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| MacRobert T.M. — Spherical Harmonics an Elementary Treatise on Harmonic Functions with Applications |
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| Предметный указатель |
Addition theorems see “Bessel coefficients and functions” “Zonal
Analytical continuation 304 307
Associated Legendre functions of the first kind 122 314—345
Associated Legendre functions of the first kind, asymptotic expansions of 322 323
Associated Legendre functions of the first kind, definite integral expressions for 124 323
Associated Legendre functions of the first kind, Dougall's formulae for 326—328
Associated Legendre functions of the first kind, expansion in terms of 336—339
Associated Legendre functions of the first kind, Ferrer's 125 126 314—343
Associated Legendre functions of the first kind, integrals of products of 126 142
Associated Legendre functions of the first kind, Laplace's integrals for 127 129
Associated Legendre functions of the first kind, Mehlor — Dirichlat integral for 323
Associated Legendre functions of the first kind, recurrence formulae for 124 142 143 329—331
Associated Legendre functions of the first kind, relations between 124 323
Associated Legendre functions of the first kind, scries for 123 315
Associated Legendre functions of the second kind 122 316—345
Associated Legendre functions of the second kind, asymptotic expansions of 322
Associated Legendre functions of the second kind, definite integral expressions for 124
Associated Legendre functions of the second kind, recurrence formulae for 124 142 143
Associated Legendre functions of the second kind, relations between 125 316
Associated Legendre functions of the second kind, scries for 123 316 318 365
Asymptotic expansions see “Bessel functions” “Ilypergeometric
Attraction, gravitational 144
Attraction, of a plane lamina 152
Axes of spherical harmonics 234
Axes of zonal harmonics 131 137
Bessel coefficients 250
Bessel coefficients, addition theorems for 252
Bessel coefficients, expansions in terms of 251
Bessel functions 241 312 313 342—344
Bessel functions, addition theorems for 252 260 271
Bessel functions, applications of 286—297
Bessel functions, asymptotic expansions of 272
Bessel functions, definite integral expressions for 255 265 271 272 342—344
Bessel functions, expansion for  in terms of 253
Bessel functions, integrals involving 275 285 302 304 311 343. 367
Bessel functions, modified 269 342—344
Bessel functions, of the first kind 244
Bessel functions, of the second kind,  248 257
Bessel functions, of the second kind,  246 248
Bessel functions, products of 364
Bessel functions, recurrence formulae for 245 259 270
Bessel functions, relations between 259 271
Bessel functions, whose order is half an odd integer 250 285
Bessel functions, zeros of 277—283
Bessel's equation 241
Bessel's equation, solution of 242 255
Bessel's integral 251
Beta junction 83
Binomial expansion, remainder in 80
boundary conditions 29
Boundary problems 168 170 228
Calory 24
Christoffel's summation formulae 100 115 335
Clerk Maxwell's theory of spherical harmonics 231
Coaxal circles 219
Coefficients, Laplace's see “Laplace's coefficients”
Coefficients, Legendre see “Legendre coefficients”
Conduction of heat 23
Conduction of Heat, equation of 27 30
Conduction of heat, in circular cylinder 255 294 295 296
Conduction of heat, in finiterod 33 35 39 41 49 50
Conduction of heat, in Fourier's ring 36
Conduction of heat, in infinite rectangular solid 41
Conduction of heat, in infinite rod 46 50
Conduction of heat, in infinite slab 31
Conduction of heat, in infinite solid 48 49
Conduction of heat, in rectangular parallelepiped 43 45
Conduction of heat, in semi-infinite solid 48
Conduction of heat, in sphere 292 290
Conduction of heat, in wedge 295
Conduction of heat, with radiation at surface 34
Conductivity, thermal 23
Conductivity, thermometric 28
Conductor, distribution of electricity on 179—194 203 222—226
Convection of heat 23 29
Convergence 313
Cross-cut 307
Curvilinear co-ordinates 195
Cylinder functions 242
Cylinder, flow of heat in 288 294 295 296
Cylindrical harmonics 242
