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

Pirmais vāks
Cambridge 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
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
65
Symmetrical Solution of Equations with Two Unknowns 6773
67
Johnsons Method 7375
73
Symmetrical Solution of Equations with Three Unknowns 7580
75
Subtraction and Division 8082
80
Solutions of Existential Expressions with One Unknown
91
Equations and Existential Propositions with Many Unknowns
97
Symbolic Equivalents of Syllogisms
103
Identification with the Algebra of Symbolic Logic
109
Propositions
115
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
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
ᎪᏒᎢ PAGES
162
CHAPTER I
171
Simple and Compound Extensive Magnitudes 177178
177
Definition of Regressive Multiplication 183184
183
Regressive Multiplication independent of ReferenceElements 190191
191
107108
199
115
206
PlaneEquation of a Quadric
213
129
219
CHAPTER V
229
First Type of Linear Construction of the Cubic
235
Third Type of Linear Construction of the Cubic
241
Chasles Construction
247
145
254
The Identical Equation
261
Skew Matrices
269
THEORY OF METRICS
273
Recapitulation of Formulæ
275
Lines in Involution
281
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
Formulæ connected with Reciprocal Systems 303304
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 P a 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
Enumeration of Types of Latent Systems and SemiLatent Groups 326338
326
Transformation of a Quadric into itself 338339
338
Direct Transformation of Quadrics 339342
339
Skew Transformation of Quadrics 342346
342
Note
346
THEORY OF DISTANCE
349
Spatial Manifolds of Many Dimensions
355
Antipodal Form
361
Parabolic Geometry
367
229231
404
Vector Systems 406407
406
Vector Systems and Parallel Lines 407408
408
Further Properties of Parallel Lines 409411
409
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 418
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
CHAPTER VIII
496
Intensities 498499
498
Congruent Transformations 500502
500
BOOK VII
503
CHAPTER I
505
Points at Infinity 506507
506
Vectors 507508
507
Linear Elements 508509
508
Vector Areas 509511
509
Vector Areas as Carriers
511
Planar Elements 512513
512
Vector Volumes
513
Product of Four Points
514
Interpretation of Formulæ 515516
515
Vector Formulæ
516
Theory of Forces 518520
518
Graphic Statics 520522
520
Note
522
VECTORS continued
523
Planes containing the Central Axis
530
Small Displacements of a Rigid Body
537
Locus of Centre of Curvature
543
PURE VECTOR FORMULÆ ART PAGES 353 Introductory 548549
548
Lengths and Areas
549
The Origin
550
System of Forces
551
A Continuously Distributed Substance 552554
552
Hamiltons Differential Operator 554555
554
Conventions and Formulæ 555557
555
Polar Coordinates 557558
557
Cylindrical Coordinates 558560
558
Orthogonal Curvilinear Coordinates 560562
560
Volume Surface and Line Integrals
562
Moving Origin 563565
563
Transformations of Hydrodynamical Equations
565
Curl Filaments of Constant Strength 567569
567
Carried Functions 569570
569
Clebschs Transformations 570572
570
Flow of a Vector 572573
572
Note
573
Index 576586
576
233235
580

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