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| Forsyth A.R. — Theory of differential equations. Part 4. Partial differential equations (Vol. 6) |
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| Предметный указатель |
Finite rank, linear equations of the second order having integrals of vi 64
First class of equations of the second order, after Ampere vi 16.
First method is a generalisation of Hamilton's results in theoretical dynamics v 382
First method, how modified by assignment of initial conditions v 387 390
First method, Jacobi's v 371 380
First method, statement of general process v 386
First method, when the dependent variable occurs v 391.
First order, any system of partial equations can be changed so as to contain only equations of the v 8
First order, are included in those of second order vi 395
First order, Cauchy's theorem for a single irreducible equation of the v 27 33 35 36.
First order, characteristics of, possessed by equations of the second order vi 394
First order, connected with intermediate integrals vi 401.
First order, geometrical interpretation of vi 397
FOURIER vi 109.
Fredholm vi 582.
Frobenius vi 73.
Functions, group of see Group of functions.
Fundamental system of integrals of a complete system of homogeneous linear equations v 86
Fundamental system of integrals of a complete system of homogeneous linear equations can be used to express any integral v 87.
General integral and singular integral, relations between v 254 255
General integral in hyperspaee derived from characteristics v 2S9
General integral of equations of the second order, as defined by Ampere vi 4 8
General integral of equations of the second order, as defined by Darboux, after Cauchy vi 5
General integral of equations of the second order, character of vi 13
General integral of equations of the second order, characteristic properties of the arbitrary elements in vi 21
General integral of equations of the second order, comparison of two definitions of vi 6—8
General integral of equations of the second order, number of arbitrary functions in vi 22.
General integral of system of equations v 195
General integral, as related to the equations of the characteristic v 211 212 215
General integral, classes of v 168 171
General integral, contact of, with singular integral v 310
General integral, deduced from complete integral by variation of parameters v 164
General integral, derived through contact transformations v 327 831 334 339.
General integral, how related to Cauchy's integral v 218
General integral, of homogeneous linear equation is completely comprehensive v 57
General integral, of non-homogeneous linear equation is not completely comprehensive v 65 68
General integral, range of, in the case when a non-homogeneous equation has been made homogeneous v 71
General integral, the most comprehensive class of v 169 171
General method for constructing intermediate integrals (if any) of an equation of the second order vi 220
General method for constructing intermediate integrals (if any) of an equation of the second order, applied to the Monge — Ampere equations vi 226
General method for constructing intermediate integrals (if any) of an equation of the second order, subsidiary equations in, coincide with Boole's vi 227
General method for constructing intermediate integrals (if any) of an equation of the second order, when based upon Darboux's method vi 314.
General order, Cauchy's theorem for integrals of systems of equations of v 43
General order, equations of, in two independent variables vi 487.
General order, limitation upon the form of equations of, and its importance v 48
Generalisation in case of Laplace's linear equation vi 379.
Generalisation of integrals of equations of the second order vi 361
Generalisation of intermediate integrals vi 377
Generalised form of Cauchy's theorem for equations of the first order v 33 36
Generalised form of Cauchy's theorem for equations of the second order v 42.
Geodesies as characteristics, equations having v 248.
Geometry of space and relation between different kinds of integrals v 186
Geometry of space and relation between different kinds of integrals, illustrated by means of the characteristics v 205
Geometry of space and relation between different kinds of the various integrals of an equation of the first order v 224
Goursat v 26 55 72 100 164 180 205 223 243 248 314 vi 7 27 39 91 94 129 159 198 261 301 303 328 333 334 344 388 397 418 424 425 434 441 454 455.
Goursat's theorem on primitive of equation of second order to be deduced from intermediate integral vi 406.
Graindorge v 370 397.
Group of functions v 314
Group of functions, applied to integrate a system of equations v 367.
Group of functions, canonical form of v 355
Group of functions, connected with complete Jacobian system of equations v 347 349
Group of functions, definition of, as applied to partial equations v 345
Group of functions, group reciprocal to, or polar of v 349
Group of functions, highest order, and construction of, a sub-group in involution v 364
Group of functions, how affected by contact transformation v 346
Group of functions, limit to order when group is in involution v 346 349
Group of functions, order, sub-group, involution, defined v 345
Group of functions, properties of indicial functions of v 350
Group of functions, relation between order of, and number of indicial functions v 355 359 360 366
Group of functions, two invariants of, under contact transformation v 364
Group of functions, when in canonical form, can be amplified into another group v 361
Guichard vi 130.
Hamburger v 407 408 428 455 474 vi 303 336 456.
Hamburger's method of constructing equations compatible with an equation of the second order vi 336
Hamburger's method of constructing equations compatible with an equation of the second order, applied to equations of the third order in two independent-variables vi 482.
