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Forsyth A.R. — Theory of differential equations (Part 1. Exact equations and pfaff's problem) 


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Algebraical substitutions 274
Algebraical substitutions, changed into differential substitutions 281
Arbitrary element in integral equivalent of single equation 146
Arbitrary element in integral equivalent of single equation of nonintegrable system 316
Arbitrary element in integral equivalent of single equation used to modify nonintegrable system 319
Arbitrary element in integral equivalent of single equation used to modify single equation 157
Arithmetic mean of two invariantive integers in transformation of simultaneous forms is a sufficient invariant 278
Baabe 33 193
BACH 33
Baecklund 230
Bertrand 18 33 109 227
Bertrand’s integration of an exact equation in three variables 18 109 226 227
Bertrand’s integration of an exact equation in three variables generalised to exact equation in n variables 20—24
Biermann 299 312 316 321
Biermann’s determination of minimum number of equations in equivalent of non—integrable system 312 316
Bilinear differential covariant 273 294
Bilinear differential covariant changed to algebraical covariant 274
Binet 148
Binet’s simultaneous variations 148 (note)
Boole 67 299 305 307
Bour 1
Brioschi 211
Canonical form, of system of exact equations 43
Canonical form, of system of exact equations, reduction of two inexact equations in four variables to 325—827
Cauchy 86 118 183 211 294
Cauchy substitutions 59 86
Cauchy substitutions used by Lie to transform conditioned expression 258 269
Cauchy substitutions, do not affect character of normal form 258 269
Cayley 81 99 115
Cayley’s symbolical solution of equations subsidiary to first reduction of expression 100
Character of a normal form, persistent 249
Character of a normal form, persistent, conditions which determine the 253—255
Character of a normal form, persistent, conditions which determine the, and inferences from conditions 255—257
Character of a normal form, persistent, inferred from results of Natani and Frobenius 278 (note)
Character of a normal form, persistent, unaffected by Cauchy substitution 258 269
Christoffel 274
Class of an expression 290
Class of an expression, examples relating to 293
Clebsch 72 76 78 82—86 117 168 173 175 179 181 194 233 244 245 252 257 266 298 321
Clebsch’s methods, note on complete systems of partial differential equations in 76 78
Clebsch’s methods, note on complete systems of partial differential equations in, abstract and general character of 85 86 181
Clebsch’s methods, note on complete systems of partial differential equations in, equations inferred from Lie’s method 266 (note)
Clebsch’s methods, note on complete systems of partial differential equations in, number of integrations required in 79
Clebsch’s methods, note on complete systems of partial differential equations in, of treating Pfaff’s problem Chap. viii see
Collet 25 34
Collet’s determination of integrating factor 25
Complete reduction of expression, after Gauss, to contain minimum number of terms 112—114
Complete systems of linear and homogeneous partial differential equations in generalisation of Bertrand’s method
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method coextensive with a system of exact equations 68
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method construction of solutions by successive derivation 68
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method construction of solutions by successive derivationby general method of Mayer 72
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method construction of solutions by successive derivationby simplest form of Mayer’s method 74
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method in Clebsch’s methods 76 78 209 225 229
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method in Frobenius’ conditions for exact equations 52
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method Mayer’s derivation of single solution of, from any integral of subsidiary system 76
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand’s method number of independent solutions of 69
Completely integrable systems of equations, Chap. ii 300
Condition of consistency, of first subsidiary system when the number of variables is odd 107
Condition of consistency, of first subsidiary system when the number of variables is odd, effect of this condition on the first transformation 109
Condition of consistency, of first subsidiary system when the number of variables is odd, verification of, from Natani’s method 164
Conditioned expression, transformed by Cauchy substitutions into an unconditioned expression having normal form of same character 258 269 297
Conditioned expression, transformed by Cauchy substitutions into an unconditioned expression having normal form of same character, normal form of, deduced from that of the transformed unconditioned expression 259—266
Conditions, for existence of integrals fewer in number than the general minimum, in form of interrupted products 123
Conditions, for existence of integrals fewer in number than the general minimum, in form of interrupted products with numerical interpretation 124—127
Conditions, for expression in even number of variables 166 167 207
Conditions, for expression in odd number of variables 168 169 207
Conditions, given by Lie to determine character of normal form 253
Conditions, given by Lie to determine character of normal form, inferences from 255
Conditions, that partial differential equation of second order possess an intermediary integral 309
Condorcet 33
Consistency of subsidiary equations when number of variables is odd, condition for 107
Consistency of subsidiary equations when number of variables is odd, condition for effect of this condition 109
Consistency of subsidiary equations when number of variables is odd, condition for verification of, from Natani’s method 164
Construction, gradual, of solution and normal form by Clebsch 198 199 208 215 223
Construction, gradual, of solution and normal form by Grassmann 141
Construction, gradual, of solution and normal form by Jacobi 120
Construction, gradual, of solution and normal form by Lie 257 268 270
Construction, gradual, of solution and normal form by Natani 151 157 160
Construction, gradual, of solution and normal form in Pfaff’s reduction 112
Corollaries from cylindrical tangential transformation used by Lie 242 248
Corollaries from cylindrical tangential transformation used by Lie, derivable from Clebsch’s results 244 245
Covariant, bilinear differential, associated with expression 273
Covariant, bilinear differential, associated with expression, multilinear, not associated with expression 274 (note)
Covariant, bilinear differential, associated with expression, no other similarly formed, exists for expression 274
Cylindrical tangential transformation 239
Cylindrical tangential transformation, equations of 241 242
Cylindrical tangential transformation, Lagrange’s examples of 242
Cylindrical tangential transformation, Lie’s corollaries from 242 248
Darboux 88 181 227 238 294
Darboux’s investigations 88
Darboux’s investigations, abstract of Chap. xii see
De Morgan 20
Deahna 51
Deahna’s conditions of exactness of system of equations 51
Definition of tangential transformation 232
Derivation, of exact equation from one integral 1
Derivation, of system of exact equations from system of integrals 37
Determinants, in algebraical transformation of bilinear form 276 et seq.
