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| Forsyth A.R. — Theory of differential equations. Part 4. Partial differential equations (Vol. 6) |
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| Предметный указатель |
One integral common to subsidiary system for Monge — Ampere equations vi 253
One integral common to subsidiary system for Monge — Ampere equations, causes the intermediate integral to lead to two equations of the second order vi 258.
One integral common to subsidiary system for Monge — Ampere equations, conditions for vi 256 257
Order of contact of integral surfaces see contact.
Order of group of functions v 345
Order of group of functions, exceeds number of indicial functions by even integer v 355.
Order of group of functions, how limited, when the group is in involution v 346 349
Order of group of functions, unaffected by contact transformation v 346
Parabolic linear equations of the second order vi 44
Parabolic linear equations of the second order, cannot have a general integral in finite terms free from partial quadratures vi 103.
Parabolic linear equations of the second order, form of, when the invariants vanish vi 100—102
Parabolic linear equations of the second order, invariants of vi 98
Parameters for equations of the second order vi 361
Parameters for systems of equations v 419 464
Parameters, number of essential, in an integral of an equation v 192
Parameters, variation of v 165
Partial quadratures as affecting character of integrals vi 17
Partial quadratures, Ampere's test for equations having integrals free from vi 17
Partial quadratures, equations of the second order having integrals in finite form free from vi 159.
Partial quadratures, in Borel's expression for integrals of linear equations vi 106
Partial quadratures, integrals with or without, as determining classes of equations of the second order vi 16
Partial quadratures, must occur in general integral of parabolic linear equations of the second order, if in finite terms vi 103
Particular integrals of equations of the second order vi 8.
Pfaff's problem v 55.
Picard v 24.
poisson v 113 370 397.
Polar groups of functions v 349.
Potential, equation satisfied by, in free space vi 15 37 576.
primitive see complete integral general
Primitive of equations of the second order vi 8
Primitive of Monge — Ampkre equations, when the subsidiary system possesses three common integrals vi 232—235
Primitive of systems of equations v 410
Primitive, deduced from intermediate integral vi 406
Primitive, how constructed vi 284.
Primitive, how limited in use of Ampere's method vi 267
Primitive, kinds of v 419
Quadratures, character of integral, affected by vi 17 see
Quadratures, presence or absence of partial, as determining classes of integrals of equations of the second order vi 16
Raabe vi 10.
Rank (rang) of a differential expression or an integral vi 60
Rank (rang) of a differential expression or an integral, Goursat's theorem on vi 90.
Rank of a self-adjoint equation as affected by Moutard's substitution vi 141 145.
Real variables and reduced forms of linear equations of the second order vi 43.
Reciprocal groups of functions v 349
Reciprocal groups of functions, effect of contact transformations upon v 349
Reciprocal groups of functions, relations between v 353.
Reducibility of differential expressions, conditions for vi 61.
Regression, edge of, of integral surface v 237 see
Regular integrals of systems of equations of the first order, conditions for v 11 21
Regular integrals of systems of equations of the first order, for equations of any order v 43
Regular integrals of systems of equations of the first order, of single irreducible equation of first order v 27
Regular integrals of systems of equations of the first order, of single irreducible equation of the second order v 37 42
Regular integrals of systems of equations of the first order, of systems of linear equations v 88.
Regular integrals of systems of equations of the first order, uniqueness of v 11 18 20 22
Resolubility of characteristic invariant, effect of vi 550 553.
Riemann vi 111 120 527.
Riemann's use of adjoint equations vi 119
Riemann's use of adjoint equations, examples of vi 124.
Riquier v 23 53.
Routh v 370.
Schwartz, A. vi 42.
Schwarz, H.A., yi 360.
Second method of Jacobi, applied to the integration of a single equation v 137.
Second method of Jacobi, as developed by Mayer v 117
Second order, characteristics of, possessed by equations of the second order vi 395
Second order, equations of Cauchy's theorem for integrals of, in two independent variables v 37 42 vi 2
Second order, geometrical interpretation of vi 397.
Second order, having intermediate integrals vi 199
Second order, in any number of independent variables vi 490
Second order, include those of the first order vi 395
Selected edges of regression v 240
Selected edges of regression, contact of, with integral surface v 241.
