there must be an intermediate value for which A becomes equal to B, and the value which produces this result is a root of the equation, since it verifies A-B=0, or the proposed equation. Hence, the proposition is proved. In the preceding demonstration, p and q have been supposed to be positive numbers; but the proposition is not less true, whatever may be the signs with which p and q are affected. For we will remark, in the first place, that the above reasoning applies equally to the case in which one of the numbers p and q, p for example, is 0; that is, it could be proved, in this case, that there was at least one real root between 0 and q. Let both p and q be negative, and represent them by -p' and -g. If, in the equation x+Pxm-1+Qx-3+... Tx+U=0, we change x into -y, which gives the transformation (−y)+P(—y)m-1+Q(−y)m2+... T(−y)+U=0, it is evident that substituting -p' and -q' in the proposed equation, amounts to the same thing as substituting p' and q' in the transformation, for the results of these substitutions are in both cases and (−p')TM+P(—p')m-1+Q(−p')m-2 + ... T(−p')+U, m (-q')+P(-q′)TM-1+Q(−q′)TM-2+ ., . T(−q')+U ; Now, since p and q, or -p' and -q', substituted in the proposed equation, give results with contrary signs, it follows that the numbers p' and q', substituted in the transformation, also give results with contrary signs; therefore, by the first part of the proposition, there is at least one real root of the transformation contained between p' and q'; and in consequence of the relation x=- -y, there is at least one value of a comprehended between -p' and -q', or p and q. This demonstration applies to cases in which p=0 or q=0. Lastly, suppose p positive and q negative or equal to -g: by making x 0 in the equation, the first member will reduce to its last term, which is necessarily affected with a sign contrary to that of p, or to that of -q'; whence we may conclude that there is a root comprehended between 0 and p, or between 0 and -q', and consequently between p and -q. Second Principle. 311. When two numbers, substituted in place of x, in an equation, give results affected with contrary signs, we may conclude that there is at least one real root comprehended between them, but we are not certain that there are no more, and there may be any odd number of roots comprised between them. We therefore enunciate the second principle thus. When an uneven number (2n+1) of the real roots of an equation, are comprehended between two numbers, the results obtained by substituting these numbers for x, are affected with contrary signs, and if they comprehend an even number 2n, the results obtained by their substitution are necessarily affected with the same sign. To make this proposition as clear as possible, denote those roots of the proposed equation, X=0, which are supposed to be comprehended between p and q, by a, b, c, ., and by Y, the product of the factors of the first degree, with reference to x, corresponding both to those real roots which are not comprised between them and to the imaginary roots; the signs of p and q being arbitrary. The first member, X, can be put under the form (x—a) (x—b) (x—c)... ×Y. Now substitute in X, or the preceding product, p and q in place of g x; we shall obtain the two results (p—a) (p—b) (p−c)... ×Y', Y' and Y" representing what Y becomes, when we replace x by p and q; these two quantities are necessarily affected with the same sign, for if they were not, by the first principle Y=0 would give at least one real root comprised between p and q, which is contrary to the hypothesis. To determine the signs of the above results more easily, divide the first by the second, we obtain Now, since the roots a, b, c,. ⚫ are comprised between p and q, hence, since p - a and ト -0. -a are affected with contrary signs, as well as p-b and q-b, p-c and q-c. . ., the partial quotients Y' are all negative; moreover is essentially positive, since Y' and Y" are affected with the same sign; therefore the product will be negative, when the number of roots, a, b, c · .., compre hended between p and q, is uneven, and positive when the number is even. Consequently, the two results (p-a) (p—b) (p—c). X Y! and (q—a) (q—b) (q—c) · XY", will have contrary or the same signs, according as the number of roots comprised between p and q is uneven or even. Limits of the real Roots of Equations. 312. The different methods for resolving numerical equations, consist generally in substituting particular numbers in the proposed equation, in order to discover if these numbers verify it, or whether there are roots comprised between these numbers. But by reflect. ing a little upon the composition of the first member, the first term being positive, and affected with the highest power of x, which is greater with respect to that of the inferior degree in proportion to the value of x, we are sensible that there are certain numbers, above which it would be useless to substitute, because all of these numbers would give positive results. 313. Every number which exceeds the greatest of the positive roots of an equation, is called a superior limit of the positive roots. From this definition, it follows that the limit is susceptible of an infinite number of values; for when a number is found to exceed the greatest positive root, every number greater than this, is, for a still stronger reason, a superior limit. determine the simplest possible limit. one of the limits, when we obtain a place of x renders the first member positive, and which, at the same time, is such, that every greater number will also give a positive result. We will determine such a number. But it may be proposed to Now we are sure of having number, which, substituted in 314. Before resolving this question, we will propose a more simple one. viz. To determine a number, which, substituted in place of x in an equation, will render the first term x" greater than the arithmetical sum of all the others. Suppose that all the terms of the equation are negative,except the first, so that It is required to find a number for x which will render xm>Рxm-1+Qxm-?+ +Tx+U. Let k denote the greatest co-efficient, and substitute it in place of the co-efficients; the inequality will become n-2 ... It is evident that every number substituted for x which will satisfy this condition, will for a stronger reason, satisfy the preceding. Now, dividing this inequality by ", it becomes k or 1 plus a series Making x=k, the second member becomes of positive fractions; then the number k will not satisfy the ine. quality; but by supposing x=k+1, we obtain for the second mem. ber the series of fractions which, considered in an inverse order, is an increasing geometrical k the last term- ; hence the expression for the sum of all the k+1 |
