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| Eisenhart L.P. — Transformations of surfaces |
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| Предметный указатель |
Nets O, transformations R of 174—182 195—197
Nets O, with isothermal spherical representation 215—218 229
Nets p, O, conjugate to congruences, p, I 160
Nets p, O, definition 156
Nets p, O, parallel 191
Nets permanent in deformation, definition 139
Nets permanent in deformation, fundamental coefficients 139—140
Nets permanent in deformation, special 142
Nets permanent in deformation, spherical representation of 154 155
Nets permanent in deformation, transformations  of 141—146
Nets R, and nets 254
Nets R, definition 107Nets R
Nets R, equations 107
Nets R, in line coordinates 255
Nets R, Laplace transforms of 108
Nets R, on a quadric 112 367
Nets R, on deforms of a quadric 327 328
Nets R, reciprocally derived 118
Nets R, transformations F of 124 125
Nets R, transformations W of 115—123 125
Nets R, with tangents in one family meeting a fixed line 284
Nets with equal point invariants 16 32 67 68 95 106 125
Nets with equal point invariants, and equal tangential invariants 150
Nets with equal point invariants, associate 59 67
Nets with equal point invariants, characteristic property 124
Nets with equal point invariants, on a quadric 366—368. See Transformations K
Nets with equal point invariants, planar 123
Nets with equal point invariants, which are O nets 206
Nets with equal tangential invariants, and equal point invariants 150
Nets with equal tangential invariants, characteristic property 153
Nets with equal tangential invariants, definition 128
Nets with equal tangential invariants, on a quadric 366—368. See Transformations
Nets with equal tangential invariants, which are permanent in deformation 139
Nets with same spherical representation as isothermic O nets 302 303
Nets with same spherical representation as isothermic O nets, transformations R of 303 319 320
Nets, characteristic property 4
Nets, conjugate to a congruence 12 14 19 32 159
Nets, definition 3
Nets, harmonic to a congruence 22 24 27 30 34 200
Nets, in 3-space 6 100—103 105
Nets, in homogeneous coordinates 72
Nets, in relation O 63 146 154
Nets, in tangential coordinates 127 128
Nets, minimal 32 66
Nets, of plane curves 152. See Applicable nets Derivant Derived Isothermal-conjugate
Nets, of translation 32 66
Nets, parallel 7—12 124
Nets, periodic 76 94
Nets, planar 9 96 123
Nets, point 5 28 30
Nets, tangent planes of 4
Normal parameters of a net 5
Orthogonal determinant, definition 163
Orthogonal determinant, in 3-space 194
Orthogonal determinant, in 5-space 165 279
Osculating planes of a net 11
Osculating planes of a net, and Laplace transforms of a derived net 32
Osculating planes of a net, and nets  286
Osculating planes of a net, conjugate to a congruence 23
Osculating planes of a net, in homogeneous coordinates 71
Parallel, congruences 15 28
Parallel, hyperplanes 2
Parallel, lines 1
Parallel, nets 7—12 32
Pentaspherical coordinates, angle between displacements in 282
Pentaspherical coordinates, equation of a sphere in 236 281
Pentaspherical coordinates, equation of a surface in 283
Pentaspherical coordinates, fundamental spheres 235 281
Pentaspherical coordinates, of a point 235
Pentaspherical coordinates, of a point of the plane at infinity 281
Pentaspherical coordinates, of an isothermic net 237
Pentaspherical coordinates, of an O net 237 279
Pentaspherical coordinates, special 237
Pentaspherical coordinates, transformations R in 283
Perpendicular, hyperplanes 2
Perpendicular, lines 1
Plane, in cartesian coordinates 2
Plane, in homogeneous coordinates 71
Plane, tangent, of a net 4
Plane, tangent, of a surface 3
Pluecker line coordinates 247
Point equation of a net, in cartesian coordinates 3
Point equation of a net, in homogeneous coordinates 72
Point nets, congruences harmonic to 28 29
Point nets, definition 5
Point nets, harmonic to a congruence 30
Polar transformation 100 125 153
Poles of a circle, definition 266
Poles of a circle, points on an envelope of spheres 266 274
Poles of a cyclic system 285 286
Principal curves form a net 245
Principal curves on an envelope of spheres 238 239 243
Problem of order n—2 165
Projective transformations, of asymptotic lines 100
Projective transformations, of isothermal-conjugate nets 104
Projective transformations, of nets in 2-space 97
Projective transformations, of nets in 3-space 103 128
Projective transformations, of nets in relations F 93
Projective transformations, of space 70
Quadrics, conjugate in deformation 365 371
Quadrics, deformation of 291 293 305 326 360
