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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Forsyth A.R. — Theory of differential equations (Part 1. Exact equations and pfaff's problem) |
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| Ïðåäìåòíûé óêàçàòåëü |
Mayer 1 59 62 67 72 76 81 86 200 215 230 233 257 258 294 331
Mayer’s establishment of equations of tangential transformation 233 238
Mayer’s establishment of equations of tangential transformation will not apply to systems of Pfaffians 331
Mayer’s establishment of equations of tangential transformation, modification and simplification of Lie’s theorem 237
Mayer’s method of integrating system of exact equations 59
Mayer’s method of integrating system of exact equations in its simplest form 61
Mayer’s method, of solving complete system of linear homogeneous partial differential equations 72—76
Mayer’s method, of solving complete system of linear homogeneous partial differential equations of obtaining at least one integral 76—78
Memoirs on systems of Pfaffians 299
Minimum number of integrals, general, for non-integrable system 312—316
Minimum number of integrals, general, for single equation 89 113 114 145 146
Minimum, conditions that the number of equations in the integral equivalent be less than the general by Clebsch 207
Minimum, conditions that the number of equations in the integral equivalent be less than the general by Lie 255
Minimum, conditions that the number of equations in the integral equivalent be less than the general in Grassmann’s symbolical form 123—127
Minimum, conditions that the number of equations in the integral equivalent be less than the general obtained by Natani 166 168
Modification of incompletely ihtegrable system by use of its exact integrals 311
Monge 80 102 307
Monge’s view of inexact equations 80
Monomial equation of Grassmann, if extensive, includes a system of simultaneous equations in several dependent variables 122
Monomial equation of Grassmann, if numerical, includes a single partial differential equation of the first order 122
Monomial equation of Grassmann, relation of, to integrals of Pfaffian 123
Monomial equation of Grassmann, transformation of 137 141
Multipliers of system of exact equations 40
Multipliers of system of exact equations, partial differential equations satisfied by 42
Multipliers of system of exact equations, quotient of two is a solution, with other properties 41 43
Natani 27 30 32 55 56 59 78 82 83 85 86 143 180 187 190 191 200 208 278 312
Natani’s method of integration, of exact equation 27
Natani’s method of integration, of exact equation, compared with du Bois — Reymond’s 32
Natani’s method of integration, of exact equation, compared with Euler’s 30
Natani’s method of integration, of exact equation, developed by Mayer 59—62
Natani’s method of integration, of exact equation, of system of exact equations 56
Natani’s method of treatment of Pfaff’s problem Chap. vi see
Natani’s method of treatment of Pfaff’s problem, abstract and character of 85 86
Natani’s method of treatment of Pfaff’s problem, application of, to solution of single partial differential equation 187 188
Natani’s method of treatment of Pfaff’s problem, generalised to system of Pfaffians to determine number 313
Natani’s method of treatment of Pfaff’s problem, inapplicable to determine integrals 322
Natani’s method of treatment of Pfaff’s problem, to solution of system of partial differential equations 190
Non-integrable systems 301
Non-integrable systems, arbitrary integrals used to modify 319
Non-integrable systems, minimum number of equations in integral equivalent of 312—316
Non-integrable systems, number of arbitrary integrals in equivalent of 316
Non-integrable systems, three fundamental problems relating to 311
Normal form, character of, to be inferred from results of Natani and Frobenius 278 (note)
Normal form, conditions which determine character of 253
Normal form, construction of see Gradual construction
Normal form, equations of Frobenius which determine 291 292
Normal form, invariantive persistence of character of 249
Normal form, involving even number of functions 194
Normal form, involving even number of functions, Clebsch’s generalisation of 196
Normal form, involving odd number of functions 221
Normal form, involving odd number of functions, Clebsch’s generalisation of 222
Normal form, of conditioned expression deduced from that of transformed un conditioned expression 259—266
Normal form, reduction of expression to a, by Lie 248
Normal form, relations between elements of equivalent 252
Normal form, unaltered in character when Cauchy substitution is applied 258 269
Normal form, used by Frobenius to make expressions equivalent 285 289
Number of equations in most general integral equivalent of non-integrable system 312—316
Number of equations in most general integral equivalent of non-integrable system, application of result to simultaneous partial equations in several dependent variables 317 329
