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Forsyth A.R. — Theory of differential equations. Part 4. Partial differential equations (Vol. 6) 


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Action, equations of, in theoretical dynamics v 381.
Adjoint equations vi 111
Adjoint equations are reciprocally adjoint vi 114
Adjoint equations, Biemann's use of vi 119
Adjoint equations, construction of vi 117
Adjoint equations, effect of Laplace transformations upon vi 114
Adjoint equations, relation between invariants of vi 115
Adjoint ordinary equations, used to construct integrals of doublyfinite rank when these are possessed by linear equations of the second order vi 89.
Ampere v 206 282 vi 1 8 16 17 21 200 201 266 302 307 376 418.
Ampere's definition of a general integral vi 4
Ampere's definition of a general integral, compared with the Darboux — Cauchy definition vi 6—8.
Ampere's first class of equations of the second order, as those having integrals without partial quadratures vi 16.
Ampere's method of integration vi 266
Ampere's method of integration and to equations of the second order in any number of independent variables vi 530.
Ampere's method of integration, and to special equations of the second order vi 281
Ampere's method of integration, applied to equation of minimal surfaces vi 277
Ampere's method of integration, compared with Darboux's method vi 303
Ampere's method of integration, construction of the primitive in vi 284
Ampere's method of integration, extended to equations of the third order in two independent variables vi 474
Ampere's method of integration, illustrations of vi 272
Ampere's method of integration, significance of, compared with Monge's and with Boole's vi 290
Ampere's test for equations having integrals free from partial quadratures vi 17.
Ampere's theorem on intermediate integral of an equation of the second order vi 248.
Arbitrary elements, characters of, in general integrals vi 21.
Arbitrary elements, in integrals of equations of order higher than the first, modes of occurrence of vi 17
Arbitrary function in integral of equation of second order, the equation for argument of, is invariantive for every compatible equation vi 320.
Arbitrary functions, arguments of vi 27
Arbitrary functions, number of, in the integral in Cauchy's theorem is the same as the order of the equation v 47 vi 22
Argument of arbitrary function in integral of equation of the second order, equation for, is invariantive for every compatible equation of that order vi 320.
Arguments of arbitrary function in general integrals, equation characteristic of vi 29 35 37
Arguments of arbitrary function in general integrals, number of vi 27—29
Arguments of arbitrary function in general integrals, various instances of, for equation of second order vi 32.
Arguments of arbitrary functions in integral made the independent variables in integration by Ampere's method vi 271.
Asymptotic curves on singular integral v 251.
Asymptotic lines as characteristics, equations which have v 245.
ausfjezeichnete Function v 350.
Baeckhand v 408 vi 42—45 432.
Baecklund transformations for equations of the second order vi 432
Baecklund transformations for equations of the second order, applied to linear equations vi 441
Baecklund transformations for equations of the second order, connected with simultaneous equations of the first order vi 450.
Baecklund transformations for equations of the second order, kinds of vi 433
Bateman vi 16 580 581.
Bertrand v 370 397 405 vr 139.
Bianchi vi 328 377 454.
Boehm vi 260.
Boole vi 199 201 266 303 307.
Boole's method for equations possessing an intermediate integral vi 208—212 215
Boole's method for equations possessing an intermediate integral in practice can be included in Darboux's method vi 201
Boole's method for equations possessing an intermediate integral, compared with Darboux's method vi 303.
Boole's method for equations possessing an intermediate integral, compared with Monge's method vi 209 212
Borel's expression, as a definite integral, for integralsof linear equations vi 104.
Boundary values, Kiemann's use of, for adjoint equations vi 119
Bour vi 201 B67.
Bourlet v 52 53 vi 334.
Bromwich vi 581.
Burgatti vi 55 129.
Canonical constants, Bertrand's theorem on, deduced from properties of contact transformation v 405.
Canonical form, how constructed v 355—359
Canonical form, of complete linear system v 81
Canonical form, of group of functions v 355
Canonical form, of linear equation of the second order in two independent variables vi 47.
Canonical forms of equations of dynamics v 373
Canonical forms of equations of dynamics, and (for general systems) only under contact Canonical forms of equations of dynamics, transformations v 399 403
Canonical forms of equations of dynamics, and as giving Bertrand's theorem on canonical constants v 406.
Canonical forms of equations of dynamics, as affected by contact transformations v 398
Canonical forms of equations of dynamics, conserved under contact transformations v 399
Canonical forms of equations of dynamics, significance of, as contact transformation v 405
Cauchy v 26 205 206 219 282 371 381.
Cauchy's construction of the equations of the characteristics v 206
Cauchy's construction of the equations of the characteristics and derivation of integrals v 210
Cauchy's construction of the equations of the characteristics as modified by Darboux v 212.
Cauchy's integral, and to the equation for twodimensional potential vi 110.
