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| Eisenhart L.P. — Transformations of surfaces |
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| Предметный указатель |
Adjoint equation 9 22 88 93
Angle, of two hyperplanes 2
Angle, of two lines 1
Angle, of two spheres 234
Angle, of two surfaces in pentaspherical coordinates 282
Applicable nets, and congruences of circles 272
Applicable nets, and cyclic systems 274
Applicable nets, definition 61
Applicable nets, parallel to given pair 62
Applicable nets, transformations F of 61 275 290 325 334
Applicable nets, which are 2, O 298. See Nets C and Nets permanent in deformation.
Applicable nets, which are O 286
Asymptotic lines, definition 98
Asymptotic lines, determination of 356
Asymptotic lines, in Lie line-sphere transformation 252
Asymptotic lines, parametric 99 123
Axis congruence, definition 125
Axis curves, definition 125
Axis curves, form a net 125 153
Axis curves, property of 126
Beltrami 272 299 310 312 314
Bianchi 112 138 142 143 180 209 210 212 214 215 216 229 230 292 298 301 305 308 310 312 316 317 320 327 344 345 352 356 360 362 363 366 371 372
Bonnet 295 317
Calapso 225 230 232 286 298 319 320 331
Calo 318
Chieffi 316
Christoffel transformations and transformations  229
Christoffel transformations of isothermic surfaces 229 286
Christoffel transformations of special isothermic nets 230 291
Christoffel transformations, nets determined by 319
Circle-planes of a transformation R, coincident for two transformations 206 211
Circle-planes of a transformation R, definition 198
Circle-planes of a transformation R, envelope of 199 213 285 286
Circle-planes of a transformation R, pass through a point 200 230
Circles K, definition 202
Circles K, form a congruence 263
Circles K, normals to nets at points of 204
Circles K, of the first kind 206
Circles K, of the second kind 206 229
Congruence K, cyclic 206
Congruence K, definition 203
Congruence K, focal points of 229
Congruence of rolling 316
Congruence of rolling, developables of 317
Congruence of rolling, focal surfaces of 317
Congruence of rolling, normal 321
Congruence of rolling, parallel 321
Congruence of rolling, pseudospherical 321
Congruence of rolling, with coincident developables 317
Congruence of rolling, with constant focal segment 321
Congruence R, definition 113
Congruence R, focal surfaces 115
Congruence, conjugate to a net 12 15 34
Congruence, conjugate to O nets 167
Congruence, conjugate, of a transformation F 34
Congruence, definition 11
Congruence, developables of 11
Congruence, direction equation of 13 153
Congruence, focal points of 11
Congruence, focal surfaces of 11
Congruence, harmonic to a net 22 24 25 29 80 81 153
Congruence, harmonic to a point net 28
Congruence, harmonic, of a transformation F 40 153
Congruence, middle surface 16 21 32
Congruence, normal to a net 168
Congruence, parallel 15
Congruences 2, I, normal 159 294
Congruences I and p, I, definition 158
Congruences of circles, a cyclic system 272 274 278 285
Congruences of circles, characteristic property 243
Congruences of circles, definition 241
Congruences of circles, envelope of planes of 258 259 269 270 271 274
Congruences of circles, focal points of 243
Congruences of circles, focal spheres of 241
Congruences of circles, in cartesian coordinates 243
Congruences of circles, parallel transformations of 272
Congruences of circles, transformations F of 269 270
Congruences of spheres and circles, conjugate 261—263 270
Congruences of spheres and circles, harmonic 255—259 264 265 267 269—271
Congruences of spheres and circles, orthogonal 266 267 285
Congruences of spheres, chords of contact of 239
Congruences of spheres, cutting a fixed sphere under constant angle 283
Congruences of spheres, definition 238
Congruences of spheres, derived 240 243
Congruences of spheres, focal circles of 239 241
Congruences of spheres, in cartesian coordinates 243
Congruences of spheres, null-spheres 243 284
Congruences of spheres, principal curves of 238
Congruences of spheres, with applicable central nets 271
Congruences R of spheres, definition 246
Congruences R of spheres, direction-equation with equal invariants 255
Congruences R of spheres, parallel 246
Congruences R of spheres, permanent in deformation 298 308 310
