsolutely anduniformly convergent within thesphere So, andthey represent respectively. v, WU we., ax” oz?” . From this itfollows that AU=Yau, a and since AU, =0, K=1,2-+-+,0@, 158) TWO-LAYER SURFACES 303 there results finally AU =0; that is,Uisharmonic inside ofSp,since itisevidently continuous and has continuous derivatives ofthe first two orders; and therefore, harmonic within theentire domain B. Itispossible togofurther andshow that iftheseries UtUrt Ust ++ +Unt s* isconvergent atany point pwhatever within thesphere S, the series xfiesmy$)dis xaids isstill convergent, and therefore, the function Uisharmonic. Let2,y,zbethecodrdinates ofthepoint p;then Eq. (1) becomes DA)=frapUbfa From Fig. 97itisevident that Psa+r, abeing theradius ofthesphere. Hence, ; a-+UO)>spatetnefEDs and fre1,D)deo>NU,(p) where ymHae? isaconstant which isindependent ofx.From this itfollows that &fuendrde<v&vA. x 4 But since LYVir) % isconvergent byhypothesis, itfollows that theseries &freeate AJs also isconvergent. 304 THE THEORY OFTHE POTENTIAL Suppose p;isanyother point within thesphere. Then Up)=fSapUGDd; but since p2za-n, itisevident that atn _. Dd<spate [OMGnD or, Blo)<Mf4G90, F where _ath M= gala ryt isaconstant which isindependent of«.Hence, theseries (Py UAE, 9, SeoYvon<ME[0snna also isconvergent. That istosay, ifthefunctions Us,for «x=1,..., ©,areeach positive andharmonic within agiven domain B,and iftheseries v=yu a converges atanypoint within asphere which lieswholly within B,theserieswillconverge ateverypointwhichlieswithinthesphere. Itisasimple matter now toshow thatthisseries converges at every point ofthedomain B,assuming, ofcourse, that thedomain Bis connected. Suppose B(Fig. 98)issuch adomain andthat theseries converges atthepoint Ps. Itisdesired toshow that theseries converges atthepoint P,Construct asphere S; which contains thepoint Ps,andthen aseries ofoverlapping spheres, thelastofwhich contains thepoint P.Intheregions which overlap mark thepoints P;,Ps,Ps,+--+. Since the point P;liesinthesphere S;,theseries converges atP,,since, by hypothesis, itconverges atPs. ButP;liesalsointhesphere S,; therefore, theseries Uconverges atthepoint P:,which alsolies inSq;therefore, itconverges atP;;andsoon,until after afinite 168) TWO-LAYER SURFACES 305 number ofsteps theconclusion isreached that Uconverges at. thepoint P. Itfollows also from Sec. 158 that Uisharmonic inthe entire domain B. , CEES. ©)“eD.a«5 a 159. Case I—The Constant CisZero.—(A) The Exterior Function.—If theequations inthefirst column ofEqs. (157.1) ‘and (157.2) areadded, and theresult ismultiplied by-1, it isfound that Oe Wyo —Wid =ose WO AP WA. (0) Byhypothesis, thelimit ofW,? asninereases, which isthe eonstunt C,iszero, Hence - Mas, a) andsinee, Ka.(156.3), |Wa? |SSods, (2) theleftmember ofEq. (1)converges like ageometric series. ‘Thefunction Wistheexterior potential ofacertain magnetic layer distributed over thegiven surface S.Itistherefore har- monic everywhere outside ofS,andithasthelimiting value W, asthepoint atwhich itisevaluated approaches thesurface S.Since itisharmonic, itsmaximum and minimum values ‘ocours onS;thatis,itsmaxirnum andminimum values arethe maximum andminimum values ofW,; and since thesum of 306 THETHEORY OFTHEPOTENTIAL these maximum values converges, byEq. (2),itfollows that the function w=-yW. (3) converges absolutely and uniformly everywhere outside ofS, and itslimit isequal tothegiven funetion fonS. Equation (3)eanalso bewritten W=YS —W)—YSar. a a ‘The function S:d* —W,ispositive everywhere outside ofS, itisharmonic, and the sum Yar -W) Ft converges absolutely and uniformly. ‘This sum is,therefore, anharmonic function, byHarnack’s theorem. Furthermore, the term ys also isharmonic, since itismerely aconstant. Consequently w-=->W, @ im isthe exterior function which was sought, foritisharmonic outside ofS,vanishes atinfinity, and takes the preseribed set ofvaluesfonS. (B)TheInteriorFunction—Ittheequations inthesecondcolumn ofEqs. (157.1) and (167.2) aretaken alternately positive and negative and then added, itisfound that Wi —WO $WO = =(HO =f (-W.; and since, limW.( =20=0, itisevident that, WO WO LW O-WO... =f 159] TWO-LAYER SURFACES 307 theleft member converging like ageometric series, Eq. (157.4). The function W,this time isthepotential ataninterior point ofacertain magnetic mass distributed over S,and W,° isthe limit ofW,asthe point P,atwhich thepotential isevaluated, approaches S$from theinside. Since Weisharmonic inside ofS,itsmaximum andminimum values arethemaximum and minimum values ofW.'°, and since [We] <28a forevery point ofthesurface $(IEq. (156.4)), itfollows that the function W=Wi- Wit Wi— Wet + 6) converges absolutely and uniformly everywhere inside ofS, and takes thelimitfonS. This expression for Wcanalso bewritten W=Y@sar' +(-1)W) —Ysa, a a The functions WA +(“We eA... areharmonic and everywhere positive inside ofS,and their sum converges absolutely anduniformly. Therefore, Lesa! +(-1)W) isharmonic, byHarnack's theorem. Theconstant term E2S»\*-? also isharmonic. Itfollows, therefore, that W(Eq. (5)), istheinterior function sought, foritisharmonic inside ofS,and takes thepreseribed ‘set ofvalues onS. Dirichlet’s principle is,therefore, established rigorously for the case C=0. 160. The Interior and Exterior Functions as Potentials ofthe Same Simple Layer.—In order todistinguish between functions interior and functions exterior, the exterior function W,Eq.(159.4), willbedenoted byWs,andtheinterior function W,Eq.(159.5) willbedenoted byW;. Also ifpismeasured from anexterior point itwillbedenoted bype,andif itismeasured 308 THE THEORY OFTHE POTENTIAL from aninterior point itwillbedenoted byp:.Hence, 2. al 2a(td=ans) ord=ai) according asthe exterior orinterior function isunder con- sideration, Ifthevalues ofW,,inEq.(159.4), arereplaced bytheir values from Eqs. (156.1), itisseen that =i o op aDWemgefotmio+wie+weet«3 (Bas, andsimilarly fromEq.(159.5), Wimgfe-ee)+ars—Wey+--2(!)a Now byEq. (157.2), W. +Wes =We, WE —Wes =We, ‘Hence theexpressions forWsand W,canalso bewritten --2 0. vo og OLWe=Afiom+Ws)+WO+a(Banwo =-A foreswesweoge.- a W.==}fare+wee+wie4-92 ()ao, sothat W,and W;arerepresented asthepotentials ofcertain double layers onS. The functions W. and W, arethesurface values ofthe external andinternal potentials ofacertain double layer onS. ‘These potentials areharmonic intheir respective domains. Furthermore, 1/p;isharmonic outside of§and1/p.isharmonic inside ofS.Hence, byGreen's theorem, Eq.(57.4), afi 1aWwoA(2)_1), Sm3G)-5.Jeo w»2(1\_ 1aweSmo) ~528m=o ‘That is afl OW da 08(1)gy = [He de,Sree) =[25% afl OW) deo08 (Day = fm? de,Sire 25-8 160] TWO-LAYER SURFACES 309 sothat qs. (1)can also bewritten Wee=Bf,gare+W04wae+fa do(2) W.=-dfdare+WO+Wi+E Since thenormal derivative ofthe potential ofatwo-layer surface iscontinuous across thesurface, Eq. (153.1), itfollows that wo we,HE Me, gw, 2,3,06> Iftherefore the common value of 2 Po Cen ey Ac) “opn +Ws+ d=5,hie+WO+) bedenoted by—2ra, itisseen that Wee[fas WeaSige andthefunctions WxandW,aretheexternal andinternal potentials ofthesame simple layer ¢onS. 161. The Constant CisNot Zero.—If the constant Cis different from zero, itsvalue canbesupposed known asthelimit: ‘ofthesequence offunctions W.. Let ent-% and letthefunctions V.,analogous tothe functions W,, be defined byrelations similar toKgs. (156.0) and(156.1): 1 a,aan =«/f de, nee vingefioass 1Vemfrevda, . Itisevident that vio=pf(f=Ode=Wi—C, and, ingeneral, Vo <9 = Sinee We =0, itisevident that V. iszero, andthetheory ofthepreceding ease isapplicable. s10 THE THEORY OFTHE POTENTIAL ‘Thefunction, Eq. (159.4), Vr=- DV a isharmonic outside ofS,vanishes atinfinity, andisequal to¢ ‘onS.The function Vr+Cisharmonic outside ofS,and is ualtoon etCns onS,butitdoes notvanish atinfinity, since itisequal toCat infinity. Itisnot, therefore, theexterior function which is sought. ‘Thefunction, Eq. (159.5) V,=Vi-VatVi-Wt--- isharmonic inside S,and isequal togon S. Hence thefunction Vite isharmonic inside ofS,since ismerely aconstant, andisequal tofonS, Since there isonly onefunction that cansatisfy these conditions, V;+Cistheinterior function sought. Since C eanberegarded astheinterior potential ofadouble layer of constant moment, C/2, onS,itisclear that theinterior funetion canalways beregarded asthepotential ofadouble layer onS, ‘Theexterior function, however, cannot beexpressed asthe potential ofadouble layer onSinthecase that theconstant C isnotzero, Forthetotal mass inadouble layer isalways zero, andtherefore, ifVisthepotential, limpV=0, forevery potential ofatwolayer surface. ForaNewtonian potential inwhich the mass Misnot zero limpV =M#0. Itiscleartherefore thatexterior Newtonian potential cannot be represented asthepotential ofadouble layer. Suppose however asimple layer isdistributed onSinsuch a waythatitisalsoalevellayer. Ifthepotential ofthislevel layerVsisequal toConS,itwillbeequal toCeverywhere within S,andwillvanish atinfinity. Consequently thefunction Va+Vo 161] TWO-LAYER SURFACES 31 isharmonic outside ofS,vanishes atinfinity and isequal tof onS;and thefunction VitVo isharmonic within Sand isequal tofonS. Therefore, if ‘theconstant Cisnotzero, both theexterior and interior functions ‘canberepresented asthepotentials ofacombination ofadouble layer and asimple level layer onS. ‘But, just asinSec. 160, thepotentials ofthedouble layer, Vzand V;canberepresented asthepotentials ofasimple layer ‘ofwhich the total mass iszero. The equations analogous to Eqs. (160.2) are 1 a des-- ©$V EVOEo, VeedeFlV20$V+VSO %e --l a o fo oF deoY=1[do VOEVE VE Just asbefore 2,0 EV + 2)=20.6 +0+ceon* “ an . , and ifthe common value ofthese two normal derivatives is denoted by—2re, theabove expressions forVzand V;become simply, VemfiZavem[2a1spe 5pi ‘That isVeand V;aretheexterior and interior potentials ofthe same simple layer onS. Henee the functions which solve the exterior and interior Green problems can berepresented always asthe potentials ofthesame simple layer onS. Itremains tobeshown how thelevel layer above mentioned can beconstructed for the class ofsurfaces considered by Neumann. 162. The Construction ofaSimple, Level Layer onS.— Suppose Oisany point within Sand poismeasured from the point 0. Starting with, Sec. 156, 1 seh 312 THE THEORY OF THE POTENTIAL build uptheseries ofexterior potentials 1, Ws,Ws, «--,and Wi, W., Ws, +++. The limit ofthe functions W. as ‘nincreases’ isacertain constant which will bedenoted byT; Wo =7, Form the function U=-Wi-Wi-Ws---, which istheexterior potential ofacertain double layer onS, and therefore, vanishes atinfinity. Itslimiting value onS is,Eq. (159.0) Us= WO -We Wo. atop, ps where posisthe value ofpoonS. Now, consider the function 1 oat-u. 1vend Oo) Itisharmonic everywhere outside ofS,anditvanishes atinfinity. ‘Therefore, since impsVo =+1, itisthe Newtonian potential ofsome distribution ofaunit amount ofmatter. Since Your onS, Sisancquipotential surface ofthedistribution. Therefore, bySec. 115, itispossible todistribute aunit amount ofmatter onSinsuch awaythat Sbecomes alevel layer. ‘Thedensity inthisdistribution is,Eq.(115.0), 18% onEfTa, The constant [isnotzero. Ifitwere, thefunction Vo would vanish onSandatinfinity. Since itisharmonic, it would vanish identically outside ofS,and vel me ‘Thisisimpossible, however, sinceUisthepotential ofadoublelayer and1/p¢is«Newtonian potential ofaunitmass (See. 161). 162] TWO-LAYER SURFACES 313 Ifitisdesired tohave thepotential ofthelevel layer equal to ConS,itisnecessary, merely, totake c(1w=$(5-2); and therefore, thetotal mass.of the level layer is Cc POINCARE’S METHODE DU BALAYAGE 163. The Balayage ofaSphere.—The balayage ofavolume consists inreplacing thematter within thevolume byanequiv- alent layer upon thesurface: equivalent inthesense that the potential ofthelayer isthesame asthepotential ofthevolume atpoints which lieoutside ofthesurface. InSecs. 108, 133, and134,itwasshown that ifaunit particle isplaced atapoint0;withinasphere, itspotential atpointsexterior tothesphere isjust thesame asthepotential ofaunit mass spread over the surface insuch away that thedensity onthesurface isinversely proportional totheeube ofthedistance from thepoint O:. Ifthere aretwoparticles ofmasses m;andmswithin thesphere, each particle can bereplaced byanequal and equivalent mass distributed over thesurface, and thesurface distribution m,+mz will have thesame potential atallpoints exterior tothesphere astheindividual particles m,and ms. This result, evidently, canbeextended toany number ofparticles, and therefore any distribution ofmatter within thesphere can bereplaced byan equivalent distribution ofthe same amount ofmatter upon the surface ofthesphere. Itisalways possible, therefore, tobalaye asphere. Itwill beobserved that inthis process negative masses are never introduced, and that the potential atoutside points is unaltered. Atallinterior points, however, the potential is diminished. Inorder toprove this, letUbetheinterior potential oftheequivalent surface distribution foraparticle forwhich theexterior potential is1/p, Green’s function is o=}-u; > Bit THETHEORY OFTHEPOTENTIAL that is,itis thedifference between thepotential oftheinterior particle and theinterior potential oftheequivalent distribution onthesurface. But, bySec. 130, theGreen function isalways positive. Hence atevery point oftheinterior u<}, ? and intheprocess ofbalaying thesphere thepotential atanyinteriorpointisdiminished whenever aparticleisreplacedbyanequivalent surface layer. Itistherefore diminished forthe final result whatever thedistribution ofmatter within thesphere may have been. 164, Existence ofaLevel LayeronaGivenSurface.—Suppose there isgiven aclosed surfaceSwhichmayhaveafinitenumberof 2 Peers Cec rsBeAr SS 7/2 ee CAE CCee N ZOO eerrrrees To hier comtiE PT KOON vane OAS TAT SCORESSE eer CSERECEEEPEEEErrr yr} DSEREEE Eas TSEC eerCASERSEEEEESEH Fu. 99. conical points and edges, butwhich atother points ofthesurface admits adefinite tangent plane and two definite principal radii ofcurvature. Itwill be shown that there exists afunction V(z, y,2)which isequal to+1atallpoints ofthesurface S,is continuous outside ofS,exeept possibly along theedges, which vanishes atinfinity, and which satisfies theequation ofLaplace. LetthesurfaceSbesurrounded byaregionRyintheformof 1shell bounded bytwo surfaces, ByandBs,Fig. 99,and letthe 164) TWO-LAYER SURFACES 315 minimum distance between thesurface Sandthesurface B,be Se,Imagine aninfinity ofsuch regions +++, Ruy Rus, Rui, Ro,Ri,Ra,Ray+++ with thecorresponding minimal distances s+ Bay Busy Baty BoyBisByBy where5»<8-4;tendstowardzeroas.nincreases, and5,>x1tends toward infinity asnincreases; and letthe successive regions overlapsothatevery point ofspace exterior toSliesin ‘atleast oneregion R.Let«,beslightly smaller than 3,,for every integer «. Lettheregion R,bedivided upintocubes byplanes which areparallel toagiven setofreference planes andatadistance «./V3apart;sothatthediagonals ofthecubesareequalto«.‘Acubewillberegarded asbelonging totheregionR,ifanypartofthecube iscontained inR,. Letasphere beconstructed about thecenter ofeach cube, thediameters ofthese spheres being equal to8,.‘Then each ofthese spheres lieswholly outside ofthesurface S,andevery point oftheregion R,lieswithin at least oneofthese spheres. Furthermore thenumber ofsuch spheres fortheregion R,isfinite. Ifspheres areconstructed forevery region Rinthemanner justdescribed, there willbeconstructed adefinite, infinite setof spheres which hasthefollowing properties. (a)Every sphere lieswholly outside ofS. (b)Every point outside ofSliesinside ofatleast onesphere. (©)Thesetofspheres isdenumerably infinite; that is,the spheres canbeputintoaonetoonecorrespondence with the positive integers. Forexample, thespheres inRycanbe numbered, andthenurabering continued intoRs,then into R1, then Ry,and Rs, and soon. Itistheproperties ofthissystem ofspheres that isimportant, and notatalltheparticular system which has been defined. ‘Anyother system ofspheres which possesses these three prop- erties willserve equally well; butitwasnecessary toshow that there exists atleast onesuch system. Construct asphere 2ofradius Awhich contains thesurface S initsinterior. Letthemass Mbedistributed uniformly on3. ‘Then theinterior potential of2isconstant andequal toM/A, andifMischosen equal toAtheinterior potential ofMisequal tounity. Inparticular, itisequal tounity within endonS. 316 THE THEORY OF THE POTENTIAL Assuming that the sphere 2passes through the sphere S: which was numbered one, letthesphere S;bebalayed; then let ‘8;bebalayod; then S;again, then Sz,then Ss;then S;athird time; and soon,thespheres being taken intheorder $1,82;81,Sa,S23Si,Sx,Ss,Sa;Si,Ss, >>+ @ 0that each sphere isbalayed infinitely many times. ‘The matter which originally was uniformly distributed on5 isredistributed, atleast inpart, ineach operation; unless the sphere which was balayed was already empty, inwhich case no alteration ofany kind occurs. Atthebeginning, thepotential was V= 1 insideof 2, v=4outisideot >. Whenever asphoro S,isbalayod thepotential atallpoints outside ofS,remains unaltered, but atallinside points itis diminished. Hence the potential Visnever increased atany point byany operation, but isdecreased everywhere within one ofthe numbered spheres byeach non-vacuous operation. Since negative masses arenever introduced, necessarily Vremains positive always. Hence, asaresult oftheoperations indicated inEq, (1), atany fixed point outside ofS,Vtends toward a limit. Within and onS,the potential Vremains unaltered and equal to+1. Let V,bethevalue ofVafter then"operation. ‘Then at any point poutside ofS 12V2V.>0. Asthe point precedes toward infinity, V,which isequal to A/p, tends toward 0,and therefore V,tends toward zero at infinity. Atafixed point pthepotential V,tends toward alimit, asa result ofthe operations indicated inEq. (1). Suppose the point pliesinthesphere S,. This sphere isbalayed infinitely ‘many times, thenumbers fortheoperations onthis sphere being @, ax,ax,+++ Consequently the sequence ofpotentials Vay Vay Vay ++tends toward thesame limit which will bedenoted byV.. Hence Vn=Ve,+(Var —Ve.) +(Va ~Vas) +(Vay— Vas) toes 164] TWO-LAYER SURFACES 317 Each oftheterms inthisseries iszero ornegative except the first, and thesum converges tothelimit V.. Since thesphere S;contains only empty space after ithasbeen balayed, itfollows that Vey j=1,2, ++, %,satisfies theequation ofLaplace intheneighborhood ofthepoint p.Therefore, each term ofthis series isharmonic. Itfollows then from Harnack’s theorem (Sec. 158) that theseries andallofitsderivatives areuniformly convergent. Consequently V.,iscontinuous and satisfies the equation ofLaplace inside ofthesphere $,;and since every point ofspace isinside ofsome sphere &%,itfollows that V, isharmonic everywhere outside ofS. Itremains only toprove that V., iscontinuous across S. Inside ofSthevalue ofthe potential hasremained steadilyat+1.Outside 2ofS,V.,isharmonic; butitisconceiv- é able that V,might nottend toward YA +1 asthe point papproached the surface S$.Itwillbeshownthatthis oy, (p isnot the case. LetP,Fig. 100, beapoint onSat which Shas adefinite tangent plane and two definite radii ofcurvature. It ispossible, then, toconstruct asmall sphere 7’ofradius rwith itscenter Fic,100. at0,which istangent toSatP,and which lieswholly within 8.Letpbeanypoint outside ofS,andlet Op=p, Pp=s. ‘The function r/p isharmonic outside of7,and isequal to +1on 7. The function V,isthepotential ofpositive masses which lieoutside ofS. Itmay, therefore, have maxima outside ofS,but itcannot have minima (Theorems Iand IV, Sec. 75). Itisequal to+1 onS,and vanishes atinfinity. The funetion u=v.-5 ’ isthepotential ofpositive masses outside ofSand anegative mass equal to—ratthe center of7. Outside ofSitean have maxima but notminima, Itisequal tozero on$and atinfinity. 318 THE THEORY OFTHE POTENTIAL Itcan, therefore, bepositive orzerooutside ofS,butnotnega~ tive. Itfollows therefore that r 12 Vaz s Now letthepoint papproach thepoint Palong thelines,Fig. 100. Itisevident then, that root leVeerery and V.,tends toward unity ass tends toward zero. Hence, V..iscontinuous across S$atanyordinary point ofthesurface. The proof asgiven above fails ifthe point Pisat comer oronanedge, because atacorner oronanedge itisnotpossible toconstruct asphere 7which lies wholly within S. Poincaré modified theproof soastoinclude conical points byreplacing thesphere 7byasurface which hasaconical point which can bbefitted inside oftheconical point onS.Theproof istoolong tobegiven here and thereader isreferred toPoincaré’s paper in The American Journal ofMathematics, Vol. XII, p.228, (1890). Doubtless, points which lieonedges alsocould beincluded byan appropriate modification ofthesurface 7. Theabove results show thatpartofthematter, which originally wason2,hasbeen deposited upon Sinsuch away astomake‘8alevelsurfaceofpotential +1,andpartofithasbeendispersedtoinfinity. The space outside ofS$isentirely empty. The proof fortheexistence ofasolution ofGreen’s problem follows at ‘once from theexistence proof forlevel layers byvirtue oftherela-tionsgiveninSee.129,butadirectproofcanbegivenbythemethod ofbalayage.* 165.Application ofHarnack’s Theorem.—Suppose there is given #closed surface Sandacontinuous funetion &which is defined on8.Itissupposed, also, that M>#>m>0, onS,where Mandmaretwopositive numbers. Let2beaspherewhichcontainsSwhollywithinitsinterior,‘Then itisalways possible tofindasequence ofpolynomials P,such that 0<P<e *Ponscant, “Theorie duPotential Newtonien,” p.283. (1899.) 166] TWO-LAYER SURPACES 319 within and onZ,where e,isasequence ofpositive numbers for which the series ateatatortedes isconvergent, and such that thefunction UsPitPitPits++Pat ee isequal to&onS. ‘Suppose that there exists, forevery index n,afunction V. which isharmonic inside ofS,and equal toP,onS. ‘Then, if 8, and 6, arethemaximum andminimum values ofP,onS, 0< 5 SMS HM Se within S,and thefunction VaVitVatVatlabo isabsolutely and uniformly convergent. Furthermore, itis equal to onS,andbyvirtue ofHarnack’s theorem itishar monic inside ofS. Asolution ofGreen's problem, therefore, exists ifthere always exists afunction V,which isharmonic within Sandequal toagiven polynomial P,onS. ‘The restriction that #bepositive andnon-vanishing onSwas necessary inorder toapply Harnack’s theorem, butthisrestrie- tion can beremoved. Suppose la<c, where Cissome positive number, onS.Then. e=C-(C-4), and C-4>0. ItWisafunction which isharmonic within §and equal to C—@on, theconditions ofHarnack’s theorem aresatisfied. ‘Then the function v=c-W isharmonic within Sand isequal to#onS. 166. Construction ofanInfinite System ofSpheres within S— Inorder toshowthatthere exists function Vwhich isharmonic within agiven closed surface Sandwhich isequal togiven polynomial PonS,letthere beconstructed firstaninfinite system ofspheres which hasthefollowing properties: 320 THETHBORY OFTHEPOTENTIAL (a) Every sphere ofthe system lieswholly within S. (b)Every point inside of liesinatleast oneofthese spheres. (c)Thesystem ofspheres isdenumerably infinite. These are the same conditions for the interior domain that were required inSec. 