Self-adjoint equations vi 131
Self-adjoint equations, construction of, when of finite rank vi 133
Self-adjoint equations, examples of vi 152
Self-adjoint equations, integrals of, when of finite rank vi 136 147
Self-adjoint equations, Moutard's theorem on vi 139
Self-conjugate characteristics v 251.
Sersawy vi 82 156 303 490.
Several dependent variables, simultaneous equations in v 407.
Simultaneous equations of the first order in several variables v 407
Simultaneous equations of the first order in several variables as affected by Backlund transformations vi 450 452.
Simultaneous equations of the first order in several variables as derived from primitives v 409
Simultaneous equations of the first order in several variables in general lead to equations of higher order in one dependent variable v 477
Simultaneous equations of the first order in several variables with general result v 424
Simultaneous equations of the first order in several variables, different kinds of integrals of v 419
Simultaneous equations of the first order in several variables, Hamburger's linear systems of v 428
Simultaneous equations of the first order in several variables, Hamburger's non-linear systems of v 456
Simultaneous equations of the second order vi 325
Simultaneous equations of the second order, integration of vi 327
Simultaneous equations of the second order, of order higher than second vi 353.
Simultaneous equations of the second order, systems of, in involution vi 330
Simultaneous equations of the second order, Valyi's process for, included in Darboux's method vi 328
Simultaneous integrals of different resolutions of a subsidiary system are compatible, and in Hamburger's method vi 339.
Simultaneous integrals of different resolutions of a subsidiary system are compatible, in Darboux's method vi 315 339
Simultaneous integrals of different resolutions of a subsidiary system are compatible, in Monge's method vi 205 227
Singular integral for equations of the second order vi 3 9.
Singular integral of an equation of the first order can be deduced from the differential equation v 182
Singular integral of an equation of the first order is the exception for the equations of the characteristic v 212 217
Singular integral of an equation of the first order of system of equations v 196
Singular integral of an equation of the first order, additional limitations and conditions for existence of v 184
Singular integral of an equation of the first order, but does not occur usually when the dependent variable is explicitly absent from the equation v 331 338.
Singular integral of an equation of the first order, can be affected by changing the form of the equation v 178
Singular integral of an equation of the first order, conditions for v 111 171
Singular integral of an equation of the first order, derived through contact transformations v 328 336
Singular integral of an equation of the first order, equations of, deduced from geometrical properties, though with exceptions v 253
| Singular integral of an equation of the first order, exceptional case of v 112
Singular integral of an equation of the first order, geometrical properties of v 249
Singular integral of an equation of the first order, how deduced from complete integrals v 166 170
Singular integral of an equation of the first order, how related to the general integral v 254
Singular integral of an equation of the first order, in hyperspace and the equations of the characteristics v 291
Singular integral of an equation of the first order, its contact with other integrals v 302 310—313
Singular integral of an equation of the first order, order of contact of, with the other integrals v 255—261
Singular integral of an equation of the first order, represent envelope (if any) of complete integrals v 251
Singular integral of an equation of the first order, when deduciblc from the system v 203
Singular integral of an equation of the first order, with appropriate tests v 201
Singular integrals v 65 see
Singularities of characteristics in ordinary space v 261—280.
Sommerfeld vi 527.
Sonin vi 303.
Special integrals v 65 67 97 172
Special integrals are connected with some of the non-ordinary positions on the characteristic v 212 217
Special integrals in hyperspace and the equations of the characteristics v 292.
Special integrals of equations of order higher than the first vi 9.
Special integrals of system of equations v 200
Special integrals, do not arise under the merely formal theory of contact transformations v 343.
Special integrals, general process for construction of equations which possess v 190
Special integrals, geometry of v 188
Special integrals, illustrations of v 69 177 188 200 see
Special integrals, which may arise in the use of Ampere's method, one kind of vi 276.
Special intermediate integrals of equations of second order, not included in other classes vi 261
Special intermediate integrals of equations of second order, with examples vi 264.
Speckman v 408 vi 303.
Sub-group of a group of functions v 345
Sub-group of a group of functions, highest order of, when in involution v 365
Sub-group of a group of functions, with mode of construction of the most extensive subgroup v 366.
Subsidiary equations in Ampere's method, agree with Monge's, in application to equations vi 283
Subsidiary equations in Ampere's method, how obtained vi 282
Subsidiary equations in Ampere's method, their significance vi 289.
Subsidiary system in Darboux's method, and simplified expression of vi 318
Subsidiary system in Darboux's method, compared with subsidiary system in Hamburger's method vi 338.
