AN ELEMENTARY TREATISE ON FOURIER SERIES AND HARMONICS
ANALYTICAL TABLE OF CONTENTS CHAPTER I PAGES INTRODUCTION 1–29 ART 1 LIST OF SOME IMPORTANT HOMOGENEOUS LINEAR PARTIAL DI?ERENTIAL EQUA- TIONS OF PHYSICS—ARTS 2–4 DISTINCTION BETWEEN THE GENERAL SOLUTION AND A PARTICULAR SOLUTION OF A DI?ERENTIAL EQUATION NEED OF ADDITIONAL DATA TO MAKE THE SOLUTION OF A DI?ERENTIAL EQUATION DETERMINATE DE?NITION OF LINEAR AND OF LINEAR AND HOMOGENEOUS—ARTS 5–6 PARTICULAR SOLUTIONS OF HOMOGENEOUS LIN- EAR DI?ERENTIAL EQUATIONS MAY BE COMBINED INTO A MORE GENERAL SOLUTION NEED OF DEVELOPMENT IN TERMS OF NORMAL FORMS—ART 7 PROBLEM: PERMANENT STATE OF TEMPERATURES IN A THIN RECTANGULAR PLATE NEED OF A DEVELOPMENT IN SINE SERIES EXAMPLE—ART 8 PROBLEM: TRANSVERSE VIBRATIONS OF A STRETCHED ELASTIC STRING A DEVELOPMENT IN SINE SERIES SUGGESTED—ART 9 PROBLEM: POTENTIAL FUNCTION DUE TO THE ATTRACTION OF A CIRCULAR RING OF SMALL CROSS-SECTION SURFACE ZONAL HAR- MONICS (LEGENDRE’S COE?CIENTS) EXAMPLE—ART 10 PROBLEM: PERMANENT STATE OF TEMPERATURES IN A SOLID SPHERE DEVELOPMENT IN TERMS OF SURFACE ZONAL HARMONICS SUGGESTED—ARTS 11–12 PROBLEM: VIBRATIONS OF A CIRCULAR DRUM- HEAD CYLINDRICAL HARMONICS (BESSEL’S FUNCTIONS) RECAPITULATION—ART 13 METHOD OF MAKING THE SOLUTION OF A LINEAR PARTIAL DI?ERENTIAL EQUATION DEPEND UPON SOLVING A SET OF ORDINARY DI?ERENTIAL EQUATIONS BY ASSUMING THE DEPENDENT VARIABLE EQUAL TO A PRODUCT OF FACTORS EACH OF WHICH INVOLVES BUT ONE OF THE INDE- PENDENT VARIABLES ARTS 14–15 METHOD OF SOLVING ORDINARY HOMOGENEOUS LINEAR DI?ERENTIAL EQUATIONS BY DEVELOPMENT IN POWER SERIES APPLICATIONS—ART 16 APPLICATION TO LEGENDRE’S EQUATION SEVERAL FORMS OF GENERAL SOLUTION OBTAINED ZONAL HARMONICS OF THE SECOND KIND—ART 17 APPLICATION TO BESSEL’S EQUA- TION GENERAL SOLUTION OBTAINED FOR THE CASE WHERE M IS NOT AN INTEGER, AND FOR THE CASE WHERE M IS ZERO BESSEL’S FUNCTION OF THE SECOND KIND AND ZEROTH ORDER—ART 18 METHOD OF OBTAINING THE GENERAL SOLUTION OF AN ORDINARY LIN- EAR DI?ERENTIAL EQUATION OF THE SECOND ORDER FROM A GIVEN PARTICULAR SOLUTION APPLICATION TO THE EQUATIONS CONSIDERED IN ARTS 14–17 CHAPTER II DEVELOPMENT IN TRIGONOMETRIC SERIES 30–55 ARTS 19–22 DETERMINATION OF THE COE?CIENTS OF N TERMS OF A SINE SERIES SO THAT THE SUM OF THE TERMS SHALL BE EQUAL TO A GIVEN FUNCTION OF X FOR N GIVEN VAL- UES OF X NUMERICAL EXAMPLE—ART 23 PROBLEM OF DEVELOPMENT IN SINE SERIES TREATED AS A LIMITING CASE OF THE PROBLEM JUST SOLVED—ARTS 24–25 SHORTER TABLE OF CONTENTS