Number of exact integrals of incompletely integrable system 304
Number of independent solutions of a complete system of linear and homogeneous partial differential equations 69
Odd invariantive integer in Frobenius’ theory 287
Odd number of variables, unconditioned expression involving, treated by Clebsch’s method 219
Odd number of variables, unconditioned expression involving, treated by Lie’s method 268
Odd number of variables, unconditioned expression involving, treated by Natani’s method 157—165
Odd reduction 111
Odd reduction can always be applied to expression involving odd number of variables 111
Odd reduction, not unique 111
Osculational transformations 230
Pais 35
Partial differential equation of first order, a single, by Natani 187 188
Partial differential equation of first order, a single, integration of, by Pfaff 181—183
Partial differential equation of first order, a single, only one integration necessary in Natani’s method 154
Partial differential equation of first order, a single, Pfaff’s integration improved by Jacobi 183—187
Partial differential equation of first order, a single, special case of Pfaffian equation 122
Partial differential equation of second order, conditions for intermediary integral of 307
Partial differential equation of second order, nature of integral equivalent of general 320
Partial differential equations, solution of simultaneous system of, by Natani 190
Partial differential equations, solution of simultaneous system of, by Natani, application to solution of Chap. vii see 238
Partial differential equations, system of, coextensive with system of exact equations 67 69 71
Persistence of character of normal form 249
Pfaff 81—83 89 180
Pfaff’s method of reduction Chap. iv see
Pfaff’s method of reduction, additions to, by Gauss and Jacobi 81 82
Pfaff’s method of reduction, general theory 81 90
Pfaff’s Problem, why so called 81
Pfaff’s solution of partial differential equation of first order 181
Pfaff’s solution of partial differential equation of first order, simplified by Jacobi 183
Pfaff’s theorem as to transformation of an expression, applied to deduce a later theorem of Jacobi’s 83 115
Pfaff’s theorem as to transformation of an expression, completion of proof of 112
Pfaff’s theorem as to transformation of an expression, simplified by Jacobi 117—120
Pfaff’s theorem as to transformation of an expression, stated 89
Pittarelii 35
Principal integrals, of subsidiary system used for transformation by literal transcription 120 152 153 157 170 171
Principal integrals, of system of equations 118 151
Pujet 33
Reciprocation a particular case of tangential transformation 230 232
Reduced form, integral equivalent derived from 116 117
Reduced form, integral equivalent derived from, containing even number of independent functions 203 255
| Reduced form, integral equivalent derived from, containing odd number of independent functions 219 268
Reduced form, integral equivalent derived from, containing odd number of independent functions, generalisation of 220
Reduced form, integral equivalent derived from, containing odd number of independent functions, normal form of 222
Reduced form, integral equivalent derived from, equivalence of, to another reduced form 195 219 249
Reduced form, integral equivalent derived from, generalisation of 194
Reduction of expression by Pfaff’s method, subsidiary equations for 89—94
Reduction of expression by Pfaff’s method, subsidiary equations for, completed by Gauss for all cases 112—114
Reduction of expression by Pfaff’s method, subsidiary equations for, procedure in, when number of variables is odd 106—111
Reduction of expression by Pfaff’s method, subsidiary equations for, three cases of 95 100 103
Reduction of two equations in four variables to canonical form 325—327
Reduction, even 105
Reduction, even, neither unique 105 111
Reduction, even, odd 111
Reduction, even, simplified by use of principal integrals 117—120
Sarrus 33
Scott 81 95 106 275
Second Method, Clebsch’s 209—218
Second Method, Clebsch’s, applies only to equation in even number of variables, if unconditioned 210
Second Method, Clebsch’s, or if Pfaffian determinant alone vanishes 214
Second order, partial differential equation of conditions that it may possess intermediary integral 307—310
Second order, partial differential equation of nature of integral equivalent of 320 321
Simplification of Pfaff’s reduction, by Jacobi 117—120
Simplification of Pfaff’s reduction, by Jacobi, of process when some coefficients vanish 153
Simultaneous partial differential equations of first order, in a single dependent variable, conditions of coexistence 191 and note
Simultaneous partial differential equations of first order, in a single dependent variable, solution of, by Natani 190—193
Simultaneous partial differential equations of first order, in several dependent variables included in extensive monomial equation 122