Determinants, in differential transformation 94 101 103 106 166 169 201 214 254
Determination of character of normal form 253—255
Determination of character of normal form, inferences from conditions which lead to 255
Differential covariant, bilinear associated with expression 273 294
Differential covariant, bilinear associated with expression none but bilinear so associated 274
Differential relation characteristic of tangential transformation 231
Differential relation characteristic of tangential transformation, can be treated as a Pfaffian equation 233
Differential relation characteristic of tangential transformation, Mayer’s establishment of relations among functions in 233—238
Diminution, of number of equations in equivalent of a single expression when conditions are satisfied 165—169
Diminution, of number of equations in subsidiary system, when more than two are known 179
Diminution, of number of equations in subsidiary system, when one integral is known 173
Diminution, of number of equations in subsidiary system, when two integrals are known 176
Direct construction by Clebsch of equations which determine all the integrals of an expression 213
Dirksen 33
du Bois—Beymond 31 32
du Bois—Beymond’s integration of exact equation 31 32
du Bois—Beymond’s integration of exact equation compared with Natani’s 32
Effect, equations determining the, constructed direct 213
Effect, of integrals already determined, on the remainder 200 215
Effect, of vanishing determinant on result of first transformation 102 105
Engel 230 299 305 306
Engel on canonical form of two equations in four variables 326
Equations, determining the first integral of an expression, in an even number of variables, determining all the integrals 213 291
Equations, determining the first integral of an expression, in an even number of variables, some properties of the simultaneous, in Clebsch’s method 215
Equations, determining the first integral of an expression, in an even number of variables, when conditioned 203—207
Equations, determining the first integral of an expression, in an even number of variables, when normal form involves an odd number of functions 224
Equations, determining the first integral of an expression, in an even number of variables, when unconditioned 200—203
Equivalence of two expressions through normal forms 252 285 289
Equivalent, integral, minimum number of equations in, of a nonintegrable system 312—316
Equivalent, integral, of expression deduced from normal form 116 117
Equivalent, integral, of general partial differential equation of second order 320
Equivalent, integral, of irreducible differential relation 250 (note)
Equivalent, integral, of system of exact equations 38
Equivalent, integral, reduced forms 195
Equivalent, integral, relations between elements of 252
Equivalent, integral, solutions, of an exact equation 2
Eronecker 35
Euler 13 30 33 55 56 67 80 109
Euler’s, integration of an exact equation 13 109
Euler’s, integration of an exact equation, compared with Natani’s 30
Euler’s, integration of an exact equation, generalised to system of exact equations 55
Euler’s, integration of an exact equation, view of inexact equations 80
Even reduction 105
Even reduction can always be applied to expression in 2n variables 105
Even reduction, not unique 105
Exact equation, all solutions of a single, are equivalent 3
Exact equation, methods of integration of a single, Bertrand’s for three variables 18 109 226 227
Exact equation, methods of integration of a single, Bertrand’s generalised for n variables 20
Exact equation, methods of integration of a single, by integrating factor 14 25
Exact equation, methods of integration of a single, comparison of Euler’s and Natani’s 30
Exact equation, methods of integration of a single, du Bois—Beymond’s 31
Exact equation, methods of integration of a single, Euler’s 13 109
 Exact equation, methods of integration of a single, inference from Lie’s general process 270
Exact equation, methods of integration of a single, Natani’s 27
Exact equations, system of, any combination of solutions of, is a solution 38
Exact equations, system of, any combination of solutions of, is a solution, are coextensive with original system 67 69 71
Exact equations, system of, any combination of solutions of, is a solution, integrating factors of 39
Exact equations, system of, any combination of solutions of, is a solution, multiplier of 40
Exact equations, system of, any combination of solutions of, is a solution, partial differential equations satisfied by solution of 38
Exact equations, system of, methods of integration of Euler’s generalised 55
Exact equations, system of, methods of integration of Euler’s generalised, by solution of partial differential equations 67 71
Exact equations, system of, methods of integration of Euler’s generalised, Mayer’s 59