METHOD OF SOLVING THE PROBLEM OF DEVELOPMENT IN SERIES INVOLVING SINES OF WHOLE MULTIPLES OF THE VARIABLE WORKING RULE DEDUCED RECAPITULATION—ART 26 A FEW IMPORTANT SINE DEVELOPMENTS OBTAINED EXAMPLES—ARTS 27–28 DEVELOP- MENT IN COSINE SERIES EXAMPLES—ART 29 SINE SERIES AN ODD FUNCTION OF THE VARIABLE, COSINE SERIES AN EVEN FUNCTION, AND BOTH SERIES PERIODIC FUNCTIONS— ART 30 DEVELOPMENT IN SERIES INVOLVING BOTH SINES AND COSINES OF WHOLE MUL- TIPLES OF THE VARIABLE FOURIER’S SERIES EXAMPLES—ART 31 EXTENSION OF THE RANGE WITHIN WHICH THE FUNCTION AND THE SERIES ARE EQUAL EXAMPLES—ART 32 FOURIER’S INTEGRAL OBTAINED CHAPTER III CONVERGENCE OF FOURIER’S SERIES 56–69 ARTS 33–36 THE QUESTION OF THE CONVERGENCE OF THE SINE SERIES FOR UNITY CONSIDERED AT LENGTH—ARTS 37–38 STATEMENT OF THE CONDITIONS WHICH ARE SU?CIENT TO WARRANT THE DEVELOPMENT OF A FUNCTION INTO A FOURIER’S SERIES HIS- TORICAL NOTE ART 39 GRAPHICAL REPRESENTATION OF SUCCESSIVE APPROXIMATIONS TO A SINE SERIES PROPERTIES OF A FOURIER’S SERIES INFERRED FROM THE CONSTRUCTIONS— ARTS 40–42 INVESTIGATION OF THE CONDITIONS UNDER WHICH A FOURIER’S SERIES CAN BE DI?ERENTIATED TERM BY TERM—ART 43 CONDITIONS UNDER WHICH A FUNCTION CAN BE EXPRESSED AS A FOURIER’S INTEGRAL CHAPTER IV SOLUTION OF PROBLEMS IN PHYSICS BY THE AID OF FOURIER’S INTE- GRALS AND FOURIER’S SERIES 70–135 ARTS 44–48 LOGARITHMIC POTENTIAL FLOW OF ELECTRICITY IN AN IN?NITE PLANE, WHERE THE VALUE OF THE POTENTIAL FUNCTION IS GIVEN ALONG AN IN?NITE STRAIGHT LINE; ALONG TWO MUTUALLY PERPENDICULAR STRAIGHT LINES; ALONG TWO PARALLEL STRAIGHT LINES EXAMPLES USE OF CONJUGATE FUNCTIONS SOURCES AND SINKS EQUIPOTENTIAL LINES AND LINES OF FLOW EXAMPLES—ARTS 49–52 ONE-DIMENSIONAL ?OW OF HEAT FLOW OF HEAT IN AN IN?NITE SOLID; IN A SOLID WITH ONE PLANE FACE AT THE TEMPERA- TURE ZERO; IN A SOLID WITH ONE PLANE FACE WHOSE TEMPERATURE IS A FUNCTION OF THE TIME (RIEMANN’S SOLUTION); IN A BAR OF SMALL CROSS SECTION FROM WHOSE SURFACE HEAT ESCAPES INTO AIR AT TEMPERATURE ZERO LIMITING STATE APPROACHED WHEN THE TEMPERATURE OF THE ORIGIN IS A PERIODIC FUNCTION OF THE TIME EXAMPLES—ARTS 53–54 TEMPERATURES DUE TO INSTANTANEOUS AND TO PERMANENT HEAT SOURCES AND SINKS, AND TO HEAT DOUBLETS EXAMPLES APPLICATION TO THE CASE WHERE THERE IS LEAKAGE—ARTS 55–56 TRANSMISSION OF A DISTURBANCE ALONG AN IN?NITE STRETCHED ELASTIC STRING EXAMPLES—ARTS 57–58 STATIONARY TEMPERATURES IN A LONG RECTANGULAR PLATE TEMPERATURE OF THE BASE UNITY SUMMATION OF A TRIGONOMETRIC SERIES