Cauchy's integral, application of this property to the equation for the conduction of heat vi 107
Cauchy's integral, of a homogeneous linear equation v 58
Cauchy's integral, of a nonhomogeneous linear equation v 73
Cauchy's integral, of a system of homogeneous linear equations v 88
Cauchy's integral, of equations of the second order vi 3
Cauchy's integral, of linear equation can be represented by a definite integral involving only a single arbitrary function vi 104
Cauchy's problem for equations of the second order and their characteristics vi 388.
Cauchy's theorem, based upon Kowalevsky's existencetheorems v 11 Chapter
Cauchy's theorem, examples when it cannot be applied to linear equations v 69 70.
Cauchy's theorem, exceptions to, for equations of general order v 51.
Cauchy's theorem, exceptions to, for equations of the first order v 32 36 158
Cauchy's theorem, exceptions to, for equations of the second order v 42
Cauchy's theorem, for a single equation of the first order v 27 33 36
Cauchy's theorem, for an equation of the second order v 37 42 vi 2
Cauchy's theorem, for equations of any order v 43
Cauchy's theorem, for equations of the second order, as described by Darboux vi 305
Cauchy's theorem, in particular, as applied to the Monge — Ampere equations vi 307
Cauchy's theorem, similarly for any number of independent variables v 33 35 36
Cauchy's theorem, with any number of independent variables vi 528.
Characteristic developable v 227.
Characteristic equation satisfied by arguments of arbitrary functions in integrals of equations and Characteristic equation satisfied by arguments of arbitrary functions in integrals of equations with any number of independent variables vi 37.
Characteristic equation satisfied by arguments of arbitrary functions in integrals of equations of any order vi 35
Characteristic equation satisfied by arguments of arbitrary functions in integrals of equations of the second order vi 29 320
Characteristic equations in dynamics, due to Hamilton v 371 376 381.
Characteristic invariant vi 532
Characteristic invariant of Laplace's equation vi 571.
Characteristic invariant, influence of, when resoluble vi 550
Characteristic number for equations having integrals of doubly finite rank vi 70
Characteristic number for equations having integrals of selfadjoint equation of finite rank vi 133.
Characteristics and are the same as the equations in Charpit's method v 208
Characteristics in hyperspace, equations of v 284
Characteristics in hyperspace, equations of, constructed from geometrical properties v 293.
Characteristics in hyperspace, kinds of integrals derived from v 288—292
Characteristics in hyperspace, method of v 282
Characteristics in hyperspace, with vise made of their integrals v 285—288
Characteristics of equations of the second order vi 388
Characteristics of equations of the second order, geometrical interpretation of vi 397
Characteristics of equations of the second order, primitive as locus of vi 393
Characteristics of equations of the second order, used to classify the equations vi 400.
Characteristics of equations of the second order, when the equation is of the Monge — Ampere type vi 394
Characteristics, and connected with the general integral v 230
Characteristics, conjugate v 251
Characteristics, envelope of, is edge of regression v 337
Characteristics, equations having asymptotic lines as v 245
Characteristics, equations having geodesies as v 248.
Characteristics, equations having lines of curvature as v 246
Characteristics, equations of, as constructed by Cauchy v 206
Characteristics, equations of, as constructed by Darboux v 212
Characteristics, equations of, deduced from geometrical properties v 228
Characteristics, equations of, in terms of surface parameters v 233
Characteristics, integral equivalent of these equations v 209
Characteristics, of equations of first order in two independent variables v 205
Characteristics, properties of v 224
Characteristics, selfconjugate v 251.
Characteristics, singularities of, in ordinary space v 261
Characteristics, their relation to integral surfaces v 224
Characteristics, used by Lie as basis of classification of equations v 244
Charpit v 157 vi 303.
Charpit's equations for integration of intermediate integral of a system vi 217 248 285.
Charpit's method for equations in two independent variables v 150
Charpit's method for equations in two independent variables, the equations in, are the eqviations of characteristics v 208.
Charpit's method for equations in two independent variables, used to integrate equations not subject to Cauchy's theorem v 1—58
Chrystal v 70 190.
Clairin vi 425 433 441.
Classes of integrals of simultaneous equations of the first order v 419 424
Classes of integrals, but are not entirely comprehensive vi 261.
Classes of integrals, by use of groups of functions v 369
Classes of integrals, derived by use of theory of contact transformations v 326 331 334 338
Classes of integrals, how far the customary classes are comprehensive v 198
Classes of integrals, of a complete system v 193
Classes of integrals, that are intermediate for equation of the second order vi 8
Classification (Lie's) of eqviations of the first order, according to the nature of the characteristics v 244
Classification (Lie's) of eqviations of the first order, and of equations of the second order similarly vi 400.
Coexistence of equations of the second order vi 339.