Diffusivity 28
Dipolar co-ordinates 219 221
Dipolar co-ordinates, Laplace's equation in terms of 226
Dipolar co-ordinates, reciprocal of distance in terms of 222
Dirichlet's conditions for a Fourier expansion 6
Dirichlet's integrals 7
Discontinuity, point of 7 11
Domain 299
Duhamel's theorem 38
Eccentric spheres 191 222 297
Electrostatics 179
Ellipsoid of revolution, oblate 194 198 212
Ellipsoid of revolution, prolate 198 202 203 209 210
Elliptic co-ordinates 199 213
Elliptic co-ordinates, reciprocal of distance in terms of 204 216
Emissivity 29
Equivalent distributions of masses 167
Euler's constant 82
for Legendre functions 98 112 113 124 142 143 329—332
Fourier series 1
Fourier series, convergence of 11 12
Fourier series, differentiation of 12
Fourier series, sine and cosine 12
Fourier series, summation of 10
Fourier — Bessel expansions 283
Fourier's double integral 16 48 49
Fourier's expansion 1 13
Fourier's expansion, at a point of discontinuity 11
Fourier's expansion, Dirichlet's conditions for 6
Fourier's expansion, for odd and even functions 3 4
Fourier's expansion, validity of 5
Fourier's ring 36
Gamma function 82 340 349
Gauss's equation 77 298
Gauss's theorem for the hypergeometric function 83 300
Gauss's theorem on attractions 146
Generalised hypergeometric functions 310 313
Green 72
Green's function 186
Green's theorem 27
Harmonics 59
Harmonics, cylindrical 242
Harmonics, sectorial 132
Harmonics, spherical see “Spherical harmonics”
Harmonics, tesseral 132
Harmonics, zonal 131 137
Harp string, vibrations of 60
Heat see “Conduction” “Convection” “Radiation”
Heat, specific 24
Hypergeometric equation 77 298
Hypergeometric equation, relations between the integrals of 305
Hypergeometric function 77 295—313
Hypergeometric function, asymptotic expansion of 80 309
Hypergeometric function, four forms of 79 303
Hypergeometric function, Gauss's theorem for 83
Hypergeometric series 77 313
Indicial equation 78 109 243
Initial conditions 29
Inverse points 161
Isothermal surfaces 24
Jacobi's proof of Rodrigues' formula 89
Kelvin 73
Kelvin's theorem 74
Krammer's function 311
Lamp's functions 218
Laplace 72 73
Laplace's coefficients 73 75 136
Laplace's coefficients, addition theorem for 137
| Laplace's equation 74 147
Laplace's equation, transformation of 74 195 199 213 226 230 241 297
Laplace's integrals 101 102 127 129
legendre 72
Legendre coefficients 72 76 85 131
Legendre coefficients, applications to potential theory of 144
Legendre coefficients, as derivatives 105 238
Legendre coefficients, axes of 131
Legendre coefficients, ChristofTel's summation formula, for 100 115
Legendre coefficients, expansions of 86 87 88 89 104 106
Legendre coefficients, expansions of functions in scries of 100 102 338 339
Legendre coefficients, expression for potential in terms of 148 150
Legendre coefficients, expression for solid angle in terms of 153
Legendre coefficients, expressions for polynomials in terms of 95 97 106
Legendre coefficients, integrals involving 91 93 94 96 104 105 106 107 117
Legendre coefficients, Laplace's integrals for 101 102
Legendre coefficients, poles of 131 238
Legendre coefficients, potential of circular lamina in terms of 152
Legendre coefficients, potential of magnet in terms of 155
Legendre coefficients, potential of magnetic shell in terms of 156
Legendre coefficients, recurrence formulae for 98
Legendre coefficients, Rodrigue's formula for 88
Legendre coefficients, summation formulae for 100 115
Legendre coefficients, zeros of 91 131
Legendre factions of the second kind 108 116 117 118
Legendre factions of the second kind, addition theorem for 355
Legendre factions of the second kind, recurrence formula for 113
Legendre factions of the second kind, relations connecting with Legendre functions of the first kind 114 115 116 117 118
Legendre functions of the first kind 111 119
Legendre functions of the first kind, addition theorem for 328
Legendre functions of the first kind, integrals involving 117
Legendre functions of the first kind, recurrence formulae for 112
Legendre functions, associated “Associated Legendre functions”