164fortheexterior domain. Inorder to show that there exists atleast onesuch system, letaninterior region Rybedefined astheregion which liesintheinterior ofa closed surface which lies wholly within Sand forwhich the minimum distance tothe surface Sisfo. ‘Then construct an infinite series ofoverlapping shells Ra,Ra,Ray, forwhich the minimal distances tothe surface Sare BybyBay ty ofsuch magnitude that fn<buoy and which have thelimit zero asnincreases, sothat theouter boundary ofthe shell Rytends toward coincidence with the surface Sasnincreases. ‘Then every point inthe interior ofS liesintheinterior ofatleast one domain R,. Let¢,beslightly smaller than 6,forevery «,and lettheregion R,be divided upinto cubes byplanes which areparallel toagiven setofreference planesandatadistance «,/+/3apart,sothatthe diagonals ofthecubes areallequal to«.Ifasphere ofradius 4,isconstructed about the center ofeach cube, there will be ‘afinite number spheres each ofwhich lieswholly within S,and every point of2liesinatleast oneofthese spheres. Tfspheres areconstructed inthis manner forevery region Rg,theinfinite system ofspheres soconstructed certainly satisfies conditions (a)and (b). Italso satisfies condition (c)for, since ‘the number ofspheres ineach region R,isfinite, the spheres inRocan benumbered and thenumbering continued into Ri, ‘then into Rs, and soon. Itwill beassumed, hereafter, that thesystem ofnumbering has been carried, sothat each sphere isdefinitely located inthesequence ofspheres 167. The Existence ofthe Required Harmonic Function.— LetPbethegiven polynomial. Form theLaplacian AP, and ‘suppose, atfirst that oP <0 167) TWO-LAYER SURFACES 32 everywhere within asphere 2which contains §wholly within its interior. Let AP=—4r0, where oispositive, and let Wef=sp 80that Wsisthepotential ofavolume distribution ofpositive masses and is,therefore, positive everywhere. Furthermore AW, =—4ro =AP inside of2. ‘Regardingoasthedensityofattractingmatter,letthespheres SS,inside ofSbebalayed intheorder SySe;Si,Sty$3}81,SeSySj + sothat each sphere isbalayed infinitely many times. IfW. isthepotential oftheattracting matter after thenoperation, itisevident that W.>0, and Was Wor ‘The sequence ofpotentials Wi,We,Wa, ‘therefore, hasalimit which will bedenoted byW. Inside ofS 0<W< Wey outside ofS W= We, since thebalayage ofasphere docs notchange the potential outside ofthesphere. ‘The sphere S,has been balayed infinitely many times. If thenumbers ofthese operations are, inorder, then the functions Way Way Way °° fare harmonic inside ofS,,since they represent. the potentials ofattracting matter inempty space; and they have theseme limit W. Hence the series W=Way+(Way—Way)+Way~Way)+=+ isconvergent. Allofitsterms, except thefirst, arenegative orzero, andallofitsterms areharmonic. Hence, byHarnack’s 322 THE THEORY OF THE POTENTIAL theorem (Sec. 158), Wisharmonic everywhere inside ofS,since every point inside of$isalso inside ofsome sphere S,. Allof the matter which was within Shasbeen deposited upon Sby these operations, and theinterior ofSisempty. Inorder toshow that, Wiscontinuous across Satevery point atwhich §has adefinite tangent plane and two principal radii ofcurvature, letQbe point ofthe surface atwhich these conditions are satisfied. Let Sobeasmall sphere outside of$ and tangent toSat Q. Since theGreen function forthesphere isknown itispossible toeffect asurface distribution ofmatter onSoforwhich the potential coincides with the values ofWs onSo. IfUisthe potential ofthis distribution ofmatter, U isharmonic outside ofSc, equal toWo onSo,and vanishes at infinity. ‘The function W—Ucan beregarded asthepotential ofa positive distribution ofmatter and anegative distribution on So. Itcan have maxima outside ofS,but not minima. It vanishes atQand atinfinity. Hence, atany point gwithin § UsWsW. Asthepoint qapproaches Qthefunction Utends toward Wo. ‘Therefore, the function Walso tends toward Wy and iscon- tinuous across S. Take now the function V=W-W+P. Itisharmonic inside ofS,since AW=0, and AW, =AP; and itisequal toPonS,since W—Wovanishes onS. Itis, therefore, the function which was sought; and sine Green's problem canbesolved forapolynomial P,itcanbesolvedforany function Pit PatPates $Pateee bytheprinciples ofSec. 165. The restriction that APshall benegative everywhere within 3 iseasily removed. Any polynomial, whatever itsvalues in> may be,canberegarded asthedifference between two poly- nomials, P=P:-Py 167] TWO-LAYER SURFACES 323 thepolynomials P;and P;being chosen sothat P,<0, AP: <0, everywhere within Z.Inaccordance with theabove analysis there exist two functions V;and V2each ofwhich isharmonic within S,andequal respectively toP:andP:on S.The desired function Vthen is Vevri-Vi Problems 4,Show directly bythemethod ofbalayage that onany given closed surfae 8there exists one and oaly one distribution of«unit mass on which iscentrobaric with respect to«given pointPwhichlieswithinS.2,SupposethefunctionVsatisfiesthefollowingconditions:(a) Visharmonic inevery region that lies wholly within orwhollywithoutagiven elosed surface S.()Visregularetinfinity,andactstherelikeaNewtonianpotential.(©)Vand tsnormal derivative have definite Limits atany point Pofthe surface both from the inside and from the outside. Ifthese limite are denoted by a, a VyGr and then or :1(aVs_avs onte) ae-hin- v0 sre continuous functions on8. Show that Visthesum ofthepotentials ofasimple layer and ofadouble layer on5. 3.Given aclosed surface $and acontinuous function &,such that far=0. Show that there exists function Vwhich sutisfes the conditions ‘thins, % av=0 withing, =o ons. 4.1£Sisaclosed,everywhereconvexsurfasewhichhastwofiniteprincipal ‘radii ofcurvature ateach point, and ifRisthe maximum value oftheradii otcurvature, show that 088g<A fimM ape where pismeasured from apoint ofthe surface, gisthe angle which » sakes with theinterior normal, and Aisthe aren ofthe surface. 5.Ashell offinite thickness separates space into three regions, Athehollowoftheshel,BthespacewithintheshellCthespaceoutsideofthe 324 ‘THE THEORY OF THE POTENTIAL shell,IfV,isthepotentialduetomatterwithinsndVeisthepotentialduetomatterlyinginCandiatallpointsofBthereexistslinearrelationwith constant cooficionts eV +eVe +0 =0,inwhich o,b,and ¢are distinct from zero, then Y= Ve=-F 6.Verify thefollowing formulas fortheconfiguration constant A: forthe circle, 1.aah fortheellipse 1/2) foranyclosed,conver,planecurveoflengthLforwhichRisthemaximumvalue ofthe radius ofcurvature, asi-ges forany closed, conver, surface forwhich Aisthearea and Rthemaxiraum ‘value which oocurs fortheradius ofcurvature, YS1eres CHAPTER VII SPHERICAL HARMONICS 168. Definitions.—A function issaid tobeharmonic within a region Rifthefunction and itsfirst derivatives arecontinuous inR,andif,also, itsatisfies theequation ofLaplace, av av, av a=SatatGene ‘There isaninfinite variety offunctions which satisfy these conditions, and therefore, aninfinite variety offunctions which areharmonie, Ingeneral, they aretranscendental; that is,they cannot beexpressed bymeans ofelementary functions which are related byafinite number ofcombinations ofthefundamental operations ofarithmetic. ‘They arefrequently expressed by series; andforthis purpose itisdesirable tohave anormal set harmonic functions interms ofwhich such series canbeexpressed. No one setofnormal harmonic functions isbest adapted to every expansion, butthesimplest, and therefore thebest: known and most commonly used, setofnormalized harmonies arethe Spherical Harmonies, towhich thepresent chapter isdevoted. Asecond normalized setofharmonic functions istheEllipsoidal Harmonics ofLamé, andathird setistheToroidal Harmonies of Hicks. Thefollowing chapter willbedevoted totheEllipsoidal Harmonies ofLamé. Aspherical solid harmonic isanharmonic function which is homogeneous intheletters 2,y,2. ‘Thedegree ofitshomogeneity, may berealorcomplex, Ofcourse, itsatisfies theequation ofLaplace,and,sinceitishomogeneous, italsosatisfiestheequationofBaler. Hence, ifVisaspherical harmonic ofdegree n,it satisfies thetwo equations av, a ov. ot et ee ateeae+an, 205 326 THE THEORY OF THE POTENTIAL thefirst ofwhich states that Visharmonic, and thesecond that itishomogeneous. ‘Aspherical surface harmonic isthesetofvalues which ©solid spherical harmonic takes onthesurface ofaunit sphere which has itscenter atthe origin. Itisobtained asafunction ofthe polar angles byreplacing therectangular coordinates bypolar coordinates, theradius veetor being taken equal tounity. Acomplete spherical harmonic isaspherical harmonic which is finite and single valued forallfinite values ofthecoordinates. ‘A.partial spherical harmonic isaspherical harmonic which either does notsatisfy theequation ofLaplace everywhere, or, which isnotasingle valued function. 169.Examples ofSpherical Harmonics.—The following exam- plesofspherical harmonies aretaken from thetable given in ‘Thompson &Tait’s “Treatise onNatural Philosophy,” Vol. I,p.172. —2, 2; Stant¥, Zogtte_ 2, Degree—2,i pata log ES -1, 2, Ltant¥, Ligtte. Degree1,5}tant; FlogTS, rte- ra(?—y") De rr? a¥. y"), eres, og tet De : 4, rte, 2).earee+1,ajston; (oste+ea) Degree +2, at—yt; tat yt; ay, ‘Thefunctions z/r*,1/r,a*~y*,ete.arecomplete sphericalharmonies; while 2tant, ths, onath (log224Fip)sete. arepartial spherical harmonies, sincetheyarenotsingle valued. ‘Thegeneral,homogeneous, polynomial oftheseconddegree,act+by?+ca+eye+fer+gry, isharmonic if atdteno, compo geneous, Harmonic Polynomials.—Consider theiste, homogeneous, polynomial ofdegree ninz,y,2.It 170] SPHERICAL HARMONICS 327 contains (n+1)(n +2)/2 coefficients which, atthe moment, canberegarded asarbitrary. Ifitissubstituted intheequation ofLaplace, there results acomplete, homogeneous, polynomial ofdegree n—2which hasn(n —1)/2 coefficients, each ofwhich must vanish ifthe given polynomial isharmonic, Hence the (n+ 1(n +2)/2 coefficients aresubject ton(n —1)/2 condi- tions, leaving 2n+1 ofthe coefficients arbitrary. Itwould naturally beexpected, therefore, that there exist 2n+1inde- pendent spherical harmonies which are homogeneous poly- nomials ofdegree n. Itwill beshown inSecs. 174 and 176 ‘that this isactually thecase. Asan example, letn=3. Then the polynomial is P=a2!+by?+ce!+Bdy'e +Beate +3fsty +hye? +Bisx* +Sjzy* +Gkzye; and AP=Gatet sz+OO+S+My+66+d+ide IfPisharmonic, thethree conditions atet+j=0, b+ft+h=0, c+dti=0, must besatisfied. After solving these equations for, say, a, 2,and c,and substituting inP,there results P=(8y%z —2)d +(Bete —xe +(Bx'y —y*)f +(8y2*— yh +Bex? —A+Bay? —aj +keys, since thisexpression isharmonic whatever d,e,f,...may be, each ofthe binomials, which are their coefficients, must be harmonic, There areseven ofthese binomials, and itisevident that they are linearly independent; that is,ifL,M,and Nare any three ofthese binomials, notwo ofwhich arethesame, and ifI,m,and nare any three constants, there does not exist a relation ofthe form WL+mM +nN =0, other than l=m=n=0. Any such linear combination, however, will beahomogeneous polynomial ofthe third degree which isharmonic; and every homogeneous, harmonic polynomial ofthethird degree can be represented asalinear combination oftheabove seven harmonics. 328 THETHEORY OFTHEPOTENTIAL 171. Relation between Certain Harmonics.—Let ¢:and #2beanytwofunctions ofz,y,and2,andform theLaplacian oftheir product. Itisfound that = BerAer, 81Bes,derdes) Alvi92)=ede:+eater+2StRStES Now let aa@+y team, ge=Hy where Hyisaspherical harmonic ofdegree n.Itisreadily verified that Ar" =m(m +I), 20:de,O619e2,Berdyr\_1%a2"bydy+38) of0H»,8H.,OH. Hymre(SE++*)2mnr™tHy; and, since AH, =0, itisseen that A(r™Hy)=m(m+2n+Ir, ) Asidefrom thetrivial value m=0,thisexpression vanishes, ifand only if, m= —(n+1). ‘That is,ifH,isaspherical harmonic, then H,/r**! alsoisa spherical harmonic. This can bewritten Ha Hy_Hoos pert Hoosy, orFP=TaD; oragain, if m+n =—1, He_Hm, which shows therelation between any two spherical harmonics which give rise tothesame surface harmonies. 172. The Expansion ofaPotential—If ¢(z, y,2)satisfies the equation ofLaplace, then 3/8 alsosatisfies it;for,onchanging the order ofdifferentiation, 22=8(4g)=20)= a8=Zae)-2@-0. 172) SPHERICAL HARMONICS 329 Inlikemanner, itisshown that every derivative of¢satisfies Laplace’s equation. That istosay, ifisahomogeneous fune- tion ofdegreenwhichsatisfiestheequation ofLaplace, attitty Betoyaat isahomogeneous function ofdegree n—i~j~k(provided it does notvanish identically) which also satisfies it. Inparticular neeyo isharmonic ofdegree —(i-+ J+ +1)outside ofany small sphere sbout thepoint &n,¢,where =VE-P TU We ‘The function 1/p can beexpanded inpowers off,9,and ¢by means ofTaylor's theorem, viz. 1(Hye att(LYeinige,>>het*pavoyioals)*‘|a Suppose ,1,¢are the coordinates ofaparticle ofmass dmof abody Bthat lieswholly within asphere 2which hasitscenter attheorigin ofthecoordinate system. Ifz,y,2isapoint which liesoutside ofZ,theseries, ofEq. (1)isuniformly convergent. Hence dm_(=i ate(1)0, vefz-PiTGUETaxeyiow\r)Jg@ram.(2) ‘The expressions Tosa=finittam fare the inertial integrals which were discussed inChap. II. Their coefficients inEq. (2)arespherical harmonies ofdegree -G@+jtk+D. Let Hage =DOM ate (1)Sik STIR Sxeyar\r Then Vi=DSAisalisn @)a 330 THE THEORY OFTHE POTENTIAL isanexpansion ofthepotential function interms oftheinertial integrals andspherical harmonies. Itshould benoted that the spherical harmonics Hj. arealtogether independent ofthe body B. 173.Rotation about anImaginary Axis.—In Sce.26itwas shown that theequation ofLaplace isinvariant under arotation oftheaxes defined bytheequations. z= ak+amfab) y=BE+Bn+Bat, azandtrn tat) provided thefollowing relations aresatisfied bythecoefficients Be 7 at+B tnt= 1, aor +Bie +172 =0, att srtrt=1, east ib tr =0,) (2) at+Bttt =1, aor +Bi +rm =0. If@and8aretwoparameters andidenotes ~/—1, itwill be found that ayet Gerla te, ge,7 EF 7 26 ns ain’ iem, altete =-% 7& & oe Be et n= n= ai, m= +1, satisfies thesixconditions ofEq.(2);andtherefore, Eqs. (1) with these values ofthecoefficients defines arotation ofthe coordinate system about animaginary axis. The axis ofthe rotation isthelocus oftheinvariant points ofthetransformation, andisdefined bytheequations Exattan tat, (a— E+an+at=0,a=BE+B+Pt,—BE+(Br—I)n+Bat=0, Femnétrantnt nétrm +da-De=0. Since thelastsetofequations ishomogeneous, itsdeterminant must vanish, anditwillbefound thatactually itdoes vanish. ‘Thesolution oftheequations then gives bapigh orepSah astheequations oftheaxis ofrotation. 113) SPHERICAL HARMONICS 331 Iftheparameters arechanged bytaking =. ooe=y4y FRiyy sothat wa1-M@=D, yoy -MW#=D],. 21+2)” * 20+%) =f{,x-NO@ED), 5-14 ¢+), &chyse} Bratt30H)’ @ we Oe naTEy uate as=— Bs=—iMy y= +1, the direction cosines ofthe axis ofrotation uty +1 depend upon yalone, and \measures the amount oftherotation about this axis. 174, Harmonics Which Depend uponrandzAlone—Suppose His anharmonic which depends only upon rand z,where reatty tet Since itsatisfies theequation ofLaplace intheletters 2,y,2, itwillalso satisfy theequation ofLaplace intheletters &,1,&,afterthetransformation Eq,(178.8)ismade,SinceEqs.(173.2) aresatisfied,thefunctionrisinvariant underthistransformation;that is Peatty tatectr te Theletter 2,however, must bereplaced by z= f+ a(t +in). Suppose further that H=H,is©homogeneous, harmonic, polynomial ofdegree nintheletters 2,y, When expressed asafunction ofrand 2,itisarational integral function ofr* and2.After thetransformation, itbecomes Hylr', 2)=Hylr*, ¢+af+)), which canbearranged inpowers ofa,thus: _A qa 4at 2H Hy=Ha+of+ine+See+PE +...eegyPHO +t MSE @ 332, THETHEORY OFTHEPOTENTIAL Since thispolynomial isharmonic whateveramaybe,thecoefficient: ofeach power of«separately isharinonic, Butthese coefficients ‘arecomplex; therefore, therealpartofeach coefficient ishar monic byitself, andthepurely imaginary part also, ‘Therefore, thepolynomial represented byEq.(1)contains 2n++1separate harmonic functions. That they are linearly independent is evident atonce bytaking Et in=pe, sothat (E+in) =p¥(cos k0+¢sink8), andbearing inmind that H,ishomogencous inr,&,and f. Itfollows, therefore, that ifthere exists ahomogencous polynomial inx,y,2ofdegreenwhichisexpressible asafunction ofr?andzalone andwhich also isharmonic, there exist 2n+1 linearly independent, homogeneous, harmonic polynomials of degree n.ItwillbeshowninSec.176that,asidefromaconstant factor, forevery positive, integral value ofnthere exists one, and only one, homogeneous polynomial ofdegree nwhich isexpressible asafunctionofrtand2alone. 175. The Equation ofLaplace forSurface Harmonics.— IfH,isasolid spherical harmonie ofdegree n,then isasurface harmonic ofdegree n;thatis,itreprosents the values which H.,takesonthesurface ofasphere ofradius unity. Ttisafunction ofthepolar angles, which canbetaken tobethe longitude andpolar distance onthesphere. IfLaplace's equation istransformed from rectangular topolar coordinates (Sec, 57)bythesubstitution 2=rin¢cos0, y= resingsin6, 2=cosy, there results° eGV) 2av) 1ayr+ 3 1a _ Inow V=H.=ns, 175) ‘SPHERICAL HARMONICS 333 ishomogeneous ofdegree n,s0that Syisafunction of»and@ alone, thedifferentiation with respect torinEq. (1)can be performed; for PCV) _,art*Ss)ro=ae=nlnt+1s. After removing thecommon factor r,Eq. (1)reduces to 1_ #8, 1 a as, - anégoF*Sing(sa2)FnEDS=0.2) Every spherical surface harmonic ofdegree nmust satisty this differential equation; and conversely, every solution ofthis equation isasurface harmonic ofdegree n. Since the integral over any closed surface ofthe normal derivativeofanyharmoniefunctioniszero(Eq.(68.5)),itfollows that forevery&[tee=o.san Onthe surface ofasphere MyarS, and Wt=pris, Hence ‘oH 1 _[ities=kr!fisiae=0. Therefore, ifk¥0, fiside =03 ® that is,theintegral over the sphere ofany spherical surface harmonic ofdegree kiszero, except when iiszero. Since Ss is constant, fSede =4ra%So. 176. Zonal Harmonics.—A spherical harmonic which can beexpressed asafunction ofrand 2alone isasolid zonal har- monic. Asolid zonal harmonic ofdegree ndivided byr*isa surface zonal harmonic ofdegree n. Evidently, itisindependent ofthe longitude. Hence, ifP,isasurface’ zonal harmonic, ‘itdoes notcontain @,and Eq. (175.2) becomes 1 @(. oP.hgE(sn2)+nentPs=0, ® 334 THE THEORY OFTHE POTENTIAL which isthedifferential equation ofazonal harmonic ofdegree n. Forthesake ofnotation, let = 008 9; then Eq, (1)becomes a(q— ye =dawih!)+nln+DP,=0, on, @) PP» dP,=phe—24+ =0. =ayBb=BP2+mln+UP, Ifthesolid zonal harmonic isapolynomial inr?and z,the corresponding surface zonal harmonic isapolynomial incosg ofdegree n;andtherefore, apolynomial inwofdegree n.Ifitis assumed that Py=aut+aut!+owt+--+, itisfound that, aside from aconstant factor, theconstant coef cients a,areuniquely determined byEq. (2). Hence, there exists one, andbut one, zonal harmonic which isapolynomial inpofdegree n.Inorder tonormalize them, itiscustomary to choose theconstant factor insuch away that Pe= +i when p= $1, With thisunderstanding, then, every standard zonal harmonic which isapolynomial inuisequalto+1atthenorthpoleofthe sphere. Since Eq.(2)isadifferential equation ofthesecond order, a complete solution contains twoarbitrary constants; andsince it islinear andhomogeneous, thecomplete solution hastheform P,=CPW) +0), where C,andC;arethetwoconstants ofintegration. Ifitisassumed thatPisthepolynomial solution, andthat thissolution isknown, itwillbefound that au =Pfao- Pasar alsosatisfies Eq.(2).Thissolution evidently contains loga-rithms andfractions, andisknown asazonal harmonic ofthe second kind. tT) SPHERICAL HARMONICS 335 177.ThePolynomials ofLegendre.—Let pbemeasured from fapointonthezaxis atadistance 2)fromtheorigin. The function a oVery tema satisfies theequation ofLaplace andisexpressible interms of 1and zalone, since piart—Dee+20% @M Ontaking V=1/pinEq.(175.1) andbearing inmindthat1/p doesnotdepend upon 4,itisfound that @(r) a na(1))_EC)+io-maG))-© Since z= 7c0s e=Thy Equation (1)canalsobewritten pia rt—Qrem +2%. After dividing through by#s%,andthentaking r pai awZ=1 1=mh-+ht=a=Ey itisfound that 1H, rahH, dr=2adh poe > and Bq. (2)becomes OH), ad nat)OD+dla-it=0. @) ‘The function HWe=(1—2h+he? -) =(La ben tay? 1 1 =(1eh) 21—eh)? isevidently expansible asapower series inh,andsince lei]=|e] =1, thisexpansion isconvergent ifhi<1. Itcanbewritten He=1stphpak+pohttobet 336 THR THEORY OFTHB POTENTIAL therefore REL=htpik?+pak+Dah+otaah ts, and ian =1-2psh+2-Bpah?+ +++nlntIpwr+> ht Also, since yoccurs only inthecoefficients oftheseries, afaH)=¥Afa-on gfeoe|=Sala—ttf ‘Hence, Eq.(3)expanded inpowers ofhbecomes (a 7| = d(qo- 9%+nln+ip)=0. (A) Since Eq. (4)holds forallvalues ofh<1, itfollows that each coefficient separately iszero, and therefore, 40—yyPe)4n ==1,2,--° é{awtf]4(n+Vpe=0.2=1,2,-++500.(5) Equation (5),however, isidentical withBq.(176.2), andsince p. isevidently apolynomial iny,from thedefinition ofh,itfollows thatp,andP,candiffer only byaconstant multiplier, But, for» =1, Ha pnt thget twee. ‘Therefore P(+I)=+1m=1,2,--+, 0, andpsisidentical with P,,thestandard zonal harmonic of degree n. 1 Thecoefficients p,oftheexpansion of(1—Quh+ht)? inpowers ofhareknown asthepolynomials ofLegendre, or,sometimes, Legendrians. Itfollows, therefore, thatthestandard zonalharmonics aresimply thepolynomials ofLegendre, and Halt YP. ©) 177) SPHERICAL HARMONICS 337 If|A|>1,thefunction Hcanbeexpanded inpowers ofthe reciprocal ofh.Thus y-——— -11TVR BT i,i1Quy+ip 1 SPs LS pppoenitZan BP, where Pyisunderstood tobeunity. Itwill beobserved, thatifnischangedto—(n+1)inEq,(5), theequation remains unaltered, andtherefore Py=Pens: 178.TheExpansion inTaylor’s Series.—The expansion of anyfunction g(2—20)inpowers of2»according toTaylor's theorem is S(—aypdl'e 20 o@—2)=9)+x!aeaT Inthepresent case o-nteoe, oeeee ee) == tee Gat Vi titat ‘Therefore 1_e(-Da/sl),.axntia) | or aLl sia & (ayPLOT EOEY | Since 1/psatisfies theequation ofLaplace, whatever ¢maybe,it follows that aL a@v\;, isharmonic and isexpressible interms ofrand 2alone. Hence aay &(L 2)cw BC) o 338 THE THEORY OF THE POTENTIAL isazonal harmonic ofdegree n.Since, for«=+1, pL+G)+G)+G) +} art tl) +) + cach coefficient ofEq. (1)reduces tounity forz=1.Therefore, these ccefficients arestandard zonal harmonies, and ttde(1 Pra“ay(0) ® 179. The Expansion inLagrange’s Series.—Let xbedefined asafunction of«and hbytherelation hae=1~ VI Bh+R, a) sothat deS- yar ® IfBy,(1)isrationalized, itis found that peut(25). This expression isadmirably suited toexpansion byabeautiful theorem ofLagrange.’ If2isdefined asafunction ofwby means oftheequation 2=w+tay(w), where aisaparameter and (2) isanyfunction which isdevelop- able inpowers of2—w,then zcanbeexpanded asapowerseries ina,andtheform ofthis expansion is Ssaett dh srwtec)+Deepa" —@ Bymeans ofthis formula, itisfound that cent Sd aerate,raat M+Dag mate )i andondifferentiating with respect toythere results ae ye 5bedt(ut=1)"PET 2y: ‘Witttamson's “Differential Calculus,” p.151, or,Goursat-Heprick, “Mathematical Analysis,” Vol I,p.40 179] SPHERICAL HARMONICS 330 Since, byEq. (177.6), H=1+ DP, rst itfollows from acomparison ofthe coefficients ofthese two expressions that -1 (#1,Pemaeles) “ aformula which isdue toRoderigues (1815). Itisevident from this expression that P,,contains only even powers ofu,ifniseven, and only odd powers ofu,ifnisodd. ‘This isevident also from thefact that ifuischanged into —1 and hinto ~Ainthe equation 1 H=Faas emDP ) theequation remains unaltered. 