A Treatise on Universal Algebra: With Applications, Volume 1

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Contents

CHAPTER I
3
CHAPTER II
13
CHAPTER III
18
Equivalence I819
19
Addition 2122
21
Principles of Subtraction 2224
22
The Null Element 2425
24
Steps
25
The Conjugate Invariant of a Triplo Group 310312
310
Transformations of O p p and G 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

Orders of Algebraic Manifolds 2728
27
The Null Element 2829
28
BOOK III
30
Classification of Special Algebras 2932
31
Note
32
BOOK II
33
Formal Laws 3537
35
Supplements
36
Reciprocity between Addition and Multiplication 3738
37
Interpretation 3839
38
Elementary Propositions 3941
39
Classification 4142
41
Incident Regions 4244
42
CONTENTS XV
43
CONTENTS xxi
44
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
CHAPTER III
83
Umbral Letters 8689
86
Elimination 8991
89
Solutions of Existential Expressions with One Unknown 9192
91
Existential Expressions with Two Unknowns 9394
93
Equations and Existential Expressions with One Unknown 94 96
94
Booles General Problem 9697
96
Equations and Existential Propositions with Many Unknowns 9798
97
Note
98
CHAPTER IV
99
Exclusion of Nugatory Forms 100101
100
Syllogism 101103
101
Symbolic Equivalents of Syllogisms 103105
103
Generalization of Logic 105106
105
CHAPTER V
107
Equivalent Propositions 54 Symbolic Representation of Complexes
108
Identification with the Algebra of Symbolic Logic
109
Existential Expressions 57 Symbolism of the Traditional Propositions
111
Primitive Predication 59 Existential Symbols and Primitive Predication
113
Propositions Historical Note
115
AST PAOK8 61 Introductory
119
Things representing Different Elements 121122
121
Fundamental Propositions 122125
122
Subregions 125128
127
Loci 128130
128
Surface Loci and Curve Loci 130131
130
Note
131
CHAPTER II
132
Homographic Ranges
133
BOOK VI
135
Elementary Properties 136
136
ReferenceFigures 138 139
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 151163
151
Closed Quadrics 153155
154
Conical Quadric Surfaces 156157
156
Reciprocal Equations and Conical quadrics 157161
157
Note
161
ABT FAOEB 85 Defining Equation of Intensity 162163
162
Locus of Zero Intensity 163164
163
Plane Locus of Zero Intensity 164166
164
Quadric Locus of Zero Intensity
166
The Intercept between Two Elements 167168
167
Note
168
BOOK IV
169
CHAPTER I
171
Invariant Equations of Condition 172173
172
Principles of Combinatorial Multiplication 173175
173
Derived Manifolds 175176
175
Extensive Magnitudes 176177
176
Simple and Compound Extensive Magnitudes 177178
177
Fundamental Propositions 178180
178
Note
180
CHAPTER II
181
Definition of Regressive Multiplication 183184
183
Pure and Mixed Products 184185
184
Rule of the Middle Factor 185188
185
Extended Rule of the Middle Factor 188190
188
Regressive Multiplication independent of ReferenceElements 190191
191
Mullens Theorems 192195
192
Applications and Examples 195198
195
Note
198
108 108
199
Extension of the Definition of Supplements 201202
201
111
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 AST 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 Rented 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 Formula 274275
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
Formula for Systems of Forces 282283
282
ART PACKS 168 Sixscifications 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
Formulae connected with Reciprocal Systems 303304
303
Systems Reciprocal to a Dual Group
304
The Polo and Polar Invariants of a Triplo Group 305306
305
Conjugate Sets of Systems and the Polo and Polar Invariants 306307
306
Interpretation of I1 x and PX 307308
307
Relations between Conjugate Sets of Systems 308310
308
Direct Transformation of Quadrics 339342
339
Skew Transformation of Quadrics 342346
342
Note
346
Theory of Distance
349
ABT PAGES 197 Axioms of Distance 349351 j 198 Congruent Ranges of Points 350351
350
Cayleys Theory of Distance 351353
353
Comparison with the Axioms of Distance
354
Division of Space 355356
355
Elliptic Space
356
Length of Intercepts in Polar Form 358361
358
Antipodal Form 361362
361
Hyperbolic Space 362
362
The Space Constant 3633C4
363
Law of Intensity in Elliptic and Hyperbolic Geometry 364
364
Distances of Planes and of Subregions 365 367
365
Parabolic Geometry 367368
367
Law of Intensity in Parabolic Geometry 368369
368
Historical Note 369370
369
CHAPTER II
371
Further Formula for Triangles 374375
374
Points inside a Triangle 375376
375
Oval Quadrics 376378
376
Further Properties of Triangles 378379
378
Planes Onesided 379382
379
Angles between Planes
382
Stereometrical Triangles 382383
383
Shortest Distances from Points to Planes 385
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 PAGK8 230 Intensities of Forces 399400
399
Relations between Two Forces 400401
400
Axes of a System of Forces 401404
401
NouAxal Systems of Forces
404
Parallel Lines 404406
405
Vector Systems 406407
407
Vector Systems and Parallel Lines 407408
408
Further Properties of Parallel Lines 409411
411
Planes and Parallel Linos 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 Linos 418419
418
Distances of Subrcgions
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 Subregious 430433
430
Rectangular Rectilinear Figures 433436
433
Parallel Lines 436438
436
Parallel Planes 439440
439
CHAPTER V
441
Intersection of Spheres 444147
444
LimitSurfaces 447448
447
Great Circles on Spheres 448451
448
AST PAQES 263 Surfaces of Equal Distance from Subregions
451
Intensities of Forces
452
Central Axis of a System of Forces 454465
454
NonAxal Systems of Forces
455
CHAPTER VI
456
Elementary Formulas 458459
458
27C 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 Formula 481482
481
Velocity and Acceleration 482484
482
The Circle 484187
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 498
498
Congruent Transformations 500
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 613
513
Product of Four Points
514
Interpretation of Formulm 515516
515
Vector Formulae 616
516
Theory of Forces 518520
518
Graphic Statics 520622
520
Note 622
522
Vectors continued ART PAGES 320 Supplements 523524
523
Rectangular Normal Systems
524
Real SelfNormal Sphere 525526
525
Geometrical Formula 526527
526
Taking the Flux 527528
527
Flux Multiplication
528
Geometrical Formula
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
AVork 537538
537
CHAPTER III
539
Osculating Plane and Normals
540
Simplified Formula
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 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 Formula 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

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