Coexistence of equations, Jacobian relations for see Jacobian conditions.
Combinants of two functions, cannot be generalised for equations in several dependent variables v 474.
Combinants of two functions, commonly called the Poisson — Jacobi v 113
Combinants of two functions, in connection with the canonical equations of dynamics v 392
 Combinants of two functions, properties of v 112
Compatible equations, and by Hamburger's method vi 336
Compatible equations, of order higher than second vi 353 539.
Compatible equations, of the second order, how constructed by Darboux's method vi 314
Complete integral v 171
Complete integral as related to the equations of the characteristic v 211 212 215
Complete integral in hyperspace, derived from characteristics v 290
Complete integral, can be particular cases of distinct general integrals v 176 182
Complete integral, contact of two, along a characteristic v 299
Complete integral, contact of, with singular integrals v 312
Complete integral, derived through contact transformations v 326 331 334 338
Complete integral, derived through groups of functions v 369
Complete integral, of simultaneous equations in several dependent variables v 419.
Complete integral, of system of equations v 194
Complete integral, relations between, general integral and singular integral, with limitations v 172 251 255
Complete integral, tests for v 178
Complete integrals of equation of second order vi 8
Complete integrals of equation of second order, subjected to variation of parameters vi 361.
Complete intermediate integrals of equations of second order generalised vi 377.
Complete linear system of equations v 79 see
Complete systems of equations and groups of functions, how related v 347 349.
Complete systems of linear equations, canonical form of v 81
Complete systems of linear equations, conditions of coexistence of v 98
Complete systems of linear equations, integration of v 99.
Complete systems of linear equations, Jacobi's method of integrating v 91
Complete systems of linear equations, Mayer's method of integrating v 89
Complete systems of linear equations, number of independent integrals of v 83
Complete systems of linear equations, or when the independent variables are changed v 80
Complete systems of linear equations, remain complete, when replaced by an algebraic equivalent v 79
Complete systems of linear equations, that are homogeneous v 76
Complete systems of linear equations, that are not homogeneous v 97
Complete systems of nonlinear equations v 109
Complete systems of nonlinear equations, classes of integrals of v 193.
Complete systems of nonlinear equations, number of independent integrals of, in Mayer's method v 122
Complete systems of nonlinear equations, when in involution v 120
Completely integrable equations with general result v 424.
Completely integrable equations, conditions for v 416
Completely integrable equations, different kinds of integrals of v 419
Completely integrable equations, integration of v 419
Completely integrable equations, Koenig's v 411
Completely integrable equations, various cases v 416
Cones associated with the geometrical interpretation of an equation of the first order v 221
Cones associated with the geometrical interpretation of an equation of the first order, and to the characteristics v 224.
Cones associated with the geometrical interpretation of an equation of the first order, how related to integral surfaces v 223
Conjugate characteristics v 251.
Contact of integrals in hyperspace v 297—306 310.
Contact transformations v 131 315
Contact transformations and canonical equations in dynamics v 370
Contact transformations and upon reciprocal groups v 349
Contact transformations, and also are infinitesimal v 322
Contact transformations, applicable only when certain relations are satisfied unconditionally and must be replaced by theory of groups of functions when the relations are satislied only conditionally v 344
Contact transformations, applied to the integration of an equation or eqviations v 324
Contact transformations, arising from Imschenetsky's variation of parameters applied to Laplace's linear equation vi 382.
Contact transformations, can be translated into each other v 405
Contact transformations, classes of integrals thus derived v 326 331 334 338
Contact transformations, definition and specific equations of v 315—317
Contact transformations, effect of, on equations of the second order possessing two intermediate integrals vi 295
Contact transformations, effect of, upon group of functions v 346
Contact transformations, general relation of, to the integration of equations v 343
Contact transformations, invariants of group of functions under v 364
Contact transformations, relations between v 398—404
Contact transformations, which are homogeneous v 323
Contact transformations, which are infinitesimal v 317
Contact transformations, which do not involve the dependent variable v 318
Contact, of edge of regression of general integral with complete integral v 240
Contact, of integral surfaces v 225
Contact, of integrals in general v 306—310.
Contact, of selected edge of regression with integral surface v 241
Contact, of singular integral with other integrals v 255—261
Cosserat vi 159 161.
Cosserat's proof of Moutard's theorem on eqviations of the second order having integrals in explicit finite form without partial quadratures vi 161
Cosserat's proof of Moutard's theorem on eqviations of the second order having integrals in explicit finite form without partial quadratures, summary of results in vi 195.
Coulon vi 490.
Critical relation for transformation of equations of the second order vi 428
Critical relation for transformation of equations of the second order, significance of vi 429 430 436—441.
Curvature, lines of, as characteristics v 246.