Subsidiary system in Darboux's method, construction of vi 316
Subsidiary system in Darboux's method, property of vi 309
Subsidiary system in Darboux's method, simultaneous integrals of distinct resolutions of vi 315 339
Subsidiary system in Darboux's method, utility of integrable combinations of the equations in vi 313
Subsidiary system of equations, in Boole's method involving differential coefficients vi 211
Subsidiary system of equations, in Monge's method involving differential elements vi 203
Subsidiary system of equations, in the general method, coincides with Boole's vi 227
Subsidiary system of equations, integrals of, how used vi 207 215
Subsidiary system of equations, relation between sets of, in Monge's method and in Boole's method respectively vi 212 215
Subsidiary system of equations, when it possesses three common integrals vi 230
Subsidiary system of equations, when it possesses three common integrals or none vi 259.
Subsidiary system of equations, when it possesses three common integrals or one vi 253
Subsidiary system of equations, when it possesses three common integrals or two vi 242
Successive reduction, Jacobi's method of, for the integration of complete linear systems v 92.
Surfaces representing integrals v 221
Surfaces representing integrals, edge of regression of v 237 see
Surfaces representing integrals, properties of v 225
Surfaces representing integrals, their characteristics v 224
System of partial equations, can be made to depend upon an equation or equations of higher order in one dependent variable only v 10.
System of partial equations, can be transformed so as to contain equations of first order only v 8
System of partial equations, in general, must contain the same number of equations as of dependent variables v 6
Systems of linear equations, complete, that are homogeneous v 76
Systems of non-linear equations that are complete v 109
Tangential transformation see contact transformations.
Tanner vi 159 197 456 460 462 464 469 490 509.
Teixeira vi 453.
Theoretical dynamics, equations of v 370
Third order, Ampere's method extended to vi 474
Third order, classes of, which possess intermediate integrals vi 457
Third order, Darboux's method extended to vi 478
Third order, equations of the, and of higher orders vi 456
Third order, Hamburger's method extended to vi 482.
Three integrals common to subsidiary system for Monge — Ampere equations vi 230
Three integrals common to subsidiary system for Monge — Ampere equations, conditions for vi 231
Three integrals common to subsidiary system for Monge — Ampere equations, construction of such equations vi 236.
Three integrals common to subsidiary system for Monge — Ampere equations, lead to a primitive vi 232—235
Transformation (Legendrian) of dependent variable so as to construct a primitive v 127
Transformation of equations of the second order, applied to linear equations vi 441
Transformation of equations of the second order, critical relation for vi 428
Transformation of equations of the second order, general theory of vi 425
Transformation of equations of the second order,forms of, after Baecklund vi 432
Transformations of contact see contact transformations.
Transformed equations, by application of Levy's transformations vi 94.
Transformed equations, Darboux's expression of vi 85
Transformed equations, invariants of vi 52
Transformed equations, possessing any vanishing invariant lead to integration of original equation vi 56
Transformed equations, series of, by application of Laplace-transformations to linear equation of second order vi 51
Tresse v 53.
Two integrals common to subsidiary system, conditions for vi 243 244
Two integrals common to subsidiary system, construction of such equations vi 246 252.
Two integrals common to subsidiary system, for Monge — Ampere equations vi 242
Two integrals common to subsidiary system,use made of vi 245 249
Uniqueness of regular integrals, for equations of any order v 45
Uniqueness of regular integrals, for equations of second order v 37 42
Uniqueness of regular integrals, for systems of linear equations v 88.
Uniqueness of regular integrals, is a part of Cauchy's theorem v 28 35
Uniqueness of regular integrals, of systems of equations of the first order v 11 18 20 22
Valyi vi 328 333.
Vanishing invariants of linear equations of the second order, effect of vi 46 56 100.
Variation of parameters v 165
Variation of parameters applied to Laplace's linear equation vi 379
Variation of parameters applied, being a contact transformation vi 382.
Variation of parameters applied, to integrals of equations of the second order vi 361
Variation of parameters for systems of equations in several variables v 419
Variation of parameters in general v 464.
Variation of parameters with special examples v 425
Vivanti vi 490 509 522 525.
von Weber v 53 407 vi 328 490.
Weierstrass v 432 vi 280.
Whittaker v 371 vi 15 16 576 580 581.
Winckler vi 81 156 303 360.
Zajacrowski v 407.
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