Lessons introductory to the modern higher algebra:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Chelsea Publ.
1964
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| Edition: | 5. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 376 S. |
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| 100 | 1 | |a Salmon, George |d 1819-1904 |e Verfasser |0 (DE-588)116766719 |4 aut | |
| 245 | 1 | 0 | |a Lessons introductory to the modern higher algebra |
| 250 | |a 5. ed. | ||
| 264 | 1 | |a New York, NY |b Chelsea Publ. |c 1964 | |
| 300 | |a XV, 376 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Determinants | |
| 650 | 4 | |a Forms, Binary | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-005701191 | ||
Record in the Search Index
| _version_ | 1804122992111779840 |
|---|---|
| adam_text | CONTENTS.
LESSON I.
DETERMINANTS.—PKKUMIJTABT ILLUSTRATIONS AND DEFINITIONS.
HOE
Bale of Bigng ....... 6
Sylvester s umbral notation . 8
LESSON II.
REDUCTION AND CALCULATION OF DETERMINANTS.
Minors [called by Jacobi partial determinants] . . .10
Examples of reduction ...... 13
Product of differences of n quantities expressed as a determinant . 14
Keduction of bordered Hessians . 1?
Continuants . . . •¦ • • 18
LESSON HI.
MULTIPLICATION OF DETERMINANTS.
The theorem stated as one of linear transformation ... 21
Extension of the theorem . . . . ¦ .22
Examples of multiplication of determinants ... 23
Product of squares of differences of n quantities . . .23
Radius of sphere circumscribing a tetrahedron . . 26
Relation connecting mutual distances of points on a circle or sphere . 26
Of five points in space .... 27
Sylvester s proof that equation of secular inequalities has all roots real . 28
LESSON IT.
MINOR AND RECIPROCAL DETERMINANTS.
Relations connecting products of determinants ... 29
Solution of a system of linear equations . . . .29
Reciprocal systems ...... 80
Minors of reciprocal system expressed in terms of those of the original . 30
Minors of a determinant which vanishes .... 32
Forms for expanding a determinant of the fourth order . . 32
LESSON V.
SYMMETRICAL AND SKEW SYMMETRICAL DETERMINANTS.
Differentials of a determinant with respect to its constituents . . 84
If a symmetric determinant vanishes, the same bordered is a perfect square 37
Skew symmetric determinants of odd degree vanish . . .37
Of even degree are perfect squares .... 38
Nature of the square root [see Jacobi, Crelle, II. 354; XXIX. 236] 39
Orthogonal substitutions . . . . . .41
Number of terms in a symmetrical determinant ... 45
X CONTENTS.
LESSON VI.
DISCRIMINATING SYMMETRICAL DETERMINANTS.
PAOE
New proof that equation of secular inequalities has all its roots real . 48
Sylvester s expressions for Sturm s functions in terms of the roots . ¦ 49
Borchardt s proof ..... i 54
LESSON VII.
SYMMETRIC FUNCTIONS.
Newton s formulae for sums of powers of roots . .57
Improvement of this process ..... 57
Determinant expression for sums of powers . . .57
Rules for weight and order of a symmetric function ... 58
Formula for sum of powers of differences of roots . . .60
Differential equation of functions of differences ... 61
Symmetric functions of homogeneous equations . . .62
Differential equation when binomial coefficients are used . ¦ 64
Serret s notation . . . . • .65
LESSON VIII.
ORDER AND WEIGHT OF ELIMINANTS
Eliminants denned ..... 66
Elimination by symmetric functions . . . .67
Order and weight of resultant of two equations ... 69
Symmetric functions of common values for a system of two equations . 72
Extension of principles to any number of equations ... 75
LESSON II.
EXPRESSION OF ELIMINANTS AS DETERMINANTS.
Elimination by process for greatest common measure . . .76
Euler s method ...... 77
Conditions that two equations should have two common factors . . 78
Sylvester s dialytic method ..... 79
Bezout s method . . . . . . .81
Cayley s statement of it . . . . . • 83
Jacobians defined . . . . . . .84
Jacobian and derived equations satisfied by common system which satisfies
equations of same degree ..... 84
Expression by determinants, in particular cases, of resultant of three equations 85
Cayley s method of expressing resultants as quotients of determinants . 87
LESSON X.
DETERMINATION OF COMMON BOOTS.