Quadrics, isothermal-conjugate nets on 112 366—370 372
Quadrics, nets R on 112 367
Quadrics, nets with equal point invariants on 367 368
Quadrics, nets with equal tangential invariants on 367 368
Quadrics, of revolution 247 293—296 305 337 360
Quadrics, osculating 124
Quadrics, permanent nets on 323—326 329—334 353 370 372
Quadrics, referred to generators 323—325
Quadrics, rolling 291—296 317
Quadrics, tangent to the circle at infinity 317
Quadrics, transformations F of 322 329—334 352 353 368
Quadrics, transformations W of 368—370 372
Quatern of transformations F, definition 48
Radial transformations, and transformations F 40—42 67 153
Radial transformations, definition 31
Radial transformations, harmonic congruences of 31
Radial transformations, of every net on a surface 124
Radial transformations, of Laplace transforms 33
Radial transformations, of Levy transforms 33
Radial transformations, of nets 193
Radial transformations, of nets O 175
Ray congruence, definition 105
Ray congruence, focal points 124
Ray curves, definition 105
Ray curves, property of 124
Relation, F 34
Relation, isotropic 157
Relation, K 58
Relation, O 63 146
Relation, R 174
Ribaucour 16 89 174 239 245 285 286 312 323
Rolling surfaces, and cyclic systems 289
Rolling surfaces, axes of rotation 315
Rolling surfaces, definition 288
Rolling surfaces, kinematically conjugate directions on 314 320
Rolling surfaces, on a quadric 291—296 317
Rolling surfaces, on a sphere 317
Rolling surfaces, on a surface of revolution 293—298 321
Rolling surfaces, plane of contact 288
Rolling surfaces, ruled 314 315
Rolling surfaces, spheres associated with 318
Ruled surfaces, applicable 320
Ruled surfaces, applicable to a non-ruled surface 314 316
Ruled surfaces, infinitesimal deformation of 372
Ruled surfaces, osculating quadrics of 124
Ruled surfaces, referred to asymptotic lines 123
Ruled surfaces, rolling 314
Ruled surfaces, W congruences whose focal surfaces are 372
Segre 372
| Sequences of Laplace, definition 74
Sequences of Laplace, derived nets of 83—85 93
Sequences of Laplace, equations of 75 76 92
Sequences of Laplace, invariants of 75 76 92
Sequences of Laplace, periodic 76—80 93 94
Sequences of Laplace, transformations F of 94
Servant 325 359 362 363 365 372
Spheres, coordinates of 233
Spheres, equations in pentapherical coordinates 236
Spheres, inversion of 281
Spheres, null- 234 289 294
Spheres, point- 234
Spheres, power with respect to 234
Spheres, rolling 317. See Congruences of spheres
Stetson 286
Surface of rolling, definition 301
Surface of rolling, equations of 313
Surface of rolling, lines of curvature on 320
Surface of rolling, one sheet of envelope of congruence R a 302
Surface, definition 2
Surface, determination of asymptotic lines on 356—358
Surface, determination of nets on 5—7
Surface, element 252
Surface, of Voss 143
Surface, R 111
Surface, tangent planes 3
Surfaces applicable to a quadric see Deforms of a quadric
Surfaces applicable to a surface of revolution 293—298 321
Surfaces applicable to a surface of revolution, complementary 292 294—296 321
Surfaces applicable to a surface of revolution, conjugate in deformation 371
Surfaces conjugate in deformation, characteristic property 362
Surfaces conjugate in deformation, definition 360
Surfaces conjugate in deformation, determination of 362—365 371
Surfaces conjugate in deformation, quadrics 365 372
Surfaces of constant mean curvature, cyclic systems associated with 286
Surfaces of constant mean curvature, determination of 360
Surfaces of constant mean curvature, generation of 294 317
Surfaces of constant mean curvature, kinematically conjugate lines on 320
Surfaces of constant mean curvature, special isothermic nets on 230
Surfaces of constant mean curvature, transformations R of 229 232 295 310 335
Surfaces of constant total curvature, equations of 232
Surfaces of Guichard, are surfaces 225 232
Surfaces of Guichard, associated surfaces 225 228 232
Surfaces of Guichard, definition 225 232
Surfaces of Guichard, isothermic surfaces associated with 286
Surfaces of Guichard, transformations R of 227 231 232 28G
Tangential coordinates, Laplace transforms in 129
Tangential coordinates, of a net 127
Tangential coordinates, of a net O 166
Tangential coordinates, of a surface, definition 98
Tangential coordinates, referred to asymptotic lines 100
Tangential coordinates, transformations F in 129—134 153
Tangential coordinates, transformations R in 178—182
Terracini 370
Theorem of permutability, extended 55—57 68 224 230
Theorem of permutability, for parallel nets 50 67
Theorem of permutability, of transformations F 45—51 90 132
Theorem of permutability, of W congruences 124 138 143.
