Number of Imaginary Roots; of Real Positive Roots. reduces the given first member to its last term, m, and this result is therefore negative in the present case. Comparing this with the above results, we see that there must be a real root between 0 and + co, and also one between 0 and 00; that is, the given equation has two real roots, the one positive and the other negative. 286. Corollary. Since the number of real roots of an equation of an uneven degree is uneven, and that of an equation of an even degree is even, the number of imaginary roots of every equation, which has imaginary roots, must be even. 287. Theorem. The number of real positive roots of an equation is even, when its last term is positive ; and it is uneven, when the last term is nege tive. Proof. The substitution of gives, for the first member of the given equation, a positive result; while the substitution of X = 0 reduces the first member to its last term. Hence if this last term is positive, the number of real roots contained between 0 and on, that is, of positive roots, must, by art. 283, be even; and if this last term is negative, the number of these roots must be uneven. 288. Theorem. If a function vanishes, that is, is equal to zero for a given value x' of its variable x; the function and its derivative must have like signs for a value of the variable which exceeds x' by Variation and Permanence. an infinitely small quantity, and unlike signs for a value of the variable which is less than x' by an infinitely small quantity. Proof. Let the given function be u, and its derivate U, and, as in art. 176, when the variable is increased by the infinitesimal i, the function becomes u + Ui. This value of the function, when u=0 is reduced to Ui, which has, obviously, the same sign with U. In the same way when the variable is decreased by i, the function becomes Ui, which, when u = 0, is reduced to - Ui, having the opposite sign to U. 289. Definition. A pair of two successive signs, in a row of signs, is called a permanence when the two signs are alike, and a variation when they are unlike. 290. Sturm's Theorem. Denote the first member of the equation 20n + a r*-1 + &c. = 0 by u and its derivative by U. Find the greatest common divisor of u and U, and, in performing this process, let the several remainders which are of continually decreasing dimensions in regard to x, be denoted, after reversing their signs, by U', U", U", &c. Theorem. 15 years Sturm's 15 yames onze dienny Find the row of signs corresponding to the values of U, U, U', TJ", &c., for any value р of the variable, and also for a value 9 of the variable. The difference between the number of permanences of the first row of signs, and that of the second, is exactly equal to the number of real roots of the given equation comprised between p and q. Proof. The method in which U', U", &c., are obtained gives, at once, by denoting the successive quotients in the process by m, m', &c. U = m U UI U" &c. U' = m U" U'I gives U' = 0; and, in the same way, it is shown that U=0 and u = -0, so that the function and the derivative are both zero at the same time, which, by art. 278, corresponds to the case of one of the equal roots of the equation. Secondly. If any term of the series, except the first or last, has a different sign in the row corresponding to the value р of the variable from that which it has in the row Storm's Theorem. corresponding to the value q of the variable, it must, by art. 281, vanish for some value of the variable contained between p and But for this value of the variable, the preceding term must have a different sign from the succeeding term; thus, when U" = 0 the equation U' = m U" UIII gives U' = U". By the change of sign which the term undergoes in vanishing, therefore, it can only change from forming a permanence with one of its adjacent terms to forming one with the other of these terms, and the change of sign of a term, which is neither the first nor the last of the series, does not increase or diminish the number of permanences of the row of signs. Thirdly. When the first term u of the series, in changing its sign, vanishes, while the second term U does not vanish, the corresponding value of the variable is, by art. 278, a root of the equation which is not one of the equal roots. If, moreover, the variable is decreasing in value, the signs of these two terms constitute a permanence before the change and a variation after the change. When the variable, therefore, in decreasing passes through a value which is one of the unequal roots of the equation, the number of permanences in the row of signs is increased by unity. Fourthly. When the given equation has no equal roots, u and U can, by art. 278, have no common divisor, and therefore the last term of the series will not contain the variable ; it must, therefore, be of a constant value and no change of sign can arise from it. In this case, then, the number of permanences must by the preceding division of the proof be greater in the row which corresponds to the greater Sturm's Theorem. of the two limits p and q, than in the row which corresponds to the less of these two limits, by a number which is exactly equal to the number of real roots contained between р and 9. Fifthly. When the given equation has equal roots, u and U must, by art. 278, have a common divisor which will be the last term of the series. This divisor must also, by art. 59, be a divisor of all the other terms of the series ; and if the series is divided by it a new series V, V, V', V", &c., is obtained, which has in all cases either the same signs as the given series or the reverse signs, so that each pair of successive signs is of the same name, whether permanence or variation, in each series. And by dividing the equations before found by this same common divisor, they become 0 = m V V &c. The first term of the new series has, by art. 278, the same roots with the given series except that it has no equal roots, and the last term is unity. The reasoning of the preceding, portion of this article may, therefore, be applied to the new series; and it follows that the theorem is applicable to the case of an equation which has equal roots, as well as to one which has unequal roots. 291. Corollary. If infinity is substituted for p and negative infinity for q in the series of divisors, the resulting rows of signs show at once the whole number of real roots of the given equation. |