180. Zonal Harmonics Given Explicitly—Although the for- mulas s0fargiven arenotthebest ones forcomputing thezonal harmonics, itwill add totheclarity ofthought tohave afew zonal harmonies, orLegendre's polynomials ifpreferred, set forth explicitly. The following are the values given byEq. (186.4) Po=1, Pian 3,1 58Prat Pray h 215, 53, 81Pomye—yetp OT 75s, 58 Pomgoat—gol+aahPymMTys9°75 4T5881"Dao" ~rae Raat ~Fae 1811-9, 1-9-7, 9-7-5, 7-5-8Pre ae! ~aaa" +raat ~age pyaBG, IBL-9-7, |11-9-7-5,,SS°Seo8 Fae R"Taaa" =2758 4T5381Pree TTaees 340 THE THEORY OF THE POTENTIAL p,=EIB, 15-18-11-9, ,1B-11-9-7,,CS EEGs h Peea kT aaa 11-9-7-5 , 9-7-5-3 ~pone tree” Pyy=WAT AS-IB-11 yg_1715-18-119) 5us 246-810 * Pees. ” 15-13-11-9-7 ,_13-11-9-7-5er ae 11-9-7-5-8 ,_67-5-3-1 +a2 r68" ~24-68-10" Ifthecoefficients inthese expressions arereduced totheir lowest terms, thedenominators will contain only powers of2. 181. The Zeros of the Zonal Harmonics are All Real.— Although Eq. (179.4) isnotparticularly well adapted toderiving ‘the explicit forms ofthepolynomials, itisofgreat advantage inshowing that thevalues of«forwhich these polynomials vanish are allreal and liebetween —1 and +1. Itwill beobserved that theequation (- 1=0 has nroots equal to+1 and nroots equal to—1. Itsfirst derivative isanequation ofdegree 2n—1,which has n—1 roots equal to-+1, n~1roots equal to—1, and oneroot equal tozero. The second derivative isanequation ofdegree 2n—2which has n—2 roots equal to+1, and n—2roots equal to1; and since the first derivative vanishes for»=0,the second derivative, byRolle’s theorem, must vanish atleast once between 4=—Landy =0,andatleastoncebetween» =Oandx=+1. Since allofthe2n—2roots have already been accounted for, itcannot vanish more than once inthe intervals mentioned. Let these two roots besnand —ps. ‘The third derivative has 2n— 3roots, ofwhich n—3are equal to+1, n—3equal to—1, one equal tozero and one ineach ofthe intervals (—1, —a) and (+s, +1). Let these last two roots beusand —ys. ‘The fourth derivative has 2n—4roots, ofwhich n—4are equal to+1, n—4equal to1, and one ineach ofthe four intervals (—1, —ss), (Hs, 0),(0,+s) and (+ms, +1). 181) SPHERICAL HARMONICS 341 Ifthe (i 1)derivative has¢—1 roots u.1% different from +1, theiderivative must vanish once ineach ofthe iintervals intowhich thei—1roots yu: separate theinterval WW. \\e wag > MBN / Fie, 101, (-1, +1). Since itisapolynomial ofdegree 2n~iwith n—#rootsforx=—1,n—irootsfor»=+1,andiroots VY > VfAY SAS | Fea, 102, lying between these limits, allofitsroots arereal andliebetween thelimits »=+1, thelimits included. Henee then!derivative has noroots 1, and nroots in theinterval (—1, +1). ‘There are,therefore, ndistinet latitudes 342, THETHEORY OFTHEPOTENTIAL i i shes, thesphereforwhichthezonalharmonic ofdegreenvanishes, tndthewlatitudes arosymmetrically situated. withrespect totheequator. 182.CertainUsefulRelations.—If differentiations withrespect to#aredenoted byaccents, itfollows from Eq. (179.4) that 1 atts 11 Platt—Pls=pestFIar’~1) 1 a ie yet 7qa ~Y 1 te yet -ararmndelgee —D =Ania+DG?o] 1 a typ =rermae +DU=1) FAn(n +Dutt—1)=ann+Gt—v=] Qnt1d. ,earae—DS from which itfollows that Prost—Poa=OntPy. a Tf,successively, nissetequalto1,2,3,... ,n,inEq.(1) ‘andtheresults arethenadded, itwillbefound that Plait Pia=1+Sk +1)Py; @) coal andifEq.(1)isintegrated fromuto+1,theformula Par—Pea==n+1){Pad @) isobtained; andtheintegral from—1to+1vanishes, exceptforn=0.SinceP,isequalto+1fory=+1,itfollows,ifpissufficiently closeto+1,thatP,isPositive, andtherefore,therightmemberofEq.(3)isnegative Hence,foreveryn, Prat<Pai<1, ifwissufficiently close to+1. 182) SPHERICAL HARMONICS 343 From thedifferential equation, Eq. (176.2) there isobtained byintegration n(n+1)f'Padu =(1—0°)P Oncomparing thisequation with Eq.(3),itisseen that 1 1 p' PaPor=(Gtepi)t—wPy —& which has acurious resemblance toEq. (1). 183. The Zonal Harmonics are Orthogona! Functions.—It hasbeen proved that: 1 « tS Pas, 1)ee” EPH @ and = SrVinee” 2,7" Ifthese twoexpressions aremultiplied together and then inte- grated with respect to»from »=—1to»=+1, there results faedjaaVI=Suh+VT—ak+[UESrrawe- FZfMome |? =inky = y id. -1 imo foie J- ‘The indefinite integral oftheleftmember is 1-sal KT=2;W)+Vi=uk +); Vig08VE BahEW)+VREBak+7) and thedefinite integral is 1ggLtMVE+4+VK Vik (l= tk +(= Bi aLogVEEVIG+VIDAjogLtVRE,Vik (VE+VIG=Vik) VikOO1=Vik Since 1 l+z = =aSiad22 +1 344 THE THEORY OF THE POTENTIAL equation (2)canbewritten =one {" = ike PPidu. Acomparison ofthe coefficients ofthetwo members ofthis equation shows that +fPPdu=0 ix, @) aa and +8[leet aay a) That is,the zonal harmonies areorthogonal functions, which isafundamental property. The method ofproof here given isdue toLegendre. Ifthe formula 1 1 SpyKOTae h~DP isdifferentiated with respect tou,there results, after dividing through by, 1 1 = he weees EP. © BOWiser Pak ©) Also Lo LS pyHOVi=2h+ePat Multiply these two expressions together, and then integrate with respect to». There results Hide Ba (MpJos =LE [Prat © ‘The indefinite integral oftheleftmember is a Z,HR?~(E—W)0 =i)’ and thevalue ofthedefinite integral is +dy 2 “Joie =cama azpee 183) SPHERICAL HARMONICS 345, Acomparison oftheright members ofEqs. (6)and (7)shows that mmf'PPidu=0, iffSi,orifi+jiseven, a1 an ® f'PPldu=2,iff>i,andi+jisodd. 184. AGeneralization ofthe Preceeding Formulas.—To simplify thenotation, let>aPPro Po = Consider theintegral +Gi=[a~wypanr.irds a a1 ‘The function P,satisfies thedifferential equation, Eq. (176.2), (1=WA)PAO =24PAY +n(n+Py =0. Ifthisequation isdifferentiated successively j—1times, there results (1=2DPA =uPA? +(Dl =F+WPA? =0, which, after multiplication by(1—«*)t, eanbewritten Ha=P] ==+N—F+DIAPOM. @) IfEq,(1)isintegrated byparts, there results G=[Parva -wore]at Ho pd +— (Pee 2ia— pyPldn.Jl PrGe— Pele ‘The first term intheright member vanishes, since (1—#*) iszero atboth limits. The second term canbetransformed by Eq.(2),80that Eq.(3)becomes 1[ia=wppaesnda = at pet (tam —5+DfMO—rPemPseran, 346 THE THEORY OF THE POTENTIAL Equation (4)canbewritten G=@tDm-J+VGr3 andsequentially, Gia =(W+5- Vln —G+ 20%, G.=(n+ Inde, Onmultiplying these equations together, itisfound that (n+p!G,=SENG; 1@aa or,written outinfull, 1 (m+ er=w)POP,dy=MED! . fio-wppaersoda =GED[Prada Itfollows from Eqs. (183.3) and (183.4) that, ifjisany positive integer less than, oratmost equal ton, +f(1=)POP8dn =0men, o) -1 +htgp=2(nti! f"a—eMPatan=5ABAH @ ‘These results can beregarded asageneralization ofEqs. (183.3) and (183.4). 185. ARecursion Formula for Zonal Harmonics.—If Eq. (183.1)isdifferentiated withrespecttoh,thereresults tah, =Sapien, (= uh they At or,onmultiplying through by(1—2uh +h*) wraft+>Pa]=(1=2uh+)YnPet=0.a Fal Ifthis expression isarranged according topowers ofh,there results aPrt ¥[Qn+uP, —nPoa —(n+1)Papslh™ =0, co inwhich, ofcourse, P)=1.From thisexpression isderived the recursion formula (n+ Past ~n+ Py +Pps =0, @ 185) ‘SPHERICAL HARMONICS 347 which holds forn=1,2,3,-->. Ifthesame formula, Eq. (183.1), isdifferentiated with respect tou,there isobtained ——! =SP; (1—2uh +h?)2 so or, AYPak™—(1—2uh+b>)Pie =0. Em ao This expression arranged according topowers ofhbecomes DPa—Plats +2uP's —Plead =0; Eon consequently Plast —2uP'a +Plat =Pru @ 186. The General Formula for Zonal Harmonics.—Consider the formula ee 1, ”(P=2uh+hy ® inwhich pisany number, notnecessarily aninteger. Itcan be expressed asapower series inh, He=D Hh, = which isconvergent if|h|<1. Iteanalso bewritten 2Pom - -yy’=[rae )] andexpandedbythebinomialtheoreminpowersof21(=») thus: aoSOFI=D of1), B=Op arOMe ah provided, ofcourse, pisaninteger. Ifpisnot aninteger the factorials which depend upon pbecome gamma functions, and 7=FEO wyy:(,—Ih). Hxareey2M'(#=gh ms THE THEORY OF THE POTENTIAL ‘This formula isvalid provided only that p>0and [2uh —ht)<1, ‘This last conditionissatisfied forevery«from—1to+1provided [hl<V2-1=4141-- >. Again, bythebinomial theorem, LY 2S (-yth yes),(2) =2-5): ‘sothat SoF@+). i!(') He=SPO IWS(ayL(Y pes 2roy Pella) # oe r+) =SSvyPOD espe, 328? Gare" ‘This series canberearranged asapower series inhbytaking i+j—en, oo ian—j. With this change itbecomes Sak aTptn—3) He=SeS(yen DyessZeXO i=are where t=n/2 or(n—1)/2, according asniseven orodd. ‘Therefore ‘ 1Tp+n—= J) Ha=S$(-1yew2!a aR @) which isapolynomial inyofdegree n, Forp=}4;this gives thegeneral expression forthe zonal harmonic ofdegree n. Ifthe notation [Qn] =2°4-6+ +++ -2n, Bn] 1-8-5-7+ +++ Gna), isused, thegamma functions canbedispensed with, and 5 [2n-2§—1) Hy =Py=(-1PR Ms, BOOM w=an” © Since: (m2!=fn—Bln—3)—1h, oneofthebrackets intheright member containing theproduct 186) SPHERICAL HARMONICS 349 ofalloftheeven numbers and theother theproduct ofallofthe odd numbers, this expression forP,can also bewritten ‘ . [an—2§-1) P==1) un, 4) Ow ain=F=De" “ which, while notassimple inappearance asEq. (3),isactually simpler forcomputing purposes. Itisreadily verified that this expression satisfies the recursion formula (Eq. (185.1)). Itis from this formula that the explicit forms given inSec. 180 were computed. 187. The General Expression For H.’—The general expres- sion for H,*” becomes particularly simple for p=1.Let Ha =Qn so that a2 1 -¥ ". T=ae *%Oo. If2uisreplaced bye*+e~‘*, which can always bedone, since #=cos¢,the expression forH*becomes . 1=Ghee (r=heme) - 1 (—*. - ew TH hee ~The 1—meine-~feShteine ~waa*Deorder | =FtVep,, 2,sine from which itfollows that sin(n+l)eQB te. @ Also =Onn=M+ Ve—sin(a—Yo, Q.=Que ato which reduces to Qn=Ques =2c08ny. ®@ 350 ‘THETHEORY OPTHEPOTENTIAL Likewise Quer =Ques =208 (n=Ney Qr=Qo=2cos2, Q:=208 ¢, Qo= 1. From these formulas itfollows readily that Qu=1+25)cosje, @it Qua=0+2'9) cos(2)—1)e, [O} it and Qe+Qn=1425 008je. (3) at Now rope Leute @i=o.co ‘Therefore, theexpansion explicitly is lth S*il ScosjiToaneR it=P(1+2cos:ie): @ Bymerely squaring theseries forHitisfound also that Qn=DPPH 188, Zonal Harmonics Expressed ByCosines ofMultiples oftheArgument ¢.—The general expression forH**, Eq. (186.1) ‘can also bewritten HY=(1—he'*)-7(1 —he), im1. Since, bythebinomial theorem, >=SPO+I) 1=her=SEOLDps0 fe y%jing and = hete)-o=FLOHByrytig (1—he-‘#y &Er) Mes 188] SPHERICAL HARMONICS 351 there follows bymultiplication SSL +AT(p +&), >=Bx DOpsstgid-be, He=BSRPG)To) This double series can berearranged asasingle power series in Abytaking jtken and therefore jrk=n-%, jen-k The series then becomes toBREMEN BLOtHia-soope ca22, Ein—B@) Py Hence aP(p+n—k)(p+ k) =HEH A=W-TRtH ang, HSD>ila—BIT)-TO)© Theexponent of¢isi[(n—k)—kg.Ifkandn—k are interchanged theexponent: of¢changes sign, but thecoefficient oferemains unaltered. The terms can, therefore, begrouped inpairs, thus: aPEPAMADLOtMgentig4term) Hi?=2Ge=Hire) Fe) ere) or @ ‘Te+n-kh)T@+h or=ayPOtnWT- - Hy2Man=BITTY (n—2k)y, where t=n/2 ort=(n—1)/2, according asniseven orodd. ‘The factor 2must be omitted from the term for which nis even and k=n/2, This term isindependent of»and there isonly one such term, not two. For the zonal harmonics p=1/2. The gamma functions can beeliminated, just asinSec. 186, and thegeneral expression forthe zonal harmonics can bewritten ayolen=2k=U2k=1) _ P.22[in=BaaKT—cs—Bey (2) with theexplanation that thefactor 2istobeomitted from the 352 THE THEORY OFTHE POTENTIAL term forwhich 2k =n,and [~1] =[0]=1. This formulagivesexplicitly Py=1, P,=cosy, 1:3 1 P,=2ag0820+5 1-3°5 1-3-1 Pym2-75 cosBe+2-52cos L:B+5:7 1:3-5+1 1:3-1-3 Pom27Tgp8He+25Fey0082p+Fo7 1:3-5-7-9 1:3-5-7-1 PamBegeBrig0085+2:Tyg 008Be 1-3-5-1-3 RidererterySia (1:3°5:7-9- 11 1:3-5-7-9-1 PeDGB IO138OF+BgTESyrcosde 1:3-5-7-1-3 1:3-5-1-3°5SaOLSererr ererererdD 3:5-7-9-11-13 3-5+7-9-11-1 PreSpe 10s1ae1d 86+FGgcqor daeOF 1:8-5-7-9-1-3 L:B-5-7-193-5 FOEGB10BdO088HFEEagag8 189.Powers of«Expressed inTermsofZonalHarmonics.— Itisevident fromthetables given inSec.180thatitisalways possible toexpress agiven power ofuinterms ofthezonal harmonics. From thistable, obviously, pe Ps, w= Py, =2p,42=2p,+3; wet Py=24by, andsoon.From thefactthat thezonal harmonies with odd subscripts areoddfunctions of#,andthosewithevensubscripts areevenfunctions ofx,itisevident thattheexpression fory** ‘interms ofthezonalharmonics willcontain onlyzonalharmonics withevensubscripts; andtheexpression for4****willcontain onlyzonalharmonies withoddsubscripts.Inorder toobtain thegeneral formula, let w= oPetaPi +P+++++aPy 189) SPHERICAL HARMONICS B53 where thea;areconstant coefficients which aretobedetermined. Ifthis expression ismultiplied through byP.and integrated, there results 4 2pt fWP»du=Saf PuPmdsy a1 6S and, byEqs. (183.3) and (183.4), this reduces to +Padu=920m;foePade =onai whence +an=ay wPada. a1 Also, bysubstituting thevalue ofP,,from Eq.(179.4), Qm +1 6+) amaMTC Ge ye, on=Basta [ga—9 ‘Suppose fisany function ofuandf™isthekderivative offwith respect tox.Then _foorea- canbeintegrated byparts stimes, with theresult +1 nl ptt[letede =oreay,‘eke provided ¢islessthan either morn,andprovided also that fim, fort, for®, ... vanish atboth limits. Ifm>n,the expression reduces to JOeta =ayn fFfond =pment? a0, IfmSn,itreduces to + actfoered=Cagtig[ese The properties which have been assumed forthefunction farepossessed bythefunction (u*— 1)". Hence, ifm>n, am+1(** amt1ame, . ed eed WS 354 THE THEORY OFTHE POTENTIAL and ifm Sn, +an=amg Pad _ _m+1)n!fer—m(1—ytd. =n =my,MG~HY ‘Integrating again byparts stimes, iaNeePate—tfarte—woreda Live get aJe » Bymeans ofthis formula itisseen that ifmSn, —2m+1__n! ponny(* nnd,2 =mallinem=TJ which iszero ifn+misodd, butifm+nis even =2m+1)nlin —_m— 1),= mlnEm+ Consequently, by setting m=n—2k and then summing with respect tok,itisseen that we5On=4k+1)nt[2k—1) ”zTn—2F+1]@RT where ¢=n/2ifniseven, ands=(n—1)/2 ifnisodd; [2n—1]=1-8-5- +--+ Qn— 1);and[—1] =1.Since (2k)! =[2k[2k —1], theformula issimplified bythissubstitution, andbecomes 5)(Qn—4k+1)nt weoP. a) Zon =2k+1)BE} Itfollows, atonce, thatanypolynomial in«ofdegree m,Qn;canbeexpressed intheform Qn=APo+AP+APrt +++tAnPm (2) wherethecoefficients A,areindependent ofy,andare,therefore,constants. Thetheorem thatanyfunction ofwwhich isfinite andhas onlyafinitenumberofdiscontinuities betweenthelimits»=+1 ‘and#=—1canberepresented inthisinterval byaninfinite seriesofzonalharmonies isincluded inthemoregeneral theorem.ofSec. 206. 190) SPHERICAL HARMONICS 355 190. ADefinite Integral RepresentationofZonalHarmonics.— Other simple expressions forthe zonal harmonies arepossible. Forexample, itcanberepresented asadefinite integral. For this purpose, consider the integral In—Jo2ibcosa where ¢=\/—I, aandbarerealconstants, anda >0.Ifthe numerator and denominator oftheintegrand aremultiplied by 4+ibcosw,theintegral becomes deo *coswdeo toelapis +* [aE Since cos(x—«)=—cos«itis seen that “coswd_ Joa®+b*cos? w ’ and therefore, that 3b de T=2aObcote Feet wd=2aasecoOP Bymeans ofthe transformation ere ane=EB ‘the integral reduces, since aispositive, to 7-2 fe~VeFooT+8"Vath Consequently, ifa>0, "de_ie @ JoBtCOGVarEOE Ifawere negative the sign ofthe second member would be reversed. Now let anl-hy b=hvi=e, inEq. (1). Itthen becomes L 1s asHew = SgVI=the+htrhimment —u?cos«)® 356 THE THEORY OP THE POTENTIAL It Osh<1, -lSes+1, itisfoundthatthemodulus ofA(u+iVT—xcosw)is hVT— (1—#9)sin?@Sfh]<1. ‘The expansion oftheintegrand inEq.(2)isabsolutely and uniformly convergent with respect touw.Itcan, therefore, beintegrated term byterm. Hence n-ly wf+iVT=Hcosw)rdw=Pah,72, Jo aa @) Piaif(+iVT=H008u)ds, anequation which isduetoLaplace. 191. AnImportant Property ofZonal Harmonics—The definite integral representation ofthezonal harmonies exhibits theproperty that forafixed utheharmonic P,tends toward zero asnincreases, provided only that uf<1. Now Pal<if+ivI=Heow|de; or, [Pal<f0=(=)sin?»)iao, For brevity ofnotation take Q-)=<1, and write SpeeLoL oS: Evidently f(-aint)ido<fdw=8; f(:~Bsin?«)ae<f(1—Bsin?a)ido= (—28)(1 —k*sin?8)?; 191] SPHERICAL HARMONICS 357 and f(1=#sin?a)¥de<f.ds=3. ‘Therefore, bytaking thesumandthen dividing by™, »,|<By(:-*yaisin?8) Suppose ¢isapositive number given inadvance, O<e<1, and s-7 Then Bi,7 and, since 1=isin5<1, itisclear thatforevery 5>0,there existe aninteger N.,such that forevery >Ne 1asin?%)3<} (:x0:=Bsin7)<p and therefore Pal<e Itfollows, therefore, thatthelimit ofP,(u), forafixed1,asn increases iszero,if|u|<1. If,however, |a|=1,then|P,|=1 for every n. ‘Since themodulus of«+i/T=#¥eoswis1—(1—x2)sin*w, whichneverexceedsunity,if0SuS1,itisevidentalsofrom Eq. (1903), Peal [atin om«ae, that joaitt Pj3,do,7Jo or, [P.| $1. Henee thenumerical value ofazonal harmonic never exceeds unity forvalues of«whichliebetween—1and+1. 102,Expansion ofsinmginaSeries ofZonal Harmonics.— Itisproposed torepresent sinmybymeans ofzonal harmonies. 358 THETHEORY OFTHEPOTENTIAL Since itisanirrational function ofu,therepresentation willhave theform ofaninfinite series, thus: sinme=DCP. q@) i ‘That such arepresentation exists isproved inSec.208. Forthe present, thevalidity oftheseries will beassumed. IfEq. (1)ismultiplied byPadu andthen integrated from —1 to+1, there results si Cn,fiP,sinmodu=5 since, bySec. (183), +4 + 2fPPadu=0,andfiPode= Hence 0,=EEfp,sinme-sinode, or, @ C=meetPaleos(m—1)e—cos(m+Ielde. Oninserting thevalue ofP,from Eq.(188.2), thisexpression for Cxbecomes C=etl’{eos(m~1p—c08(m+1)¢] S$:2[2n—2k—1][2k—1] _xe =BE]ae]08—BWledy. (8) The product ofthetwofactors oftheintegrand isasum ofa finite number ofcosines ofintegral multiples ofy.Themultiplesarealloddifm+niseven,andtheyareallevenifm+nisodd.Since Ssjede=0,iff#0, (& andisequaltor,ifjiszero,itfollowsthatafterintegration all‘theterms ofEq.(3)vanish except theterm which carries the cosine ofthezero* multiple of¢asafactor; andthezero” multiple isaneven multiple. Inorder thatsuch aterm may 192] SPHERICAL HARMONICS 350 exist, itisnecessary that m+n shall beodd. Consequently, ifmisodd, Eq, (1)contains P’s with even subscripts only; and ifmiseven, Eq. (1) contains P's with odd subscripts only. Even though m+n isodd, nosuch term will exist unlessnSm~1,asiseasilyverified. Hence,Haq,(1)canbewritten sinme =>)CP; @) jee Ifn=m—1 there isjust one term inEq. (8)that carries the cosine ofthe zero® multiple of¢asafactor, namely, that one for which &iszero. Hence Cay=21n=I[2m~Ir, har’[an [2m=2]4 Ifn> m—1, there are two terms inEq. (8)that carry the cosine ofthe zero multiple ofyasafactor, namely, those for which m-l=n-% and m+l=n—2k For these terms ,=22+1(_[n +m—2]In=m]a Nnem1m+1) _ [nt mln -m=2} emt liem1)” =2n+1[n+m—2%I[n —m—2 [atm—ila—m— i] om atm nm +l atm tly” and therefore, finally, C.=—mn281im+m=2IIn—m=3.™ 2” In-Fm+In—m+1] ‘The expansion forsinmy asaseries ofzonal harmonics is therefore sin me = (2m=r_meSs[n+m—2Iln=m=2p, Bma2) PDem ln=mwFO+Po Since, however, nisslways even oralways odd, itisbetter to set n=m-143, 360 THE THEORY OF THE POTENTIAL and then theexpression becomes sinmy= ® [2m=Mr_mes.(Bi+2m—311%—Bas+om—1)Pryyess m=35~PR Fam)yt aan where Pi]=2-4-6---.2%, [-1] =[0]=1, (+1]=1-3-5-- ++G+), Form=1,thisformula reduces to ing=F—7M —Togs,yp 7 eeFe te @ 198. The Potential ofaSolid ofRevolution.—If thez-axis istaken astheaxisofasolid ofrevolution, andifforagiven value of2thedensity isafunction ofz*+y*,itisevident that thepotential ofthebody atexterior Points isafunction ofz andralone, where r?=2'+y?+2%;andtherefore, itcanbe expressed bymeans ofthesolid zonal harmonics. If Zn=Pay q@) itisevident thatZ,satisfies theequation ofLaplace andisa ‘solidzonal harmonic ofdegree n.BySee.171, Zn _Pa pie =jel (2) alsosatisfies theequation ofLaplace, andisazonal harmonic whichvanishes atinfinity. Forpointsintheneighborhoodoftheorigin, provided theoriginisinempty space, thepotentialcanbeexpressed intheform Vi=Yate =YawmP., (3) En) = andintheneighborhood ofinfinity intheform S80,fe_SsCn VmSenn =Sone, (a)Ea ae 193] SPHERICAL HARMONICS 361 Ontheaxisofrevolution P,=1foreveryn,andtherefore, Visafunction ofralone, namely, Vaw= Sor, otVan SSK;neo a0” or,since along theaxis r=2, Vase=Sour", or, Van=SLR ©) n=O aso” Conversely, iftheexpansion along theaxis isknown, Eq. (),thecorresponding expansion forpoints notlying onthe axis canbederived from itmerely bereplacing 2*byPy, iftheexpansion isinascending powers ofz,orbyreplacing 2-0 byPyr-+0 iftheexpansion isindescending powers of2.Since themodulus ofP,isalways lessthan unity, orat most equal tounity, theseries (3)or(4)willcertainly converge iftheseries (5)converges. That is,ifriskept fixed, theseries ‘will converge everywhere onthesphere ifitconverges atthe poles. If,asisfrequently thecase, theexpansion, Eq.(5),along the axis iseasily obtained, the general expansion isobtained with equal ease. Inordertobesurethatthefunction sodetermined actually represents thepotential function itwillbesufficient toprove ‘thefollowing theorem. Theorem—If Visananalytic function ofrand 2which is regular intheneighborhood oftheorigin (that is,isexpansible ‘inpowers ofrand2)andwhich satisfies theequation ofLaplace anv),a ater)2 =) 2BOD+2fa—u9@P)] =o, where ru=2;andifVvanishes foru=1forallvalues ofr<ro, then Vvanishes identically. Inorder toprove this theorem itisconvenient tochange thevariables bytaking Ue=rv, l-yp=» Forthese newvariables Laplace’s equation becomes BU,afiyaU]_Sh+2a-9]<0. © 362 THETHEORYOFTHEPOTENTIAL ‘Since Uvanishes with r,andalsowith »byhypothesis, it carries rvasafactor; and itsexpansion asapower-series in rand vhastheform . U=S Save, where thecoefficients ai;areconstants. Itisfound easily that eu=LYLie-Yarns,or imtint Shoo-9]=BS26+Wreues 5G+Doster, | ” intj=0 Therefore, ontaking thesum, YEG +Waser +6G—1)HF+Vlawlrot=o, from which itfollows that AVS =D,, [F201 -+1, ne Crs) esi Since Uvanishes with »,every a=0.Hence ‘every coeffi- cienta=0,andtheneveryaj:=0,andsoonsequentially,andU=0. Therefore V=0,sinceitdiffers from U‘onlybythe factor r. Forexpansions intheneighborhood ofinfinity, let 1 rel w Then Laplace's equation is wv 8 avwatgfe-a]=0, whichhasprecisely thesameformasEq.(6),andtheargumentisrepeated unaltered. Itfollows, therefore, thatifVandWaretwofunctions of and2which areharmonic ingivenregion through which‘thez-axispasses,andiftheytakethesamevaluesontheaxis,thentheytakethesamevalues everywhere, sinceV—Wsatisfies theconditions oftheabovetheorem. Consequently V-Ws0. 194] SPHERICAL HARMONICS 363 194. The Homogeneous Oblate Spheroid.—A convenient example ofthis mode ofdevelopment isfurnished bythehomo- geneous oblate spheroid, sinee itspotential asafunction of ,y,andzhasalready been given inSee. 32,and Sec. 39. The value oftheexterior potential along theaxis, according toEq. (39.2), is 3M Vow"7A z Ve-e VY#) 1+2.)tonVERE _VF, Gy where Iisthemass ofthespheroid, aistheequatorial axis, and cisthepolar axis. The eccentricity ¢ofameridian section istherefore ¥/a?—c?/a. Since tan=Sense,2M aay theexpansion ofEq. (1)asapower series is SE Get Youu=3MComINOW#8)Gamat Consequently, theexpansion forthepotential ofahomogeneous oblate spheroid atany exterior point forwhich thedistance from theorigin isr>aandforwhich thepolar angle is»(that is,the angle which rmakes with theaxis) is < (-1)" (ae)™, vau,BaFynFs)rei @ Ifahomogeneous shell isbounded bytwo eoncentrie, co-axial, spheroids ofrevolution, itsconter liesinempty space, and the interior potential eanbeexpanded interms ofzonal harmonics. Leta:and a;betheequatorial radii oftheouter and inner bounding surfaces, and czand c;thecorresponding polar radii. Then thepotential atany interior point is,byEq. (32.13), U=(4r- A) +Bs BG +y)+C—O), @) where 4:,Bi,C;arecertain definite integrals which depend upon theconstants a,and¢,,and A,Bs,C2arethesame functions oftheconstants a:and ¢;. ‘The level surfaces, therefore, are also surfaces ofrevolution ofthe second order. 