THE THEORY OF EQUATIONS: WITH AN INTRODUCITON TO THE THEORY OF BINARY ALGEBRAIC FORMS (Google eBook)

Front Cover
1881
0 Reviews
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Selected pages

Contents

Graphic representation of a polynomial
14
CHAPTER II
19
12 13 14 Theorems relating to the real roots of equations 15 Existence of a root in the general equation Imaginary roots 16 Theorem determining the...
25
Imaginary roots enter equations in pairs
27
Descartesrule of signs for positive roots
28
Art Page 20 Descartes rule of signs for negative roots
30
Theorem relating to the substitution of two given numbers for the variable
31
Examples
32
CHAPTER III
35
Applications of the theorem
36
25
41
Depression of an equation when a relation exists between two of its roots
42
26
43
Symmetric functions of the roots Examples
47
Remark referring to symmetric functions Miscellaneous examples
54
CHAPTER IV
60
Eoots multiplied by a given quantity
61
Beciprocal roots
62
Reciprocal equations
63
To increase or diminish the roots by a given quantity
64
Removal of terms
67
Binomial coefficients
68
The cubic
71
The biquadratic
73
Homographic transformation
75
Transformation in general
76
Equation of differences of a cubic
77
Criterion of the nature of the roots of a cubic
80
Formation of the equation whose roots are any powers of the roots of the proposed equation
82
Miscellaneous examples
84
SOLUTION OF RECIPROCAL AND BINOMIAL EQUATIONS Art Page 45 Reciprocal equations
88
4652 Binomial equations Propositions embracing their leading general properties
90
The special roots of the equation xn 1 0
93
Solution of binomial equations by circular functions
96
Miscellaneous examples
98
CHAPTER VI
101
The algebraic solution of the cubic equation
105
Expression of the cubic as the difference of two cubes
107
Rule with regard to signs
108
Solution of the cubic by symmetric functions of the roots
109
Examples
111
Minor determinants Definitions
113
Development of determinants
114
Homographic relation between two roots of a cubic
115
First solution by radicals of the biquadratic Eulers assumption
116
Second solution by radicals of the biquadratic
121
Eesolution of the quartic into its quadratic factors Ferraris solution
122
Linear homogeneous equations
123
Reciprocal determinants
124
Skew symmetrical determinants
125
Eesolution of the quartic into its quadratic factors Descartes solutioi
126
Transformation of the biquadratic into the reciprocal form
128
Solution of the biquadratic by symmetric functions of the roots
132
Equation of the squares of the differences of the roots of a biquadratic
135
Criterion of the nature of the roots of a biquadratic
137
Miscellaneous examples
138
CHAPTER VII
145
Theorem relating to the maxima and minima values of a polynomial
146
Eolles theorem Corollary
148
Theorem relating to multiple roots
149
Determination of multiple roots
150
Theorems relating to the passage of the variable through a root of the equation
152
Miscellaneous examples
154
CHAPTER VIII
156
Limits of roots Prop II
157
Practical applications
159
Sturms theorem
174
Sturms theorem Equal roots
179
Application of Sturms theorem
182
Art Page
184
Conditions for the reality of the roots of an equation
185
Criterion of the nature of the roots of the biquadratic
187
Miscellaneous examples
188
CHAPTER X
189
Theorem relating to commensurable roots
190
Newtons method of divisors
191
Application of the method of divisors
192
Method of limiting the number of trial divisors
195
Determination of multiple roots
197
Newtons method of approximation
199
Horners method of solving numerical equations
201
Principle of the trialdivisor in Homers method
205
Contracted method of applying Horners process
209
Application of Horners method to cases where roots are nearly equal
212
Lagranges method of approximation
215
Numerical solution of the biquadratic by Descartes method
216
Miscellaneous examples 219
219
Props I
220
Page 221
224
Newtons theorem relating to the sums of the powers of the roots Prop I
266
Expression of a rational symmetric function of the roots in terms of the coefficients Prop II
268
Further proposition relating to the expression of sums of powers of roots in terms of the coefficients Prop III
270
Expression of the coefficients in terms of the sums of the powers of the roots
271
Definitions of order and weight of symmetric functions and theorem relating to the former
274
Calculation of symmetric functions of the roots
277
Brioschis equation connecting the sums of the powers of the roots and the coefficients
279
Derivation of new symmetric functions from a given one
282
Operation involving the sums of the powers of the roots Theorem
285
Miscellaneous examples
286
CHAPTER XIII
291
Elimination by symmetric functions
292
Properties of the resultant
293
Eulers method of elimination
295
Sylvesters dialytic method of elimination
296
Bezouts method of elimination
297
The common method of elimination
302
Discriminants
304
Determination of a root common to two equations
306
Symmetric functions of the roots of two equations
307
Miscellaneous examples
308
Definitions
311
Formation of covariants and invariants
313
Properties of covariants and invariants
314
Examples
316
Formation of covariants hy the operator D
317
Roberts theorem relating to covariants
319
Homographic transformation applied to the theory of covariants
320
Reduction of homographic transformation to a double linear transforma tion
322
Properties of covariants derived from linear transformation
323
154157 Propositions relating to the formation of invariants and covariants of quantics transformed by a linear transformation
326
Miscellaneous examples
332
CHAPTER XV
339
Number of covariants and invariants of the cubic
342
Thequartic its covariants and invariants
344
Expression of the quartic itself by the quadratic factors of the sextic
347
CHAPTER
357
Removal of the second third and fourth terms from an equation of
363
CHAPTER XVII
370
Variation of the argument of the function corresponding to the descrip
376
Proof of the fundamental theorem that every equation of the nth degree
379
Determinants
387

