CHAPTER Ⅰ POLYNOMIALS AND THEIR MOST FUNDAMENTAL PROPERTIES 1
1．Polynomials in One Variable 1
2．Polynomials in More than One Variable 4
3．Geometric Interpretations 8
4．Homogeneous Coordinates 11
5．The Continuity of Polynomials 14
6．The Fundamental Theorem of Algebra 16
CHAPTER Ⅱ A FEW PROPERTIES OF DETERMINANTS 20
7．Some Definitions 20
8．Laplace＇s Development 24
9．The Multiplication Theorem 26
10．Bordered Determinants 28
11．Adjoint Determinants and their Minors 30
CHAPTER Ⅲ THE THEORY OF LINEAR DEPHNDENCE 34
12．Definitions and Preliminary Theorems 34
13．The Condition for Linear Dependence of Sets of Constants 36
14．The Linear Dependence of Polynomials 38
15．Geometric Illustrations 39
CHAPTER Ⅳ LINEAR EQUATIONS 43
16．Non-Homogeneous Linear Equations 43
17．Homogeneous Linear Equations 47
18．Fundamental Systems of Solutions of Homogeneous Linear Equations 49
CHAPTER Ⅴ SOME THEOREMS CONCERNING THE RANK OF A MATRIX 54
19．General Matrices 54
20．Symmetrical Matrices 56
CHAPTER Ⅵ LINEAR TRANSFORMATIONS AND THE COMBINATION OF MATRICES 60
21．Matrices as Complex Quantities 60
22．The Multiplication of Matrices 62
23．Linear Transformation 66
24．Collineation 68
25．Further Development of the Algebra of Matrices 74
26．Sets，Systems，and Groups 80
27．Isomorphism 83
CHAPTER Ⅶ INVARIANTS．FIRST PRINCIPLES AND ILLUSTRATIONS 88
28．Absolute Invariants；Geometric，Algebraic，and Arithmetical 88
29．Equivalence 92
30．The Rank of a System of Points or a System of Linear Forms as an Invariant 94
31．Relative Invariants and Covariants 95
32．Some Theorems Concerning Linear Forms 100
33．Cross-Ratio and Harmonic Division 102
34．Plane-Coordinates and Contragredient Variables 107
35．Line-Coordinates in Space 110
CHAPTER Ⅷ BILINEAR FORMS 114
36．The Algebraic Theory 114
37．A Geometric Application 116
CHAPTER Ⅸ GEOMETRIC INTRODUCTION TO THE STUDY OF QUADRATIC FORMS 118
38．Quadric Surfaces and their Tangent Lines and Planes 118
39．Conjugate Points and Polar Planes 121
40．Classification of Quadric Surfaces by Means of their Rank 123
41．Reduction of the Equation of a Quadric Surface to a Normal Form 124
CHAPTER Ⅹ QUADRATIC FORMS 127
42．The General Quadratic Form and its Polar 127
43．The Matrix and the Discriminant of a Quadratic Form 128
44．Vertices of Quadratic Forms 129
45．Reduction of a Quadratic Form to a Sum of Squares 131
46．A Normal Form，and the Equivalence of Quadratic Forms 134
47．Reducibility 136
48．Integral Rational Invariants of a Quadratic Form 137
49．A Second Method of Reducing a Quadratic Form to a Sum of Squares 139
CHAPTER Ⅺ REAL QUADRATIC FORMS 144
50．The Law of Inertia 144
51．Classification of Real Quadratic Forms 147
52．Definite and Indefinite Forms 150
CHAPTER Ⅻ THE SYSTEM OF A QUADRATIC FORM AND ONE OR MORE LINEAR FORMS 155
53．Relations of Planes and Lines to a Quadric Surface 155
54．The Adjoint Quadratic Form and Other Invariants 159
55．The Rank of the Adjoint Form 161
CHAPTER ⅩⅢ PAIRS OF QUADRATIC FORMS 163
56．Pairs of Conics 163
57．Invariants of a Pair of Quadratic Forms．Their λ-Equation 165
58．Reduction to Normal Form when the λ-Equation has no Multiple Roots 167
59．Reduction to Normal Form when ψ is Definite and Non-Singular 170