364 THE THEORY OF THE POTENTIAL Along theaxis thevalue ofthepotential is Unie =(Aa —As) +(C2—Cr)e*. ‘Therefore, atany interior point thepotential is U=(Az—Ay)+(C2—Ci)r'P 8 1=1A)+Ceomcost»—) ® =a=Ad+C00[28Heervi and thelevel surfaces aredefined bytheequation ESM 2const. ) ‘These surfaces arehyperboloids ofoneortwo sheets according as the constant isnegative or = positive; allofwhich areelwyTyasymptotictoaconewhose KKSQ\,__ generating angle aisdefined Ky \\\bytherelation tana=2, J \ora=54°45'. Itisremark-N ||ablethatthese hyperboloids\\ [)sreallofthesameeccentricity,\ kybutitismoreremarkable thatNXle "theeccentricityisindepen- ST dentofthebounding surfaces,aa provided, ofcourse, thatthey Fro. 108, are spheroids ofrevolution. ‘There isanexception only in casebothsurfaces arespheres, inwhich casetheinterior potential isconstant. Oncomparing Eqs. (4)with (3),itisseen that Br~By=~:~ C)). © Itisasimple matter toprove thisrelation directly from the definitions ofBs,Ba;Os,Cyasdefinite integrals. For otef(t pctstlates perste tit esi)Veroe raere “a 1 =~2ate{Ger ——— JoAVF OF ayepH) =42 194) SPHERICAL HARMONICS 365 Hence, onsetting 6equal toainthis equation, there results arefy<ptesy? lemetjo(at+Ver+8) jo(+Ner+yveta thatis, 2B, =2x0 —C2, and also 2B, =2no —Cy. From thedifference ofthese two expressions, Eq. (6)follows at ones. 195. The Apparent Size ofaPlane Circular Disk.—It was shown inSec. 8that thez-component, ofthe attraction ofany plane area inthezy-plane atany point 0isproportional tothe apparent size ofthe plane area asseen from the point 0. That is,ifVisthepotential ofthearea, Zisthez-component ofthe attraction, and @istheapparent size ofthe area, orthesolid angle subtended bythearea atO,then (Eq. 18.1)), aveZeaa7 «2, SinceVsatisfies theequation ofLaplace, andalllofitsderiva~tives likewise, since av acre=a4”) =0, itfollowsthattheapparent size0isanharmonic function ofthecoordinates ofthepoint 0. Iftheplane area isacircle inthezy-plane about theorigin, itisevident that theapparent size ofthecircle isindependent of thelongitude ofthepoint Oand isafunction ofrand¢alone. Itcan, therefore, beexpressed bymeans ofthezonal harmonics. Itwasfound inSec. 9that thesolid angle ofaright circular cone, forwhich thegenerating angle isa,is Q=2r(1 —cos a). Ifaistheradius ofthecircular base and (0,0,2)thecoordinates ‘oftheapex, theapparent sizeofthebase asseen from anypoint ‘onthe z-axis is = ot Gan=2r(1ara 366 THE THEORY OFTHE POTENTIAL ‘Therefore, if2<a, VELS(ayant Wepra) au=2ef1YE+Ecof ey and if2 >a Moe=aBY) inFONz, Consequently, iftheline+which joins theorigin tothepoint 0 makes anangle ¢with thez-axis, theapparent sizeofthedisk atOis _ _t<_-[2n+1]p\intt a=aftPit DoER(APoa} ifr<a;andifr>aitsvalueis woe(Hntua)" 9=ZCWore ar)Pate 196. The Potential ofaZonal Distribution ofMatter on a ‘Spherical Surface.—Suppose thedensity ofasurface distribution ofmatter onasphere isproportional toazonal harmonic, sothat @=oP,(u). ‘Thepotential ofthisdistribution atapoint onthez:axis is vef@=2rae,f**Pats,> m1 where, Fig. 104, p=VFataFa; andthetotal mass iszero, except when n=0. The expansion of1/pis Llisp(2)>Pa) »if<a and 1_1l< ‘a\* ,iidrar’), if>a 196} SPHERICAL HARMONICS 307 Hence =e +tvated (2)[Purdy if<a, oho) Ja or S /q\ne pttVerm3(2)"("Pat ifpoeFea) 1 The surface integral oftheproduct oftwo zonal harmonies vanishes, however, unless the two zonal harmonies areofthe same degree, Eqs. (183.3) and(183.4), Hence dro (2\" ros (a), veani?) yoo Vege iQ) according as|2|<aor|2|>a. Inaccordance with theprinciple ofSec. 193, thepotential at any point, p,whose radius veetor rmakes anangle ywith the axis is 4roy_(r\" 4roo_(a\""*Voor (3)%oY=£e() Puy according asthepoint pliesinside oroutside ofthe sphere. Hence the potential ofazonal harmonic distribution ofmatter ofdegree n ‘onthesurface ofasphere isasolid a] zonalharmonic ofdegree nwithin ’thesphere, and asolid zonal har- monic ofdegree —(n +1)outside ofthesphere. From this result and the fact that: thepotential ofasum ofdistribu- tions isthesum ofthepotentials of theindividual distributions, itfol- lows that ifthedensity onthesur- face ofasphere isrepresented by the series nen ee aes thepotential atanypoint inside ofthesphere isrepresented by the series <ca r\* ve“>Rt(2)Py 368 ‘TETHEORY OFTHEPOTENTIAL ‘andatanypoint outside ofthesphere bytheseries <=_¢»a)” Ve“dxtn(’)Pa Ifthecooffcients o,aredefined asfunctions ofa,thedensity isdefined over aspherical volume, and thepotential ofsuch avolume distribution can beobtained byintegrating these series with respect toa. 197.Tesseral Harmonics.—The general expression forazonal harmonie ofthefirstkind ofdegree nis,Eq.(186.4), ‘ s(2n—2j—1) =ite Des, 1) Palu)z!YG=ae ro} ‘There exists, therefore, one, andonly one, solid zonal harmonic of degree n,aside from aconstant multiplier, which isarational integral function ofrand z,namely, <{2n—25—1]* (7,2)=Pa(u)=Dy(—1 ITpatipts;(2) Ile,8)=Pal)=(Ieae) andfurthermore, H,canbeexpressed always asahomogeneous polynomial ofdegree ninz,y,and 2.This isevident from Eqs.(2);forifn iseven H(z, y,2)contains onlyeven powers of 2,y,and2,while ifnisoddthepolynomial isoddin2,buteven ingandy. Itcanalways bewritten H,(r’, 2). Itwasshown inSec,174thatifthereexists suchahomogeneous polynomial ofdegree ninrand2,then2n+1linearly inde- Pendent, homogeneous, harmonic, polynomials ofdegree nin2,y,andzcanbeobtained from it‘byarotation about an imaginary axis. Inthisrotation, 1"isreplaced by PettyteaBigts, andzisreplaced by tattaCEtin, tava Ifthepolynomial soobtained isdeveloped inpowers ofa itbecomes, Eq.(174.1) Gu=Mu+alt+igHae4ag420H Felt NSE+letiE + PHO +7 to 197] SPHERICAL HARMONICS 369 where H, istheexpression forH,forthe value a=0;or, returning totheletters z,y,2, sat «ON5) Gules8)=De +ah Now HAO _ar*P x)yus9Pn, de "aru ant and, ingeneral, PH Ee @) ‘Therefore . OP, a=See+yee Ce cy Since aisanarbitrary constant, thecoefficient ofeach power ofaseparately isharmonic. Consequently rote+the © 7 isanharmonic which issaid tobeofdegree nand oforder k. Ifachange tospherical coordinates ismade bythesubstitution z=rsin ¢cos 6,° y=rsin gsin6, cosy=K, t=reese, sing=VI=W and ifforsimplicity ofnotation OP, om2s, Pr Out theabove harmonic, Fq. (5), becomes: ret# sink yPAO, Onremoving thefactor r*,itisseen that oOsin! gP.O isasurface harmonic ofdegree nand order k.Since itiseom- plex, itsreal and itspurely imaginary parts separately are harmonic, sothat Cu=sint¢P,™-cosk8, 6) Su =sint ¢P.® -sink6, 370 ‘THE THEORY OFTHE POTENTIAL aretwo distinet types ofsurface harmonies ofdegree nand ‘order k.‘These particular surfacc harmonies arecalled tesseral! surface harmonies, and their products byr*arecalled solid tesseral harmonics. Ifk=0,Cyoissimply Ps,and S,ovanishes identically; sothatthereare2n+1tesseralharmonies ofdegreenIfk=,P,®ismerelyaconstantandthecorrespondingharmonics Cun=[2n—1]sin”ycosn6, @Sua =[2n —1]sin® gsin6, are called sectorial harmonies. Itwill beobserved that the coefficients ofcos k@and sin k# inthe expressions for the tesseral harmonies, (Eq. (6)), are functions ofg,orofu,alone. Ifthey aredenoted byTas, itis seen that, explicitly, : Ta=sintPL?=(1—yt)?SPs,a @) tS[Qn—2)—1 =D (yiPeta ty, aPxYB=Fay where I=(n—k)/2 or(n— k~1)/2 according asn—kis even orodd. Itisevidentnowthatonthesurfaceofaunitspherethevalue ofanyhomogeneous, harmonic, polynomial ofdegree ninz,y, and2isrepresented bytheformula Hale,6)=SY(ACns+BiSua)a 5 @) =Y(Ascos86+Bysink)Puya or, Fue, &)=YDiTxcos(ko—4), (10) fo provided A,andBy,orD;and6,aresuitably ch . 7i " lychosenconstants; andthatitsvalueonthesurfaceofanyothersphereofradius 7canbeobtained bymultiplying theseexpressions byr", *Tessera,asquareorrectangle. 197] SPHERICAL HARMONICS 371 The Equation ofLaplace for the Tesseral Harmonics.—The equation ofLaplace forany surface harmonic S,ofdegree n (Eq. (175.2)) is a ndSs 1dS,E(a- 8)+7=aeTn+DS,=0. IfS,isatesseral surface harmonic, Sa= Tmcosk8, or Sa=Tuasin8, Ineither case aS, 2pe=~PSe ay and thedifferential equation forthefactor Tus, which depends upon ¢alone, is a Tw __k .Ha-#rt)+(nie+)-ipa)=0.(12) 198.—Examples ofSolid, Tesseral Harmonics.—The solid tesseral harmonics aregiven byEq. (197.5) interms ofz,y,2} foritisevident from Eq, (197-1) that wo SS(yyLM=BMnicer4yttaty Zag=oexWoes ape et+e) ‘The expansion of(x+iy)* is @+iy =Xet+ iYy where XeSew, iBeBO Mone ar Y ‘ kt Yo2OVGE HES De Consequently the solid tesseral harmonic, corresponding to Eq. (197.6) are Ony =XiZu, and Say =Vaan uptoand including n=4they areasfollows: 372 THE THEORY OF THE POTENTIAL rw=, |“ere r€u =2, Sn =ys Hie==fat=Byte, 18:0=0, 190m =Bez, 78 =Seu, Cag =3:~By, Sis =62y, C9=foyet, |AS=0, 3.3. 3 3, On=33 2|Sy=—Baty—By+Byst, Can=—Set—Bay?+Gast, |S=—Feta—gy+OY °C =5c —Ty's, PSs. =30242, 1s =152" —45zy%, PS =452% —15y%, 3.3 3 22g? Cao=Sat+yt at+Sty?—Byte?—Sze, Ca=-Bere-Boys+10z2* Ha=—Bat+yt4abot—454/24, 1Cuy =1052'2 —315zy*%, Cu =10524 —6302*y* +105y4, Sa=—Patye —Pyte+ye, Sq =—L5sty —Lizy! +90zy2%, P'S =4152¢ys —105y%2, rSu =4202¢y —4202y*. 199, The Zeros oftheTesseral Harmonics.—It was proved inSec. 181 that the zeros ofthe zonal harmonics P,are all real and Lebetween p= —1 and y= +1, and that they are symmetrically situated with respect to4»=0. Acontinuation cftheargument ofSec. 181shows that the same statements aretrue also forthederivatives P,, which have n—kzeros inthe interval »=—1 ton= +1, and none elsewhere It follows, therefore, that onthesurface ofasphere azonal har- monic P,vanishes along ncircles oflatitude, one ofwhich is theequator itself ifnisodd, and theothers aresymmetrically situated with respect totheequator inthenorthern and southern hemispheres. Similarly P,® vanishes along n—k circles 199] SPHERICAL HARMONICS 3738 oflatitude which are symmetrically situated inthe two hemispheres. ‘The function . Tax=(1—oP obviously haszeros oforder k/2ateach poleandn—kzeros oforder 1along certain circles oflatitude which aresymmetrically situated with respect totheequator, making nzeros altogether inlatitude. Finally thetesseral harmonic ofdegree nandorder k, Taxcos(kB—4), Fra. 105. hasthesame zeros inlatitude nsthefunction Ty1,andinaddition, itvanishes along themeridians, orgreat circles through the twopoles, forwhich cos(k@—#4) vanishes, thatis, atte Ly, = 22k —oaBERSsim, 820,12 062BL ‘Theanglebetween anytwosuccessive meridians forwhich cos (ko—6) vanishes isx/k,sothatanytesseral harmonic of order &vanishes inlongitude 2ktimes. Thezeros areevenly spaced inlongitude, butonlysymmetrically spaced inlatitude,‘Thecircles oflatitude andlongitude forwhich C11,« vanishes aredrawn inFig, 105. 200,TheSurface Integral oftheProduct ofTwo Spherical Harmonics ofDifferent Degrees.—Let VandV,betwosolid spherical harmonies ofdegree mandnrespectively, andSm 374 THE THEORY OF THE POTENTIAL and S,bethetwo corresponding surface harmonies; sothat Va=TSm Va=Vn Q) Let2beasphereofradiusawithitscenterattheorigin.‘Thensince V,, and V,and all oftheir derivatives are continuous within 3,itfollows from Green’s theorem that Vm av. fivare=VeaVajar=fi(v¥s-vs). ® Since Vmand Vqareharmonic within ,the left member of Eq. (1)iszero, and therefore onava, fi(es-veda =0. @) Onthesurface ofthesphere ov, on Or’ sothat, byEq. (1) OV man OV _ getSEE=may, =na1S. Furthermore du=a*dudd; therefore Eq. (3)becomes yen (TE0% (m=namin (7(8,Sidud0 =0. Since, byhypothesis, m>n,itfollows that 1 peLUG sesuaua =0. ‘That is,theintegral over thesphere oftheproduct ofany two spherical harmonics ofdifferent degrees iszero (compare with Eq.(183.3)). Ifm=n,noconclusion canbedrawn from this argument; another investigation isnecessary. 201. The Surface Integral oftheProduct ofTwo Spherical Harmonics oftheSame Degree.—Let S,and Z,betwo spherical surface harmonies ofdegree n.Then, byBq,(197.10) Sq=YYAssn’ gPy!cosi(8—4), . O) Zn=SYBysin’oP,cosj(0—6;%),i 201) SPHERICAL HARMONICS 375 wheretheA;,By,0,and6;aresuitably chosenconstants, arethe expressions forS,andZinterms oftesseral harmonics. The surface integral oftheproduct ofthese twoharmonies is (fSotado =zyAGB)f"sin! 9°POPs Pay Xcosi(0—04)cos58—0%)d0.(2) ‘Theintegral with respect to@iseasily evaluated, for LP00si(@—04°)008(0—9,)0 = 0, ifivi, cos i( —8%), ifi=7 0, 2r, ifi=jmo. Hence theintegral reduces tothose terms inEq.(2)forwhich i=j. That is, [Su2sde =WeAoBof7'Patdn = am—9)fa—parc)’ +rQAuBicos(00—06»)[0=OPY'du. ByEq. (183.4), HO}Jiret al and byEq. (184.6), Ma 2(ntifiC=BPO) =Ga hence de feta=agiAeBe Peeg pt! Osean—gyn +mri ABeyi0081—4),@) which may, ormay not, vanish. Suppose S,andZ,aretesseral harmonics ofthesame degree nbut ofdifferent orders. Then either Aiszero orBiszero forevery index i(see Eq. (1)). Hence, thesurface integral oftheproduct oftwo tesseral harmonies iszero notonly if 376 THE THEORY OFTHE POTENTIAL they areofdifferent degrees, bySee,200, butalsoifthey are thesame degree butofdifferent orders, byEq.(2). IfS,andZ,aretesseral harmonics areofthesame degree nandthesame order i,then Eq.(2)reduces tothesingle term argMD. gm—9.00 [isezate =appTAB GyC08HOP—84), ‘This expression vanishes if i—0)=5 Hence thesurface integral vanishes, even though thetwohar- monies areofthesamedegreeandthesameorder,butofdifferent types, Eq.(197.6), Using thenotation ofSec.197,inwhich thetesseral harmonics are Cuz=sin‘g-P,-cos 18, Sui=sin’g-P,(9 sini, theabove proofs canbesummarized intheequations SiCuCnpte=[.CuSmdo =f‘SaSndeo=0,men. fenOude =[(CuSudo =[Sueno =0,inj. fienSueto =0, tio= [S.tao=< 27 +)! |fiewseejsut RIG io td=, tde=fcwtte=Aefisutae=0. — 202.The Expansion of[(z—§)*+ (y—9)? +(e—94)? JnaSeries ofTesseral Harmonics—If x,y,2and&1,f, arethecoordinates oftwopoints andRisthelength oftheline which joins them, then Ree @— B+ y—a +@—Oe Itisfrequently desirable tohave theexpansion of1/Rasa series, anditisthepurpose ofthepresent section toshow how ‘thisexpansion canbeobtained interms ofthetesseral harmonics. 202) SPHERICAL HARMONICS 377 If Pasty te, ea Bte te, then a RO VE- PFU PTE —— a1 VitaBipco a ee TVi heh + where na? cosratitmte, r tp Inpolar coordinates a2=rsin cos61, ==psin¢CosO, yersing:sin&, 1=psing sin6s, 2=100sv1, 5=pcosos, sothat cosh=cosgicosys+sing:singrcos(8;—63). (1) ‘Thefunction 1/Risexpansible inpowers ofh,andthisexpan- sionisconvergent atallpoints forwhich h<1. That is,by Eq. (180.5), L_LS pee SRRo7,BokPa inwhich Ry=P.(u) for«=cosh. Ifthis expression iswritten 1 _ssRaptrRoZee ms itisseen that thenumerator R,p"r" canberegarded asahomo- geneous polynomial ofdegree 2ninthecoordinates z,y,2) &n,$;andthatitishomogeneous ofdegree ninz,y,2andalso homogeneous ofdegree nin£,9,¢.Itisobviously symmetric inthese twosets ofvariables, and isasolid spherical harmonic ineither set. 378 ‘THE THEORY OFTHE POTENTIAL Itfollows, therefore, that Ry=P,(cos d) when expressed interms of¢1,¢,0,and0,isaperfectly definite surface harmonic ofdegree n,andtherefore, expressible intermsofthetesseralharmonies. ‘Thatis Ry=DAcsint x-Py'(us)-e0s i(0s—6) & where Ba =008 1, ‘andA;and°° areproperly chosen constants, which inthis ‘easemust depend upon g:and6:(oruz=cosvx). Itisevident, however, that when thesubstitution k=cosyicosg+sinyisinycos(0,—62) ismade inP,(4), thefunction P,(cos X)isapolynomial in cos(8,—6)which, when rearranged, eancontain only cosines ofmultiples of(6;—6:). Hence 0)=6,forevery index i. Also, since Rissymmetric ing;andgs,aswell asin6;and 4, itsform must be Ry=SSBudsin! voPui(us)) % (Gin! g-Pa(u.)) 0816, —62), (2) inwhich theBy: areconstants which donot depend upon ¢:, 2}01,OFOa. Inorder toobtain thevalues oftheconstants By theargu- ments ¢192}81,and 2can begiven particular values. Itis convenient, then, totake Oh m=m=aH; andtodenote thisparticular R,byR,*, Consequently Ra*=D)Ball —2°¥(P,)? 008iw, a and cos }becomes 08 h=w+(1—p*)cosw. Forthese particular values, itisseen that a SRAVint =Oee ZR 203] SPHERICAL HARMONICS 379 Multiply this equation bydy,and then integrate from u=—1 tou =+1. Since 4 2firs‘du=ay byEa.(183.4), and aid yup.coyy 2(nti!f(1=4)(P.)'dp =an+iG-al byEq.(184.6), itisfound that: N+1CHo 4, wp, (ntl.ae Retde=DBu eos.) ‘The integral oftheleftmember is enSoVI—2hcosw+A?—2A —cosw)u? se cin|2M=cos)VIR =0080)8°NT=Bhcoso+HP Therefore, theresult ofintegrating Eq. (3)with respect toxis a int, [AT 005 LS peRsVoi —cosw)vi=Bheosw+it2”| ‘Now multiply thisbyV/A,andthendifferentiate withrespect toh. ‘The result is Fh ELBE Oeay, Vi t=theo FR7a*feaus(8) or,onmultiplying again byV/h, 1th smtii rtp, Th cosoFRya fFatdu. (8) ‘Itwas shown inEq. (187.7) that lth _ Se . ‘[other 14Bat+23cae)fo) Hence, onsubstituting Hqs. (4)and (7)inEq, (6),itisfound that, < Sscoeie)=See ptPad+2cos*)=2Pac=apieesia.(8) 380 THE THEORY OF THE POTENTIAL Since Eq. (8)isanidentity inA,itfollows that - a2 FO gaa... Bu=1,Bui=ey+i! i=], any nels,%@ ‘These values ofthe coefficients By, substituted inEq. (2) xkive thecomplete expression forthecoefficients F,,namely, SnD. peo, fig=Pals)Polis)+23Geyin#0PalCo)X(sin!y+Pa'(us)) 08#61—63).(10) Onexpanding cosi(6, —#2),viz., cosi(8;—#3)=cos18;cosiM.+sin#9,sin102, and setting, asinSec. 197, forthetesseral harmonics Cui =sin gx Px(gr) +008 18s, Su)=sinoxPx(ea)+sinHO, Kq. (10) becomes Eq.(11), = Pals) $=Diere+SaPSac Ry=Pals)»Pa(us)+2>GrlOnPCas+SalSalPI and, finally 1_1¢ ‘oY.notBa(i): ‘The surface harmonics R,areknown asLaplace’s coefficients. Itisevident from the relation Ra=P,(cos 2) that R,isazonal harmonic with respect toapole which lies onthelinewhich joins theorigin tothepoint x,y,z. Equation (11) isitsexpression interms ofthetesseral harmonics ofthe original pole ofthesphere. Inother words, Eq. (11) can be regarded merely asanequation oftransformation. 203. The Expansion ofthePotential ofaFinite Body ina Series ofTesseral Harmonics.—The potential ofany finite body Bisdefined astheintegral ‘dm Ve|>fa 203) SPHERICAL HARMONICS 381 Ir,inFig. 106,£,n,£arethecoordinates ofapoint ofthebody, x,y,2thecoordinates oftheattracted point, and R= (e+=a?+@-H4 Perttytte, Peete te, theexpression forthepotential becomes, onusing theresults ofthepreceeding section, =1 Va Dda[rvrram.Peak ‘ q@ : esxst (ko, Fra, 100, ‘The expression forRyisgiven inHq. (202.11). Let mPa(us) =Pa, vy2)y PPalus) =Pa®(G 0,2), 1Cul(@ry 81)=Cul(2Ys2),eMC’(pa6s)=Cul(E,1Ey 75S8xi€(os, &s)=Sal(2, Uy#)y ASas(G2,8s) =Sui™(E, my2) BySec. 197, Py, Cx, Sac? and P.O, Cys, Sy? arepoly- nomials in2,y,2and &,1,¢respectively, which aresolid tesseral harmonics. Hence frptR,dm=Pao[Pavan2 2 3MHD EwCoun ©fsen EG Filcu flonan+suof'suteam]. @) ‘This result can bestated inwords asfollows: The coeficient ofthesolid tesseral harmonic ofdegree —(n +1) and order i ‘intheexpansion ofthe potential ofabody inpowers of1/r is 382 THE THEORY OF THE POTENTIAL equal totheintegral taken over thebody ofthecorresponding solid atesseralharmonic ofdegreenandorderimultiplied byfe exceptthat,when4iszero,thefactoris1insteadof2. ‘The integrals fficvdm and f/Sdm will bereferred toasthetesseral harmonic integrals ofthebody. ‘The above expansion is,therefore, aseries arranged according tothesolid tesseral harmonies oftheattracted point (2,y,2); or,equally well, according tothetesseral harmonic integrals ofthebody, 204. The Expansion ofthe Potential ofaFinite Body As aSeries ofInertial Integrals.—A tesseral harmonic integral ofabody canbenegative aswell aspositive. For example, forahomogeneous parallelopiped ofwhich theedges are2a,2b, ‘and2c,thetesseral harmonic integral (Sec. 198) eapteph rt SSSeesam=30f2"f°fe=waeanar =Mie —b5, ispositive ornegative according asa2b. Every tesseral harmonic integral obviously isresolvable into the sum ordif- ference ofanumber ofinertial integrals (Sec. 172). Asthere is nodirect method ofcomputing thetesseral harmonic integrals, thegeneral expansion interms oftheinertial integrals will begiven, The formula given inSec. 172, Ly Dae amie ¥=3Sitarazopaa(;)ferrem, issymbolical only. Itisdesired tofind thecoefficients ofthese integrals explicitly. Retaking theequation ofSec.202, 1 SsTp"RsRoDoe a) itwillberemembered that R,=P,(cos \)where cosy=tnt zt,os 204] SPHERICAL HARMONICS 383 andsince, Eq.(186.3), 1AF (H-NetPaw= &(1)"Bal@ =a5!" itisevident that 1_Fepena28=1, antag Poke=DO a eetntIe, 2) where|isn/2or(n—1)/2according asnisevenorodd.The symmetry andhomogeneity with respect tothetwosetsof variables z,y,2and&,1,¢isevident. Bythemultinomial theorem ofalgebra ent aaa @tb+ey~Zager itjtken Accordingly a =2)ayint«Byit eetbane=Beeb itjthk=n—2, te22a0 8!ssapagper,pen(Pt+o)=3aanwt,atBty=e Therefore Get yt ME att=(n=2s)laltyet, oPeo eeaegaateen, ‘This expression canberearranged bytaking i=p—2%, j=9-2%8, kar—%y “ptotren ‘The result is Geto tate tot toy _ (n—28)telar-ty-ar-7 =22Daapip= 2a—a= BI andthisresult, substituted inEq.(2)gives 1oR,=F (yen 2aeRe=2(Dna =BT (a =28)shateyetar xBDalatei@ =2a)=I 384 THE THEORY OFTHE POTENTIAL where 1, ptatr=n, atbty=s I=jn orJn-2), [Qn] =2-4-6--+-2n [Qn 1]=1-3-5+++-Qntl),and fo=1. Since [26]=2%, thefactors (n~24)!ands!canbecancelled, leaving Lyyt 10Rn=(-})[en—25—ap ameyetey g x%Davri@-Bwig— wera © Onsubstituting thisresult inEq.(1),multiplying bydm, andthen integrating, thore results =1 1aye Vedead(3(-F)er-a- yj Bo Maier Orn) ritieats arty tag et ’ xD aay =a)1GBESan)fewram, athe, Which istheexpansion ofthepotential function with respect totheseries ofinertial integrals. Ifthegeneral expression fortheinertial integrals ofthegiven body canbegiven, then thegeneral term oftheexpansion ofitspotential canbewritten down. This canbedone fortheclass ofbodies which isdis- cussed inSec. 50,andperhaps forothers also. From thesymmetry relations between thetwosetsofvariables 2,y,2nd &,»,f,itispossible toderive from Eq.(4)theexpansion forVaccording topowers ofz,y,and2,viz., « 1 :-Eea =PVan—25— veZREmef[R(-A)m-2—n Wien Eattie 5) x3aaie= tale leaf © hfe, 206.Laplace's Integral Equation.—It hasalready been shown (Sec.200)thatthesurface integral oftwospherical harmonics of 205) SPHERICAL HARMONICS 385 different degrees iszero, and even when thedegrees arethesameitmayvanish,asisshowninSec.201.Aparticularly interesting ease when the two surface harmonics are ofthe same degree isthat inwhich one ofthe harmonics istheLaplacian coefficient ‘Ry,(Bq. (202.11). LetS,beany surface harmonic ofdegree ninthevariables 16;Then, ifA,; and 6" aresuitably chosen constants, Sx=DAnesingr»Px(ui) +c08i(6—04), (1) Byitsdefinition By=BaaPals)+S)BacSin‘o1~Pu(ui)-€08i(0:—62), om where (n—i)! Boo=Palit), Bue=ATP sinkexPuMGe), ‘Ha=COSy2, and 62isindependent ofi. ‘The surface integral ofthe product ofthese two harmonics is given inEq. (201.2), 41 poe 7POPsetaanto,=Besaa+ 2 (+o! aProst2AB cosi(6—0”). Ifthevalues ofB,;ofEq. (2)aresubstituted inthis formula, it becomes 41 par de[OGstata,=EEAoPabes)+ an , ‘FEETDyAnsinos-PeM(n) 08Hs—81) 4 . . = pease) byBa.(D3 or, es poe® 1Sibu,6)=E20 (sson eda 1 Jo which isone ofthe earliest examples ofanintegral equation. 386 THE THEORY OF THE POTENTIAL Since Rissymmetrical inthesubseripts 1and 2,this equation could also bewritten 1 poeSuan,0)=EAL FreSstin8am,— 206. The Expansion ofanArbitrary Function inaSeries ofSpherical Harmonics.