Hamburger's method of constructing equations compatible with an equation of the second order, subsidiary system of equations in, compared with those in Darboux's method vi 338
Hamburger's systems of simultaneous equations, applied to the case with any number of dependent variables v 442
Hamburger's systems of simultaneous equations, applied to the special case of two dependent variables v 435
Hamburger's systems of simultaneous equations, general result v 459
Hamburger's systems of simultaneous equations, generalisation of Jacobi's process not generally effective for v 474.
Hamburger's systems of simultaneous equations, integrable also by means of partial equations v 449
Hamburger's systems of simultaneous equations, special method for v 467
Hamburger's systems of simultaneous equations, the method limited to the case of two independent variables v 430
Hamburger's systems of simultaneous equations, transformed so as to be linear equations in an increased number of variables v 456
Hamburger's systems of simultaneous equations, when linear v 42S
Hamburger's systems of simultaneous equations, when non-linear v 458
Hamburger's systems of simultaneous equations, with examples v 439
Hamilton v 370.
Hamilton's characteristic equations in dynamics v 371 376 381.
Hamilton's theorem on integrals of a dynamical system v 379
Hamilton's theorem on integrals of a dynamical system is the basis of Jacobi's first method v 380 382.
Harmonic equations and their integrals vi 157.
hilbert v 230.
Hill, M.J.M. v 249.
Homogeneous contact transformations v 323.
Homogeneous linear equations v 56
Homogeneous linear equations, Cauchy's integral of v 58
Homogeneous linear equations, most general integral of v 57
Homogeneous linear equations, number of independent integrals of v 57
Homogeneous linear equations, systems of v 76
Hyperbolic case of linear equations of the second order vi 44 see
Imschenetsky v 100 164 370 vi 1 8 10 21 46 68 199 237 266 361 376.
Imschenetsky's generalisation of sub-complete integrals of Monge — Ampere equations vi 366
Imschenetsky's generalisation of sub-complete integrals of Monge — Ampere equations and is a contact-transformation vi 382.
Imschenetsky's generalisation of sub-complete integrals of Monge — Ampere equations, applied to Laplace's linear equations vi 379
Independence of linear equations in a homogeneous complete system v 77.
Independent integrals, of homogeneous linear equation v 57
Independent integrals, of system in involution, number of v 122.
Independent integrals, of system of homogeneous linear equations v 83
Indicial functions of a group v 350
Indicial functions of a group, other properties of v 352 354
Indicial functions of a group, relation between number of, and the order of the group v 355 366.
Indicial functions of a group, their number is invariantive under contact transformations v 351
Infinitesimal contact transformations v 317
Infinitesimal contact transformations, determination of all, is equivalent to integrating an equation v 324
Infinitesimal contact transformations, determined by energy of a dynamical system v 405
Infinitesimal contact transformations, form of v 318
Infinitesimal contact transformations, significance of, leads to Bertrand's theorem on canonical constants v 405.
Infinitesimal contact transformations, which do not involve the dependent variable v 322
Infinitesimal transformation, invariant equation for v 73.
Infinitesimal transformation, invariant for v 72
Integrability, conditions of, of a single differential expression v 101
Integrability, conditions of, of a system of simultaneous equations v 103.
Integrable equations, and examples v 425.
Integrable equations, conditions for v 416
Integrable equations, construction of integral equivalent of v 418
Integrable equations, kinds of integrals of v 419
Integrable equations, Koenig's completely v 411
Integrable equations, with general result v 424
Integrable equations, with various systems v 416
Integral curves v 238
Integral curves can always be obtained as an edge of regression v 239
Integral curves, equations of v 239.
Integrals of an equation of first order, complete, general, singular, special, exceptional see under these titles respectively
Integrals of an equation of first order, deduced by method of characteristics v 210 214 288—292
Integrals of an equation of first order, different kinds of, and relations between v 164et
Integrals of an equation of first order, of a complete system, classes of v 193
Integrals of an equation of first order, particular kinds v 171
Integrals of an equation of first order, relations of different, to one another v 297
Intermediate integrals of equations of order higher than the first vi 8
Intermediate integrals of equations of order higher than the first, general, and complete vi 10
Intermediate integrals of equations of order higher than the first, not necessarily possessed vi 10.