ISOTHERMAL LINES AND LINES OF ?OW EXAMPLES—ART 59 TABLE OF CONTENTS POTENTIAL FUNCTION GIVEN ALONG THE PERIMETER OF A RECTANGLE EXAMPLES—ARTS 60–63 ONE-DIMENSIONAL ?OW OF HEAT IN A SLAB WITH PARALLEL PLANE FACES BOTH FACES AT TEMPERATURE ZERO BOTH FACES ADIATHERMANOUS TEMPERATURE OF ONE FACE A FUNCTION OF THE TIME EXAMPLES—ART 64 MOTION OF A STRETCHED ELASTIC STRING FASTENED AT THE ENDS STEADY VIBRATION NODES EXAMPLES—ART 65 MOTION OF A STRING IN A RESISTING MEDIUM—ART 66 FLOW OF HEAT IN A SPHERE WHOSE SURFACE IS KEPT AT A CONSTANT TEMPERATURE—ARTS 67–68 COOLING OF A SPHERE IN AIR SURFACE CONDITION GIVEN BY A DI?ERENTIAL EQUATION DEVELOPMENT IN A TRIGONO- METRIC SERIES OF WHICH FOURIER’S SINE SERIES IS A SPECIAL CASE EXAMPLES—ARTS 69–70 FLOW OF HEAT IN AN IN?NITE SOLID WITH ONE PLANE FACE WHICH IS EXPOSED TO AIR WHOSE TEMPERATURE IS A FUNCTION OF THE TIME SOLUTION FOR AN INSTANTA- NEOUS HEAT SOURCE WHEN THE TEMPERATURE OF THE AIR IS ZERO EXAMPLES—ARTS 71–73 VIBRATION OF A RECTANGULAR DRUMHEAD DEVELOPMENT OF A FUNCTION OF TWO VARIABLES IN A DOUBLE FOURIER’S SERIES EXAMPLES NODAL LINES IN A RECTANGULAR DRUMHEAD NODAL LINES IN A SQUARE DRUMHEAD MISCELLANEOUS PROBLEMS 135–143 I LOGARITHMIC POTENTIAL POLAR CO¨ORDINATES—II POTENTIAL FUNCTION IN SPACE III CONDUCTION OF HEAT IN A PLANE—IV CONDUCTION OF HEAT IN SPACE CHAPTER V 143–195 ART 74 RECAPITULATION SURFACE ZONAL HARMONICS (LEGENDRIANS) ZONAL HARMONICS OF THE SECOND KIND—ARTS 75–76 LEGENDRIANS AS COE?CIENTS IN A POWER SERIES SPECIAL VALUES—ART 77 SUMMARY OF THE PROPERTIES OF A LEGENDRIAN LIST OF THE ?RST EIGHT LEGENDRIANS RELATION CONNECTING ANY THREE SUCCESSIVE LEGENDRIANS—ARTS 78–81 PROBLEMS IN POTENTIAL POTENTIAL FUNC- TION DUE TO THE ATTRACTION OF A MATERIAL CIRCULAR RING OF SMALL CROSS SECTION POTENTIAL FUNCTION DUE TO A CHARGE OF ELECTRICITY PLACED ON A THIN CIRCULAR DISC EXAMPLES: SPHEROIDAL CONDUCTORS POTENTIAL FUNCTION DUE TO THE ATTRACTION OF A MATERIAL HOMOGENEOUS CIRCULAR DISC EXAMPLES: HOMOGENEOUS HEMISPHERE; HETEROGENEOUS SPHERE; HOMOGENEOUS SPHEROIDS GENERALISATION—ART 82 LEG- ENDRIAN AS A SUM OF COSINES—ARTS 83–84 LEGENDRIAN AS THE MTH DERIVATIVE OF THE MTH POWER OF X2 ??1—ART 85 EQUATIONS DERIVABLE FROM LEGENDRE’S EQUATION—ART 86 LEGENDRIAN AS A PARTIAL DERIVATIVE—ART 87 LEGEN- DRIAN AS A DE?NITE INTEGRAL ARTS 88–90 DEVELOPMENT IN ZONAL HARMONIC SERIES INTEGRAL OF THE PRODUCT OF TWO LEGENDRIANS OF DI?ERENT DEGREES INTEGRAL OF THE SQUARE OF A LEGENDRIAN FORMULAS FOR THE COE?CIENTS OF THE