Legendre's associated equation 76 122 314 316
Legendre's associated equation, relations between the integrals of 316—321
Legendre's equation 76 90
Legendre's equation, solution of 108 in
Legendre's expansion 72 85
Lommel integrals 275
Loops of vibrating strings 59
Magnet 155
Magnetic shell 156
Membranes, vibrations of, see “Vibrations of membranes”
modes, fundamental 58
Modes, normal, of vibrating strings 56
Neighbourhood of a point 6
Neumann's addition formula for  252
Newton's Law of attraction 144
Nodal lines 71
Nodes of vibrating strings 59
overtones 59
Piano string, vibrations of 65
Point souuv of heat 69
Poisson's equation 145 195
Poisson's integrals 103 169
Poles oi spheiieal harmonies 234 238
Poles oi zonal harmonies 131 137 23s
Polynomials, Legendre see “Legendre coefficients”
Potential 72 145 146 157 158
Potential, derivative of, in terms of spherical harmonics 150
Potential, in terms ol spherical harmonics 148
Potential, of circular lamina 154
Potential, of circular wire 151
Potential, of concentric spheres 169
Potential, of eccentric spheres 191 222
Potential, of ellipsoid of revolution 203 209 210
Potential, of hemispherical shell 152
Potential, of magnet 155
Potential, of magnetic shell 156
Potential, of singular point 235
Potential, of sphere 166 180—188
Potential, of spherical shell 158—167 165 185—191
Potential, of spheroid 172 175 191—194
Potential, of spheroidal shell 177
Radiation of heat 23
Radiation of heat, at surface 29 34 37 39 41
Recurrence formulae, for Bebscl functions 248 . 270
Relations between Bessel and Legendre functions 339—345 363 366
Remainder in binomial expansion 80
Ring functions 230
Ring surfaces 225
Ring, Fourier's 36
Rodrigues' formula 88
Rodrigues' formula, Jacobi's proof of 89
Rodrigues' formula, the extended 334
Rodrigues' general formula 89
Sectorial harmonics 132 238
Singular points 235
Source of heat, point 49
Specific heat 24
Spheres concentric 169
Spheres, boundary problems for 168 170
Spheres, conduction of heat in 292. 290
Spheres, eccentric 191 222 297
Spheres, potentials of 160 169 180—188
Spherical harmonics 73
Spherical harmonics, axes of 234
Spherical harmonics, Clerk Maxwell's theory of 231
Spherical harmonics, expansions in terms of 138 141
Spherical harmonics, expression of potential in terms of 148
Spherical harmonics, expressions of 236
Spherical harmonics, in terms of zonal, tesseral, and sectorial harmonics 133
Spherical harmonics, integrals involving 135 137
Spherical harmonics, poles of 234
Spherical harmonics, potential of shells and spheies in terms of 158 166
Spherical harmonics, solid 74 77
Spherical harmonics, surface 73 75 77 131 132 133 135 236
Spherical shells, potentials of 157—168 188—191
Spheroidal shells, potentials of 177
Spheroids, potentials of 172—178 191—194
Stirling's formula 83
Strings, vibrations of see “Vibrations of strings”
Surface conditions 29
Tait 73
Tesseral harmonics 132 238
thermal conductivity 23
Thermometric conductivity 28
Thomson 73
Tone, fundamental 59
Vibrations of strings 52
Vibrations of strings, differential equation of 52 55
Vibrations of strings, energy of 54 60
Vibrations of strings, for semi-infinite string 55
Vibrations of strings, for string fixed at ends 56
Vibrations of strings, for unlimited string 55
Vibrations of strings, forced 54 66
Vibrations of strings, free 54
Vibrations of strings, fundamental tones of 59
Vibrations of strings, harmonics of 59
Vibrations of strings, loops of 59
Vibrations of strings, nodes of 59
Vibrations of strings, normal modes of 56
Vibrations of strings, of harp 60
Vibrations of strings, of piano 65
Vibrations of strings, of violin 63
Vibrations of strings, overtones of 59
Vibrations, circular 286 364
Vibrations, of membranes 69
Vibrations, rectangular 70
Violin string, vibrations of 63
Waves 55 56 61
Wedge, conduction of heat in 295 296
Whitehead, R.F. 340
Zeros of Legendre coefficients 91 131
Zeros, of Bessel functions 277 283
Zonal harmonics 131
Zonal harmonics, addition theorem for 137
Zonal harmonics, axes of 131. 137
Zonal harmonics, general 137
Zonal harmonics, poles of 131 137 238
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