Sturm's Theorem. 15 years owner discovery Find the row of signs corresponding to the values of u, U, U', U", &c., for any value p of the variable, and also for a value of the variable. q 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. First. Two successive terms of the series cannot vanish at the same time, except for a value of x which is one of the equal roots of the given equation. For when U" and U", for instance, are zero, the equation 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 P of the variable from that which it has in the row Sturm's Theorem. corresponding to the value q of the variable, it must, by art. 281, vanish for some value of the variable contained beBut for this value of the variable, the pre tween p and q. ceding term must term; thus, when the equation gives 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 and զ. Р Fifthly. When the given equation has equal roots, u and U must, by art. 278, have the last term of the series. a common divisor which will be 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 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. Sturm's Theorem. 292. EXAMPLES. 1. Find the number of real roots of the equation and also the number contained between 1 and 2. When, therefore, x∞ the row of signs is there are then two real roots. Again when x = 2, the row of signs is the two real roots are therefore both between 1 and 2. Number of Real Roots. 2. Find the number of real roots of the equation x2 + ax + b = 0. Solution. In this case, we have u = x3 + a x + b σ = 2 ax 3 b U- 4 a3 27 62. First case. When a is such that U" is negative, that is, when — 4 a3 < 27 b2, or — a3 < † b2, or (— } a)3 < (} b)2 -,,(like a), —, so that there is only one real root. a< The row of signs when x = 0 is, when b is positive, +, ±(like a), —, —, so that, in this case, the real root is negative. This row, when b is negative, is so that, in this case, the real root is positive, which agrees. with art. 284. Second case. When a is negative and of such a value that (} b)2 = — (} a)3, that is U" = 0, in which case the equation has the equal root, obtained from the equation U' 2 ax-3b=0 == or 36 -2 ai |