SERIES— ARTS 91–92 INTEGRAL OF THE PRODUCT OF TWO LEGENDRIANS OBTAINED BY THE AID OF LEGENDRE’S EQUATION; BY THE AID OF GREEN’S THEOREM ADDITIONAL FORMULAS FOR INTEGRATION EXAMPLES—ARTS 93–94 PROBLEMS IN POTENTIAL WHERE THE VALUE OF THE POTENTIAL FUNCTION IS GIVEN ON A SPHERICAL SURFACE AND HAS CIRCULAR TABLE OF CONTENTS SYMMETRY ABOUT A DIAMETER EXAMPLES—ART 95 DEVELOPMENT OF A POWER OF X IN ZONAL HARMONIC SERIES—ART 96 USEFUL FORMULAS—ART 97 DEVEL- OPMENT OF SIN N? AND COS N? IN ZONAL HARMONIC SERIES EXAMPLES GRAPHICAL REPRESENTATION OF THE ?RST SEVEN SURFACE ZONAL HARMONICS CONSTRUCTION OF SUC- CESSIVE APPROXIMATIONS TO ZONAL HARMONIC SERIES ARTS 98–99 METHOD OF DEALING WITH PROBLEMS IN POTENTIAL WHEN THE DENSITY IS GIVEN EXAMPLES—ART 100 SURFACE ZONAL HARMONICS OF THE SECOND KIND EXAMPLES: CONAL HARMONICS CHAPTER VI SPHERICAL HARMONICS 196–219 ARTS 101–102 PARTICULAR SOLUTIONS OF LAPLACE’S EQUATION OBTAINED AS- SOCIATED FUNCTIONS TESSERAL HARMONICS SURFACE SPHERICAL HARMONICS SOLID SPHERICAL HARMONICS TABLE OF ASSOCIATED FUNCTIONS EXAMPLES—ARTS 103– 108 DEVELOPMENT IN SPHERICAL HARMONIC SERIES THE INTEGRAL OF THE PRODUCT OF TWO SURFACE SPHERICAL HARMONICS OF DI?ERENT DEGREES TAKEN OVER THE SURFACE OF THE UNIT SPHERE IS ZERO EXAMPLES THE INTEGRAL OF THE PRODUCT OF TWO ASSO- CIATED FUNCTIONS OF THE SAME ORDER FORMULAS FOR THE COE?CIENTS OF THE SERIES ILLUSTRATIVE EXAMPLE EXAMPLES—ARTS 109–110 ANY HOMOGENEOUS RATIONAL INTEGRAL ALGEBRAIC FUNCTION OF X, Y, AND Z WHICH SATIS?ES LAPLACE’S EQUATION IS A SOLID SPHERICAL HARMONIC EXAMPLES—ART 111 A TRANSFORMATION OF AXES TO A NEW SET HAVING THE SAME ORIGIN WILL CHANGE A SURFACE SPHERICAL HARMONIC INTO ANOTHER OF THE SAME DEGREE—ARTS 112–114 LAPLACIANS INTEGRAL OF THE PRODUCT OF A SURFACE SPHERICAL HARMONIC BY A LAPLACIAN OF THE SAME DEGREE DEVELOPMENT IN SPHERICAL HARMONIC SERIES BY THE AID OF LAPLACIANS TABLE OF LAPLACIANS EXAMPLE—ART 115 SOLUTION OF PROBLEMS IN POTENTIAL BY DIRECT INTEGRATION EXAMPLES—ARTS 116–118 DI?ERENTIATION ALONG AN AXIS AXES OF A SPHERICAL HARMONIC—ART 119 ROOTS OF A ZONAL HARMONIC ROOTS OF A TESSERAL HARMONIC NOMENCLATURE JUSTI?ED CHAPTER VII CYLINDRICAL HARMONICS (BESSEL’S FUNCTIONS) 220–238 ART 120 RECAPITULATION CYLINDRICAL HARMONICS (BESSEL’S FUNCTIONS) OF THE ZEROTH ORDER; OF THE NTH ORDER; OF THE SECOND KIND GENERAL SOLUTION OF BESSEL’S EQUATION—ART 121 BESSEL’S FUNCTIONS AS DE?NITE INTEGRALS EXAMPLES— ART 122 PROPERTIES OF BESSEL’S FUNCTIONS SEMI-CONVERGENT SERIES FOR A BESSEL’S FUNCTION