Intermediate integrals of equations of second order and the characteristics vi 401
Intermediate integrals of equations of second order and the characteristics, can lead to primitive vi 406
Intermediate integrals of equations of second order and the characteristics, general theory of vi 403
Intermediate integrals of equations of second order and the characteristics, various cases and examples vi 409
Intermediate integrals of equations of the second order in any number of independent variables vi 490
Intermediate integrals, Ampere's theorem on integration of vi 248
Intermediate integrals, and for Boole's argument vi 208
Intermediate integrals, assumption of particular type of, necessary for Monge's argument vi 203
| Intermediate integrals, Boole's method of obtaining vi 210
Intermediate integrals, construction of, after Darboux's method vi 314
Intermediate integrals, equations of the second order and the Monge — Ampere type which possess vi 200
Intermediate integrals, equations of the second order possessing two, are reducible by contact transformations to s = 0 vi 295
Intermediate integrals, general method for see general method
Intermediate integrals, general theory of vi 470
Intermediate integrals, generalised by variation of parameters vi 377.
Intermediate integrals, Monge's method of obtaining vi 201
Intermediate integrals, of equations of the third order in two independent variables vi 457
Intermediate integrals, simultaneous, can exist vi 205 227
Invariant, and invariant equation, for infinitesimal transformation v 72 73.
Invariant, characteristic vi 532 see
Invariants of an equation of the second order, significance of, when equal to one another vi 131
Invariants of linear equation of the second order vi 44
Invariants of linear equation of the second order of equations arising through Laplace-transformations vi 52
Invariants of linear equation of the second order, Darboux's expressions for successive vi 83.
Invariants of linear equation of the second order, used to construct canonical forms of the equation vi 47
Invariants of linear equation of the second order, when they vanish for a transformed equation, the original equation can be integrated vi 56
Invariants of linear equation of the second order, when they vanish, the equation can be integrated by quadratures vi 46
Invariants of parabolic linear equations of the second order vi 98
Invariants of parabolic linear equations of the second order, effect of their vanishing upon the form of the equations vi 100—102.
Involution, equations of the second order in vi 330.
Involution, function in, with a group v 345
Involution, highest order of a sub-group in v 365.
Involution, limit to order of a system in v 346 349
Involution, number of independent integrals of v 122.
Involution, system of functions in v 345
Involution, systems in v 82 120
Irreducibility, significance of, for equations of first order v 33.
Irreducible differential expressions vi 60 73.
jacobi v 100 113 137 157 370 380 382 397 407 417 432 vi 302.
Jacobi — Hamiltonian method v 371
Jacobi — Hamiltonian method, constructed by Jacobi on Hamilton's theorem on dynamical equations v 380 382
Jacobi — Hamiltonian method, general result stated v 386
Jacobi — Hamiltonian method, modification of, when the dependent variable occurs v 391.
Jacobi's method of integrating complete linear systems v 91
Jacobi's method of integrating complete linear systems is a method of successive reduction v 92.
Jacobi's methods of integrating equations of the first order see first method second
Jacobi's second method, applied to a single equation v 137
Jacobi's second method, as developed by Mayer v 117
Jacobian conditions of integrability, of a single differential expression v 101
Jacobian conditions of integrability, of a system of simultaneous equations v 103
Jacobian conditions of integrability, sufficient as well as necessary v 104
Jacobian process of combination of equations in one dependent variable not effective for equations in several dependent variables v 474
Jacobian process of combination of equations in one dependent variable not effective for equations in several dependent variables, form of v 476.
Jacobian system of equations of the first order see complete systems.
Jacobian system of linear equations v 82 see complete
JORDAN v 26 164.
Kapteyn vi 261.
Kinds of integrals of an equation of the first order v 164
Kinds of integrals of an equation of the first order, as connected with the characteristics v 210 214.
Kinds of integrals of an equation of the first order, geometrical illustration of v 186 see
Kinds of integrals of an equation of the first order, tests for v 178
Koenig v 408 411 vi 303 335.
Koenig's systems of completely intograble equations v 411
Koenig's systems of completely intograble equations, conditions to be satisfied by v 416
Koenig's systems of completely intograble equations, different kinds of integrals of v 419
Koenig's systems of completely intograble equations, integration of v 418
Koenig's systems of completely intograble equations, with general result v 424.
Koenigsberger v 407 419 425 428 439.
Kowalevsky v 11 26 48.
Lacroix v 157.
Lagrange v 131 164 370 vi 9 111 159 361.
Laplace vi 39.
Laplace transformations of linear equations vi 49
Laplace transformations of linear equations, applied to adjoint equations vi 114.
Laplace transformations of linear equations, as affecting integrals of finite rank vi 57 70
Laplace transformations of linear equations, Goursat's theorem on vi 91
Laplace transformations of linear equations, how affecting Levy transformations vi 96
Laplace transformations of linear equations, successive applications of vi 51
Laplace transformations of linear equations, the two are inverses of each other vi 50
Laplace's equation for potential vi 571
Laplace's equation for potential, and its relation to the Cauchy integral vi 578.
Laplace's equation for potential, integral of, provided by extension of Darboux's method vi 573
Laplace's equation for potential, Whittaker's integral of vi 576
Laplace's linear equation vi 160 297
Laplace's linear equation and by E.Liouville vi 384.
