A Treatise on Universal Algebra: With Applications, 1. sējums

Pirmais vāks
The University Press, 1898 - 586 lappuses
 

Saturs

The Null Element 2829
28
Classification of Special Algebras 2932
29
POSITIONAL MANIFOLDS
30
Note
32
BOOK II
33
CHAPTER I
35
SUPPLEMENTS
36
Reciprocity between Addition and Multiplication 3738
37
Interpretation 3839
38
Elementary Propositions 3941
39
Classification 4142
41
Incident Regions 4244
42
CHAPTER II
45
Elimination 4755
47
Solution of Equations with One Unknown 5559
55
On Limiting and Unlimiting Equations 5960
59
On the Fields of Expressions 6065
60
Solution of Equations with More than One Unknown 6567
67
Johnsons Method 7375
73
Symmetrical Solution of Equations with Three Unknowns 7580
75
Subtraction and Division 8082
80
CHAPTER IV
99
Exclusion of Nugatory Forms
100
CHAPTER V
107
Primitive Predication
113
CHAPTER I
119
Things representing Different Elements 121122
121
Fundamental Propositions 122125
122
Subregions 125128
125
Loci 128130
128
Surface Loci and Curve Loci 130131
130
Note
131
CHAPTER II
132
Homographic Ranges
133
BOOK VI
135
Elementary Properties 136137
136
ReferenceFigures 138139
138
Perspective 139142
139
Quadrangles 142143
142
CHAPTER III
144
Poles and Polars 145147
145
Generating Regions 147148
147
Conjugate Coordinates 148151
148
Quadriquadric Curve Loci 151153
151
Closed Quadrics 153155
153
Conical Quadric Surfaces 155157
155
Reciprocal Equations and Conical quadrics 157161
157
Note
161
Defining Equation of Intensity
162
CHAPTER I
171
Simple and Compound Extensive Magnitudes
177
Pure and Mixed Products
184
Regressive Multiplication independent of ReferenceElements
191
107108
199
Extension of the Definition of Supplements 201202
201
Different kinds of Supplements
202
Mutually normal Regions 203204
203
Selfnormal Elements 204206
204
Selfnormal Planes
206
Inner Multiplication
207
Elementary Transformations
208
Inner Multiplication of Normal Regions
209
Quadrics 210212
210
PlaneEquation of a Quadric 212213
212
CHAPTER IV
214
Illustration of Method
215
Grassmanns Constructions 219223
219
Projection 224228
224
CHAPTER VI
228
CHAPTER V
229
Further Transformations 231233
231
Linear Construction of Cubics
233
Linear Construction of Cubic through Nine arbitrary Points 235237
235
Second Type of Linear Construction of the Cubic 238239
238
Third Type of Linear Construction of the Cubic 239244
239
Fourth Type of Linear Construction of the Cubic 244246
244
Chasles Construction 246247
246
MATRICES ART PAGES 140 Introductory
248
Sums and Products of Matrices 250252
250
Associated Determinant
252
Latent Points 254255
254
SemiLatent Regions
256
The Latent Region of a Repeated Latent Root 257258
257
The First Species of SemiLatent Regions 258259
258
The Higher Species of SemiLatent Regions 259261
259
The Identical Equation
261
Symmetrical Matrices 262265
262
Symmetrical Matrices and Supplements 265267
265
Skew Matrices 267269
267
BOOK V
271
CHAPTER I
273
Recapitulation of Formulæ 274275
274
Inner Multiplication 275276
275
Elementary Properties of a Single Force
276
Condition for a Single Force
277
Null Lines Planes and Points
278
Properties of Null Lines 279280
279
Lines in Involution 280281
280
Reciprocal Systems 281282
281
Formulæ for Systems of Forces 282283
282
GROUPS OF SYSTEMS OF FORCES ART PAGES 168 Specifications of a Group 284285
284
Systems Reciprocal to Groups
285
Common Null Lines and Director Forces
286
Quadruple and Dual Groups 287290
287
Anharmonic Ratio of Systems 290292
290
SelfSupplementary Dual Groups 292294
292
Triple Groups 295298
295
Conjugate Sets of Systems in a Triple Group 298299
298
CHAPTER III
300
The Harmonic Invariants of a Dual Group 301302
301
Further Properties of Harmonic Invariants 302303
302
181
303
Systems Reciprocal to a Dual Group
304
The Pole and Polar Invariants of a Triple Group 305306
305
Conjugate Sets of Systems and the Pole and Polar Invariants 306307
306
Interpretation of Px and PX 307308
307
Relations between Conjugate Sets of Systems 308310
308
The Conjugate Invariant of a Triple Group 310312
310
Transformations of G p p and G P P 312315
312
CHAPTER IV
316
Enumeration of Types of Latent and SemiLatent Regions 317321