Exact equations, system of, methods of integration of Euler’s generalised, Natani’s 56
Exact equations, system of, methods of integration of Euler’s generalised, special method for two equations in four variables 62
Exact integral of system of equations, what is meant by 300 301
Exact integral of system of equations, what is meant by, conditions that it may exist 304 305
Exact integral of system of equations, what is meant by, satisfies a system of partial differential equations 302
Exact integral of system of equations, what is meant by, used to modify the system 311
Exact ordinary equations, note on and memoirs relating to 33
Exactness, Frobenius’ form of 51—54
Exactness, necessary conditions are sufficient 7—12 45—51
Exactness, necessary conditions of, for a single equation 4
Exactness, necessary conditions of, for a system 44
Exactness, number of necessary conditions independent 6
Expression in 2n variables can always be subjected to an even reduction 105
Expression in 2n+l variables can always be subjected to an odd reduction 111
Expression, completely reduced 113 114
Extensive equation, representing single equation 122
Extensive equation, representing single equation, representing system of equations 122 329
Extensive equation, representing single equation, solved 136 137
Extensive equation, representing single equation, subsidiary, transformed 132 135
Extensive variable, introduction of 121
Factor, form of, removeable from expression after transformation when it does vanish 101 105 203
Factor, form of, removeable from expression after transformation when the Pfaffian determinant does not vanish 90 99
First integral of expression if conditioned 207
First integral of expression if normal form involve odd number of functions 224
First integral of expression if unconditioned 203
First Method, Clebsch’s, of treating Pfaff’s problem 199—209
Frisiani 82
Frobenius 51 54 87 88 181 272 294 299
Frobenius, on conditions of exactness of system of equations 51—54
Frobenius’ method of treatment of Pfaff’s problem Chap. xi see
Frobenius’ method of treatment of Pfaff’s problem, abstract and general character of 87 272
Frobenius’ method of treatment of Pfaff’s problem, connection of Darboux’s method with 88
Functions, number of that can be obtained without affecting invariance of determinantal integer 282
Fundamental problems for nonintegrable systems 311
Fundamental problems for nonintegrable systems, two of the three yet unsolved 329 330
Gauss 81 112 251
Generalisation of Natani’s process for a single Pfaffian to determine number of equations in equivalent of system 313
Generalisation of Natani’s process for a single Pfaffian, inadequate to determine the equations in equivalent of system 322
Generalisation of normal forms in even number of functions 197 252
Generalisation of normal forms in odd number of functions 222 253
Generality of integral equivalent, characteristic properties of 145 311
Gradual construction of solution and normal form by Clebsch 198 199 208 215 223
Gradual construction of solution and normal form by Grassmann 141
Gradual construction of solution and normal form by Jacobi 120
Gradual construction of solution and normal form by Lie 257 268 270
Gradual construction of solution and normal form by Natani 151 157 160
Gradual construction of solution and normal form in Pfaff’s reduction 112
Grassmann 82 83 86 121 251 329
Grassmann, on system of equations 122 329
Grassmann’s method of treatment of Pfaff’s problem Chap. v see
Grassmann’s method of treatment of Pfaff’s problem, abstract and general character of 83 84
Hamburger 85 331
Hamburger’s comparison of methods of Natani and Clebsch 85
Hamilton 86 118 183
Hesse 35
Historical summary of methods of treating Pfaff’s problem Chap. iii see
Homogeneous tangential transformations 244
Homogeneous tangential transformations applied by Lie to Pfaff’s problem 252
Homogeneous tangential transformations, equations characteristic of 245
Homogeneous tangential transformations, infinitesimal, a special case of 246
Imschenetsky 33 299 307
Imschenetsky on intermediary integral of partial differential equation of second order 307
Incompletely integrable system of equations 301
Incompletely integrable system of equations modified by use of exact integrals 311
Incompletely integrable system of equations, example of, in subsidiary equations for partial of second order, together with conditions for two integrals 307—310
Incompletely integrable system of equations, number of exact integrals of 304
Independent solutions of a complete system of linear and homogeneous partial differential equations, number of 69
Inexact equations, views of, before Pfaff’s memoir 80
Infinitesimal homogeneous transformation 246
Infinitesimal homogeneous transformation, equations characteristic of 247
Initial values of variables, used by Jacobi 82 183 183