Expression of roots common to a system of equations by the differentials of
the resultant ...... 91
Equations connecting these differentials when the resultant vanishes . 91
Expressions by the minors of Bezout s matrix ... 93
General expression for differentials of resultant with respect to any quantities
entering into the equations . . . . .96
General conditions that a system may have two common roots . 97
CONTENTS. XI
LESSON XL
DISCRIMINANTS.
Order and weight of discriminants * . . . .99
Discriminant expressed in terms of the roots . . . 101
Discriminant of product of two or more functions . . . 101
Discriminant is of form a,cp + a^xfr .... 102
Formation of discriminants by the differential equation . . . 103
Method of finding the equal roots when the discriminant vanishes . 104
Extension to any number of variables .... 106
Discriminant of a quadratic function .... 107
LESSON XII.
LINEAR TRANSFORMATIONS.
Invariance of discriminants ..... 108
Number of independent invariants. . . . . 110
Invariants of systems of quantics . . . . .112
Covariants .... .. 114
Every invariant of « covariant is an invariant of the original . .114
Invariants of emanants are covariants .... 116
Contravariants ...... 117,120
Differential symbols are contragredient to variables . . . 119
a£ + yi + ^c* absolutely unaltered by transformation . . . 120
Mixed concomitants ...... 121
Evectants . . . . . . .122
Evectant of discriminant of a quantic whose discriminant vanishes . 123
lesson xni.
FORMATION OP INVARIANTS AND COVARIANTS.
Method by symmetric functions ..... 124
Concomitants which vanish when two or more roots are equal . 125
Method of mutual differentiation of covariants and contravariants . . 126
Differential coefficients substituted for the variables in a contravariant give
covariants ...... 127
For binary quantics, covariants and contravariants not essentially distinct . 127
Invariants and covariants of second order in coefficients ¦ . 129
Cubinvariant of a quartic ...... 129
Every quantic of odd degree has an invariant of the 41* order . . 129
Cubicovariant of cubic ...... 130
Method of the differential equation .... 130
Weight of an invariant of given order .... 130
Binary quantics of odd degree cannot have invariants of odd order . 130
Coefficients of covariants determined by the differential equation . 131
Skew invariants ...... 131
Investigation of number of independent invariants by the differential equation 132
Source of product of two covariants is product of their sources . .134
Cayley s definition of covariants . . . . . 135
Extension to any number of variables ... . 136
xii CONTENTS.
LESSON XIV.
SYMBOLICAL REPRESENTATION OF CONCOMITANTS.
PAOB
Method of formation by derivative symbols . . . 137
Order of derivative in coefficients and in the variables . . . 140
Table of invariants of the third order .... 141
Hennite s law of reciprocity ..... 142
Derivative symbols for ternary qnantics .... 144
Symbols for evectants ... ... 146
Method of Aronhold and Clebsoh .... 147
LESSON XT.
CANONICAL FORMS.
Generality of a form examined by its number of constants . * 150
Bednction of a quadratic function to a sum of squares . . 151
Principle that the number of negative squares is unaffected by real substitution 151
Seduction of cubic to its canonical form .... 152
Discriminant of a cubic and of its Hessian differ only in sign . 153
General reduction of quantic of odd degree . . ¦ .153
Methods of forming canonizant ..... 154, 155
Condition that a quantic of order 2n be reducible to a sum of «, 2»u powers , 156
Canonical forms for quantics of even order . . . 157
Canonical forms for sextic and octavic .... 159
For ternary and quaternary cubics . . . 160
LESSON XVI.
8YSTEM8 OF QUANTICS.
Oombinants defined, differential equation satisfied by them . . 161
Number of double points in an involution .... 162
Geometrical interpretation of Jacobian . . . .162
Factor common to two quantics is square factor in Jacobian . . 162
Order of condition that u + kv may have cubic factor . . . 163
Nature of discriminant of Jacobian .... 164
Discriminant of discriminant of u + ko . . . . 165
Proof that resultant is a combinant .... 166
Discriminant with respect to x, y, of a function of «, v . . 167
Discriminant of discriminant of u+ ir for ternary quftntics . . 169
Tact invariant of two curves . . . . .169
Tact invariant of complex curves .... 170
Osculants . . . . . . .171
Covariants of a binary system connected with those of a ternary . 172
LESSON XVII.