317
Matrices and Forces 322323
322
Latent Systems and SemiLatent Groups 323326
323
Kleins Theorem 353354
353
Comparison with the Axioms of Distance
354
Spatial Manifolds of Many Dimensions 354355
355
Elliptic Space
356
Length of Intercepts in Polar Form 358361
358
Antipodal Form 361362
361
Hyperbolic Space 362363
362
The Space Constant 363364
363
Law of Intensity in Elliptic and Hyperbolic Geometry 364365
364
Distances of Planes and of Subregions 365367
365
Parabolic Geometry 367368
367
Law of Intensity in Parabolic Geometry 368369
368
Historical Note 369370
369
CHAPTER II
371
Further Formulæ for Triangles 374375
375
Oval Quadrics 376378
376
Further Properties of Triangles 378379
378
Planes Onesided 379382
379
Angles between Planes
382
Perpendiculars 383385
383
Shortest Distances from Points to Planes 385386
385
Common Perpendicular of Planes
386
Distances from Points to Subregions 387388
387
Shortest Distances between Subregions 388391
388
Spheres 391396
391
Parallel Subregions 397398
397
EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY ART PAGES 230 Intensities of Forces 399400
399
Relations between Two Forces 400401
400
Axes of a System of Forces 401404
401
NonAxal Systems of Forces
404
Vector Systems 406407
406
Vector Systems and Parallel Lines 407408
408
Further Properties of Parallel Lines 409411
411
Planes and Parallel Lines 411413
412
CHAPTER IV
414
Intensities of Points and Planes 415416
415
Distances of Points 416417
416
Distances of Planes 417418
417
Spatial and Antispatial Lines 418419
418
Distances of Subregions
419
Geometrical Signification
420
Points on the Absolute
422
Properties of Angles of a Spatial Triangle 424425
424
Stereometrical Triangles 425426
425
Perpendiculars 426427
426
The Feet of Perpendiculars 427428
427
Distance between Planes 428429
428
Shortest Distances 429430
429
Shortest Distances between Subregions 430433
430
Rectangular Rectilinear Figures 433436
433
Parallel Lines 436438
436
Parallel Planes 439440
439
CHAPTER V
441
Intersection of Spheres 444447
444
LimitSurfaces 447448
447
Great Circles on Spheres 448451
448
ART PAGES 263 Surfaces of Equal Distance from Subregions
451
Intensities of Forces
452
Central Axis of a System of Forces 454455
454
NonAxal Systems of Forces
455
CHAPTER VI
456
Elementary Formulæ 458459
458
Simple Geometrical Properties 459460
459
Translations and Rotations 460462
460
Locus of Points of Equal Displacement 462463
462
Equivalent Sets of Congruent Transformations
463
Commutative Law
464
Small Translations and Rotations 465466
465
Associated System of Forces
466
Properties deduced from the Associated System 467468
467
Work 468469
468
Characteristic Lines
470
Surfaces of Equal Displacement
472
Associated Vector Systems of Forces
473
Small Displacements 476477
476
CHAPTER VII
478
Curvature and Torsion 479481
479
Planar Formulæ 481482
481
Velocity and Acceleration 482484
482
The Circle 484487
484
Motion of a Rigid Body 487488
487
Gauss Curvilinear Coordinates 488489
488
Curvature of Surfaces 489490
489
Lines of Curvature 490493
490
Meuniers Theorem
493
Curvilinear Coordinates
494
Plane Equation of the Absolute
496
CHAPTER I
505
Vector Areas as Carriers
511
VECTORS continued ART PAGES 320 Supplements 523524
523
Rectangular Normal Systems
524
Real SelfNormal Sphere 525526
525
Interpretation of Formulæ
526
Taking the Flux 527528
527
Flux Multiplication
528
Geometrical Formulæ
529
Planes containing the Central Axis
530
Invariants of a Dual Group
531
The Cylindroid 532533
532
The Harmonic Invariants
533
The Pole and Polar Invariants 534535
534
Equation of the Associated Quadric
535
Small Displacements of a Rigid Body 536537
536
Work 537538
537
CHAPTER III
539
Osculating Plane and Normals
540
Simplified Formulæ
541
Locus of Centre of Curvature 542543
542
Gauss Curvilinear Coordinates 543544
543
Curvature 544545
544
Lines of Curvature 545546
545
Dupins Theorem 546547
546
Eulers Theorem
547
Introductory
549
Hamiltons Differential Operator
555
Vector Formulæ
558
Volume Surface and Line Integrals
562
372
569
Operation of Taking the Vector
575
Index
577
111112
578
512513
586

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