Initial values of variables, used by Jacobi, introduction of, leads to principal integrals 118
Integers associated with determinants in transformation of bilinear quantics are invariantive 277
Integers associated with determinants in transformation of bilinear quantics are invariantive, arithmetic mean of the two 278
Integral equivalent, minimum number of equations in, of single equation 145 146
Integral equivalent, minimum number of equations in, of system of equations 312—317
Integral equivalent, of Pfaffian expression deduced from normal form 116 117
Integral equivalent, of Pfaffian expression deduced from normal form of general partial differential equation of second order 320 321
Integral equivalent, of Pfaffian expression deduced from normal form of irreducible differential relation 250 (note)
Integral, an exact, of system of equations, conditions for existence 304 305
Integral, an exact, of system of equations, must satisfy partial differential equations 302
Integral, an exact, of system of equations, used to modify system 311
Integral, an exact, of system of equations, what is meant by 300 301
Integrals of Pfaffian equation, effect of, on subsidiary system when one is known 173
Integrals of Pfaffian equation, effect of, on subsidiary system when two are known 178
Integrals of subsidiary system used to diminish the system when more than two are known 179
Integrals of subsidiary system used to diminish the system when one is known 173
Integrals of subsidiary system used to diminish the system when two are known 175
Integrals of subsidiary system used to diminish the system when two are known with alternative inferences 177
Integrating factor of single exact equation, by Collet’s equations 25
Integrating factor of single exact equation, by De Morgan’s partial equation 20
Integrating factor of single exact equation, determined, by linear equations 14—17
Integrating factor of single exact equation, general form of 4
Integrating factor of single exact equation, quotient of two is a solution 4 16
Integrating factors of system of equations 39
Integrating factors of system of equations, determinant of sets of, a multiplier 40
Integrating factors of system of equations, special method for determination of, when there are four variables 62—67
Integration of extensive equation 136 137
Intermediary integral, conditions that partial differential equation of second order possess an 307—310
Interpretation, numerical, of Grassmann’s forms 124 125 133
Interrupted products 123
Interrupted products, conditions in form of, for existence of integrals fewer than general minimum 123—127
Interrupted products, interpretation of 124—127
Invariantive integers in transformation of bilinear quantics 277
Invariantive integers in transformation of bilinear quantics, arithmetic mean of two, a single sufficient invariant 278
Invariantive integers in transformation of bilinear quantics, persistence of, sufficient to ensure equivalence of two expressions 286 290
Invariantive persistence of character of normal form 249—251
Irreducible differential relation, integral equivalent of 250 (note)
jacobi 1 40 41 76 78 79 81—83 86 115 117 118 120 138 153 173 181 183 209
Jacobi’s additions to Pfaff’s theory 81 82
Jacobi’s additions to Pfaff’s theory, introduction of initial values of variables 82
Jacobi’s additions to Pfaff’s theory, simplification of Pfaff’s reduction 117—120
Jacobi’s additions to Pfaff’s theory, simplification of Pfaff’s solution of differential equation 183—187
Jacobi’s additions to Pfaff’s theory, theorem relative to fourtermed expression 83 115 226
Joachimstahl 33
JORDAN 239 298
Jordan, on multipliers 40
Jordan, on multipliers on tangential transformations 298
Kowalevski 331
Lagrange 33 242
Lagrange’s examples of cylindrical tangential transformation 242
laurent 35
Le Pont 59
legendre 242
Legendrian transformation is a tangential transformation 242
Lemmas of transformation in Clebsch’s theory 210
Lexell 33
Lie 78 82 84 86—88 168 181 183 230 248 278 294
Lie’s corollaries from tangential transformations 242—244 248
Lie’s corollaries from tangential transformations, derivable from Clebsch’s results 244
Lie’s method of treatment of Pfaff’s problem Chap. x see
Lie’s method of treatment of Pfaff’s problem, abstract and character of 86 87
Lie’s method of treatment of Pfaff’s problem, connection herewith of Darboux’s method 88
Lie’s theorem for functions in tangential transformation 237
Lie’s theorem for functions in tangential transformation, modified and simplified by Mayer 237
Linear transformations, introduced into treatment of Pfaff’s problem 274
Linear transformations, introduced into treatment of Pfaff’s problem, applied to bilinear form give two invariantive integers 277
Lipschitz 274
Mansion 183 230 306
Maximowitch 58



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