APPLICATIONS TO BINARY QUANTICB.
Invariants when said to be distinct ..... 175
Number of independent covariants .... 175
Cayley s method of forming a complete system . . 177
The quadeic ..... 178
Resultant of two quadrics ...... 180
CONTENTS. Xlii
System op three or more quadrics .... 181
Extension to qualities in general of theorems concerning quadrics , 182
The cubic ....... 183
Geometric meaning of covariant cubic .... 184
Square of this cubic expressed in terms of the other covariants . 186, 192
Solution of cubic . . . , . . .186
System op cubic and quadkio .... 187, 225
Geometrical illustrations ...... 189
The quartic ....... 189
Catalecticants ....... 190
Discriminant of a quartic ..... 190
Relation of covariants of cubic derived from that of invariants of a quartic . 191
Sextic covariant geometrical meaning of . . . 193
Relations connecting quadratic factors of . . , 194
Seduction of quartic to its canonical form . . . 194
Relation connecting covariants of quartic .... 195
Symmetrical solution of quartic ..... 196
Criteria for real and imaginary roots ..... 197
The quartic can be brought to its canonical form by real substitutions . 197
Conditions that a quartic should have two square factors . . 198
Cayley s proof that the system of invariants and covariants is complete . 199
Application of Burnside s method ..... 200
Covariants of system of quartic and its Hessian . . . 201
Hessian of Hessian of any quantic ..... 202
System op quadric and quartic .... 202, 267
System op two cubics ...... 204
Resultant of the system ..... 205
Condition that u + v may be a perfect cube .... 205
Mode of dealing with equations which contain a superfluous variable . 207
Jacobian and simplest linear covariants .... 209
Any two cubics may be regarded as differential coefficients of same quartic . 210
Invariants of invariants of « + v are combinants . . .211
Process of obtaining concomitant of system from concomitant of single quantic 212
Complete list of covariants of system . . . . 213
Plane geometrical illustration of system of two cubics . . . 214
System op pour cubics ..... 215
Illustration of twisted cubics ..... 216
System op quartic and cubic ..... 218, 226
System op two quartics ..... 219
Their resultant ...... 220
Condition that u + Xt Bhonld be perfect square . . . 220
Condition that u + v should have cubic factor ... 221
Special form when both quartics are sums of two fourth powers . . 223
Three quadrics derived functions of a single quartic . . . 224
Three quadrics quadric covariants of two cubics . . . 225
lesson xvni.
applications to higher binary quantics.
The quintic ....... 227
Canonical form of quintic ...... 227
Condition that two quartics be first differentials of the same quintic . 228
XIV CONTENTS.
PAOI
Discriminant of quintic ...... 229
Fundamental invariants of quintic .... 232
Conditions for two pairs of equal roots .... 233
All invariants of a quantic vanish if more than half its roots be all equal . 233
Hermite s canonical form ...... 233
Hermite s skew invariant ..... 233,282
Its geometrical meaning ...... 234
Covaiimts of quintic for canonical form .... 235
Cayley s arrangement of these forms .... 237
Cayley s canonical form ..... 239
Sign of discriminant of any quantic determines whether it has an odd or even
number of pairs of imaginary roots .... 239
Criteria furnished by Sturm s theorem for a quintic . . . 240
If roots all real, canonizant has imaginary factors . . . 241
Invariant expression of criteria for real roots . . . 241
Sylvester s criteria ....... 245
Conditions involving variation within certain limits of a constant . 246
Cayley s modification of Sylvester s method .... 248
Hermite a forme type ...... 249
The Tschirnhausen transformation ..... 260
Modified by Hermite and Cay ley . . . . 251
Applied to quartic ....... 252
Applied to quintic ...... 254
Bextic resolvent of a quintic ..... 257
Harley s and Cockle s resolvent ..... 257
Expression of invariants in terms of roots .... 258
The sextic—its invariants and simplest covariants . . 260
Conditions for cubic factor or for two square factors . . . 263
The discriminant ...... 263
Simplest quartic covariant ...... 266
Quadric covariants ...... 268
The skew invariant expressed in terms of other invariants . . 269
Functions likely to afford criteria for real roots . . . 271
System of two Quartics ...... 271
Jacobian identified with any sextic by means of a quintic . , 273
Functional determinant of three quartics .... 274
Can be similarly identified ..... 275
New canonical forms of sextic by Brill .... 276
Also by Stephanos ...... 277
Factors of discriminant of Jacobian ..... 278
Sextic covariant of third order in coefficients . . . 279
Canonical form referred to ternary system . . . .281
Condition for sextio to be sextic covariant of quartic . . 282
to be Hessian of quintic .... 282
LESSON XIX.