Common terms and phrases

Popular passages

Page 175 - Series, f(x\ f'(x), ri(x), j"2(x), . . . , rn(x), the difference between the number of changes of sign in the series when a is substituted for x and...
Page 135 - The first method which suggests itself is one similar to that usually given to determine the coefficients of the equation whose roots are the squares of the differences of the roots of any given equation.
Page 28 - This rule, which enables us, by the mere inspection of a given equation, to assign a superior limit to the number of its positive roots, may be enunciated as follows : — No equation can have more positive roots than it has changes of sign from + to—, and from — to +, in the terms of its first member. We shall content ourselves for the present with the proof...
Page 190 - We have, therefore, a fraction in its lowest terms equal to an integer, which is impossibleHence j- cannot be a root of the equation. The real roots of the equation, therefore, are either integers or incommensurable quantities. Every equation whose coefficients are finite numbers, fractional or not, can be reduced to the form in which the coefficient of the first term is unity and those of the other terms whole numbers (Art. 31) ; so that in this way, by the aid of a simple transformation, the determination...
Page 36 - The coefficient p, of the fourth term with its sign changed is equal to the sum of the products of the roots taken three by three ; and so on, the signs of the coefficients...
Page 20 - Every equation of an odd degree has at least one real root ; and if there be but one, that root must necessarily have a contrary sign to that of the last term. - 4".
Page 389 - An Outline of the Necessary Laws of Thought ; a Treatise on Pure and Applied Logic. By W. THOMSON, DD Crown Svo.
Page 226 - In the Propositions of the present and following Articles are contained the most important elementary properties of determinants which, by the aid of Cauchy's notation above described, render the employment of these functions of such practical advantage. PROP. I. — If any two rows, or any two columns, of a determinant be interchanged, the sign of the determinant is changed. This follows at once from the mode of formation (Rule (2), Art. 108), for an interchange of two rows is the same as an interchange...
Page 79 - The expression in brackets is called the discriminant of the cubic, and is represented by A ; giving the identities EXAMPLES.
Page 157 - If each negative coefficient be taken positively and divided by the sum of all the positive coefficients which precede it, the greatest of all the fractions thus formed increased by unity, is a superior limit of lhe positive roots. Let the equation be f(x) = 0, where f(x) denotes p,l'•K" + plx"~l+p,xn~13-pllx"~a + p,x"~ν + ... -px"

Bibliographic information