Tortorici 372
Transformation,  , definition 113
Transformation, , of isothermal systems on a sphere 124
Transformation, , of surfaces of constant curvature 124
Transformations  of surfaces applicable to a quadric 340—346 354—356
Transformations , permutability with transformations  346 371
Transformations , theorem of permutability of 352 371
Transformations  208 209 231
Transformations , and surfaces of rolling 301 302
Transformations , characteristic property 285
Transformations , deformable 304
Transformations , generation of 291
Transformations , induced 286 319
Transformations , isothermic nets determined by 230
Transformations , of minimal surfaces 215 304
Transformations , of special isothermic surfaces 212—214 230 291
Transformations , parallel nets of 230
Transformations , theorem of permutability of 209—211 215 230
Transformations  215 — 217 231
Transformations , and envelopes of rolling 305
Transformations , characteristic property 230
Transformations , deformable 305
Transformations , theorem of permutability of 218 230
Transformations , of nets on a quadric 329—334 353
Transformations , of surfaces applicable to a quadric 334—339 354
Transformations , theorem of pemiutability of 323 339 340
Transformations , in point coordinates 146—149
Transformations , in tangential coordinates 134—136
Transformations , of a focal surface of a W congruence 149 150
Transformations , of nets permanent in deformation 141—146
Transformations , of nets with equal point and equal tangential invariants 150—152
Transformations , polar transformation of 153
Transformations , theorem of permutability of 137 138 153 154
Transformations F, and polar transformations 153
Transformations F, and radial transformations 40—42 67 153
Transformations F, conjugate congruence of 34 42 89
Transformations F, equations in cartesian coordinates 34—37
Transformations F, harmonic congruence of 40 182
Transformations F, in homogeneous coordinates 87—91
Transformations F, in tangential coordinates 129—134 153
Transformations F, inverse of 37
Transformations F, of applicable nets 61—63
Transformations F, of congruences of circles 269—271
Transformations F, of congruences R of spheres 246
Transformations F, of cyclic systems 274—278 290
Transformations F, of deformable transformations R 306
Transformations F, of derived nets 54 66
Transformations F, of isothermal-conjugate nets on a quadric 368 372.
Transformations F, of Laplace transforms 66 94
Transformations F, of Levy transforms 66
Transformations F, of nets 2, O 189—191
Transformations F, of nets G 193
Transformations F, of nets in relation O 64 146 154
Transformations F, of nets O 168—175 185—188
Transformations F, of nets on a hyperquadric 90 94
Transformations F, of nets on a quadric 322 323
Transformations F, of nets on a sphere 197 217
Transformations F, of nets R 124 125
Transformations F, of periodic Laplace sequences 94
Transformations F, parallel 38 50 67
Transformations F, projective transformations of 93
Transformations F, theorem of permutability of 45—51 55—57 67 90 132
Transformations F, triads of 44 90
Transformations H of ruled deforms 366
Transformations H of surfaces applicable to a quadric, definition 366
Transformations H, permutability with transformations 372
Transformations H, permutability with transformations 366 372
Transformations K, characteristic property 95
Transformations K, definition 58
Transformations K, in cartesian coordinates 57—59 67
Transformations K, in homogeneous coordinates 94 95
Transformations K, of associate surfaces 68 124
Transformations K, of Levy transforms 68
Transformations K, of nets on a quadric 368—370 372.
Transformations K, of nets with equal point and equal tangential invariants 150—152
Transformations K, polar transformations of 153
Transformations K, theorem of permutability of 60 61 67 68 154 372
Transformations K, with normal conjugate congruences 206 221
Transformations R, and envelopes of rolling 305
Transformations R, and surfaces of rolling 302 318
Transformations R, central nets 174 189 196 228
Transformations R, circle-planes of see Circle-planes
Transformations R, circles K of see Circles K
Transformations R, conformal 206
Transformations R, congruences K of see Congruences K
Transformations R, cyclic systems of 197 206 285
Transformations R, deformable of the first type 296 308 310
Transformations R, deformable of the second type 298—306
Transformations R, in cartesian coordinates 173—177 195
Transformations R, in pentaspherical coordinates 283
Transformations R, in tangential coordinates 178 181
Transformations R, inverse of 178
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