number of times that the first member of the equation changes its sign, we shall have the number of real roots, and, consequently, the number of imaginary roots in the equation, since the real and imaginary roots are together equal in number to the degree of the equation. Sturm's theorem enables us easily to determine the number of such changes of sign. 453. Sturm's Functions.-Let the first member of the general equation of the nth degree, after having been freed from its equal roots, be denoted by X, and let its first derivative be denoted by X1. We now apply to X and X, the process of finding their greatest common divisor, with this modification, that we change the sign of each remainder before taking it as a divisor; that is, divide X by X1, and denote the remainder with its sign changed by R; also, divide X, by R, and denote the remainder with its sign changed by R1, and so on to Rn, which will be a numerical remainder independent of x, since, by bypothesis, the equation X=0 has no equal roots. We thus obtain the series of quantities X, X1, R, R1, R2,........ Rn, each of which is of a lower degree with respect to x than the preceding; and the last is altogether independent of x, that is, does not contain x. We now substitute for x in the above functions any two numbers, p and q, of which p is less than q. The substitution of p will give results either positive or negative. If we only take account of the signs of the results, we shall obtain a certain number of variations and a certain number of perma nences. The substitution of q for x will give a second series of signs, presenting a certain number of variations and permanences. The following, then, is the Theorem of Sturm. 454. If, in the series of functions X, X1, R, R„,.... Rn, we substitute in place of x any two numbers, p and q, either positive or negative, and note the signs of the results, the difference between the number of variations of sign when x=p and when x=q is equal to the number of real roots of the equation X=0 comprised between p and q. Let Q, Q1, Q2,.... Qn, denote the quotients in the successive divisions. Now, since the dividend is equal to the product of the divisor and quotient plus the remainder, or minus the remainder with its sign changed, we must have the following equations: 455. If, in the series of functions X, X1, R, etc., any number be rubstituted for x, two consecutive functions can not reduce to zero at the same time. For, if possible, suppose X1=0 and R=0; then, by Eq. (2), we shall have R1=0. Also, since R=0 and R1=0, by Eq. (3) we must have R2=0; and from the next equation R ̧=0, and so on to the last equation, which will give R2=0, which is impossible, since it was shown that this final remainder is independent of x, and must therefore remain unchanged for every value of x. 456. When, by the substitution of any number for x, any one of these functions becomes zero, the two adjacent functions must have contrary signs for the same value of x. For, suppose R, in Eq. (3) becomes equal to zero, then this equation will reduce to R=-R2; that is, R and R2 have contrary signs. 457. If a is a root of the equation X=0, the signs of X and X1 will constitute a variation for a value of x which is a little less than a, and a permanence for a value of x which is a little greater than a. Let h denote a positive quantity as small as we please, and let us substitute a+h for x in the equation X=0. According to Art. 449, the development will be of the form h2 X+Xh+X+other terms involving higher powers of h. Now, if a is a root of the proposed equation, it must reduce the polynomial X to zero, and the development becomes or Xh+X+other terms involving higher powers of h, 22 Also, if we substitute a+h for x in the first derived polynomial, the development will be of the form X1+X2+other terms involving higher powers of h. Now a value may be assigned to h so small that the first term of each of these developments shall be greater than the sum of all the subsequent terms. For if h be made indefinitely small, then will Xh be indefinitely small in comparison with X1, which is finite; and, since the following terms contain higher powers of h than the first, each will be indefinitely small in comparison with the preceding term; and, since the number of terms is finite, the first term must be greater than the sum of the subsequent terms. Hence, when h is taken indefinitely small, the sum of the terms of the two developments must have the same sign as their first terms, X1h and X1. When his positive, these terms must both have the same sign; and when h is negative, they must have contrary signs; that is, the signs of the two functions X and X, constitute a variation when x-a-h, and a permanence when x=a+h. 458. Demonstration of Sturm's Theorem.-Suppose all the real roots of the equations X=0, X1=0, R=0, R1 =0, etc., to be arranged in a series in the order of magnitude, beginning with the least. Let p be less than the least of these roots, and let it increase continually until it becomes equal to q, which we suppose to be greater than the greatest of these roots. Now, so long as p is less than any of the roots, no change of sign will to +, occur from the substitution of p for x in any of these functions, Art. 446. But suppose p to pass from a number a little smaller to a number a little greater than a root of the equation X=0, the sign of X will be changed from + to or from Art. 446. The signs of X and X, constitute a variation before the change, and a permanence after the change, Art. 457; that is, there is a variation lost or changed into a permanence. Again, while p increases from a number a little smaller to a number a little greater than another root of the equation X=0, a second variation will be changed into a permanence, and so on for the other roots of the given equation. But when Ρ arrives at a root of any of the other functions X1, R, R1, its substitution for x reduces that polynomial to zero, and neither the preceding nor succeeding functions can vanish for the same value of x, Art. 455; and these two adjacent functions have contrary signs, Art. 456. Hence the entire number of variations of sign is not affected by the vanishing of any function intermediate between X and Rn, for the three adjacent functions must reduce to +0- or -0+. IIere is one variation, and there will also be one variation if we supply the place of the 0 with either + or -; thus, ++ or -±+. Thus we have proved that during all the changes of p, Sturm's functions never lose a variation except when p passes through a root of the equation X=0, and they never gain a variation. Hence the number of variations lost while x increases from p to q is equal to the number of the roots of the equation X=0, which lie between p and q. Now, since all the real roots must be comprised within the limits ∞ and +∞, if we substitute these values for x in the series of functions X, X,, etc., the number of variations lost will indicate the whole number of real roots. A third supposition that x=0 will show how many of these roots are positive and how many negative; and if we wish to determine smaller limits of the roots, we must try other numbers. It is generally best in the first instance to make trial of such numbers as are most convenient in computation, as 1, 2, 10, etc. EXAMPLES. 1. Find the number and situation of the real roots of the equation x3-3x2-4x+13=0. Here we have X-x3-3x2-4x+13, and X,=3x2-6x-4. Dividing X by X,, we find for a remainder -14x+35. Rejecting the factor 7, and changing the sign of the result, we have R=2x-5. Multiplying X, by 4, and dividing by R, we find for a remainder -1. Changing the sign, we have R1 = +1. Hence we have X1, the sign of the result is +; if we substitute ∞ for x in the expression 2x-5, the sign of the result is ; and R1, being independent of x, will remain + for every value of x, so that, by supposing x-x, we obtain the series of signs Proceeding in the same manner for other assumed values of x, we shall obtain the following results: We perceive that no change of sign in either function occurs by the substitution for x of any number less than —3; but, in passing from -3 to 2, the function X changes its sign from - to, by which one variation is lost. In passing from 2 |