—Suppose there isgiven afunction of thearguments ¢and @,which isgenerally continuous intheregion -Jsest} 05052, although afinite number oflines along which thegiven function hasfinite discontinuities ispermissible. Itwas first shown by Laplace that such afunction canbeexpanded inaconvergent series ofspherical harmonics, although theproof given byLaplace was lacking inrigor. The first rigorous proof was given by Dirichlet, and the proof ofDirichlet’s was followed by& number ofothers, the simplest being those ofBonnet? and Darboux.! The argument ofDarboux will befollowed here. LetS,beageneral surface harmonic ofdegree nwith 2n+1 arbitrary constants, and letF(¢:, 6;)bethegiven function (not necessarily harmonic). Itwillberemarked first ofallthat ifF isexpansible inaseries ofsurface harmonics, S,, then that expansion isunique. For, if, Fler %)=YSw a) % and ifRa(er, 615es,@:)isLaplace's coefficient ofthe ndegree, then, onmultiplying Eq. (1)byRadu:d@; and integrating over thesphere, +1pe 2pttpie [0earaaatos=&ffRuSvdrdds. Jar Jo Aida Jo But, since +1aeffR,Sidusd6, =0,ifken, |: Jo tH tae£fBaSales,6:)dusd0s=57Saler, 01), *JournalfarMathemath, Vol.XVII,p.35(1837). *Journal deLiouville, Vol. XVII (1),p.265(1852). *Journal deLiouville, Vol. XIX (2),p.1(1874). 208] SPHERICAL HARMONICS 387 itisevident that Ae free. =Tpsen2),dw,=dud6,; and similarly 4 [pres =BETH A), don=usd, ButsincefPaPaesisaperfectlydefinitefunctionofg,and6,,it follows that theharmonic S,(y:, @,)is@perfectly definite one, and therefore theseries inEq. (1),ifitexists, isunique, and =21 Foon)=3EfRaPCeneden ® Fe ‘Asitcannot beassumed that theseries inEq. (1)does exist, letthefunction F,bedefined bythefinite series =Qn+1 Paley,1)=x=fBa+F(ex,62)deo2y @) which isperfectly definite, andseek thelimit ofF,,asmincreases. Itwill befoundthatthelimitofF.isF,and I\ ‘that theseries inEq.(1)isvalid. Letthepole where thez-axis pierces ‘theunit sphere bedenoted bythelet- terC,Fig. 107. Letthepoint whose coordinates are¢1,6:bedenoted by Cy,and thepoint whose coordinates are gs,62bedenoted byCz The angle \ismeasured bytheareofthe great circle which passes through C; Fra,107. and C,,since 008 X=608vxC08v2+sinyssingxC05(8s—1)5 and Ralor, 035264)=Pa(cos ¥), P,being thezonal harmonic ofdegree n. Ifthe function F(y2, 2)iswritten Fle, 62)=FC), as THETHEORYOFTHEPOTENTIAL and deisasurfaceelementinanysystemofcoordinates, thenthe expression ~ 1 Falen6)=322(ecaPacos do a0 is isindependent ofthecoordinate system used. Letthepoint C,betaken asthepole ofanew system of coordinates, and inthis new system letthecoordinates ofthe Point C.beysand @s. Then =oy€08h=COSvs=ds, F(CA) =Fale, 0), Py(co8 »)=Pals). Since Pa(us) isindependent of6s, mon +1ptt a Futon8)=SPE EP [Flosdee.a 1 5 ‘Theintegral 1p Glo)=a2),Palen 6s)dds Tepresents themean value ofthefunction Fs(ys, @3)along the circle oflatitude y3. Itisaperfectly well defined function of s,even though the function F;has afinite number offinite discontinuities along thecircle. Equation (4)becomes * +Felon)=Sarf. Palas)+Glus)dus. Thefunction G(u:) isindependent ofn,and, byEq.(282.2), YAn+WP.=Pla+Pos ‘Therefore upttFeaafG(Pa!+Phns)dus. ) Itwillbeassumed atfirstthatG(us)isacontinuous function Ofusintheinterval —1Su, S+1. Then Eq.(5)canbe integrated byparts, with theresult Hops Palen8)=30>a+Pad]3Pa+Pood Since7 Pa(+1) =41 and=P,(—1) =(—1)™ 206) SPHERICAL HARMONICS 389 itfollows that Upa41) +Pan(+D] =1, and BPa(-Y) +Paax(—D] =05 ‘sothat ip Pa=O41)—FfPa+Pan's © Lettheintegral intheright member beseparated intothree partsfi peut pee et(of ee aJa eras7Sia‘andconsider thesumofthefirstandlastofthese three integrals eetf+f.[e-+Pau" s atid Se eT ie ptf+fi.De+Posil*ifs<fr+fie.Since Giscontinuous thediscontinuities ofG’,ifinfinite, are oforder lessthanunity; therefore, thesumofthetwointegrals eetforeSijoieisfiniteandvanishes with6.If,therefore, eisgiveninadvance, 8can betaken sosmall that Sept y 4Joefhdpa +Pannctian <3 [Asfortheremaining integral 18bf(n+Pande’, ars letQubethemaximum value of[Px+Pmyjl/2 intheinterval, and pice rn ae! then “pe capeBfPat PawdGtdn 5Joppa PestIota 5Gm. 390 THE THEORY OF THE POTENTIAL But since, (Sec. 101), lim[Pa =0, if lal<1, thelimit ofQ,forincreasing values ofmalso iszero. Hence ‘mcanbetaken solarge that atif(Pu+PasiG'dus <be, 2J-i48 2 and therefore. wenafPat+Pmsi)G'das <6 Itfollows from Eq, (6), therefore, that: limFae, 6)=G(+1). Byitsdefinition, G(+1) isthemean value ofFs(gs, #3)along. acircle ofinfinitesimal radius about thepole; that is,itisthe value ofFatthepoint C,,orF(e,, 93). ‘Therefore sn+1 linFa=Flo8)=mE(rarerfds, which isEq. (2). Ifthe function G(u:) has finite discontinuities atafinite number ofpoints, the integration ofEq. (5)byparts isstill permissible. Neither the results nor the remainder ofthe argument isaltered, butasthetext books donotgive theproof ofthevalidity ofintegration byparts under such conditions alengthy digression would benecessary toprove it.The proof isnotdifficult, however. 207. The Representation ofaRational, Integral Function.— Suppose G(z, y,2)isagiven homogeneous polynomial ofdegree n intheletters 2,y,2,and assume forthemoment that @can be expressed intheform A 6G,y2)=Dre, Oo) Fat where H,isasolid spherical harmonic ofdegree p,and21=n orn~1according asniseven orodd. Form the Laplacian ofboth members ofEq.(1). AGisahomogeneous polynomial of degree n—2which isreadily formed; andsinee ACH) =59+2k+Vr, 207] SPHERICAL HARMONICS 391 byEq. (171.1), Eq. (1)becomes AG=YJ)2s(2n—2s+Ir. oe @) cot the harmonic H,disappearing intheprocess, since AH, =0. Aside from theconstant eoefficients intheright members, Eq.(2) issimilar toEq. (1), but itsdegree ism—2,The operation can therefore berepeated AAG =AG = A DY40(¢—Qn—28+1)@n—2s—DrPH (8)co and Eq. (8)isofdegree n—4,Ingeneral, after performing the operation &times 1= ye na et ew ad>Goifh— wee Whenkisequaltotheleftmemberisahomogeneous polynomialofdegree 1,ifnisodd,orofdegree zeroifniseven. Ineither event itisharmonic, since every polynomial ofdegree zeroor one isharmonic. The right member isreduced toasingle term which contains H,orHe. This equation determines H, orHo,asthecasemaybe,uniquely. Thepreceeding equation then determines H,(orH:),andsoon,back toEq.(1)itself, which determines H,. Thus allofthefunctions 7,areuniquely determined, andtherepresentation of@intheform ofEq.(1) ispossible. IfSyisthesurface harmonic corresponding tothesolid har- monic Hs,80that Hy=18, itisseen that Eq. (1)becomes D GG,y,2)=D Saul, 6)- o Hence theseries ofsurface harmonies which representsthevalue ofarational, integral function onthesurface of«sphere is finite series which contains noharmonies ofdegree greater than n, ‘Asanexample, letthemonomial zy*%*beexpressed inthe form yt! =He+PHe +Hs+1H (@) 302 THE THEORY OF THE POTENTIAL ‘The successive operations yield theequations A(zy%s*) =Qxz* +Gry"z =22H, +36r*H: +42rtHo, (0) At(zy%) = 24zz =36-14H, +42-20rHo, —(@) Ary?) = 0=42-20 -6Ho. ‘Thelastequation gives Ho=0.Then mm Hy=Fp fromFa.(0), KeAl-60%+lizy's+z2*),fromEq.(6), and finally, from Eq. (a), Hee spate —S6ay's—Ldze'—492y%s+161zy's*—72%"). ‘Translated into thetesseral surface harmonies bymeans of therelations inSec. 198, these results give sa 9_9 _ 1 aye=FaqylllCn +Ca—GCu—2a—GHCel- 208, Green's Problem forthe Sphere.—If acontinuous set ofvalues aredefined onthesurface ofasphere bythefunction V(&, 1,$),Green's equation, Eq. (136.2), defines afunction V(z, y,2)which isharmonic inside (oroutside, ifthe point 2,y,2isoutside) thesphere andwhich isequal toV(E, m£) onthesphere. This equation is a@ =r) 1 5 a ed @ where P= @- P+ U— m+ E-0F and déisanelement ofthesurface of unit sphere. Suppose thefunction Visexpanded inaseries ofspherical surface harmonies, Sec. 206, sothat Vem =LYSale,.F=f Itwasfound inEq.(183.5) that 15 — a SPra, Ginanen” 2, 208) SPHERICAL HARMONICS 393 and therefore 1-h pywaaay ~*+B)PtsPa, =>y(Qm+1)Pmah™, byEq.(182.1).mo Sincept=a?—2arcosh+r* and Pa(cosd)=Ry itfollowsthata@=")Sem+pae(2)" »~2 ‘, Hence, Eq.(1)canbewritten -3($sGmty a(t)” rena 3(5ee”Insel) SinceRaandS,aresurface harmonics ofdegree mandnrespec- tively, itfollows that, Sec.200, fjPoSnde =0, except when m=n. ‘Suppose thatinspherical coordinates xz=rsin go00840, y=rsin gosin00, 2=1c08 oss ‘then,byEq.(205.4), ifn=m, amt}|aS,45=Salem15 and therefore vee,1.2)=3Salem00(2) ®wo Ifthepoint=,y,zliesoutside ofthespherethecorresponding expression is - -Vee,na)=¥Sater00(2): ®oo 209,ThePotential ofaSurface Distribution ofMatter on@‘Sphere.—Suppose thereisgiven surface distribution ofmatter 304 THE THEORY OF THE POTENTIAL ofdensity ¢onasphere ofradius a.The potential ofthis distribution atanyinterior orexterior point p(2,y,2),is vef.gle,lsP where Paty -9+@-9% and &»,{arethecoordinates ofthesurface element de. Also reottyt te ae Pte Forpoints exterior tothesphere, that isr>a,theexpansion of1/pis,bySec. 202, L_1sp (a\”ao andforinterior points, that isr<a, Lid p(r\” where Ryistheparticular surface harmonic which isknown as Laplace's coefficient, Eq. (202.11). BySec. 208, thedensity 6,which isassumed tobegenerally continuous although itmay have afinite number oflines of Aiscontinuity, canbeexpanded inaconvergent series ofsurface harmonies, which, forconvenience, istaken intheform a1 Mt)0-63 EM 9. Hence als S(aynt+1 .aC ani or el SS(rVrti1rad 3()“fe according asthepoint pisexterior orinterior tothe sphere Under theassumption that = rsin g608Oe, y=7rsingosin60) 2=7005 ey 209) ‘SPHERICAL HARMONICS 395 thevalues oftheabove surface integrals are [pPobde =0,fonem and am +1"=f.“Baal, tsSales;86)s byBq.(2053). Hence atV=¥Salen(2)»ifr>a, Oy and ¥=Salen#0(5))iir<a ° ‘Thisresultshowsthatanyharmonic distribution ofmatter onasphereproduces thesame,andnoneother,harmonies inboththeinterior andexterior potentials. ‘Thetotal mass ofthedistribution is <xQn+1f~-S 1sd=aS byEq.(175.3).1theinterior andexterior potentials aregiven,andif <2m +1,2m+4s,2,ana isconvergent atallpointsofthesphere, itisreadily verified that (SeeEq.126.5) 1(avi, ave ss2m+1gn eh(eg OU) = YEN. +a(in+m)2,ae Oncomparing Eqs.(1)and(2)ofthepresentsectionwithge()and(3)ofthepreceeding section, itisseenthatGreen'scanition forthespheredefinestheinterior andexterior potentials‘ofthesamedistribution ofmatter onthesphere, &factwhich,of‘course, wasalready known. ‘Thedefinition, however, isnotbycrore’ofthesurface density, butbymeans ofthevaluesofthe potential itself onthesphere. 210,Differentiation withRespect toPoles.—It wasshown inChap,II,See,54,thatifa,8,varethedirection cosines of«line 396 THB THEORY OF THE POTENTIAL and g(z, y,2)isafunction oftherectangular coordinates, the derivative of¢inthedizection oftheline is ae ag pe ae Bo=oSe+Oe+38, and, also, that dz=ail, dy=dl, da=il. Consider asphere ofradius awith itscenter atthe origin Let«8, 7bethedireetion cosines ofaline, I,which starts atthe origin and intersects thesphere inthepoint p,‘The point pis called thepole oftheline, and differentiation inthedirection of theline1iscalled differentiation with respect tothepole p,and the symbol forsuch differentiation is 268,048greg odtod Oy) ‘The result ofthis differentiation is,ingeneral, afunction of 2,y, and 2which ean bedifferentiated asecond time with respect tothe same pole, oreven adifferent pole; and soon, Let «Bi, 71bethedirection cosines ofthe first pole, and ax,Bs,72be the direction cosines ofthe second pole. The result oftwo successive differentiations with respect tothetwo poles is a aap dar Al, 8 4984.8aur-(#3:+Og+og)(a+Bag+or) ” a a maa +aig +annghe, ae a a +Boogie +OS, +Boasts, ae a ae +nega +nPagigy +e Itwill beobserved that these differential operators obey thefundamental laws ofalgebra, namely, theassociative law, the distributive law, and thecommutative law. They canbetreated intheir combinations, therefore, just asthough they were algebraic quantities. Accordingly aaa a_Theeag2a02 Sade a7U(mge+83;+mR) {isthe general expression for differentiation with respect to1 poles, which may, ormay not, bealldifferent. 210] SPHERICAL HARMONICS 397 Itisevident that the n®derivative ofanharmonic function with respect tonpoles isitself anharmonic function, foritis merely the sum ofafinite number ofordinary derivatives multiplied byconstants; and each ordinary derivative ishar- monic (Sec. 172). If¢isanharmonic function ofdegree m,its n*derivative with respect tonpoles isanharmonic function of degree m—n. ‘Asanexample ofpolar differentiation, consider thethefourth derivative of1/rwith respect tothefour corners ofaregular tetrahedron which has itscenter attheorigin, one corner onthe z-axis, one corner lying intheyz-plane, and one edge parallel to the z-axis. The direction cosines ofthefour corners are, then, 2 [2 w= an mene aetye 2, 1 1 B=0,f=3v2 f=-3V2, Bem-3V% =1, --} =-} ~-!melo oue-p nen} w= -} ‘The fourth derivative with respect tothese four poles is a _w1/.2 a aaialabal, *U(x:+og+mo)= 8(29218-fig-2-12 aa\3 “ay~3a2, 13az3ay3az, (32_V2a 1a 3Ox 3dy 3a, =tv2_a2atav?ot2 at=—"Q"dztayae *9dz%02? *27dy'd2’9dy*ax* 27az* ‘This symbol operating onthefunction 1/rgives gal-8e —By—Bet—Grty?+2dy%2?+24e%e* —GOV222+204/2y%). Since 1/rissymmetrie with respect toalldirections, the above expression isanharmonic which issymmetric with respect tothe four lines which pass through the center and through thefour corners ofthetetrahedron respectively. This symmetry isnot inevidence inthe above expression, since the tetrahedron 398 THE THEORY OFTHE POTENTIAL itself isnotsymmetrical with respect totheaxes ofthecoordinate system. Avery simple example inwhich thesymmetry isinevidence isthethird derivative with respect toeach ofthe coordinate axes, namely a (1)__jae wat) =1 211. Derivation oftheTesseral Harmonies byPolar Differ- entiation.—It was proved inSec. 178 that the surface zonal harmonic ofdegree nis ametan(1) Pw=(wr Z() which isthen“derivative of1/rwithrespect tonpoles multiplied byr*andaconstant factor. Inthiscase thenpoles are coincident andarealllocated onthez-axis; andtheharmonic has butoneaxis ofsymmetry. If,instead oflying onthez-axis, thencoincident poles ofa surface harmonic ofdegree nlieonalinewhose direction cosines area,8,7,itisclear fromthesymmetry of1/rwith respect toall directions thatitisazonal harmonic which hasthelinewhose direction cosines area,8,yasanaxis,andthatitsexpression is ewccpt™(2 2od.aV/T R,=(-1)Taktota)(3): @ Thisisevidently Laplace’s coefficient (Sec.202), Ifitisreferred toitsown axis, itissimply azonal harmonic. Intheanalysis which follows itwillbeshown thatthetesseral harmonics ofdegree nandorderkcanbeobtainedbydifferentiat- ‘ing1/rwithrespect ton—kcoincident poles which lieonthe zaxis andksimple poles which lieinthezy-plane andwhich are uniformly distributed inlongitude. ‘Thelongitude ofthese ksimple poles willbetaken tobe Ce aeee thelatitude, ofcourse, being zero. Thesymbol fordifferentia- tionwithrespect tothese poles, Eq.(210.1),is oe 2ir\a 2jr\aa7con) +anos42Eoa(2) 211] SPHERICAL HARMONICS 399 Differentiation with respect tozdoes notoccur since thedirection cosine ofeach ofthese poles with respect tothe saxis iszero. Let thevariables bechanged bytaking f=2+iy, v=2-iy, where i=/-i, sothat a a a a (2 aaaEtay ayi-ay The derivative with respect tothepole p;then becomes a Qi eat 2jn\] aae[oss+=)+iin+E)Ise +[cos(s.+2)sin(°6+A)leeo(usa, (ne) a, aeOs, A aa aL s(t)a-i(u+88)a|) aha”“aodilate an This somewhat complicated expression issimplified bytaking anend, peenn?= ee ‘on and becomes es aa a_ Tobe sist a2 =He+0).Te.« =eee(ae+0). The first factor inthis expression, e~@+!, isequal to+1 if kisodd, and isequal to—1if#iseven. Ifkisodd, say k=28+1,the derivative isBb/din aaaL et 5) aa**aia7It(«+) ©) ait ait The two groups ofcomplex numbers e**? ande”*1, j=1, 2,--++, 28+, are identical except forthe order inwhich thenumbers occur; for det ai FHL on, 400 THETHEORYOFTHEPOTENTIAL ifn=2mandmS5;or,ifn=2m—(28+1)ands<ms 2s+1, Hence aa Ee adaa Fai”Ul44) since theright members ofEqs. (5)and (6)differ only inthe ‘order inwhich the factors occur. Itisknown from the theory ofequations, however, that Bit; Gab ue Bett aitaatitlHGaeBH)=ii(:+&)cost int Onreplacing xinthis equation byt/a and then multiplying through bya+", there results mei at T(-*+’)=asstHatt; @ and, therefore, from Eq. (6), aa a 1 21,aac agree, or ®) 98 8 py pnandl,aha aT get k= 2et1 Ifkiseven,sayk=2s,theexpression fortheoperator becomes 27 ait aaa a Wakaeli+} and since Bit 2U-toe! Trae, each ofthefactors oftheright member isrepeated. Therefore, aa a oted I"hahan-[(~+) (9) int If8isodd Eqs. (7)and (8)show that aa a wg ginOhhy*al~-(«agt¢“i)aein ad 9af OOBB a BE 211) SPHERICAL HARMONICS 401 If¢iseven,sothatkisamultipleof4,sayk=2s=41,Eq. (9)becomes ay fei aa a a Now 2 me) i the two expressions differing only inthe order ofthefactors. Hence m7 intT(!+)=b*—a%, int aa a9982(qupape ayaf,al,"lg=~~HYae ou ae a =Oeoee ‘ant+aga Itisevident from the equations oftransformation from x,y tof 9that ae, et _at ant ap=*atom Hence, theLaplacian operator becomes e ee ya ieoat+aye+at™4Geaq+a Ifthefunction operated upon isharmonic, itisclear that ea 4g +ja7 and therefore, when applied toharmonie functions, a La atdn~~4at a2) With the symbolism thus developed, itisnow possible to differentiate 1/r, which iscertainly harmonic, with respect tonpoles, n—kofwhich arecoincident and lieonthez-axis, while the remaining &are inthe zy-plane and are uniformly distributed with respect tothe longitude, 402 THETHEORY OFTHEPOTENTIAL ‘Thedifferentiation withrespect tothen—kpolesontheF-axis isgiven byEq.(178.3), ae (12oa)=(=a =DIOP, (4); andsince, Eq.(186.3), A . - [en=2~2-1) 13 ProvaSo pe) where2¢isequalton—korn—k—1 according asn—kiseven orodd, itcanalsobeexpressed ane (y -.(Qn—2k—9j—1]ee aeG)=p Gory(a ace Theletters£and7enterthisexpression onlyimplicitly through theletter r,and Petty teamte Ifhisanyinteger, itisreadily verified that 2A), #h)=hat?ot ae\rh ae eeae andingeneral, am(1)_(_1\"th+ 2m—2}gnaE, 72,|) Likewise an(1 1)"+2m—2]ge (2)=(-2)=a From these results itfollows that _ gent m2! (cam+om) (1)-(Gg! 4) é [2n—2k- 2-1) n-%-1 grey <2D peee Ba=e —9j1]pet X(ate +penitny, Now E=zt+iy=rsing-c@, and g=r—iy=rsing ew, hence emits 4.weit =Ortsintwcosk(O—Bade 211] SPHERICAL HARMONICS 403 Since this expression isindependent ofj,Eq. (14) reduces to a a\art(1 =(1.GBsint¢c08(0—6) : [2n- 2-1)1) antsy x2OMpia=bay Ifthis equation iscompared with Eq, (197.8), itis seen that the right member canbewritten (—1G tn008k(0—6), =m — B!which,asidefromthefactoros isthetesseral harmonie ofdegree nand order k. Itfollows also, ifkiseven, that (Eq. (12)) ae 1aTrae(-1) ator? and #1) (pil H(iFyFa)-CoBC) agar? : wtp on! =HD Pal. Ifk/2 isodd this term occurs inEq. (10) with anegative sign, while, ifk/2iseven theterm occurs inEq. (11) with apositive sign. Hence, ineither equation thesign oftheterm is(—1)". ‘Onsubstituting these results inEqs. (8), (10), and (11), itis found that aa... aatt) _y=!ahah ah=)=(“Dygeass Tat008k(—G0), ifkisodd, and 88 8eI) ln—BD! aha 6) =(1)piperTne008k(8—80) nl +(1gaPAW, ifkiseven. 404 ‘THE THEORY OF THE POTENTIAL Ttwas from thepoint ofview ofpolar differentiation that tesseral harmonies were discussed byJames Clerk Maxwell in hisTreatise onElectricity and Magnetism. Problems 1.Show that “ 8p, _16. &sinte=ish ae+are 2.Show that [rede =Setager' -en. Tene EOP—PPO. 3.Show that bo yPMGt=DS+Oe~G—ye =)dust(Cr a orsymmetrically ‘ ‘ Gt=? age—neGt=Fait—De +o at (n=a! dur 4.Show that the coofclnts of4°Ps4s) arealintegers 6.Iftheexpansion off() interms ofthe zonal harmonics is 0) = oP mo show that theexpansion forthefunction yf(s) is we)PAC:+GES")Pn inwhich 01isequal tozero. leSttattepotenti of»unierm einheformofeof MS [20—177) v MS (a(t),axear (6)Pom or MS [2n—1]/0\™,v= Say 2+ oa(G)Po accordinga8<arr>a 2Vol. 1,Chap. XU 211] SPHERICAL HARMONICS 405 1.Show that VINBAF a1PaDsaoe 8.Aspherical cap iscutfrom theaurface ofaaphere ofradius aby& cone with itsapex atthe center ofthe sphere and generating angle ,and covered uniformly with matter. Show that thepotential oftheeapoutside Ofthesphere, and at&distance rfrom the center, i oS Pans) =Pas(2)™tp yh, Vaaeeala=aftP=Pen09(2)p Gah, {eared PSS OE) ron} where 2=cos aand »=cos ¢;and inside ofthesphere So Pari—PenQ(r)*p gah V=deeaf(da)+ae Pals) fo-n¥, Matera} 9.Ifmatter isdistributed onacircular disk ofradiusainsuchawaythat thepotential onthedisk isconstant and equal tounity, show that elsewhere vert? DP ite<a ot and 2S ayyPm (ayeverecwret) itr>a, 10. The potential ofanequipotential distribution ofmatter onthesurface ofanoblate spheroid is aMPeSe_(oper yen v455-@) Paiifr>oe, MS (=1)" (ae)va¥Sai?) Preity>a8 11. The potential ofahomogeneous hemisphere ofradius aoutside the sphere ofradius aia pM sgSd(ysl—Mayes, vai(Ges DRS Es OND Inside thesphere butoutside thehemisphere, i.e,r<a,9>x/2, y-M(3_3/r r\}; St(an—Iron, v=M(B - 37), = us) 406 THETHEORYOFTHEPOTENTIAL 12, Show that H na)ta=(yeas mamth, SiPate =Depa) n+ and vanishes ifm=2n 38, Show that Patdy =LiPo may 14, Show that S2ard =mn+0. 416, Show that nl 5bey =ngwe 1 mtsinby 36,Show that, if)+546 =20, +=Pall2n —24—Ifn—25—1)f2n—2k=1) SiePa=OSaaiealoea ‘17.Ifasphereofradiusrhasitscenterattheorigin,andifitsdensityis «=af+by+f, show thatthepotential atanyexterior point ,zat distance Rfrom thecenter is betV=tepalae+by+e2). 18.If inanyrealpostive quantity Sette <0 —1(+eet and Sits =(-9 2,Li mF 18.Foranygiven harmonic ofdegree ntheproblem offinding them lines ‘hich passthrough thepoles oftheharmonic hasoneandonlyanesolution, btthedirections which aretoberegarded aspostive along these linesca bereversed inpais. 20.If =cotyand »=sing,show that : at PoeS(Wiggs tia, 3 rea where 21=norn—1accordingasnisevenoFodd. CHAPTER VIII ELLIPSOIDAL HARMONICS 212. Introduction—Among the geometrical bodies theellip- soid issecond inimportance only tothesphere, and many oftheproblems whichrelatetothesphereoccuralsofortheellipsoid,such asthe attractions ofsurface distributions ofmatter, the flow ofheat through thesolid body, themotions offluids around bodies of given shape, etc. Itisnatural therefore toseek for harmonic functions which arerelated totheellipsoid inthe same manner that the spherical harmonies arerelated tothe sphere. ‘The first theory which was constructed forthese functions was byGeorge Green in1833,! using only rectangular and polar coordinates. For this reason, Cayley proposed the name Greenians for these functions. Asecond theory inwhich elliptic coordinates were introduced was published byLamé inthe Journal deLiouville for1830,* and hisfunctions arecommonly known astheFunctions ofLamé. Both the functions ofGreen and the functions ofLamé reduce tospherical harmonics ofLaplace when theellipsoid becomes asphere. The development ofthe theory ofLamé ismuch improved bytheintroduction oftheelliptic function ofWeier strass, aswas done byPoincaré,* and itisthetheory ofLamé, according toPoincaré, that will besetforth here. 213. Definition ofthe Elliptic Coordinates.—The elliptic coordinates ofapoint 2,y,2arethevalues, 91,92,935ofqwhich satisfy theequation a v 2 _qeatgeetpoacine @ +Gneex, Geonor, “On the Determination oftheExterior and Interior Attractions ofElipsoids ofVariable Densities, TransactionsoftheCambridge Philosophical Society, 1885. Seealso hiscollected works, p.187. "Lame, ‘‘Legons sur les Functions Inverse des Transcendantes etles Functions Isothermes.” Paris, (1857). *Porcané, H.,“Figures D'Equilibre d’une Masse Fluide,” (1903). 