EXAMPLES—ART 123 PROBLEM: STATIONARY TEMPERATURES IN A CYLIN- DER (A) WHEN THE TEMPERATURE OF THE CONVEX SURFACE IS ZERO; (B) WHEN THE CONVEX SURFACE IS ADIATHERMANOUS; (C) WHEN THE CONVEX SURFACE IS EXPOSED TO AIR AT THE TEMPERATURE ZERO—ART 124 ROOTS OF BESSEL’S FUNCTIONS—ART 125 THE IN- TEGRAL OF R TIMES THE PRODUCT OF TWO CYLINDRICAL HARMONICS OF THE ZEROTH ORDER TABLE OF CONTENTS EXAMPLE—ART 126 DEVELOPMENT IN CYLINDRICAL HARMONIC SERIES FORMULAS FOR THE COE?CIENTS EXAMPLES—ART 127 PROBLEM: STATIONARY TEMPERATURES IN A CYLINDRICAL SHELL BESSEL’S FUNCTIONS OF THE SECOND KIND EMPLOYED EXAMPLE: VIBRATION OF A RING MEMBRANE—ART 128 PROBLEM: STATIONARY TEMPERATURES IN A CYLINDER WHEN THE TEMPERATURE OF THE CONVEX SURFACE VARIES WITH THE DISTANCE FROM THE BASE BESSEL’S FUNCTIONS OF A COMPLEX VARIABLE EXAMPLES—ART 129 PROBLEM: STATIONARY TEMPERATURES IN A CYLINDER WHEN THE TEMPERATURES OF THE BASE ARE UNSYMMETRICAL BESSEL’S FUNCTIONS OF THE NTH ORDER EMPLOYED MISCEL- LANEOUS EXAMPLES BESSEL’S FUNCTIONS OF FRACTIONAL ORDER CHAPTER VIII LAPLACE’S EQUATION IN CURVILINEAR CO¨ORDINATES ELLIPSOIDAL HARMONICS 239–266 ARTS 130–131 ORTHOGONAL CURVILINEAR CO¨ORDINATES IN GENERAL LAPLACE’S EQUATION EXPRESSED IN TERMS OF ORTHOGONAL CURVILINEAR CO¨ORDINATES BY THE AID OF GREEN’S THEOREM—ARTS 132–135 SPHEROIDAL CO¨ORDINATES LAPLACE’S EQUATION IN SPHEROIDAL CO¨ORDINATES, IN NORMAL SPHEROIDAL CO¨ORDINATES EXAMPLES CONDI- TION THAT A SET OF CURVILINEAR CO¨ORDINATES SHOULD BE NORMAL THERMOMETRIC PA- RAMETERS PARTICULAR SOLUTIONS OF LAPLACE’S EQUATION IN SPHEROIDAL CO¨ORDINATES SPHEROIDAL HARMONICS EXAMPLES THE POTENTIAL FUNCTION DUE TO THE ATTRAC- TION OF AN OBLATE SPHEROID SOLUTION FOR AN EXTERNAL POINT EXAMPLES—ARTS 136–141 ELLIPSOIDAL CO¨ORDINATES LAPLACE’S EQUATION IN ELLIPSOIDAL CO¨ORDINATES NORMAL ELLIPSOIDAL CO¨ORDINATES EXPRESSED AS ELLIPTIC INTEGRALS PARTICULAR SOLU- TIONS OF LAPLACE’S EQUATION LAM´E’S EQUATION ELLIPSOIDAL HARMONICS (LAM´E’S FUNCTIONS) TABLES OF ELLIPSOIDAL HARMONICS OF THE DEGREES 1, 2, AND 3 LAM´E’S FUNCTIONS OF THE SECOND KIND EXAMPLES DEVELOPMENT IN ELLIPSOIDAL HAR- MONIC SERIES VALUE OF THE POTENTIAL FUNCTION AT ANY POINT IN SPACE WHEN ITS VALUE IS GIVEN AT ALL POINTS ON THE SURFACE OF AN ELLIPSOID—ART 142 CONICAL CO¨ORDINATES THE PRODUCT OF TWO ELLIPSOIDAL HARMONICS A SPHERICAL HARMONIC— ART 143 TOROIDAL CO¨ORDINATES LAPLACE’S EQUATION IN TOROIDAL CO¨ORDINATES PARTICULAR SOLUTIONS TOROIDAL HARMONICS POTENTIAL FUNCTION FOR AN ANCHOR RING CHAPTER