## An Elementary Treatise on the Theory of Equations |

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algebraical approximation auxiliary functions biquadratic equation changes of sign column consider constituents contour contrary signs cube roots deduce degree denote derived function Descartes's rule determinant divide divisible divisors double permanences equa equal roots equation f equation f(x example factor x factors Fourier's functions fraction given equation greater greatest common measure Hence highest power homogeneous function identity imaginary roots integer involves last term left-hand member less Let f(x multiply negative root Newton's method number of changes occur odd number original equation permutations polynomial positive integer positive quantity positive roots preceding Article proceed proposed equation quadratic elements quotient rational integral function real roots reciprocal equation remainder required equation result rows rule of signs shew shewn solution solve the equation square Sturm's functions Sturm's theorem substitute superior limit suppose symbols symmetrical function Theory of Equations tion transformed equation unity unknown quantities vanish zero

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Page vi - University Examination Papers, and the results have been given when it appeared necessary. In order to exhibit a comprehensive view of the subject, the treatise includes investigations which are not found in all the preceding elementary treatises, and also some investigations which are not to be found in any of them. For the Second Edition the work...

Page 29 - Or we may enunciate the laws thus : the coefficient of the second term with its sign changed is equal to the sum of the roots; the coefficient of the third term is equal to the sum of the products of...

Page 250 - If we consider the intrinsic beauty of the theorem, * • • the interest which belongs to the rule associated with the great name of Newton, and the long lapse of years during which the reason and extent of that rule remained undiscovered by mathematicians, among whom Maclaurin, Waring, and Euler are explicitly included, we must regard Professor Sylvester's investigations made to the theory of equations in modern times justly to be ranked with those of Fourier, Sturm, and Oauchy.

Page 58 - 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"~í + ... -px"

Page 113 - J. . 22 the equation in e should be of the sixth degree. But as the sum of the four roots of the biquadratic equation is zero by Art. 45, the sum of any two roots is equal in magnitude and opposite in sign to the sum of the remaining two roots ; and thus we see the reason why the equation in e only involves even powers of e, so that the values of e' can be found by the solution of a cubic equation. We may observe that when we have found e...

Page 71 - C whose elements are the terms of the expansion of and is to the effect (1) that there are no odd powers of t in the development of the circulant, and (2) that if in the said development £ be put for f 2 , the equation in £ C fa=f = o has for its roots the squares of the differences of the roots of the equation x

Page 21 - The mere approach of a wire forming a closed curve to a second wire through which a voltaic current flowed was then shown by Faraday to be sufficient to arouse in the neutral wire an induced current, opposed in direction to the inducing current; the withdrawal of the wire also generated a current having...

Page 45 - VARIATIONS of sign, nor the number of negative roots greater than the number of PERMANENCES. 325. Consequence. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences ; we shall have m=n+p. Moreover, let n' denote the number of positive roots, and p' the number...

Page 29 - Thus generally if pr denote as usual the coefficient of x"~r in the equation, (- 1)' ' pr = the sum of the products of every r of the roots. 46. It might appear perhaps that the relations given in the preceding Article would enable us to find the roots of any proposed equation ; for they supply equations involving the roots, and the number of these equations is the same as the number of the roots, so that it might be supposed practicable to eliminate all the roots but one and thus to determine that...

Page 57 - Let f(x) = 0 be the equation ; suppose it of the wth degree. Let p be the numerically greatest negative coefficient which occurs in f(x). Then if such a value be found for x that f(x) is positive for that value of x and for all greater values, that value is a superior limit of the positive roots of the equation f (x) = 0 ; now if any positive value of x make . positive, it will a fortiori make f(x) positive.