407 408 THB THEORY OF THE POTENTIAL Ifthis equation iscleared offractions, itbecomesacubicequation inqwiththecoefficient ofthehighest powerofgequalto—1. Therefore theequation eye=t+-4t+ 4-1 {0=iatptetite ©=(9=aa —9-0) @= ang=@=e) ismerely anidentity ing.Itissatisfied whatever value q may have. ‘The quantities 91,gx,gs;a2,b*,c*,arranged inthe order of their magnitude, areasfollows O<t<ga<cBh<n<ae<g The surface q:=const. isanellipsoid; ‘The surface g:=const. isanhyperboloid ofone sheet; ‘The surface 9:=const. isanhyperboloid oftwo sheets. Ifa,8,7aretheaxes oftheellipsoid, then Heataab+ patty from which itisevident that a<b<r ‘The z-axis coincides with shortest axis oftheellipsoid and the zaxis with the longest. IfEq. (2)ismultiplied successively by(g—a’), (g—b*), (q—c*)and qisthen setequal toa?,b%,and c?,there results gtaBANG: ~2°V(qs =08), GF ba —ej 2GQ—PVG—BGs—8) yO= FO ay ® a =AO ANG ~), (= ae 5) Ifthese three equations areadded, itisfound that etyt2=GQtata-@+ +e), or “Fe Qt G- + ~e. Hence theequation ofasphere whose center isattheorigin, in elliptic coordinates, is gq+G2+Qs=const. =r?+(a?+BF+c?). (6) 213} ELLIPSOIDAL HARMONICS 409 Ifthefirstequation ofEq.(8)ismultiplied by +c’), ‘thesecond by(ct+a*)andthethird by(a*+b),andthe results arethen added, itisfound that (+ zt+(ct+ay? +(at+DY=(a90+ged+900) =(a +Bet+at). Hence theequation igs+guts+oar=const, @) represents acertain family ofsimilar ellipsoids. Finally, ifthefirstequation isdivided bya*,thesecond by 0,andthethirdbyc,andtheresults areadded, itisfound that Bw ly4Beets, atet an! +ape Hence the equation aida =const. Oy alsorepresents afamily ofsimilar ellipsoids. Ofcourse, the equation a1=const. represents afamily ofconfocal ellipsoids. 214. Differential Relations.—If thepoint z,y,2isgiven & displacement ds;,inwhich qivaries while gsandgsremain fixed. then 2a[(2)4(UY4(ZYlear ans=(Ra)+Gos)*Ge)Fo =Rid, where 2a(22)4(UY4(HY a=($5)+(Se)*Gn) ot 4#4 Ho) byBa(2133)sl@a*@—tTGF _lat 5-el byEq.(213.2). Ifthesecond expression forf(q)isused, itisseenthat _%___@=a9=99) -{| oo@=aa Gs TA” 410 THE THEORY OF THE POTENTIAL Itisnotnecessary tocompute theterms which carry (¢—4) ausafactor sincetheyallvanish whenqissetequaltoqu. ‘The notation ismuch simplified bytaking At=Kaaalge—ala=0.)>0,|At=@=aq —PVG) >0, io) AS =@—(Gr —WGe— ec) <0, Af =@— eG —WG—e) >0 ‘The quantities A,A,,andA,arereal, butAyisapure imaginary: Itisthen found that Reet —_@-@m-@) AP=4@= a —IG +e)” AKG =a! and similarly, ral _@-w@-a _ at _}@ BE4GBG—GP) 7AG—a pent@—a@——) at IGHG-MG oe)~WGw Since thedisplacements ds,=Ridg, ds;=Ridg:, ds,=Ridgs, (3) aremutually orthogonal, ds,being normal totheellipsoid, the general displacement, orthearc-element, is ds*=Ry'dg;* +Ra'dgs? +Rstdgs' ‘The surface element ontheellipsoid, g:=const., is da=ReRidgedgs; @ and the element ofvolume is dr=RiReRsdqrdgedgs. (5) Itmight beobserved, ifp1,ps,psarethelengths oftheper- pendiculars from theorigin totangent planes oftheellipsoid, hyperboloid ofonesheet: andhyperboloid oftwosheets respec tively, that -3 a2 1ee ee © 216. TheEquation ofLaplace.—The equation ofLaplace in anyorthogonal system ofcoordinates is,Eq.(57.8), a(RsR:2V) ,8(RsR,aV\ |9(/RsRaV\ _ExRya)+andRyin)+ARyon)=0 215) ELLIPSOIDAL HARMONICS 4iL Using thevalues ofR,,R2,and R;from Eq. (214.2) this expression becomes 8(Aulas =43)V)40(Axl=99) aq:\ AsAs qi, ags\ AsAy 82, 8(Aslqs=03)2)_ +aad‘Aids5)=o Since theproduct AA; isindependent ofq,this equation can bemultiplied through byA.d:A;, and written aal aav(422). a99+Ae(42%)-@s— as 4aai(4a)Ga)+il9)@—a a/,a . +Agi(Agr) @~a)=05 or,again, a)’ ay co0042) 0+n(n.) V+ (a—09(4:2.) =0.(a) (Qs=Qe)"45, . Ifthree new functions w,us,usaredefined bythedifferential relations = =2%, ds du=-$fdu—3Hdu=FY 2) inwhich du:isapure imaginary, Eq. (1)takes thesimplified form av av eV oh +@-WER +@-ash=0. There exist also the following identities: (aa ~48)+ax(as ~as)+aslas —08)=0, @—a) + @—a)+ Ga) =O. Ifthefirstoftheseidentities ismultiplied by—N'Vandthesecond by—MV, where MandNareanytwoconstants, andarethen added toEq. (3),there results 7 - &[ZR-oatMv]a- 2=0, Oy wherethelettersi,j,kare1,2,3andaretobepermuted circularly. 412 THE THEORY OF THE POTENTIAL Expressed wholly interms ofthe letters qi,this equation can also bewritten, inview ofEq. (1), 1oV,Laddav __we D[ackohMEM—watInv law=0.© 216. The Elliptic Functions ofWeierstrass.—The letters ui, ‘us,Usasdefined inEq. (215.2) areevidently particular values of the general function uwhich isdefined interms ofgbythe equation =daa 1 =Tig=Ha-a= ® This would bethenormal form ofthedifferential equation of the elliptic @-function ofWeierstrass ifthe sum ofthe roots a?+8+c? were zero. Itisasimple matter, however, to change thevariables and satisfy this condition. Let sbeanew variable and h,¢,és,ésnew constants, which are defined as follows: a=sth Baath @=e+h, Caath With these letters Eq. (1)becomes —ds t=Tiga aesa) ® and if A=he+h ted, itisseen that atete=0; also 1>er>ee ‘The solution ofEq. (2)is $= Bu, provided the constant ofintegration ischosen sothat sisinfinite whenwuiszero. Therefore ga=Puth, u-a=~u—«4, G=Puth 2-%=Gu—ey, @) G=Puth, qa—CF=Pus—es, 216) ELLIPSOIDAL HARMONICS 413 Ifthe derivatives of@with respect toware denoted by ‘accents, Eq. (2)shows that! =4(P—4)(Y—&2)(9—“ ® =4G? —12 —Gs, where 92=—A(eres +ext+ern), ga=t+4ereres. IfEq, (4)isdifferentiated with respect touand thefactor 29” isremoved, andifforbrevity ofnotation Pi=O-ey Pr= Pe, Pr=P—-ey itisfound that ()} 9" =49:9:9, and 9"=APs +PH.+PiPal)=o? —|, (6)=69*—502. ‘The power series expansion of@uintheneighborhood ofthe value w=Ois me pM pO ee.GunetttSuttye @ fromwhichitisseenthatPuhasapoleofthesecondorderatthe origin. Itisadoubly periodic function, thereal period being 2u;and thepurely imaginary period 2a: For convenience of notation itiscustomary totake or ortos. Asuincreases from zero along thereal axis, @udecreases from + and atthehalf period, u=w,itsvalue is@w =e. Ifw increases tothe full period 2u:, @u, ofcourse, returns to+0. But if,instead ofcontinuing tomove along theaxis ofreals, w turns atthepoint wand moves ataright angle toit,thereby becoming complex, thefunction Pucontinues tobereal, decreases from ¢;andarrives atthevalue exwhen uarrives atthepoint w. (Fig. 108). Ifwturns again at«2and proceeds toward «along thethird side oftherectangle whose sides arethehalf-periods, @ucontinues tobereal anddecreases from ¢:toes.Along the *ForformulasrelatingtotheWeierstrass ellipticfunetions, seeScuwanz,H.A,,“Formeln undLehrsditze zum Gebrauche dereliptischen Funetionen” (1808), 414 THE THEORY OF THE POTENTIAL fourth side oftherectangle uisstill real and decreases from ¢3 ‘to—@ asumoves from wstotheorigin. Hence, @uisreal all along the rectangle whose sides arethe half periods, and its derivative isalways negative ifumoves asisindicated inthe diagram, Furthermore Pang uy Ma % ee | , 0} —Rote bs ayFae Fro, 108. Po, =e, Por =er, Pos =es. Since u-a=~u—e>0, itfollows that O<u<a; that is,u:isrealandliesonthefirstsideoftherectangle. Also ga—by=Pus—ee>0. ‘Therefore, weiscomplex and lies onthe second side ofthe rectangle. Lastly, qi—C8=Ous—€:>0, and us,also complex, liesonthethird side oftherectangle. The functions Ve =VRUR=%, VR =VOU es, V¥i=Vei=a 8) arethesigma quotients ofWeierstrass. They alsoaresingle valued elliptic functions ofuwith asingle poleattheorigin, asis evident from Eq.(7). They aredoubly periodic, but their 216} ELLIPSOIDAL HARMONICS 415 periods arenotnecessarily 2w;and 2w;. One oftheperiods is doubled inaccordance with thefollowing rule.* VGitu Fe) =+VPu i=1,2,3, VET Re)=-VRU ii, © but VOU +ha) =+VRau. Since a= Guth, Eg. (215.4) becomes av>Fae~Nut MV |x—-Pu)=0, (20) where M=Nh+¥. ‘The expressions forthe rectangular coordinates (Eq. (213.3)) become =GAX Gus ~6)(Pus~4),Cer=ea) —@9) 2=Pur =e)(Pus —es)(Gun~2), an eee =es)(G2 =63)ty=Ot=4)Pra—€)(Pus—e),(ea=ea)(@ —€2) ; and Eq. (213.4) gives Pty tear =Gut Put Gu (a2) 217. Spherical Harmonics inElliptic Coordinates.—From the point ofview ofdimensions z,y,2}a,b,and carelengths, and each has the dimension Lt;consequently g,93,92,gshave the dimension L*, Any homogeneous polynomial ofdegree nin z*,y*,and 2?canbeexpressed rationally and integrally interms oftheelliptic coordinates bymeans ofEq. (213.3). ‘That is,a homogeneous polynomial ofdegree nin2%,y?,and 2*becomes a non-homogeneous polynomial ofdegree nineach ofthethree letters q,gs,andgs;andissymmetric inthese three letters, since 2,y*,and 2*separately atesymmetric, Although notevery homogeneous polynomial inz,y,and z isanintegral function of2%,y*,and 2,itistrue that every 4Scuwanz, §23. 416 THE THEORY OF THE POTENTIAL polynomial can beexpressed asasum ofterms each ofwhich belongs tooneofthefollowing classes: P(2*, y*,24), © (@)2P(z*, y*,2), ()yPG?, y’,2), aD (©) PR v4,2), (@)zyP(z*, y*,2), ©)ve, ¥,2), (o08) (©) 22P(2*, y*,2), y2P(2*, ¥*,24) ay) Inthis table P(z?, y?,2*)means ahomogeneous polynomial in 2,y3,and 2,Polynomials ofevendegreeinz,y,and2eanberesolved intopolynomials ofclasses IandILI; andpolynomials ofodd degree can beresolved into polynomials ofclasses Ifand IV. Only polynomials ofthe first class can beexpressed rationally in terms oftheelliptic coordinates. ‘The other three classes become polynomials which aresymmetric inq1,gs,and gymultiplied byone ofthe following radicals (Eq. (213.3)) which also are symmetric inq,,gs,and 9s: Uve—a, Wve), Uva=e, IVG—AGE—H), IW@=W@e=e), IVG—VG—), UV G—AQ— We). Asolid spherical harmonic which isahomogeneous poly- nomial intherectangular coordinates ean, therefore, beexpressed inthis manner interms oftheelliptic coordinates; but itdoes not ‘cease tobeaspherical harmonic when soexpressed. ‘The trans- formation gives merely another, though interesting, expression of afamiliar harmonic. If,after thetransformation, 9:iskept fixed while g2and gsarevaried, thenew expression gives thevalue of thespherical harmonie onthesurface ofacertain ellipsoid, but theharmonic itself isinnoway related totheellipsoid. 218. The Inverse Problem.—Suppose f(q:) isapolynomial of degree ninq,,and f(g:) isthesame polynomial intheletter gs. Expressed interms ofitsfactors Slax)=(Qi=eq=aa)++(Qi=), 218) ELLIPSOIDAL HARMONICS 417 $a)=(@~ai)(qa=02)«++(Qe=a),Mqs)=a=ax)(as=02)©+(Qa=a). ‘The product ofthese three expressions is Ka)fla)“H(as)=TI(a:~a)(qe=ad(a~a)]. int Ifq1,qa,andqsaretheelliptic coordinates ofapoint, then, by Eg. (213.2), (a—a)(q —(02 —as) a r 2(glatatetates- y where C;isthe constant Cy=(ai—a*)(ay—BF)(cy—c*). ‘Therefore, theproduct Seas)-Has)(qs)omi(setgoptgce :) =Q(z’,y’,24), @ where D=TI -a)(a-(as -4}, int isapolynomial which, expressed intheq's,issymmetric in @y4,andgsandofdegree nineach; andexpressed inrectangular coordinates isapolynomial ofdegree ninx,y?,andz,which, in general, isnon-homogeneous. Ifthepolynomial f(g) ismultiplied byoneoftheradicals, and ela) =VE=FSO), ola)=Van=PY), ea) =Vin=fa), then, byEq. (213.3), ealqs)*#0(02)*e(qs)=AuxQa(z*,¥2,24),eas) «eo(as) -(ga) =AnvQules, v4,24), ap elas) -las) -eelgs) =AezQuCe', v2), where A,,As,A,arecertain constants, arepolynomials ofdegree 2n+Linz, y,and 2. 418 THE THEORY OFTHE POTENTIAL Similarly, if Vala) =VG =0)@— PYG) i=12,3, thesymmetrical products esl.) *Yoola2) *Yas(qs) =BazyQn(z?, y’,2%), Woc(qu) *Yoe(Q2) Yoe(Gs) =Broy2Qn(z?, y*,2°), e080) Weal) *Yealds) *Yea(a) =Bea2zQn(z*, y?,2*), arepolynomials inz,y,and2ofdegree 2n+2. Finally, if 0) =Va AG —PG Aa), the symmetrical product, (qs) *(a2) *(gs) =Cay2Qa(e, y%,2*), ay) isapolynomial inx,y,and zofdegree 2n+3. ‘The polynomials inz,y,and zofclasses Iand ILIareofeven degree, while those ofclasses ITandIVareofodddegree. In general, they arenothomogeneous. 219. The Functions ofLamé.—Suppose V;isafunction of 4alone and that itsatisfies thedifferential equation (see Eq. (215.4))PVi=WatMV, =1,23, ro) iata % + 2,3, whereMandNareconstants, and1;isdefined inEq.(215.2). ‘The functions V;which aredefined bythese equations arenot harmonic unless WfandNarezero, buttheproduct LAAA isharmonic, since itsatisfies the equation ofLaplace (Eq. (215.4). ‘Lamé, who was interested incertain problems relating tothe conduction ofheat inellipsoids, sought solutions ofthis type, doubtless inanalogy with the spherical harmonies, which, in polar coordinates, have theform Rr) -O(y) (8). Following the analogy further, hesought funetions V;, Vs, Vswhich arepolynomials inqs,gs,orgs,orpolynomials multiplied bytheradicals indicated inthepreceeding section. Itisevident that, ifsuch functions exist, thesymmetric product ViveVs 219) ELLIPSOIDAL HARMONICS 419 alsoisapolynomial inz,y,andz,andtherefore aparticular combination ofsolid spherical harmonics. Itwillbeshown in thefollowing sections thatsuch solutions exist foreachofthe four classes ofSec. 218. Ttwasproved byMoutard that thefunctions ofLamé present theonlycaseinwhich harmonic polynomials in2,y,and2canbe resolved into quadratic andlinear factors. 220. Determination oftheConstant V.—Omitting thesub- scripts, forthesake ofconvenience, thedifferential equation which istobesatisfied isEq.(219.1), or,preferably, theform giveninEq.(216.10), namely, aY_(gu+MV=05 i) andthesolutions sought arepolynomials in9orpolynomials multiplied byradicals. ‘That is,thesolution must beofclassL II,III,orIV. Hence, ifPisapolynomial inPu,these classes are ® VAP, v= VORP, vever, OD\V= Vem |afr=VeP, V=VORP V=VeP, AV) V= VeODP, where Pi=P-a=q-a, Pr=P-a=g-h, O=P-a= ae ®) ‘When expressed interms oftherectangular coordinates, ‘thepolynomial willbeassumed tobeofdegree m.Ifmis ‘even, saym=2n,only function ofclasses IandIIIcanoccur. Inclass I,thepolynomial Pwillbeofdegree nin@,andinclass IIIthepolynomials willbeofdegree n—1in9.Ifmisodd, saym=2n+1,onlyclassesIIandIVoccur;thepolynomials Pofclass IIwillbeofdegree nand those ofclass IVwillbeof degree n—1. Ifitisborne inmind that padate Burg hug.uw 20 28 ’ ai4(%@)oy oe VBR NE+(% a5boy itisseen that each ofthe functions inEqs. (2), ifitexists, can 420, THETHEORY OFTHEPOTENTIAL beexpanded inascending powers ofu,and that inevery case Vhastheform vate Sets. ‘The differential Eq. (1)also canbearranged inpowers ofu, and the coefficient ofeach power ofuseparately must vanish. The coefficient ofthefirst term, u~""*®,is[m(m+1)—N]a_w. Therefore N=m(m +1), @ whatever theclass may be. Expressed interms oftheletter n, however, itsexpressions fortheseveral classes are (1)N=2n(2n +1), (IIT)N=2n(2n +1), ®(DN =@n+1)Qn+2), (IV) N=(@n+)@n+2); forinclasses Iand III, m= 2n, and inclasses IIand IV, m=2n+1. 221. Existence ofSolutions forClass I.—On substituting the value ofNfrom Eq.(220.5), thedifferential equation, Eq. (220.1) i oY_nn+1)9+MIV =O, a andVisunderstood tobeapolynomial [email protected] constant MinEq. (1)isstill arbitrary and isavailable for satisfying thenecessary conditions. Since Visapolynomial in@ofdegree n,itcan bewritten V= a9, 4 where the coefficients a,are constants and must bechosen soas tosatisfy Eq. (1). Since Vdepends upon uonly through 9, av_ av,%-be ®) and ovPV _#V 2 Vy,oe-Sy" +So". ®) But, 4g 1wwog?—1 7=49-9 —9, and 9"=69-1, ) 21] ELLIPSOIDAL HARMONICS 421 sis seen from Eqs. (216. 4)and (216.6). Hence Eq.(1)becomes 49"019~095MeDesr+ (09"—I)in os fd —[2n(2n +1)+M]Saye =0. mo Arranged according topowers of(9,this expression gives >>}[2—k)(Qn+2k+Vex+Many: (5) Ft +142) E+Some+F+2NE-+ Som|e=0, inwhich the coefficients Oy =On =Ones =Aaya =0. Commencing with k=n, itisseen that the coefficient of @r*! vanishes, and that a,can betaken equal to+1. Then sequentially, 2-1: (4n—Van =—M, 2+2-(Am—B)dn2=—Mani— 3n(2n~1g, 80that ——-M m= ny we te mn=toon| Ont =35-31Gn —Gn —8) 2-2-Gn— 3)’ Itisseen from this sequence that the coefficient ays isof degree kinM,and therefore apisofdegree nin M. The coeffi- cient of9°inEq. (5), (k=—1), May+30:01+2psa»=0, a) contains noundetermined coefficient ay. Itmust therefore vanish byitself. This gives anequation inMofdegree n+1,and M must bechosen soastosatisfy it. There are n+1values ofM which will dothis, and foreach value ofMthe coefficients a,are uniquely determined. There exist, therefore, n+1functions ofLamé ofClass I. 422 THE THEORY OF THE POTENTIAL 222. Existence ofSolutions for Class II.—For class II the differential equation is $Y—(an+Qn+29+MV=0, oy and, taking thefirst oftheforms ofClass II,Eq. (220.2), v= VoP, where Pisapolynomial in@ofdegree n.Tosimplify the notation P* will beused todenote the first derivative ofPwith respect to9and P**thesecond derivative ofPwithrespectto9. ‘Then, bydifferentiating logarithmically, itisseen that av _(1.Pt), 1. 29=(we+mr-VPi(o5?+P) and wy [1 P\t_ 1, pee pet-[Go+%) mga eel =Ve(~2,+P"+pes). v0(—aptet?) Itisbetter thistime totake, Eq.(216.6), 9"=19.9.9, 9"=2.9. +9H:+9M). ‘Then Eq,(1)becomes, after removing thefactor 1/#i, (2+ Bs—n+ Gn +29 —MP +(69:93 +27.9: +29:92)P* +49:9:9:P** =0, or,byvirtue ofEq.(220.3), [-2n(2n +3)9+e: —MIP +[109 +4e.9 +(Gees —2¢,*)|P* @) +(AP?+4(exes —€:°) —dereres|P** =0. Onsubstituting P=zoe Pt=Skagh!, perakk—Nag?a Fed 4 222) ELLIPSOIDAL HARMONICS 423, inEq,(2)and then arranging according topowers of@,there results y{-20—R)(Qn+2k+3)ax+[(4k+5):—Many wet +[2(k +2)(2k +5)eses —2(k +2)(2k +3)e;"arse A+DETevened gr=0,@) inwhich Oy=1, Oy =ngs =Gays =days =0, ‘The coefficients are determined sequentially just asbefore: a, =GutDa=M neBinFly ” ayy=(ntYer=MYf(4n—3)e1—M)moe 2-4-Gn+Gn=1) 4M@n_+ Yess —@n~Ve} ‘2(4n —1) ’ The coefficient a,+ isofdegree &inM,and therefore ao isofdegree ninM. There remains, finally, the equation for which k=—1, namely, (6:—M)as +(Bests —2e:*)ai —Sevesesas =0. ‘This equation, which determines M,isofdegree n+1.There exist therefore n+1functions ofLamé ofthe form v= Ve; and similarly, n+1functions foreach oftheforms V=V@:P and V=V®P. There are, therefore, 3n+3 functions ofLamé inclass II. 223. Existence ofSolutions for Class II.—In class III, the function Vhas the form V=VO@P, 424 THE THEORY OP THE POTENTIAL where Pisapolynomial in@ofdegreen—1.Also WY_OtOps Vem, 36“ave.” VOR BYPEEAPROop4Bo4VoipiP™ 49:92)?ue 9"=49:99, B"=20:01 +PM +PPD. Since, Eq. (220.5), N=2n(2n +1), thedifferential equation becomes, after removing thefactor VPP (Pi+Gr+4Gs—2n(Qn +9—MIP +[6P2Ps +6939: +29.9:1P* +49:.9:9:P** =0.(1) Onsetting na not acl P=¥ag, Pt=Skage, P= Fkk—Yay’, mo im im replacing 1,Ps,and@;bytheir values interms of@,andthen arranging theentire expression inpowers of@,Eq.(1)becomes, Z{-20=b=1)Qn+2k+B)an—[(k+Tes+Migs +[e+ 2)(4k +Geren —(k+2)(4E +10)estJany2 ~A$DEHSeana} oH=o. Ifa, istaken equal tounity, itisfound that ag=t= Des+ aa2Gn 1)” aay=E+ (An=DesM +(An—dea}a 2-4- Gn—1)Gn —3) (2) 4(=Did—Bees—(n=fbn=Des?4Gn =3) , "Thelastcoefficient a,isofdegreen—1inM,andtheeoefcient of9°gives =(Bes+Mao +(Zeves —Besar —Seiexesaz =0. (3) 223) ELLIPSOIDAL HARMONICS 425 ‘Thisisanequation ofthen®degree in‘M.There exist, there- fore, nfunctions ofLamé foreach oftheforms V=VORP, V=VGBP, V=VOGP andtherefore, 3nfunctions inclass IIL. There arealson+1 funetions inclass I.Hence, ifm=2niseven, there are altogether(n+1)+3n=4n+1=2m+1 functions ofLamé. 224. Existence ofSolutions forClass IV.—There isonly ‘oneform inthefourth class, namely, V=VOPHP, inwhichPisapolynomial ofdegreen—1in9;and N=(2n+1)(2n +2). Itisfound that DY_Vee@)EG1ap+ Edvowel} atete)rtP ev sf Wa 1 127vere igstptBP+ ii 1 1 1, 14 1\pe4 pewewteetoe)*rata Taking 97=4919.9, 9"=2°PPs +PAP+PD), andremoving thefactor ~/@i?@2, thedifferential equation reduces to [4s +2+M2)—Qn+1)Qn +2) —MIP +6929s +PsP: +PIP +49,9.9:P** =05 andthisexpression, arranged inpowers of,gives oat >»{-20=k=Qn+2k+Sax—Maver att =HEFDCR+gear—+DETdomusloe=0. 426 THE THEORY OFTHE POTENTIAL Ifaysistaken equal tounity, theremaining coefficients are Oya =mt 240 +1)’ ec) — =n =Dg» 2Gndn1)~aGre1) Thelastcoefficient ayisofdegree n—1inM;andthecoefficient of@%,thatis,theequation inM,is 2Mao +Bgx1 +4goa2 =0. Sincethisequation isofdegree n,therearenfunctions ofLamé inclassIV. Butthereare3n.+3functions inClassII.Hence,ifm=2n+1isodd, there are Gn+3)+n=4n+4+3 =2m+41 functions ofLamé. 226.TheProducts ofLamé.—Whether ‘misevenorwhether itisodd,therearealways2m+1functions ofLamé,which,whentranslated intorectangular coordinates, arepolynomials inz,y,and2ofdegreem. ‘These functions areseparable intofour.classes asisindicated in Eq.(220.3). Suppose, forexample, V=VoiP@) isafunction ofLamé, where 9;=9—¢;,andPisapolynomialin9ofdegreem.LetR,beoneoftherootsofthepolynomial,80that V=Vou-aJ I@u-Ry. ie Then Vi= Veual@u -Ry, rast Vi=VP —eT](Pu-Ry), () ret Vi=VPus =eTTus—Ry) bet 225) ELLIPSOIDAL HARMONICS 427 ‘The product ofthese three functions ViVaVs =V(Pts —e1)(Pus —e1)(us =1) TL@u:—RB)@ur—RY@us —Ri)2) bet iscalled aproduct ofLamé. Itisexpressible asapolynomial of degree 2n+1 inx,y,and z,which satisfies theequation of Laplace. ByEq.(216.11), theradical isequal toz,aside from aconstant factor. Aswas seen inSec. 218, (a~@)(q2 ~a)(q —@) 2 v 2(cat tetsta,@) whatever amay be. If@inthis expression istaken equal to h+R,then, since @=q-h, h—-at=—e, h-B= ey hoc =e Eq. (8)becomes (Pus —RYGus —R)(Pus —R) - a eo -Cer t+RoatRoa a) ‘Hence, aside from aconstant factor which isnotimportant, mypt v eo VaVeVa=Bier +avoat act) © Itshould beremembered always, that V,,Vs,and Vsrepresent thevalues ofthesame function fortheletters q,,gz,and qs,or,if preferred, forw:,us,and us. 226. Liouville’s Proof That All ofthe Roots are Real.—The coefficients ofthecharacteristic equations inMarereal functions ofthereal quantities n,¢:,€2,and ¢;. Therefore iftheroots are not allreal, they are conjugate complexes inpairs, and the corresponding functions Vlikewise. Suppose M,andM,aretwo distinct roots ofthecharacteristic equation and U;and Usare the corresponding values ofV. Then aU,“tat (N@+M)U,, 428 THETHEORY OFTHEPOTENTIAL and PU _Us=(VP+MU Ifthefirst ofthese equations ismultiplied byUs,thesecond by—U;, andthen added, there results oils—EUs=(—MIT and therefore, byintegration, aU, aus? *[oe-ofl=Gh=wfUae. (1) ‘The left member ofthis equation assumes different forms according totheclass towhich U,and U;belong. Bearing in mind that. 2-@=-2V0.0Gs itisseen that ifVbelongs tothefirst class V=PQ) andFm-2V9GMP. Hence Usp!-USE=2VPPIPPY—PAP. IfVbelongs toclass IT, VaVoPandF=VORP+2.VPRPs, sothat OAR—02=29.FRBUPPY—PP).(1) IfVbelongs toclass III,itisfound, similarly, that aU, ay —Ugg ~UNG =—2PPV COBUPP.*—PP"); (ILD) and forclass IV, 7aU, dU: aa,Ura ~Ugg =~2PPPVPPHlP»P.* —P.P.*). (LV) Since (See. 216), Pw) =0, Pile) =0, Plo) =0, 226) ELLIPSOIDAL HARMONICS 429 and since Panditsderivatives with respect to@arefinite forall finite values of@,itfollows that ifthelimits ofintegration in Eq. (1)are a=0, b=u, of, aso, baw, theleftmember ofEq.(1)vanishes whatever theclass may be. ‘Therefore, since M;isdifferent from Mz,byhypothesis, [CUUadu=[Wau =0. IfM,and Mzareconjugate imaginaries, soalso areU;and Uysay U=WitiW, Ur Wi iWy where W;and Wsarereal. ‘Then [CUUade =[208 +Wed =0, which isimpossible, sinee W;and Wearenotboth zero. Itfollows that M;and M,cannot beconjugate complex num- bers; therefore theroots oftheequation inMareallreal. 297. Particular Examples ofLamé's Functions.