Congruences R of spheres, representation in 5-space 285
Congruences R of spheres, transformations F of 246
Congruences, nets conjugate to 159 168
Conies, pencil of 95
Coordinates, Cartesian 1
Coordinates, homogeneous point 69
Coordinates, of a sphere 233
Coordinates, pentaspherical see Pentaspherical
Coordinates, tangential 97 166 286
Cosserat 206
Curve, definition 2 71
Curve, parametric 2
Cyclic congruences, characteristic property 280 285
Cyclic congruences, definition 183
Cyclic congruences, generation of 289
Cyclic congruences, multiply 185 188 191
Cyclic congruences, parallel 191
Cyclic systems, circles pass through a point 191
Cyclic systems, definition 183
Cyclic systems, envelope of planes of 199 213 285 286
Cyclic systems, generation of 272—274 285 289
Cyclic systems, of a transformation R 197 206 285
Cyclic systems, representation in 5-space 277—279
Cyclic systems, transformations F of 275 277 278 290
Cyclides 283
Darboux 6 92 147 154 208 212 213 239 254 274 281 282 283 285 288 291 295 313 317 358
Deform of a quadric, which meets the circle at infinity in 4 points 291
Deform of revolution 293—296 305 360
Deform permanent nets on 326—329 353
Deform R nets on 327 328
Deform tangent to the circle at infinity 317
Deform transformations  of 340—352 354—356
Deform transformations  of 334—340 346 353 354
Deform transformations H of 366 371 372
Demoulin 107 108 112 113 115 119 124 134 150 202 204 220 225 229 254 284
Derivant net, definition 25
Derivant net, of two transformations F 54
Derivant net, of two transformations K 61
Derivant net, of two transformations R 265
Derived congruences, definition 21
Derived congruences, of a W congruence 108
Derived congruences, of spheres 240 241
Derived net, definition 25
Derived net, Laplace transforms of a 32 33
Derived net, of two transformations F 54
Derived net, parallel 33 53
Derived net, reciprocally 116
Derived net, transformations F of 54 66
Derived net, which is O 289
Developable surface, definition 10
Developable surface, of a congruence 11
Dini 362
Direction-parameters, of a congruence 12 14 20 30
Direction-parameters, of a cyclic congruence 183 281
Direction-parameters, of a line 1
Drach 155
Envelope of rolling, definition 305
| Envelope of rolling, lines of curvature 320
Envelopes of spheres, conformal mapping of the two sheets 206 286
Envelopes of spheres, congruence of chords of contact 239 241 242 244 245 258 259 285
Envelopes of spheres, cutting a fixed sphere under constant angle 283
Envelopes of spheres, principal curves on 238 239 243
Envelopes of spheres, with applicable central nets 272 274 298 302 307 308 310
Envelopes of spheres, with O nets corresponding 174 189 229
Envelopes of spheres, with spherical lines of curvature 285
Equation, direction, of a congruence 13 153
Equation, point, of a net 3 72
Equation, tangential, of a net 128 153
Focal circles of a congruence of spheres 239 241
Focal nets in relation F 34
Focal nets of a congruence 11 15
Focal nets of parallel congruences 21
Focal nets radial transforms 33
Focal nets, coordinates of 10 14 29
Focal nets, of a congruence of circles 245
Focal nets, of a ray congruence 124
Focal points, of first and second rank 11
Focal spheres of a congruence of circles 241
Focal spheres, meeting orthogonally 183
Focal surfaces of a congruence 11 15
Green 124 125 126
Guichard 5 12 26 - 156 165 200 255 258 262 263 266 280 285 295 296 337
Hammond 77 85 87 93 94
Homogeneous point coordinates 69
Homogeneous point coordinates, of a net in 2-space 96
Homogeneous point coordinates, of a net in 3-space 101 103
Homogeneous point coordinates, of an R net 107
Hyperplane at infinity 69
Hyperplane, definition 2
Hyperquadric, net on 90 254
Hyperquadric, transformations F of nets on 90 94
Hyperspheres, focal 183 185
Hyperspheres, in transformations R 171 174 189
Invariants equal to zero 73 92
Invariants in a sequence of Laplace 75 92
Invariants of an equation of Laplace, definition 16
Inversion, in pentaspherical coordinates 281
Inversion, in transformations M 175
Inversion, of a cyclide 283
Inversion, of a sphere 281
Isothermal-conjugate nets, and nets with equal point invariants 125 150
Isothermal-conjugate nets, characteristic property 126
Isothermal-conjugate nets, definition 104
Isothermal-conjugate nets, on a deform of a quadric 326