ON THE ORDER OP RESTRICTED SYSTEMS OP EQUATIONS.
Order and weight of systems defined .... 284
Restricted systems ....... 285
Determinant systems, k rows, k + 1 columns . . . 287
Order and weight of conditions that two equations have two common roots . 291
CONTENTS. XV
PAOE
System of conditions that three equations should have a common root . 293
Systems of conditions that equation have cubic factor or double square factor 294
Intersection of quantics having common curves .... 295
Case of distinct common curves ..... 297
Number of quadrics passing through five points and touching four planes . 298
Rank of curve represented by a system of k rows, k + 1 columns . 299
System of conditions that two equations should have three common roots . 300
System of quantics having a surface common . . . 301
Having common surface and curve . . . 303
Having common two surfaces .... 305
System of conditions that three ternary qualities have two common points 306, 309
Bule when the constants in systems of equations are connected by relations 307
Number of curve triplets having two common points . . 310
Mr. S. Boberts method . . . . . .310
LESSON XX.
APPLICATIONS OF SYMBOLICAL METHODS.
Symbolical expression for invariant or covariant . . , 314
Clebsch s proof that every covariant can be so expressed . . 315
FormuUe of transformation ..... 316
Beduction to standard forms ..... 318
Transvection ....... 320
Symbolical expression for derivative of derivative . . . 321
Forms of any order obtained by transvection from forms of lower order 323
Gordan and Clebsch s proof that the number of irreducible covariants is finite 324
Every invariant symbol has (ab)P as a factor, where p is at least half n . 326
Symbolical expression for resultant of quadratic and any equation . 326
Investigation of equation of inflexional tangents to cubic . . 3301
Application of symbolical forms to theory of double tangents to plane curves . 333
Typical exposition of an even binary quantic . . . 335
of a quantic of order 3/ . . . 337
NOTES.
History of determinants ...... 338
Gommutants ....¦• 339
On rational functional determinants .... 340
Hessians . ...... 341
Symmetric functions ...... 342
Elimination ....... 342
Discriminants ...... 342
Bezouti nts ....... 343
linear transformations ...... 343
Canonical forms ....... 345
Combinants ....... 345
Applications to binary quantics ..... 345
Table of transvectants . . . . ¦ . 346
M. Roberts table of sums of powers of differences . . . 347
Table of resultants ...... 348
Hirsch and Cayley s tables of symmetric functions . . .360
Index ....... 373
|
| any_adam_object | 1 |
| author | Salmon, George 1819-1904 |
| author_GND | (DE-588)116766719 |
| author_facet | Salmon, George 1819-1904 |
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| ctrlnum | (OCoLC)1302860 (DE-599)BVBBV008660825 |
| dewey-full | 512.87 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.87 |
| dewey-search | 512.87 |
| dewey-sort | 3512.87 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 5. ed. |
| format | Book |
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| indexdate | 2024-07-09T17:22:41Z |
| institution | BVB |
| language | English |
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| spelling | Salmon, George 1819-1904 Verfasser (DE-588)116766719 aut Lessons introductory to the modern higher algebra 5. ed. New York, NY Chelsea Publ. 1964 XV, 376 S. txt rdacontent n rdamedia nc rdacarrier Determinants Forms, Binary HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005701191&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Salmon, George 1819-1904 Lessons introductory to the modern higher algebra Determinants Forms, Binary |
| title | Lessons introductory to the modern higher algebra |
| title_auth | Lessons introductory to the modern higher algebra |
| title_exact_search | Lessons introductory to the modern higher algebra |
| title_full | Lessons introductory to the modern higher algebra |
| title_fullStr | Lessons introductory to the modern higher algebra |
| title_full_unstemmed | Lessons introductory to the modern higher algebra |
| title_short | Lessons introductory to the modern higher algebra |
| title_sort | lessons introductory to the modern higher algebra |
| topic | Determinants Forms, Binary |
| topic_facet | Determinants Forms, Binary |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005701191&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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