—It isworth while tointerrupt theargument atthis point forthepurpose of exhibiting some ofthesimplest ofthefunctions ofLamé, anda numerical computation ofsome ofthemore complicated ones. m=0. Inthisease, inwhich thefunctions areconstants itisevident that Vie=Vi=Vi=l m=1. Itisseen from Sec. 222 that V=VG, VG, of VR Therefore, since Pi=a-0, P:=g-B, macy itisfound that Vis=Vn—OVGe—BVqa—@=hi, or,. ViVaVs =Van —OVGe—V/s—8=hay, or, VWs =Va OVE —OV — =het, 430 THE THEORY OF THE POTENTIAL where __h=V@-WE-A), baVe-AP =a), hy=VETER BH. m=4. Inorder that thenumerical details may becarried out, itwill beassumed inwhat follows, that a=5, b=4 c=3, From the formulas ofSec. 216 there isobtained G=9th aah, e=a?—h, e=e—h, loge 22) h=3+b+c%), G2=—4(crea +exes +ese), Gs=+4ereres. Hence, h=163, a= 83, @=-3, a= —7h tite+exbeats=—641, exeaes=180, and 7712 4600,nats mato Inorder nottomake theproblem toosimple nor, ontheother hand, too difficult, adetermination ofharmonics ofthe fourth order will bemade. That is, m=4, and therefore, n=2. Since miseven, only functions ofLamé ofthefirst and third classesarepossible; and,since2m+1is9,therearealtogether9such functions. Functions ofClassI—Since n+1is3,therearethreefunctions ofLamé inthisclass. Using theformulas ofSec.221,itisfound that - _M Me_300at aT omKeBee, 227) ELLIPSOIDAL HARMONICS 431 ‘The characteristic equation, Eq. (21.7), Mag+Soe:+29x=0, becomes M?—52g:M +56093 =0, or a—$0144452,576,000 _ we—Su4 0, the roots ofwhich are M, =+111.933, Mi=+ 7.187, Ms =—119.090. ‘The polynomial in9, V=Gi+ a9 +a 2d M*_392eae +(&%) has three different expressions, one foreach oftheabove roots. ‘They are V(M,) =9—7.9959 +6.146 =(Y—.862)(Q —7.134), V(M:) =9?—5119 —38.417 =(@—6.459)( +5.948), V(M;) =@*+8.5069 +12.051 =(@+1.796)(~ +6.711). ‘The corresponding polynomials ingare obtained from these expressions bythesubstitution @=q-h=q- 163. ‘These arethepolynomials that were denoted byf(q) inSec. 218. Hence Hq, Mi) =(q—17.529)(q —28.801), S(, Ms) =(q—23.126)(q—10.719), @ I, M2) =(q—14.871)(q —9.956). Itwill beobserved that allofthe zeros ofthese polynomials are real and liebetween 9(=ct) and 25(=a*). They were denoted bytheletters azinSee. 218. Each ofthese polynomials isafunction ofLamé. Taking the first one, f(q, M,), 432 THE THEORY OF THE POTENTIAL ay= 17.529, a= 801, a—at= -7471, ay—at=—1.199, a—bY=$1529, ay—b*=+7.801, ay—ct=$8.52, ay—c? =+14.801. ‘The product —fe Me l(-2+4+4- (qaritism+3.291K1190+7801+Ta.8011) =+.11162! +.0839y¢ +007924 =.56272ty* +.0503y*2" —.1068*2* ) +.9675a*—.7826y*—18482"+1. isthecorresponding product ofLamé, anditisasimple matter to verify directly that itisharmonic, ‘The corresponding polynomials forf(g,M:) andf(g, Ms) will beleft tothe student asanexercise, Aside from aconstant factor, thesymmetric product (qr —17.529)(q1 —23.801)][(g2 —17.529)(q2—23.801)] {(qs—17.529)(gs —23.801)} (3) isthesame asthepolynomial inEq. (2). Forany point 2,y,2 the order ofmagnitude ofthe elliptic coordinates q1,gy92i, See. 213, UEBSHZWEZI. The product, Eq. (3),andtherefore, Lamé’s funetion, (Eq. (2)) also, vanishes atallpoints forwhich = 17.529 org: =23.801; ) that is,thepolynomial inEq. (2)vanishes atevery point ofthe twohyperboloids ofonesheet which areindicated byEq.(4),and nowhere else. ‘The function ofLamé corresponding to Hq,Ms) =(g—23.126)(q —10.719) vanishes only onthesurfaces @2=23.126 and gy=10.719, thefirstofwhichisanhyperboloid ofonesheetandthesecondanhyperboloid oftwosheets. ‘The third function ofLamé corresponding to $0, Mi) =(q—14.871)(q —9.956) 227] ELLIPSOIDAL HARMONICS 433 vanishes only onthetwo hyperboloids oftwo sheets corresponding tothe values q=14871 and—4g,=9.956. ‘On any given ellipsoid, q.=const., the lines along which the funetion ofLamé vanishes make certain patterns corresponding tothepatterns onthesphere formed bythelines along which atesseral harmonic vanishes, See. 199. InFig. 109thepattern isgiven forM,ontheellipsoid q:=40asseen from thepoint z=15, y=40, 2=10, and inFigs. 110and 111thepatterns correspondingtoM;and M, are given. ‘The student willdoubtless find itinteresting andprofitable to sketch these patterns ontheshells ofeggs. ‘They scem much less complicated when sodrawn, Functions ofClass III.—It issufficient toconsider theform V=VPP, since theothers can beobtained from itbyacyclical permutation ofthe subscripts. Pisapolynomial in@ofdegree n—1, which inthepresent ease isunity. Hence P=O+ay where, byEq. (223.2), M+Ter wat, a=tte, and, byEq. (223.3) (M+rode+30)4ose,—Goi?=0, or M? +1euM —63034 +28exe2 =0. ‘This reduces to 9M? —690M —34,727 =0, sothat M=111327 =, or M=34.660 =Ma. Hence VM) =VPPAP —4.119), and VM) =VPPx +6.309). af.vieWWJ Es ey 227] ELLIPSOIDAL HARMONICS 435 ‘The functions ingareobtained bythesubstitutions Pi=P-a=G-a, Pr=P-a=q—d, @=4-163. Henee, using thenotation ofSee. 218, Yala) =V(q —2°) —b)(g —20.786) forMy, and val) = Vig=a)(q=BF)(q—10.358) for the root Ms ‘The corresponding polynomials inx,y,and #are =m|[ 5% 4+," eHve2Gata+4786+11.786 | ==.28782'y +2000zy* +.0840ry2" —zy, and en eeve2[tae+5642+1.3581] ==.06832'y —1772zy9 +.7366zry2* —ay. The first ofthese polynomials vanishes inthe ye- and the zeplanes and onthe hyperboloid ofone sheet for which g:= 20.786. ‘The second polynomial vanishes inthe yt-and the zz-planes and onthehyperboloid oftwo sheets forwhich qs= 10.388. The other four functions inclass IIIean beobtained merely byaeyclical permutation ofthesubscripts intheabove formulas. 228. The Pattern AsaFunction ofq.—In any family of triply orthogonal conicoids, the ellipsoid represented by1 equal toaconstant increases insize asq;increases and itsshape tends toward that ofasphere. ‘The patterns represented in Figs. 109, 110, and 111 are the intersections onthe ellipsoid ofthe surfaces q:=const, or—gs=const., () which are not altered by@change inthe value of91. Inthe limit, forqinfinite the ellipsoid becomes asphere whatever values the constants a,b’,and c?may have. The surfaces represented byEqs. (1),however, depend upon these values, and the surface a=const. 436 THE THEORY OF THE POTENTIAL foroneset,a2,b%,c*,isnotingeneral asymptotic toany one of thesurfaces foranother set, a’,b*,c*. Hence, the limiting pattern upon thesphere foragiven ellipsoidal harmonic depends upon thevalues ofa*,b,and c*;and itcannot, ingencral, coincide with thepattern ofsome particular tesseral harmonic, asonemight suspect tobethecase. 229. Parametric Representations ofaSphere.—Instead of letting q:increase, leta’,b*,andc?bereplaced everywhere by ‘da*,bt,anddc*;andthen let\diminish. Then, inorder of magnitude, %>da®>ge>ABE>gs>Ac* ‘Thecoordinate g,>a*canbekeptfixed,but2andqsdiminish with\.Itisdesirable, therefore, totake ga=Ne and 93=Xp, sothat, although psandp;arefunctions of2,theorder of magnitudes, @>m>Ph>p>c, isalways maintained. Equation (213.3) becomes (ps=2°)(ps— a) z=(q,—dat)(=AN@aSHS a2~b°)(ps —B4) v=(q—nb)(Pebs ~B*) @ Cae a @) 2 (ps~(ps ~o8) a(qr—not)B=As=8), (a1aaa) and, Eq.(213.5), h =r+NatB+ctpy—py), inthelimit,forX=0,theellipsoid. g,= decor asphere, and 1¢ellipsoid. ¢,=const., mes: 1=72(P1=0*)(ps—at) a=Pia2°)(Ps~a),&=a?=e) tm y2(P2 —b)(p, —2)eS aa=a @) BaP= Np —2) bs mane sey:Which isaninteresting parametric 10 ;presentation ofa sph sineotheparameters, psandps,entersymmetrically, 229) BLLIPSOIDAL HARMONICS 437 Since @z2e2zc, @2zm zd, Bemze,newparameters ¢,7,ws,andwscanbeintroduced bytherelations b=ra?+te’, ostsi, Po=a?sin*we+b*cos?ws, ttr=1 Ps=c*sin?ws+b*cos*ws, Osrsl. Itisfound then that x?=r*cos?wall—7cos?ws), y?=r? sin? w.sin® os, @) 2=1?cos?ws[l—¢cos?w]. If8?coincides with a®,tvanishes, and a?=r?cos? wesin? ws, y?=r? sin? wssin’ ws, 2?=r?cos? ws. Ifb*coincides with c*,rvanishes, and 2?=r*cos* wa, y?=r? sin? wssin? ws, 2?=r?cos! w,sin? we. Inboth ofthese extreme cases w:and wsare ordinary polar coordinates. Inthe first case w;isthe polar distance, and in thesecond case w:isthepolar distance. Itisalso readily found from Eqs. (3)that ey wy _# Lo,nate tye 7% 6a er)reia ae ‘The first term inthefirst ofthese equations isnegative and the other two are positive. Hence, p:=constant, represents a cone the axis ofwhich coincides with the z-axis. The first two terms ofthesecond equation arenegative while the third ispositive. Hence p;=const. represents acone the axis of which coincides with the z-axis. Across-section ofeither of these cones iselliptical, unless b*coincides with a?orc?. On thesphere, therefore, either ofthe curves Ps=const, or, ps=const. 438 THE THEORY OF THE POTENTIAL isaspherical conic, and the points ps,psarethepoints ofinter- section ofthese spherical conics. 230. The Ellipsoidal Harmonics As Functions ofX.—If the constants a*,b*,and c*are replaced byda‘, Xb*, and dc, the formulas ofSec. 216 show that ¢:,¢:,and e;are replaced byder, Nes,and Nes,while gxand gsarereplaced bygz and 2°43. Iftheletters g,s,and¢alsoarereplaced bydg,ds,andA,the letter ucanbereplaced by\*u,andthedifferential equation (Eq. (220.1) which defines thefunctions ofLamé, becomes. ey . N5g~ONGu+MV =0, or eyGu(Nee+X)V=o. This shows that theroots ofthecharacteristic equation inMfare merely multiplied by,andthesame istrue oftheroots Ryofthe polynomials P(#). The function ofLamé, (Eq. (220.2)) vev@-er@—oyr@—ePT¢-a) =) where a=R+h, and a,8,y=Oorl, becomes Va=VGeG—AKG—KPTDs—dee)2) forq=gy,anexpression which ishomogeneous ofdegree m/2 in q:and);and,afterremoving afactor \?, VieVGOR—PP APT —a),1=2,3,8) forthe subscripts 2and 3.‘TheproductofLamé(See.218)becomes,asidefromaconstantfactor, - a Viveseels Satatetsta )im= ayey]( "+4 +2 -y).owl atmoat Rae » Consequently the ellipsoidal harmonic, which isofdegree ‘mwhen expressed inrectangular coordinates, becomes ViVaVs=Hola,v2)+Haale,vs2)+Hmale,Ys2) te, © 230) ELLIPSOIDAL HARMONICS 439 where H;istheensemble ofterms ofdegree k,and therefore anharmonic polynomial, oraspherical harmonic, ofdegree k. This harmonic (Eq. (5)) vanishes atevery point ofthesurface Era rea-eta-etaa r=0, © there beingn+1suchsurfaces forclasses IandIT,andnsurfaces for classes III and IV. For \=0,the ellipsoid becomes a sphere, whatever a*,b?,and c*may be,and these zero-surfaces (Eq. (6)) reduce toelliptic cones. Hence thezeros ofthesurface harmonics onthe sphere are spherical conics—intersection of the sphere and the cones—and not circles asthey are inthe case ofthe tesseral harmonics. 231, The Surface Harmonic V;V:.—If the coordinate qis fixed, soalso isV;fixed, and theharmonic V,V2Vs, apart from a constant factor, becomes asurface harmonic onthe ellipsoid 1=const. Consider theequations (Eqs. (229.2) 2 2)(P2 =0)(ps —a?) = — GS ay (ps=#)(ps =B9), vera@=ge= a 2m(q—et)PDO = #=@-MGSGaby r= git Mp2 +ps—a?—BF—c?), If,y,andzarekept fixed and 2varies, then :,p2,andps vary. But ifps,psand r*arekept fixed while \varies, then ,y, 2,and g,vary. Let a a er rr.)Van—vat Va—iF VanXe Then paP= Ps —0%),Gaya 2=P= P(p,—8) Y=Go- ae oa @) pa—M- oD, @-w@e—e) Itisevident that &,9,and¢aredirection cosines, since Ftet tel uo THE THEORY OFTHB POTENTIAL ‘Theyarefunctions onlyofpsandps,Withthesamevariables Ba(ANH) + OYSO, 2— (ps— PRGNGr—2)+Os—oA, |ay 2a(P=N@r—0)+Os—BLIB=3, from which itisevident that £,»,¢arethedirection cosines, not ofthepoint x,¥,%butofthelimiting position ofthepoint 2,y,2. ‘The functions V;and Vsdepend upon p;and psonly (Eq. (230.3)) andareindependent of\.Since thelimiting value ofa;for)=0isr3,andthelimitingvalueofV1isr™(Eq.(230.2)),itisseen bytaking \=0inEq,(230.5) that Va¥s =Half, 8). © Since Hmisahomogeneous, harmonic polynomial in£,1,and fitfollows that theproduct ofLamé VV; isasurface spherical harmonic. Itis,therefore, expressible asalinear combination coftesseral harmonics ofdegree m. Itisasimple matter toreturn from the surface harmonic =:oyitata) VaVs=Rd)rooIIsSatgintgeoa) © where RQ) =VORB TKR, Eqs. (230.3) and (230.4), tothesolid harmonic V,V:V3; for ViVaVe =RQ —2ROIR(D) xTe-r(xfotapt ata) @ Onreplacing &,9,and bytheir values from Eq.(2),itisfound that: a radet__ gu =day? (gr=dada?(ae—ar —Na?) (ar—BY =NO) [ee—(Qi —Ne) .Eas Pa a-(Geet tetate) -(—2. yt _# sGetget aSa) and since the last parenthesis isequal to1,Eq. (7)becomes Vee weet eat gta) © 231) ELLIPSOIDAL HARMONICS 441 which isHq.(230.4). Thesolid harmonies vanish onsurfaces which arehyperboloids; thesurface harmonies vanish onthe cones towhich thehyperboloids areasymptotic. 232,TheSpheroidal Surface Harmonic V.V3.—It willbeof interest toseewhat becomes ofthesurface harmonic V2V, when b?tends toward a?ortoward c*. Ifb*tends toward a®,the ellipsoid, q1=constant, becomes aprolate spheroid, and gs ceases toplaytheréleofaparameter. Ifb*tends toward c?,the ellipsoid becomes oblate, thez-axis being theaxisofrevolution, andq;ceases tofunction asaparameter. Inorder toavoid this difficulty, thetransformation ofSec.229willbemade. bt=ra?+te’, Ostsi, gz=a?—t(a*—c?)cost, O07S1, @qs=ct+r(a?—c)conto, 9Et=1 With thistransformation Vsisafunction ofw:alone, andVjisa function ofw;alone, Itisnecessary totransform thedifferential equations (Eq. (215.4), avFF mim+Det MV=O, =33 Since, Eqs. (215.2) and (214.1), @ anya etf-VEG WE=My, d ig a edBrVEE OG ey and gq:—a?=—t(a* —c*)cos? wa, g2—bY=+£(a* —c?)sin?ws, q2—ct=(a?—c*)(1—¢008?we), a5—at=—(a?~c*)(1—1cos" ws), qs—bt=—1(a? —c*)sin?ws, qs—ct=+r(a* —c*)cos? ws, itiseasily verified that é=VaiVt ayiaTearage.» 442 THETHEORY OFTHEPOTENTIAL and a ‘atO/T arcoatot da” (VERONI=rearagt Itmust notbeforgotten thatdusisapureimaginary. Hence,thedifferential equations (Eq, (2))become (ot—VTTearsdoVi=Pomszwaa) +(mom+Dla?=1(0?—c4)cos?wl+mr.=0,@) (@—Vi=reoreornOARFoFie) -(mem+Dlet+r(at~c2)costwi)+Mrs=o. Ifb*=a*,thespheroid isprolate; =0,7=1,andtheseequa-tions reduce to @Vs,mm+tat+dag+Geaee2=, na?anado(*fe) “ m(m +la?+ +(min+0-eases =0. Ifb*=ct,theepheroid isoblate,thez-axisbeingtheaxis ofrevolution;t=1,r=0,andEge.(3)reduceto 1 ead dV,maa “@)+(mom+0 ] mim+De+WA | +Saye)M=0|) BV;_mlm+lc+Mf, dog~ARE, =0. Oncomparing Bags.(4)and(5)withEq,(197.11) and(197.12), itisseenthat‘theyarethedifferential equations ofatesseralharmonio ofdegree m.InEas,(4),theorderoftheharmente mm+Dart My,“Sogo, 232) BLLIPSOIDAL HARMONICS 443 and inEqs. (5), the order ofthe harmonic is _m(m+ Der+M_yya= , wherekissomeintegernotgreaterthanm; k= 0,1,2, 5+ym. Hence, ifb*=a*,and thespheroid isprolate, there arem+ 1 Possible values oftheconstant M,viz., M=(a?—ck? mim +Lat, k= 0,1,2, °°+,m. 6) Ifbt=ct,andthespheroid isoblate, thevalues ofMfare M=—(a—c*)kt— m(m+ Uc, k= 0,1,2,+ ++,m. Each ofthe 2m+1tosseral harmonies ofdegree misthus a limiting form forone ofthe 2m+1surface harmonics V2V's of ‘Lamé, when b*tends toward a?ortoward c*. Inthefirst case, the limits are Vie sinkos, or c08kes, and Vi=Tals), inwhich w:plays the réle ofthe longitude 9,and w;the polar distance y.Inthesecond ease, thelimits are Va=Tu(w), and=Va=sin kus,or608kus, inwhich 2plays theréleofthepolar distance ¢,and a1plays the role ofthelongitude @. 233, The Roots oftheCharacteristic Equation Considered As ‘Functions ofSuppose m=2niseven. There is@character- istic equation inM,orinM=M—m(m+1)h,(Eq.(216.10)), for class I,and one foreach ofthe cases inelass III. Consider first theroots ofthe characteristic equation in forclass I.‘These roots arefunctions oft,ifb?isregarded asa function off,There aren+1ofthem, andeach gives risetoa polynomial ofLamé which has the form wht v=[@- 4; therefore nit es Ve=[1a:—a)=[](a?—a:—t(a*=&*)008?wn).ros] int 444 THE THEORY OFTHE POTENTIAL Theaj,which aretherootsofthepolynomials ofLamé ing,also arefunctions of¢.Aswasshown inthepreceeding section, the limit ofV;fort=0iscoskws,orsinkws,asidefromaconstant factor.Forsomevaluesofi,at,least,a?—oz;musthave¢asa factor, which ofcourse, canberemoved, unless V;reduces toa constant. Inanyevent V+isafunction ofcos?w,andtherefore, aneven function ofw2. Itcould notreduce tosinkwswhich isan oddfunction ofws.Also, since2cos?w:=1+cos2.ow,Viisan even function of2u:. Hence, thelimits ofthen+1functions Vsofclass Iare COSOws, c082us, 084a, sey cos2nas. ‘That istosay,kisaneven integer ifVsisofclass I. IfVsbelongs toclass III,each ofthethree characteristic equations hasnroots. The forms ofthepolynomials are Vi=V@-PMG@—AT@ —a), @ iat i= V—@= Mae A]T@ —a), ® ist Y=V-@— MG P][@ —a) ) ist Areference totheformulas ofSec.232showsthattheseexpres- ‘sionsbecome, asidefromaconstant factor, Va=VITos?aysinws X[Ite-a;-(a=c*)cos?a),(0) Va=VI=008?w2*008ws [Ite—ai—e(a*—2)008?ws),(0) Va=sinwscosws](a*—ax—t(a*—c2)cos?ws). © ‘Thelimiting values oftheseexpressions are,evidently, Vs=sinwr,sin32, vty sin(2n—1)ws, (a) Vi=cose, —c0s3us, ++, cos(2n—Iw,(b) Vi=sin2u2, sindos, soy sin2nus. © 233) ELLIPSOIDAL HARMONICS 445 Hence, theinteger kisodd forgroups (a)and (6)and even for (c). All ofthe 2m+ 1functions are accounted for, since (n+l tntntn=4nt1=m+L Asimilar analysis holds ifm=2n+1isodd. 234. The Characteristic Equation Has NoMultiple Roots.— Itwas shown inSec, 226 that allofthe roots ofthe charaeteristie equation are real. Itisalso afundamental property that the roots arealldistinct. This isevident from Eq. (232.6) when tis zero. For brevity ofnotation let a@—ct=A and——m(m+1a?=B. For classes Iand IIT(), and fort=0,theserootsare B, 2A+B, 44+B, +--+, MmA+B, A+B, 34+B,---, (2n-1A+B. am) Itwill beobserved that theroots forIIT(#) liemidway between therootsforclassI.As¢increases fromzerotoone,allofthese roots vary. Two ofthe roots ofclass Icannot become equal without first (orsimultaneously) coinciding with one oftherootsofclassIII(b).ItwillbeshownthatarootofclassIcannot coincide with aroot ofclass III(6). Suppose aroot ofclass Idoes coincide with root ofclass IIT(Q), andletU;bethecorresponding polynomial ofclass Iand Usbethecorresponding polynomial ofclass I1T(#). ‘Then Ur=P), Us=VPPiPA(G)- @) Both ofthese expressions satisfy thesame differential equation, since, byhypothesis, and Marethesame forboth, namely, @U; BUGe=NG+ MU, Ga =We+MU. Consequently@U;_yy@Us Oegut~Oras =% and therefore,aq,_yasUng —UG! =const. ) Since WsWig, Us_aay,qu age” Gu=ap?” 446 THE THEORY OF THE POTENTIAL and B= -2V PPPs this result (Eq. (2)) reduces to (Rational function of@)+/: =const., which isimpossible unless the constant iszero. But ifthe constant iszero, iq. (2)gives Us=UiXconst. which, also, isimpossible byEq. (1). Itfollows, therefore, that the roots ofthe characteristic equation ofclass Icannot coincide with anyoftheroots ofclass III), and neither ofthecharacteristic equations can have a double ormultiple root. Asimilar argument holds fortheother classes. Theroots of characteristic equation arealldistinc. 235. The Functions ofLamé are Linearly Independent— Onaccount ofthe irrationalities which are involved, there is noquestion with respect tolinear independence between the different classes. Itisnecessary toshow, however, that for thesame characteristic equation there cannot exist arelation ofthe form Vero =VO $aVO +... +a, where V© isthe function ofLamé associated with the root M; j=1,2,--+,n—1, orn,according tothe class, and a;is@constant different from zero. Suppose such arelation didexist, and that vero=ave, i<n—lorn a) ca ‘The function V®satisfies theequation nyoOY).=NOVO+MYO, @ ‘Multiply thisequation bytheconstant a;andthen sum with respecttoj.Thereresults aveLoge =NPLave +FyMave, @) 235} ELLIPSOIDAL HARMONICS 447 which byvirtue ofEq.(1)reduces to ave en ;Gar=NOVY +yMavi. @ Byvirtue ofEa. (2) nyeaE =NOVO +ManVOD, Acomparison ofEqs. (3)and (4)shows that MiyVO =¥Mave, at or 0=Far —Mav. (8) am Since theroots ofthecharacteristic equation arealldistinct, none ofthe coefficients vanishes. Itfollows that ifsuch a relation asEq. (1)existed between i+1functions ofLamé, asimilar relation would exist between ifunctions ofLamé. Arepetition oftheargument would show such arelation between i—1functions ofLamé, and soon;until, finally, anequation ofthe form 0=Oh —MyVo was reached, where b,isaconstant different from zero. As this result isimpossible, itfollows that nosuch relation asEq. (1) exists, and the functions ofLamé are linearly independent. Since V;isafunction ofgzand V;isafunction ofgs,itfollows ‘atonce that there cannot exist arelation ofthe form O=YaVvs%, isn—l,orn Hence thesurface harmonics ofLamé are linearly independent, and they form acomplete, orfundamental, system. 236. The Expansion ofanArbitrary Function inTerms ofthe Ellipsoidal Harmonics—Any given homogeneous polynomial inz,y,andzcanbeexpressed asalinear combination ofsurface tesseral harmonies ofdegree m. LetKw(o, 6),5 =1,2) --- 2m +1,bethe tesseral harmonics ofdegree m,and Lq® = 448, THETHEORYOPTHEPOTENTIAL (¥:Va)u®, j=1,2,+--+, 2m+1, bethe surface harmonics ofLamé ofdegree m. Then ant Hale, yz)=" SDAK@®, i and, also, Haley, 2)=1"YBln? Frat where A;and B;areconstants. Since both systems ofharmonics are linearly independent itfollows that there exist constants C\ and D,® such that amet KO=FCOL, and also, amtt La? = DOK, ‘That is,any harmonic inonesystem can beexpressed linearly interms ofthe other. The two systems are, therefore, ‘equivalent. ‘Suppose there isgiven afunction F(qs, g3). Bymeans ofthe relations given inEq. (231.3) itispossible totranslate this funetion into afunction ofpolar coordinates ¢,6,since E=singcos6, =singsin#, f=cosy. ‘Suppose Fg) =8). The function (y,6), under certain very general conditions, ean beexpanded inaseries ofsurface harmonics inaccordance with thetheorem ofSee. 206, 6,0 =DYSale, 9, ote where S,,isasurface harmonic ofdegree m. But, ashasjust been seen, S,can beexpressed interms ofthe ellipsoidal harmonies, Sale) =DYAwPLn®, mn 236) ELLIPSOIDAL HARMONICS 449 where the A, areconstants. Hence ptt Bo,8)=Flana)=5(FyAnnie «0he Ifthefunction (¢, 6)onthesphere ofradius r,when expressed interms oftherectangular coordinates, isapolynomial ofdegree sin 2,y,and ¢,theseries ofharmonies isfinite, and a pastFana)=(>4n%n(ass) NA 287. Surface Integrals.—The derivative ofthe harmonic function Va ViVaVs normal totheellipsoid g,=const. is(Eq. (214.3)) a_Va¥ad¥s on Rydq’ where, Eq. (214.2), R,-—VY@=oG=H) _____A__.Va@= GG) Ava ‘The surface element dwontheellipsoid is(Eq.(214.4)) da=RiRdgrdg. ‘Sincethesurface integral ofthenormal derivative ofanharmonic function over any closed surface iszero (Eq. (62.2) itfollows that ‘av ‘(?, dV;RR,Sn=LJVivagpada, A,dV: VV,=oof See~addavda,=0, and therefore, VVaee G@- 1s=0;JPG ~aodzeta or, since age gsFyeae eta S*foVa¥Gus=Pur)durdus =0. 450 THETHEORYOFTHEPOTENTIAL Again, ifU;andUzaretwoharmonic functions itfollows from Green’s theorem (Eq. (62.1)) that au;aU? f(a-vs) =0. @ that Supposetat, =(WAWaVo =VilPL®, and Us=(VV =VOL, areellipsoidal harmonies ofdegrees sand{respectively. ‘Then Eq, (1)becomes avn? av deVnygVLCpopalee (7 )fnonnge =0. “Thefactor outside oftheintegral sign isafunction ofqyalone, and therefore constant onthe ellipsoid, Itisnot zero, unless Vu=CV9; and inorder that thisshould be50,itis necessary notonly that s=4, butalo k=. That isUydiffers from Usonly bya constant factor. Assuming that U;and Usare different har- monies, though they may beofthesame degree, itfollows that dia 1.01, =9,fpRy or,since thedenominator ofR,isafunction ofg,alone, fLOLMde Vaasa a) theintegral being taken over theentire surface. Bytaking —gVG a@@—@ =v this important formula canbewritten inthemore compact form fants =0. 2) ls Itisasimple matter toshow that. dus 12Bi ® for du=-# byEq,(215.2), 237] ELLIPSOIDAL HARMONICS 451 and dn=Ridgy. Hence du, L -1du __=—t___, _byEq.(214.1),a~BARVGaaa) =-l Ageneral formula fortheintegral ficerras,le ‘apparently, hasnotbeen worked out. 238. TheCoefficients ofanExpansion inTerms ofEllipsoidal Harmonics.—It wasshown inSec. 