Isothermal-conjugate nets, on a quadric 112 367 372
Isothermal-conjugate nets, preserved by projective transformation 104
Isothermal-conjugate nets, R nets are 107
Isothermic O nets, are nets  231
Isothermic O nets, definition 206
Isothermic O nets, in pentaspherical coordinates 237
Isothermic O nets, Lie transformation of 284
Isothermic O nets, nets with the same spherical representation as 302—304 318 319
Isothermic O nets, special 212—214 291
Isothermic O nets, spherical representation of 319. See Transformations and Chiristoffel transformations
Isotropic, line 289 291 296
Isotropic, plane 289 296
Isotropic, relation 157
Jonas 42 47 110 113 119 201
Kinematically conjugate directions, auto-conjugate 815 320
Kinematically conjugate directions, axes of rotation 315
Kinematically conjugate directions, definition 314
Kinematically conjugate directions, for congruences of rolling 317 321
Kinematically conjugate directions, orthogonal 320
Kinematically conjugate directions, parametric 320
Koenigs 58 123
Laplace equations 3 12
Laplace equations, invariants of 16
Laplace equations, Moutard form 237
Laplace equations, sequence of 74
Laplace transforms, definition 17
Laplace transforms, in cartesian coordinates 17
Laplace transforms, in homogeneous coordinates 72
Laplace transforms, in tangential coordinates 129
Laplace transforms, of a derived net 32 33
Laplace transforms, of a function 19 32
Laplace transforms, of a net R 108
Laplace transforms, of a radial transform 33
Laplace transforms, of parallel nets 18
Laplace transforms, transformations F of 66 94
Laplace transforms, which are O nets 192 193 286
Levy sequences, of higher order 83—85 93
Levy sequences, of the first order 81 93
Levy sequences, periodic 85 86
Levy transforms, and transformations F 66
Levy transforms, definition 19
Levy transforms, in a hyperplane 81
Levy transforms, in homogeneous coordinates 81
Levy transforms, of parallel nets 32
Levy transforms, of radial transforms 33
Lie 249 283 284
Lie line-sphere transformation, equations of 249 283
Lie line-sphere transformation, linear complex of 249 284
Lie transformations of surfaces, definition 252
Lie transformations of surfaces, equations of 283 284
Lie transformations of surfaces, W congruence of 253
Linear complex, definition 248
Linear complex, lines conjugate with respect to 250 251 284
Linear complex, null-plane of 248 251 252
Linear complex, special 248
Linear complex, tangents to a curve lines of a 284
Martin 32
Minimal surfaces, deformable transformations R of 304 305
Minimal surfaces, focal sheets of a W congruence 230
Minimal surfaces, in relation R 296 308
Minimal surfaces, O net of 229
Minimal surfaces, transformations of 215 217
Nets , and nets R 254
Nets , definition 218
Nets , equations of 219 220
Nets , isothermic 231
Nets , on surfaces of Guichard 225 232
Nets , parallel 221
Nets , transformations R of 221—224
Nets , with isothermal spherical representation 231
Nets 2, O, applicable 298
Nets 2, O, as central nets of transformations R 171 189
Nets 2, O, associated with cyclic systems 279
Nets 2, O, definition 156
Nets 2, O, on quadrics of revolution 247
Nets 2, O, special 188 228
Nets 2, O, transformations F of 189—191
Nets C, congruences harmonic to 287 288.
Nets C, definition 200
Nets C, harmonic to a normal congruence 200
Nets C, on the envelope of planes of a cyclic system 200
Nets corresponding with orthogonality of linear elements 63 146
Nets corresponding with orthogonality of linear elements, parallel 64
Nets corresponding with orthogonality of linear elements, transformations F of 64—66
Nets G, definition 192
Nets G, Laplace transforms of 192 286
Nets G, parallel 192
Nets G, radial transforms of 193
Nets G, transformations F of 193
Nets O, congruences conjugate to 168
Nets O, conjugate to congruences I 2 1 1 160
Nets O, definition 156 194
Nets O, derived 289. See Isothermic O nets
Nets O, determination of 165—167
Nets O, generated by a rolling surface 289 291 294
Nets O, in 5-space and cyclic systems 277
Nets O, in pentaspherical coordinates 237 279
Nets O, inversion of 175
Nets O, on a hypercone 191 283
Nets O, on a hypersphere 167
Nets O, parallel 167
Nets O, radial transformations of 175
Nets O, transformations F of 168—175 185—188
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