Laplace's linear equation, integral of, generalised by Imschenetsky, through variation of parameters vi 379
Laplace's method for linear equations of the second order vi 39
Legendre's equations for minimal surfaces vi 280.
Legendrian transformation of the dependent variable so as to construct a primitive v 127 131 217 292.
Levy vi 94 96.
Levy's transformation vi 94
Levy's transformation, how related to Laplace's transformations vi 96.
Lie v 137 157 205 244 248 314 370 vi 295 324 332 424.
Lie's classification of equations of the first order according to the characteristics v 244.
Lie's theorem that equations of the second order possessing two intermediate integrals can be changed into s = 0 by contact transformations vi 295.
Linear equations in any number of dependent variables v 442
Linear equations in the parabolic case see parabolic
Linear equations of the second order in three independent variables vi 520.
Linear equations of the second order subjected to Backlund transformations vi 441
Linear equations of the second order, can be integrated when any invariant of any transformed equation vanishes vi 56
Linear equations of the second order, canonical forms of vi 47
Linear equations of the second order, having integrals of finite rank see finite rank doubly
Linear equations of the second order, its invariants vi 44
Linear equations of the second order, Laplace's method for vi 39
Linear equations of the second order, reduced to one of two alternative forms vi 42
Linear equations of the second order, three cases, when variables are real vi -13
Linear equations of the second order, transformations of, in succession vi 49
Linear equations, can be integrated (when integral exists) by simultaneous systems of partial equations v 449.
Linear equations, complete systems of homogeneous v 76
Linear equations, in several dependent variables, Hamburger's system of v 428
Linear equations, in two dependent variables v 435
Linear equations, subsidiary equations for, with the critical algebraic equation v 430
Linear equations, that are homogeneous v 56
Linear equations, that are not homogeneous v 60
Linearly distinct integrals of a linear equation of the second order, Goursat's theorem on v 90.
Lines of curvature as characteristics, equations having v 246.
Liouville (J.) v 382 vi 143 160 194 197.
Liouville's equation of the second order vi 143 160 177 194 197.
Liouville, E. vi 69 111 119 384.
Lovett v 371.
Mansion v 100 164 205 220 371.
Mayer v 55 89 100 115 117 127 137 157 316 388 417.
Mayer's development of Jacobi's second method v 117
Mayer's development of Jacobi's second method, with use of Legendre's transformation v 127
Mayer's form of Lie's general theorem on contact transformations v 316.
Mayer's method of integrating complete linear systems v 89.
Minimal surfaces, equation of, integrals of, due to Legendre, Monge, Weierstrass vi 280.
Minimal surfaces, equation of, integrated by Ampere's method vi 277
Monge v 205 237 248 199 266 280 301 302 307.
Monge — Ampere equation generalised, when there are more than two independent variables vi 511.
Monge — Ampere equation of the second order vi 200 202 208 213 226 281 307 367 433
Monge — Ampere equation of the second order, characteristics of vi 393—395.
Monge — Ampere equation of the second order, construction of classes of vi 236 246 252
Monge's method for equations possessing an intermediate integral vi 201—208 215
Monge's method for equations possessing an intermediate integral compared with Boole's method vi 209 212
Monge's method for equations possessing an intermediate integral in practice is included in Ampere's method vi 201
Monge's method for equations possessing an intermediate integral, compared with Darboux's method vi 303.
Moray v 53.
Moutard vi 111 159 160.
Moutard's theorem on equations of the second order having integrals of explicit finite form vi 160
Moutard's theorem on equations of the second order having integrals of explicit finite form with a summary of results vi 195.
Moutard's theorem on equations of the second order having integrals of explicit finite form, Cosserat's proof of vi 161
Moutard's theorem on self-adjoint equations vi 139
Moutard's theorem on self-adjoint equations also the integrals of such equations vi 147.
Moutard's theorem on self-adjoint equations and their construction in successively increasing rank vi 141
Natani v 407 vi 456.
Non-homogeneous linear equation v 60
Non-homogeneous linear equation can be made homogeneous v 71.
Non-homogeneous linear equation, a general integral of v 62
Non-homogeneous linear equation, general theorem as to integral of v 67
Non-homogeneous linear equation, special integrals of v 65
Non-linear equations of the first order Chapter IV passim v 100
Number of arbitrary functions in Cauchy's theorem is same as order of the equation v 47.
Number of equations in a system must, in general, be the same as the number of dependent variables v 6.
Number of independent integrals, of homogeneous linear equation v 57
Number of independent integrals, of system of homogeneous linear equations v 83.
Number of independent variables, equations involving any general vi 527.
Number of quadratures in Mayer's method of integrating complete linear systems compared with the number in Jacobi's method v 94 95.
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