236that afunction F(qs, 93) canbeexpanded inaseries ofellipsoidal harmonics, namely, =, [mthF(a)=(¥4.7100G009) Q@ wo 51 where thecoefficients A, areconstants. Ifthis expressionismultiplied byWL,Pdeandthenintegrated overtheellipsoidE,there results (fPbaiide =An?[1Ln0)de, alloftheother terms oftheexpansion vanishing byEq.(237.2). Hence SpiFbnPde 4,0 =, @ SUL?)*deo Itwill beobserved that these coefficients areindependent ofqi,since Ide =IRRedasdas (qs—q)= ay byEq.(214.2), ei, byBa 142) does notcontain g:.‘Thisexpression isrealsince As*isnegative. IfFa(ga, q2)isapolynomial inga,qsmultiplied bytheradicals VG eG —PMG=2) and —_ oe Vag ag =Ge 452 THE THEORY OFTHE POTENTIAL issymmetric with respect tothesubseripts 2and 3,and isof degree ming,and qs,itsexpansion interms oftheellipsoidal harmonies terminates, and thehighest degree oftheharmonies whieh occurs ism. Hence, if#>m,itisevident that, S[Palade =0, (3) L.® being anyharmonie ofdegree k>m. 239. The Roots ofLamé’s Polynomials are Real, Distinct, ‘and Lie between a’and b.—The theorem that the roots of Lamé’s polynomials (Eq. (220.2)) arerealanddistinct, and lie between a?and c#is analogous tothetheorem that theroots ofthezonalharmonies arerealanddistinet,andliebetween+Land—1.‘The theorem isimportant since these roots define thelines onthe surface oftheellipsoid along which theharmonies vanish (see Figs. 109,110,and111). ‘Thefollowing proof isduetoAppell.* InSecs. 221to224Lamé’s polynomials Pwere regarded a3 polynomials in@u,butsince (Bq. (2163)) Pu=q-h ‘theyeanberegarded alsoaspolynomials ofthesame degree ing. Itisaspolynomials ingthat theroots liebetween aand c?, It iseasily verified thataspolynomials intheroots liebetweene: and e. (a)TheRoots areAUReal—Consider firstclassI,Eq.(220.2), V=P@. Thepolynomial P() which isofdegree ncanberesolved into itslinear factors. Sincethecoefficients ofParereal,thecomplex factors, ifanyoccur, willoccur inpairs, theproducts ofwhich are real. Theproduct ofallofthecomplex factors isarealpoly- nomial, andtheproduct oftherealfactors alsoisarealpoly-nomial. ‘Therefore, Pq) =Pilg) -Pa(g) canberesolved intotwofactors, oneofwhich P,hasallofits rootsreal,andtheotherP;hasonlycomplex roots. ‘TheproductofLamé is Ln=(VaVabm=Pa(as)Pi(as)«Pa(g)Palas). *Apveut,P,“Traité deMécanique Rationnelle,” Vol,1V,p.187, 239) ELLIPSOIDAL HARMONICS 453 InEq. (238.3), let F=Puasa), which isofdegree lessthan m,unless Psisaconstant. Then fiFbndo =[iPtaP(@)Pa(qaPalqs)de =0. This formula isapplicable, ifthere areanycomplex roots, since ‘thedegree ofFislessthan m. Theresult obviously isimpossible, since the integrand isalways ofthe same sign. Itfollows, therefore, thattherearenocomplex rootsifV=PisofclassI.IfVdoes notbelong toclass I,ithastheform V=V@= eG —OG —AP =RO) +Pla) where a,8,and +areeither 0or1.Also Lm=(VV )m=BO)R()P(92)PiCO)PAQ)PAQ), where P;andP;have thesame significance asbefore. If F=Rl) R(Q)Ps)PA(a), then, ifPhasany complex roots, SPRGIRAIPAGPHIPLae)Palaide =0. Since XQ) =(ge—"(a2 —BMG —CY, RHQ) =(as—a(qs —BMG — and a>qr>BE>gs>ct, itisseen again that theintegrand does notchange sign and the surface integral cannot vanish. Itfollows, therefore, that whatever class Vmay have, theroots ofthepolynomial Pare allreal. (b)The Roots are All Distinct—Suppose the roots ofthe polynomial Parenotalldistinct. Conceivably, they may have any order ofmultiplicity, say, P=mq@- aX ‘The polynomialPwillberesolved,justasbeforeintotheproduct oftwo polynomials P=P.P: inwhich P2contains factors which occur aneven number of times and P,those which occur anodd number oftimes. Thus, 454 THE THEORY OFTHE POTENTIAL ifniseven, thefactor (g~a,)*is putinPa,butifme=2s+1 isodd, thefactor (—a:)** istaken intoPyandthesingle factor (q—«)isincluded inP;, With thisresolution ofthepoly- nomial P,thepolynomial P;does notchange sign. Ifthen Fis taken equal toP,,andifPsisnotofdegree zero, thesurface integral beoomes, justasbefore, [fips =0, which alsoisimpossible. Hence P;must beofdegree zero, and the roots arealldistinet. (©)AllloftheRoots Liebetween a*andc2—Substantially the same argument suffices toprove that alloftheroots liebetween, a®and ci. LetP,betheproduct ofthefactors which vanish between a?endc,andPstheproduct ofallofthefactors which vanish outside ofthese limits, IfFistaken equal toPy, and if thedegree ofP;isnotzero, thesurface integral again leads toa ‘contradiction, since P;does notchange sign within thelimits of integration, Itfollows therefore that thedegree ofP;iszoro and that allofthe roots liebetween aand c?, 240. Ellipsoidal Harmonics oftheSecond Kind.—Since, by Eq. (213.5), atmteart@+e+c) and @>q>h>gre itisevident, that ifris very large, g:isapproximately equal to 7,and thefunction ofLamé V3ofdegree misapproximately equal tor=, Consequently theellipsoidal harmonies (ViV2V3)s arenotsuitable fortheexpansions ofpotentials intheneighbor- hood ofinfinity, just asthe spherical harmonics which are polynomials inz,y,and 2arenotsuitable, The spherical harmonies were made suitable, however, merely bydividing bya certain power ofr.Asomewhat similar possibility exists forthe ellipsoidal harmonies. Itwillberemembered that thefunction V;VV3 ofdegree mis harmonic byvirtue ofthefact that each oftheletters Vsatisfies thedifferential equation (Eq. (220.1), bad=[mim+@+MV. re) 240) BLLIPSOIDAL HARMONICS 455 Butasthisdifferential equation isofthesecond order, ithastwo solutions. LetVbethesolution already determined, andUthe other solution. Then, also, a=[m(m+DE+MU. @) IfEq,(1)ismultiplied by—UandEq,(2)by+Vandthetwo arethen added, there results nu vo.Vie~Uae = therefore, byintegration au Vv _ vay—USE=const.=2m+1. (3) Thisconstant isarbitrary sineo, ifUisasolution ofEq.(2),CU alsoisasolution, Cbeing anyconstant, Lettheconstant be taken equal to2m+1. Byvirtue ofthischoice Ubecomes a perfectly definite function U,,which, aswillbeshown, satisfies the relation lim"HU,(q)=1. IfEq.(8)isdivided through byV,%,itcanbewritten d (Us) _2m+1 XV,ve ‘Therefore Uevf‘2m+lay @ lo Vy Thelower limit ofintegration, ofcourse, isarbitrary, butitis convenient totake ittobezero. Itwasshown inSec,220that vate temo,uwwy andinSec.216thatasu;tendstowardzero,@u;tendstoward+e. Hence, forlargevalues ofr,9:=ui+h,islarge, and %,issmall. Using onlytheprincipal term oftheexpansion of V;,itisseenthatforlargevalues ofr,approximately, ueaz,‘m+1utmdu=wnt mM"Jo 1 4156 THETHEORY OFTHEPOTENTIAL Hence lim re'U, =1. Furthermore, since U;satisfies thedifferential equation (Eq. (1)), the function U,V2Vs isharmonic. Asrtends toward infinity V2and V;remain finite, but U,tends toward zero like r+», Hence the ellipsoidal harmonies ofthe second kind (UiV:Vs)m aresuitable forexpansions ofexterior potentials. 241, The Potential ofan Ellipsoidal Harmonic Surface Distribution ofMatter.—It was shown inSec. 209 that a spherical harmonic distribution ofmatter onthe surface ofa sphere produces asimilar and similarly placed harmonic inboth theexterior and interior potentials ofthedistribution. Itwill beshown inthepresent section that ananalogous theorem holds foranellipsoidal harmonic distribution ofmatteronanellipsoid. Itwas shown inSec. 126 that ifW:isharmonic inside ofa closed surface S,W,isharmonic outside ofSand vanishes at infinity, and ifW;=W.onS;there exists one and only one distribution onSforwhich theexterior potential isW.and the interior potential isW,. Furthermore thedensity ofthe dis- tribution is 1fpaw. ,awe .Blan+onl Forthegiven ellipsoid let Wea ViOUL, Wee UOVL beellipsoidal harmonics ofdegree m,where L= (ViVs)m%, thesuperscript zero indicating thevalue atthesurface ofthe ellipsoid. Thus V\° and U, areconstants, and W.=W,on thesurface oftheellipsoid. Now aw: +aVi 78V9,on.=+0,"Lane=-W"La, and aw, aU; aurWe. yo2U - :ang7VALan,=TMMLgs ‘sothat aLes |W.) Wy Us Vs °aoe+an=(ree-oeme): 241) ELLIPSOIDAL HARMONICS 457 the parenthesis being evaluated onthe surface. Therefore, byEq. (240.3), dao=(2m +1). Thus the surface distribution differs from anellipsoidal har- monic distribution only bythe factor J,which however isa function ofgzand q;that isindependent ofm. Conversely ifthesurface density isgiven by o=AlL where Aisaconstant fry =4 ye We=Im+1"UL, and Wiom+quiVib. 242. The Potential ofan Ellipsoidal Homoeoid.—If mis sero, Vie=Vi=Vs=L=1, and, byEq. (240.4) U, =uy. Then o=Al, W,=4rAu, Wy=4A. 1e3) 1 * nar’ Consider theshell which isbounded bythe given ellipsoid forwhich theaxes aregi —a%,qi —bY,and gi—b%, and asimilar ellipsoid which isinfinitely close toit,Fig. 112. ‘Their equations are a ye an=aoetawietgwsech and 4+ +t 2,ate t geetpt ata Inthesmall triangle PP:Ps, PP, =dn, PP, =ds, OP=r. 458, ‘THETHEORYOFTHEPOTENTIAL ‘Since theellipsoids aresimilar PP; =ds=rdh, since PP, isnormal toboth ellipsoids theangle atP,isaright angle, and PP, =dn=PP; cosP=rcosPdh. ‘The direction cosines ofrare % 4% and 2 roof r ‘The direction cosines ofthe normal are tz, tv, I+. Rg —a Rg PK Rg Hence 1at y Eaa cosPaeaSatGere tgreal“ORY and dh dn=oR! or,since 1 — aH,7V@P=AG—WESH, itresults that the thickness ofthe shell is an=VG TG=PG=Har; that is,thethickness oftheshell isproportional toJ,since the remaining factors areconstant onthegiven ellipsoid. ‘The surface distribution ofmatter canberegarded asavolume distribution ofconstant density throughout the shell. Ifoo istheconstant volume density o=on =od VG =AGT = San, Consequently, if A=OVGTFG=HG—Ar, ‘thepotentials inEq. (1)represent equally well thepotentials ofanellipsoidal homoeoid, andsince (Eq. (216.1)) * ay ye+fafe,VaGQ= a) =BQ =e) 242) ELLIPSOIDAL HARMONICS 459 itisseen that We=reoGOaGTS GTSeya xfa.Va= Hq —G= A which, aside from notation, isthesame result that wasobtained inSee. 123, 243. Green's Problem onanEllipsoid.—The results already obtained with Lamé's functions furnish the solution ofGreen's problem (or, Dirichlet’s problem, asitisoften called) foran ellipsoid. Given afunction F,which isdefined onthesurface ofanellipsoid. Itisrequired tofind afunction which is harmonic within theellipsoid and equal toFonthesurface (interior problem), or,afunction which isharmonic outside oftheellipsoid, equal toFonthesurface, andwhich vanishes at infinity (exterior problem). Letthefunction Fbeexpanded interms oftheellipsoidal harmonies inaccordance with theprinciples ofSec.236. = pongtrP=>[Y4.0009|, iO) sol A where A. arecertain constants, which eanalso bewritten An? =BaiO(V OU) 9, and therefore = fensre[ESB.0(VU)aE99|® ol Vortheinterior problem take thefunetion «= fametm=>[>BaO(rv0)a220 () aol Theseries ofEq.(8)isconvergent provided theseries ofEq.(2), isconvergent, since intheinterior oftheellipsoid V;<Vi;andWisequal toFonthesurface. Furthermore, itisharmonic,since itsatisfies theequation ofLaplace. Itis,therefore, the funetion which was sought. 460 THETHEORY OFTHEPOTENTIAL Similarly thefunction = [ames We= DY]YBaP(VLU) an” Sol satisfies alloftheconditions fortheexterior problem. ‘Therefore, W,and W,aretheinterior andexterior potentials ofacertain distribution ofmatter onthesurface oftheellipsoid. The density inthisdistribution is 1paw, ,awe v=Pak+mn From theresults ofSee. 241, itiseasily seen that le ang=E YAnt) YBane |.En) i EXTENSION OF THE GENERAL THEORY 244, Fundamental Functions.—In addition tothe spherical and ellipsoidal harmonics which areassociated with thesphere and the ellipsoid, there exist other harmonic functions; for example, theToroidal Function ofHicks' which areassociated with theanchor ring. One concludes that there exists acorre- ‘sponding setofharmonics functions associated with every closed surface. ‘The equation ofLaplace which these functions satisfy ismerely ‘particular case ofthemore general equations aU, ou, aU, pu, av, aU, avoon+pa+pe+dae+oe+FE+PEew=0 inwhich imay have the values 1or2.The solution of various problems inelectricity and magnetism, elasticity, ‘acoustics and hydrodynamics depends upon the integration of such equations forwhich¢=2;andthesolutionsofmanyprob- lems inthe analytic theory ofheat are reducible tothe inte- gration ofsuch equations fori=1. For example, thevibrations ofamass ofgasenclosed ina solid vessel isdefined bytheequation aUaU|aU_|aUaetaytae ae @ +Philosophical Transtetions (1881) 244) ELLIPSOIDAL HARMONICS 461 inwhich &isapositive constant, thesolutions being subject tothe conditions that, = aLU=f@u), 5,=0 os, and g=flzy2) fort=0. Theequation ofacooling solid body leads totheequation eUaU,aU_,,aUaa?+ay?tae =ar? ® together with theconditions that U=fl,y2) for t=0, and ou B+kv=0 ons. ‘The time canbeeliminated from Eq, (1),aswas done by Cauchy, byassuming that U=Vi(z, y,2)(Aj sindst+B;cosXi). Equation (1)then reduces to AV; +%V;=0 within 5, and avy _ .Fino oS; andasimilar reduction ispossible forEq.(2). The problem isthen divisible into twoparts, (a)Todemonstrate theexistence ofaninfinity ofsimple solutionsofEq.(8),f¢.j=0,1,2,+--+,© (®)Toestablish thepossibility ofthedevelopment ofagiven function f(z,y,2)within thegiven domain inaseries which proceeds according tothefunctions V;towhich thesimple funetions reduce for¢=0. ‘Theexistence ofsuch functions forproblems ofthis type was established byPoincaré inhismemoir SurlesEquations dela Physique Mathematiquet in1804, andwere called Fundamental Functions. Similar setsoffunctions suitable fortheexpansions ofother types ofproblems have been proved byother authors +Pomcané, “Acta Mathematics,” Vol. XX; Amerizan Journal of Mathematics, Vol.XIT, and““Rendiconti diPalermo,” Val.8,p.57(1894). 462 THE THEORY OF THE POTENTIAL Itisofgreat interest toobserve that allofthese functions can bedefined bycertain integral equations oftheform Viev8)=HfOC@, vs236 OVE, mDar, whereiscertainconstant andGisageneralized Greenfunction, or,perhaps toamore general form Vilas 2)=NUE mE, wy25&myS)VAG, mar. Foranexposition ofthis extension ofthetheory thereader is referred tothe works ofPoincaré above mentioned and tothe memoir by W. Steckloff, Theorié Generale des Fonctions Fondamentales.' +Annales dolaFaculté desSciences deToulouse. (1904.) BIBLIOGRAPHY The subject ofSpherical Harmonies haditsbeginning in» paper ofLegendre which waspublished in1785, although itwas written in1782. Legendre’s paper developed only thezonal harmonies, butitinspired Laplace’s paper, also written in1782, inwhich thegeneral spherical harmonies, asfunctions oftwo angles, were developed, andinwhich alsothetheory ofthepoten- tial was founded. Several decades elapsed before thefunds- mental papers ofGreen (1828) andGauss (1841) appeared. ‘Among thevolumes which arelisted below that ofHeine on Spherical Harmonies isthemost nearly complete both astothe theory itself andalsotothehistorical references. Insome ofthe volumes, such asthose ofNeumann andofPoincaré, theauthor isconcerned only with thetheory ofthepotential. Inothers, such asthose byClerke Maxwell, Tisserand, Picard andAppell thetheory ofthepotential isincidental toalarger ficld. Inthe articles ofBurkhardt and Meyer intheGerman Enelyklopidie very complete references totheoriginal memoirs willbefound up to1896, and inthearticle ofLichtenstein references tomore recent papers upto1918. References tocertain papers which have appeared since thatdatewillbefound inapaper byO.D. Kellogg “Recent Progress with the Dirichlet Problem” in ‘volume 32oftheBulletin oftheAmerican Mathematical Society, p.601, EARLY MEMOIRS Leorxpat, Surattraction desspheroids, Memoirs deMathematique otde Physique, presentes aI'Academie royale dessciences pardivers savans, Tome X. (1785.) Lartace, P.8,Theorie desattractions desspheroids etdeIafigure des planetes, Mee. Cel,Tore Ul,Chap. IIT,AlsoOeuvres, Val.X.(1785) Gneex, Gronos, “The Mathematical Papers of”(1828), edited byN.M. Ferrers, facsimile reprint, Paris. (1903.) Gavss, C.F, Allgemeine Lehraitze inBezug aufdieimverkehrten Verhalt- rise desQuadrats derEntfernung wirkenden Anziehungs-und Abstos- sungs-Krifte, Werke, Bde 8. (1841.) 463 464 THE THEORY OF THE POTENTIAL BOOKS ON THE THEORY OF THE POTENTIAL ‘Twoursox andTarr, “Treatise onNatural Philosophy,” Vol. I,Cambridge. (1867,) Maxwett, Janes Cun, “Electricity and Magnetism,” Vol. I.(1873.)Nevwans, C,“Untorsuchungen tberdasLogarithmische undNewtonschePotential,” Leipsic.(1877).Marmzv, E,,“Theorie duPotentiel etsesApplicationsaElestrostatique ctauMagnetisme,” Paris. (1886.) Prince, B.0.,"The Newtonian Potential Function,” Boston. (1886.) Lusrowe-Dirichlet, P.G.,“Vorlesungen Uber dieimUmgokehrten Verhal- tniss desQuadrats derEntfernung wirkenden Krifte,” herausgegeben von F.Grube, Leipsic. (1887,) NeuMANs, F,"Theorie desPotentials und derKugelfunctionen,” edited by ©.Neumann, (1887,) ‘Tisaznann, F.,“Mecanique Celeste,” Vol. I,Paris. (1891.) Aprett, Pavt, “Legons surVattraction otIafonction potenticlle,” Paris (1892,) Porcang, Hexni, ‘Theorie duPotentiel Nowtonien,” Paris. (1895.) Konx, A.“Lehrbuch ber dasLogarithmische und Newtonsche Potential, Leipsic. (1880.) Porxeané, Hens, “Figures d’Equilibre d'une Masse Fluide,” rédigées pat L.Dreyfus, Paris. (1902.) Wensren, A.G.,“Tho Dynamics ofParticles, and ofRigid, Elastic and Fluid Bodies,” Leipsic. (1904.) Prcano, E,,“\Traite d’Analyse,” Vol. I,Paris. (1906.) ‘Waxaenn, A.,“Theorie desPotentials und derKugelfunctionen,” Leipsic. (1909,} Pumas, J.,“Potentialtheoretische Untersuchungen,” Leipsic. (1911.) Oscoon, W.F.,“Lehrbuch der Functionentheorie,” Vol. I. (1912.) Avprit, Pavi, “Traite deMecanique Rationelle” Vol. TIL. (1908.) Vol. IV. (a921.) Srexxseno, W., “Potentialtheorie,” two volumes, Berlin. (1025-1926.) Evans, G.C., “The Logarithmic Potential, Discontinuous Dirichlet andNeumann Problems, Colloguium Publications oftheAmerican Mathe-matical Society, New York. (1927.) Keuz090, 0.D.,“Foundations ofthePotential Theory,” Berlin. (1929.) BOOKS ON HARMONIC FUNCTIONS ‘Lasté, G.,“Legons surlescoordinees curvilignes etlesdiverses applications,” Paris, (1850.) Herve, E,,“Theorie derKugelfunctionen," 2Ed,, Berlin. (1878) Topavwree, L,“The Functions ofLaplace, Lamé, and Besse.” (1875.) Fennens, N.M,, ‘(Spherical Harmonics,” London. _(1877.) Byenuy, W.E,“Fourier Series, and Sphericsl, Cylindrical, and Ellip- soidal Harmonies,” Boston. —(1893.) BIBLIOGRAPHY 465 ON THE LITERATURE Bounxnanor, H.,andW.F.Mrven, Potentialtheorie, Encyklopadie der ‘Matematische Wissenschaften, TlAT. Licurexsren, L.,Neuere Entwicklung derPotentialtheorie, Konforme Abbildung, Encytlopadie, 1C3.Kettose, 0.D.,Recent Progress withtheDirichlet Problem, Bull.Ameri- ‘canMathematical Society, Vol.XXXII. (1926.) INDEX A Electric images, theorems relating to, 216 Andromeda nebula, 10 Bllipsoidal harmonies, 407 ‘Apparent size 6,365, ‘expansion interms of,447 Arithmetic mesn, themethodof,297secondkind,454 Arithmetio-geometric mean, 198 surface integrals, 449 distributions, 456 B eros of,484 Bllipsoid, exteriorpotential, 58 Balayageofasphere,313 interiorpotential, 45 Bodies, centrobarie, 212 Blliptic coordinates, 286, 407 functions ofWeierstrass, 412 c integrals, 58,203 Equation ofLaplace, 32 Cauchy, theory ofresidues, 117 ingeneralized coordinates, 104 Centrobirie bodies, 212 forelliptical coordinates, 410 center ofgravity, 213 forsurface harmonies, 382 detached masses, 215, fortessoral harmonies, 371 cllipsoid ofinertia, 214 forzonal harmonies, 334 Configuration constant ofasurface, Equipotential surfaces, defined, 34 293 ofellipsoids, 50 Conform transformation, 206 ofestraight rod,48 Continuity ofthepotential, 186 ‘Exhaustion ofpotential energy, 138 Contour integral, normal derivative ofasphere, 188 oflog»,122 Fr D Fieldofvectors, 96 Derivativesof«potential, 27,109 Flux across asurface, 102 atexterior points, 27 Fundamental functions, 460 atinterior points, 29 Fuzzy surfaces, 225 Aiscontinuities of,168, 174, 178, 185, 202 a normal, 04 ‘Gauss’ problem, 228 E Gauss, ‘theorem onhomogeneous bodies, 106 Edgepotential, 75 ondistribution ofmatter, 128 Electric images, 209 extension ofabove theorem, 128 forlogarithmic potential, 270 Gravitation, lawof,1 ofcentrobarie bodies, 271 Gravitational constant, 2 487 468 THEORY OPTHEPOTENTIAL Generalized coordinates, 102 Lame, produets of,428 Green's equation forthesphere, 200 Lame’s polynomials, roots of,452 forany surface, 265 formula forLaplacian, 104 function, existence of,248 Laplace, equation of,32 forthe circle, 278 Laplace's cooficients, 380 forthesphere, 254 integral equation, 384 properties of,249, 252,253 Laplacian, 33,104 theorem relative to surface Legendre's equality, 238 density, 269 polynomials, 82,385 problem, 240 Level layer, example, 231 fortheellipsoid, 459 ‘onprolate spheroid, 238 forthe logarithmic potential, layers onanarbitrary surface, 222269 defined,210‘theorem, 104 families of,221 fextension offorharmonic fune- onellipsoids, 235 tions, 111 onelipticl disks, 238, fortwo dimensions, 115 offinite thickness, 240 potential, 128 surfaces, defined, 34 Line integrals, 148, H Lines offorce, 34 Logarithmie potential, defined, 35 Hiary surfaces, 25 ‘expansion inseries, 84 Harmonie functions, definition, 111 ‘maxima and minima, 138 M surface integrals ofnormal deriva- tive, IT MacLaurin’s theorem, 60 Hammack’ theorem, 900, 318 Magnets, 285 Heat oftheaun, 139 Magnetic doublet, 288 Helmbolts’ theory, 199 rnoment, 283 sheets, 285 I N Inertial integrals, 89,91, 94,382 Integral equations, 227, 384 Normal derivatives, 97 Integrals, proper end improper, 157 Neumann's method, 208 semi-convergent, 161Inversionofellipsoidal shell, 211 © theory of,208 Jvory’s method, 82 Orthogonal coordinate systems, 102, 2 K P Kelvin, Lord, 200, 277 Perspectivty, definition, 8 L ‘theorems relating to,9 Picard’s solution ofRobin's equa Lagrange’ series, 338 tion, 228 Tame, functionsof,418 Poincaré,208 linearindependence, 446 Poincaré’s problem, 280 INDEX 469 Poincas’s méthode du balayage, Surface integrals ofnormal derive- 313 tivesofharmoniefunetions Poiston's equation, 124 ut fortwo dimentions, 126 of1/s, 120 Potential, expansion inseries, 1,328 Surfaces, attraction of153 function, defined, 24 fverage value on sphere, 182 T Characteristic properties, 190 txistence of,25 ‘Taylors series, 897 Higher derivatives, 109 ‘Tesora harmonies, 968 rtxima andminima, 235 Gistibution ofmatter onaphere, of circular diak, 41 308 fe homogeneous rod, 42 expansion of1/0,870 ofellipoidal level layers, 240 ofpotenti, 380 ofanoblate spheroid, 263 Green's problem fortheaphere, ofa solid ofrevolution, 360 302 ofephercal shel, 36, obtained bypolar differentiation, {able ofvalues 40 308 oft uniflorm hoop, 195 solid, examples of,971 ofa ronal distribution, 367 turface integral, 374 ofelit eylindre, 89 eros of,373, ‘ofhomogeneous bois, 108 ofrectangular paralelopiped, 72 v ofspheroids, 62 Ofsolid elipeid, exterior, 58 ‘Univers, infinite, 155 v R ‘iprocal radii, transformation Vootor curls, 144Rest aheowomation PsYairernof,102Residues, theory of,117 f,08Robin's integral equation, 227 ‘Volume integrals, 99 Roderiguee’ formu 399Rotation about animaginary axis, Ww ‘0 Weierstrass, elipti functions, 412 s z Singular points ofsurfaces, 199Spier,parametricpresentations, Zonalhammonie, 31,383hy harmony, 208 nitenel pram examples of,325 ssexpansion intermsof,386 ‘expansion ofsinmg,358Stokes theorem, 143 ‘general formula for,247 Surface density necessary fora property oforthogonality, 343 given potential, 244 recursion formula, 346 integrals, 99 second kind, 334 dependence oncontours, 49 aeros arereal, 340 CATALOGUE OFDOVER BOOKS Catalogue ofDover Books BOOKS EXPLAINING SCIENCE AND MATHEMATICS General Saiamare 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