Curves associated with the geometrical interpretation of an equation of the first order v 222
Curves associated with the geometrical interpretation of an equation of the first order, how related to integral surfaces v 223
Curves associated with the geometrical interpretation of an equation of the first order, integral v 238
Darboux v 205 212 226 227 243 251 282 408 vi 5 39 47 55 60 70 78 82 111 120 127 128 131 139 157 158 159 161 200 201 295 302 377 432 454.
Darboux — Cauchy definition of a general integral vi 5
Darboux — Cauchy definition of a general integral, compared with Ampere's definition vi 6—8.
Darboux's forms of linear equations of the second order, having integrals of finite rank vi 82
Darboux's method for integrating equations of the second order in two independent variables vi 302
Darboux's method for integrating equations of the second order in two independent variables and integrals of that system vi 313
Darboux's method for integrating equations of the second order in two independent variables, applied to equations f(r, s, t) = 0 vi 344
Darboux's method for integrating equations of the second order in two independent variables, applied to Laplace's equation vi 571.
Darboux's method for integrating equations of the second order in two independent variables, applied to obtain compatible eqviations of order higher than the second vi 353
Darboux's method for integrating equations of the second order in two independent variables, central aim of vi 302
Darboux's method for integrating equations of the second order in two independent variables, compared with methods of Monge, Ampere, Boole vi 303 313
Darboux's method for integrating equations of the second order in two independent variables, extended to equations of the second order in any number of independent variables vi 589 562
Darboux's method for integrating equations of the second order in two independent variables, extended to equations of the third order in two independent variables vi 478
Darboux's method for integrating equations of the second order in two independent variables, includes Valyi's process for integration of simultaneous equations of second order vi 328
Darboux's method for integrating equations of the second order in two independent variables, property of subsidiary system in vi 309
Darboux's modification of Cauchy's method of characteristics v 212
De Boer vi 343 344 351 352.
De Morgan vi 201.
Deformable surfaces, equation of, referred to minimal lines as parametric curves vi 344.
Delassus v 53 vi 104.
Developable touching an integral surface along characteristic, properties of v 227.
distinguee, fonction v 350.
Dixori vi 260.
Dominant functions and equations used v 13 14.
Donkin v 370.
Doubly finite rank, characteristic number of vi 70
Doubly finite rank, construction of the equations vi 72—78
Doubly finite rank, how affected by Laplace transformations vi 70
Doubly finite rank, linear equations of the second order having integrals of vi 69
Doubly finite rank, with Darboux's modified forms for vi 82.
Dynamics, canonical form of v 373
Dynamics, conserved by contact transformation v 399—404
Dynamics, equations of theoretical v 370
Dynamics, represent an infinitesimal contact transformation v 405.
Dziobek v 370.
Edge of regression of integral surface v 237
Edge of regression of integral surface is envelope of characteristics v 237
Edge of regression of integral surface is general form of an integral curve v 239
Edge of regression of integral surface of general integral has contact of second order with complete integral v 240
Edge of regression of integral surface with properties v 241
Edge of regression of integral surface, equations of, deduced from the differential equation v 243.
Edge of regression of integral surface, selected curves from the infinitude v 240
Element of integral in hyperspaee v 299.
Elliptic case of linear equations of the second order vi 44.
Energy is the source of the infinitesimal contact transformation represented by canonical equations v 405.
Energy, when it gives an integral of dynamical equations v 375
Envelope of characteristics on a bypersurface, equations of v 300.
Envelope of characteristics, is edge of regression of surface v 237
Equal invariants, linear equations of the second order having vi 131 see
Essential parameters, number of, in an integral of an equation v 192.
Euler vi 127 159 521.
Exceptional integrals v 185 see
Exceptional integrals, geometry of v 188.
Exceptions to Cauchy's theorem for equations of the first order in two independent variables v 158
Existencetheorems for integrals of system of equations can lead to a singular integral v 111.
Existencetheorems for integrals of system of equations, for integrals of a complete system of homogeneous linear equations v 83
Existencetheorems for integrals of system of equations, for single equation, exceptional case omitted from v 110
Existencetheorems for integrals of system of equations, of any order v 43
Existencetheorems for integrals of system of equations, of the first order and any degree in any number of independent variables v 35
Existencetheorems for integrals of system of equations, of the first order and linear v 11
Existencetheorems for integrals of system of equations, of the first order and not linear v 21
Existencetheorems for integrals of system of equations, of the second order v 37
Falk vi 166 456 469.
Finite form of general integral, characteristic property of vi 14
Finite form of general integral, equations of second order determined by vi 159
Finite rank of an equation and its adjoint vi 116
Finite rank of selfadjoint equations vi 133
Finite rank, and as affected by Moutard's theorem vi 141.
Finite rank, Goursat's theorem on vi 90
Finite rank, how affected by Laplace transformations vi 70
Finite rank, in both variables vi 69



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