CHAPTER ⅩⅣ SOME PROPERTIES OF POLYNOMIALS IN GENERAL 174
60．Factors and Reducibility 174
61．The Irreducibility of the General Determinant and of the Symmetrical Determinant 176
62．Corresponding Homogeneous and Non-Homogeneous Polynomials 178
63．Division of Polynomials 180
64．A Special Transformation of a Polynomial 184
CHAPTER ⅩⅤ FACTORS AND COMMON FACTORS OF POLYNOMIALS IN ONE VARIABLE AND OF BINARY FORMS 187
65．Fundamental Theorems on the Factoring of Polynomials in One Variable and of Binary Forms 187
66．The Greatest Common Divisor of Positive Integers 188
67．The Greatest Common Divisor of Two Polynomials in One Variable 191
68．The Resultant of Two Polynomials in One Variable 195
69．The Greatest Common Divisor in Determinant Form 197
70．Common Roots of Equations．Elimination 198
71．The Cases a0=0 and b0=0 200
72．The Resultant of Two Binary Forms 201
CHAPTER ⅩⅥ FACTORS OF POLYNOMIALS IN TWO OR MORE VARIABLES 203
73．Factors Involving only One Variable of Polynomials in Two Variables 203
74．The Algorithm of the Greatest Common Divisor for Polynomials in Two Variables 206
75．Factors of Polynomials in Two Variables 208
76．Factors of Polynomials in Three or More Variables 212
CHAPTER ⅩⅦ GENERAL THEOREMS ON INTEGRAL RATIONAL INVARIANTS 218
77．The Invariance of the Factors of Invariants 218
78．A More General Method of Approach to the Subject of Relative Invariants 220
79．The Isobaric Character of Invariants and Covariants 222
80．Geometric Properties and the Principle of Homogeneity 226
81．Homogeneous Invariants 230
82．Resultants and Discriminants of Binary Forms 236
CHAPTER ⅩⅧ SYMMETRIC POLYNOMIALS 240
83．Fundamental Conceptions．∑ and S Functions 240
84．Elementary Symmetric Functions 242
85．The Weights and Degrees of Symmetric Polynomials 245
86．The Resultant and the Discriminant of Two Polynomials in One Variable 248
CHAPTER ⅩⅨ POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 252
87．Fundamental Conceptions．∑ and S Functions 252
88．Elementary Symmetric Functions of Pairs of Variables 253
89．Binary Symmetric Functions 255
90．Resultants and Discriminants of Binary Forms 257
CHAPTER ⅩⅩ ELEMENTARY DIVISORS AND THE EQUIVALENCE OF λ-MATRICES 262
91．λ-Matrices and their Elementary Transformations 262
92．Invariant Factors and Elementary Divisors 269
93．The Practical Determination of Invariant Factors and Elementary Divisors 272
94．A Second Definition of the Equivalence of λ-Matrices 274
95．Multiplication and Division of λ-Matrices 277
CHAPTER ⅩⅪ THE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF BILINEAR FORMS AND OF COLLINEATIONS 279
96．The Equivalence of Pairs of Matrices 279
97．The Equivalence of Pairs of Bilinear Forms 283
98．The Equivalence of Collineations 284
99．Classification of Pairs of Bilinear Forms 287
100．Classification of Collineations 292
CHAPTER ⅩⅫ THE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF QUADRATIC FORMS 296
101．Two Theerems in the Theory of Matrices 296
102．Symmetric Matrices 299
103．The Equivalence of Pairs of Quadratic Forms 302
104．Classification of Pairs of Quadratic Forms 305
105．Pairs of Quadratic Equations，and Pencils of Forms or Equations 307
106．Conclusion 313
INDEX 317