IX HISTORICAL SUMMARY 267–274 1 CHAPTER I INTRODUCTION 1 IN MANY IMPORTANT PROBLEMS IN MATHEMATICAL PHYSICS WE ARE OBLIGED TO DEAL WITH PARTIAL DI?ERENTIAL EQUATIONS OF A COMPARATIVELY SIMPLE FORM FOR EXAMPLE, IN THE ANALYTICAL THEORY OF HEAT WE HAVE FOR THE CHANGE OF TEMPERATURE OF ANY SOLID DUE TO THE ?OW OF HEAT WITHIN THE SOLID, THE EQUATION DTU = A2(D2 XU + D2 YU + D2 ZU), [I] WHERE U REPRESENTS THE TEMPERATURE AT ANY POINT OF THE SOLID AND T THE TIME IN THE SIMPLEST CASE, THAT OF A SLAB OF IN?NITE EXTENT WITH PARALLEL PLANE FACES, WHERE THE TEMPERATURE CAN BE REGARDED AS A FUNCTION OF ONE CO¨ORDINATE, [I] REDUCES TO DTU = A2D2 XU, [II] A FORM OF CONSIDERABLE IMPORTANCE IN THE CONSIDERATION OF THE PROBLEM OF THE COOLING OF THE EARTH’S CRUST IN THE PROBLEM OF THE PERMANENT STATE OF TEMPERATURES IN A THIN RECTANGULAR PLATE, THE EQUATION [I] BECOMES D2 XU + D2 YU = 0 [III] IN POLAR OR SPHERICAL CO¨ORDINATES [I] IS LESS SIMPLE, IT IS DTU = A2 R2 DR(R2DRU) + 1 SIN ? D?(SIN ?D?U) + 1 SIN2 ? D2 ?U [IV] IN THE CASE WHERE THE SOLID IN QUESTION IS A SPHERE AND THE TEMPERATURE AT ANY POINT DEPENDS MERELY ON THE DISTANCE OF THE POINT FROM THE CENTRE [IV] REDUCES TO DT(RU) = A2D2 R(RU) [V] IN CYLINDRICAL CO¨ORDINATES [I] BECOMES DTU = A2[D2 RU + R DRU + R2 D2 ?U + D2 ZU] [VI] IN CONSIDERING THE ?OW OF HEAT IN A CYLINDER WHEN THE TEMPERATURE AT ANY POINT DEPENDS MERELY ON THE DISTANCE R OF THE POINT FROM THE AXIS [VI] BECOMES DTU = A2(D2 RU + R DRU) [VII] IN ACOUSTICS IN SEVERAL PROBLEMS WE HAVE THE EQUATION D2 T Y = A2D2 XY; [VIII] 1FOR THE SAKE OF BREVITY WE SHALL OFTEN USE THE SYMBOL ?2 FOR THE OPERATION D2X + D2Y + D2Z; AND WITH THIS NOTATION EQUATION [I] WOULD BE WRITTEN DTU = A2?2U INTRODUCTION 2 FOR INSTANCE, IN CONSIDERING THE TRANSVERSE OR THE LONGITUDINAL VIBRATIONS OF A STRETCHED ELASTIC STRING, OR THE TRANSMISSION OF PLANE SOUND WAVES THROUGH THE AIR IF IN CONSIDERING THE TRANSVERSE VIBRATIONS OF A STRETCHED STRING WE TAKE AC- COUNT OF THE RESISTANCE OF THE AIR [VIII] IS REPLACED BY D2 T Y + 2KDTY = A2D2 XY [IX] IN DEALING WITH THE VIBRATIONS OF A STRETCHED ELASTIC MEMBRANE, WE HAVE THE EQUATION D2 T Z = C2(D2 XZ + D2 YZ), [X] OR IN CYLINDRICAL CO¨ORDINATES D2 T Z = C2(D2 RZ + R DRZ + R2 D2 ?Z) [XI] IN THE THEORY OF POTENTIAL WE CONSTANTLY MEET LAPLACE’S EQUATION D2 XV +D2 YV + D2 ZV = [XII] OR ?2V = WHICH IN SPHERICAL CO¨ORDINATES BECOMES 1 R2 RD2 R(RV ) + 1 SIN ? D?(SIN ?D?V ) + 1 SIN2 ? D2 ?V = 0, [XIII] AND IN CYLINDRICAL CO¨ORDINATES D2 RV + R DRV +