Simultaneous partial differential equations of first order, in several dependent variables number of integral equations equivalent to 317 329
Simultaneous variations used by Binet 148
Simultaneous variations used by Darboux 295
Simultaneous variations used by Frobenius 273
Simultaneous variations used by Natani 148 149 157 174 178
Single exact equation Chap. i see
Single integration of subsidiary system sufficient for partial differential equation of first order 154
Solution of partial differential equation, connected with Natani’s general result 187
Solution of partial differential equation, improved by Jacobi 183—187
Solution of partial differential equation, Natani’s method of 188—190
Solution of partial differential equation, note on history of 183 (note)
Solution of partial differential equation, Pfaff’s method of 181—183
Solution of system of equations by Mayer 72—78
Solution of system of equations by Natani’s method 190—193
Solution of system of equations in Clebsch’s method 215
Stodockiewicz 58
Stoffel 33
Subsidiary equation for transformation of Grassmann’s monomial form 128—130
Subsidiary equation for transformation of Grassmann’s monomial form, integration of solved equation 136 137
Subsidiary equation for transformation of Grassmann’s monomial form, interpreted numerically 133
Subsidiary equation for transformation of Grassmann’s monomial form, solution of 130—136
Subsidiary equations constructed by use of simultaneous variations 148
Subsidiary equations for conditioned expression an exact system 54 170
Subsidiary equations for conditioned expression an exact system, integration of 170—172
Subsidiary equations for Pfaff’s reduction, formation of 89—94
Subsidiary equations, when number of variables is even, alternative cases 94 296
Subsidiary equations, when number of variables is even, Cayley’s symbolical form of 100
Subsidiary equations, when number of variables is even, principal integrals of 118
Subsidiary equations, when number of variables is even, solution of, in most general case 96
Subsidiary equations, when number of variables is even, still valid, if determinant alone vanish 101
Subsidiary equations, when number of variables is odd, effect of satisfaction of 109
Subsidiary equations, when number of variables is odd, formed by Natani 157—159
Subsidiary equations, when number of variables is odd, in general inconsistent 110
Subsidiary equations, when number of variables is odd, Jacobi’s condition for consistency of 107
Subsidiary equations, when number of variables is odd, principal integrals of, used to transform expression 151 160
Subsidiary equations, when number of variables is odd, solution of 160 164
Subsidiary equations, when number of variables is odd, verified by Natani’s results 164
Subsidiary system, modified by one of its integrals 173—175
Subsidiary system, modified by one of its integrals by more than two integrals 179
Subsidiary system, modified by one of its integrals by two integrals of original equation 178
Subsidiary system, modified by one of its integrals by two of its integrals 176 177
Summary of methods of treatment of Pfaff’s problem Chap. iii see
Symbolical solution by Cayley of subsidiary system 100
System of equations, completely integrable 300
System of equations, incompletely integrable 301
System of equations, non-integrable 301
System of equations, what is meant by exact integral of 300 301
System of exact equations Chap. ii see
System of Pfaffians Chap. xiii see
System of Pfaffians, memoirs relating to 299
System of Pfaffians, represented by a single extensive monomial equation 122 329
Tangential transformations Chap. ix see 298
Tangential transformations, applied by Lie to Pfaff’s problem 252
Tangential transformations, note on history of 230
Tanner 117 142 299
Tanner’s symbolical forms similar to Grassmann’s 142
Transformation of expression to monomial form 122
Transformation of expression when number of variables is even completed 113
Transformation of expression when number of variables is even three cases of 99 100—102 103—105
Transformation of expression when number of variables is odd, completed 114
Transformation of expression when number of variables is odd, two cases of 107—109 110—111
Transformation of expression, by principal integrals 118 152 157
Transformation of expression, by principal integrals, by Cauchy substitutions when it is conditioned 258 269
Transformation, gradual, of monomial form 127—141
Uniqueness, want of, in reductions even or odd 105 111
Unsolved problems, connected with non-integrable systems of equations 330
Valyi 321
Vanishing coefficients, effect of, in simplifying construction of normal form 153
Vanishing Pfaffian determinant, effect of, on transformation of expression 102 105
Variable, extensive 121
Voss 33 34 299
Weiler 32
Winckler 33
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