Sturm's Theorem.

298. The object of this theorem is to explain a method of determining the number and places of the real roots of equations involving but one unknown quantity.

Let

X=0

(1),

represent an equation containing the single unknown quantity x; X being a polynomial of the mth degree with respect to x, the co-efficients of which are all real. If this equation should have equal roots, they may be found and divided out as in Art. 269, and the reasoning be applied to the equation which would result. We will therefore suppose X = 0 to have no equal roots.

1

299. Let us denote the first derived polynomial of X by X1, and then apply to X and X, a process similar to that for finding their greatest common divisor, differing only in this respect, that instead of using the successive remainders as at first obtained, we change their signs, and take care also, in preparing for the division, neither to introduce nor reject any factor except a positive one.

If we denote the several remainders, in order, after their signs have been changed, by X, X... X, which are read X second, X third, &c., and denote the corresponding quotients by Q1, Q Q1, we may then form the equations

Since by hypothesis, X=0 has no equal roots, no common divisor can exist between X and X, (Art. 267). The last reX,, will therefore be different from zero, and inde.

mainder

pendent of x.

300. Now, let us suppose that a number p has been substi tuted for x in each of the expressions X, X1, X, ... X; and that the signs of the results, together with the sign of X, are arranged in a line one after the other: also that another number 9, greater than p, has been substituted for x, and the signs of the results arranged in like manner.

Then will the number of variations in the signs of the first arrangement, diminished by the number of variations in those of the second, denote the exact number of real roots comprised between p and q.

301. The demonstration of this truth mainly depends upon the three following properties of the expressions X, X1.. X, &c.

I. If any number be substituted for x in these expressions, it is impossible that any two consecutive ones can become zero at the same time.

For, let X, X, X+1, be any three consecutive expressions. Then among equations (3), we shall find

from which it appears that, if X-1 and X, should both become 0 for a value of x, X+1 would be 0 for the same value; and since the equation which follows (4) must be

X1 = Xn+1Qn+1 — Xn+2)

=

we shall have X+20 for the same value, and so on until we should find X,

cannot both become

-1

0, which cannot be; hence, X and X, 0 for the same value of x.

II. By an examination of equation (4), we see that if X, be comes 0 for a value of x, X-1 and X+1 must have contrary signs; that is,

If any one of the expressions is reduced to 0 by the substi tution of a value for x, the preceding and following ones will have contrary signs for the same value.

III. Let us substitute a + u for x in the expressions X and X1, and designate by U and U1 what they respectively become under this supposition. Then (Art. 264), we have

in which A, A', A", &c., are the results obtained by the sub stitution of a for x, in X and its derived polynomials; and A1, A'1, &c., are similar results derived from X1. If, now, a be a root of the proposed equation X 0, then A 0, and since A' and A1 are each derived from X1, by the substitution of a for x, we have A′ = A1, and equations (5) become

u2

U A'u + A" + &c.

2

=

(6).

Now, the arbitrary quantity u may be taken so small that the signs of the values of U and U, will depend upon the signs of their first terms (Art. 276); that is, they will be alike when u is positive, or when a + u is substituted for x, and unlike when u is negative or when a -u is substituted for z. Hence,

If a number insensibly less than one of the real roots of X = 0 be substituted for x in X and X1, the results will have contrary signs; and if a number insensibly greater than this root be substituted, the results will have the same sign.

302. Now, let any number as k, algebraically less, that is, nearer equal to co, than any of the real roots of the several equations

be substituted for x in the expressions X, X1, X2, &c., and the signs of the several results arranged in order; then, let z be increased by insensible degrees, until it becomes equal to h, the least of all the roots of the equations. As there is no

root of either of the equations between k and h, none of the signs can change while x is less than h (Art. 277), and the number of variations and permanences in the several sets of results, will remain the same as in those obtained by the first substitution.

When becomes equal to h, one or more of the expressions X, X, &c., will reduce to 0. Suppose X becomes 0. Then, as by the first and second properties above explained, neither X-1 nor Xti can become 0 at the same time, but must have contrary signs, it follows that in passing from one to the other (omitting_X1 = 0), there will be one and only one variation; and since their signs have not changed, one must be the same as, and the other contrary to, that of X, both before and after it becomes 0; hence, in passing over the three, either just before I becomes 0 or just after, there is one and only one variation. Therefore, the reduction of X, to 0 neither increases nor diminishes the number of variations;, and this will evidently be the case, although several of the expressions X1, X2, &c., should become 0 at the same time.

If xh should reduce X to 0, then h is the least real root of the proposed equation, which root we denote by a; and since by the third property, just before becomes equal to a, the signs of X and X, are contrary, giving a variation, and just after passing it (before becomes equal to a root of X1 = 0), the signs are the same, giving a permanence instead, it follows that in passing this root a variation is lost.

In the same way, increasing x by insensible degrees from x = a + u until we reach the root of X=0 next in order, it is plain that no variation will be lost or gained in passing any of the roots of the other equations, but that in passing this root, for the same reason as before, another variation will be lost, and so on for each real root between k and the number last substituted, as g, a variation will be lost until x has been increased beyond the greatest real root, when no more can be lost or gained. Hence, the excess of the number of variations

obtained by the substitution of k over those obtained by the substitution of g, will be equal to the number of real roots comprised between k and g.

It is evident that the same course of reasoning will apply when we commence with any number p, whether less than all the roots or not, and gradually increase x until it equals any other number 9. The fact enunciated in Art. 299 is therefore established.

303. In seeking the number of roots comprised between p and q, should either por q reduce any of the expressions X1, X2, &c., to 0, the result will not be affected by their omission, since the number of variations will be the same.

Should p reduce X to 0, then p is a root, but not one of those sought; and as the substitution of p+u will give X and X the same sign, the number of variations to be counted will not be affected by the omission of X = 0.

Should q reduce X to 0, then q is also a root, but not one of those sought; and as the substitution of q u will give X and X contrary signs, one variation must be counted in passing from X to X1.

n

304. If in the application of the preceding principles, we observe that any one of the expressions X1, X, ... &c., X, for instance, will preserve the same sign for all values of x in passing from p to q, inclusively, it will be unnecessary to use the succeeding expressions, or even to deduce them. For, as X, preserves the same sign during the successive substitutions, it is plain that the same number of variations will be lost among the expressions X, X, &c. . . . ending with X, as among all including X. Whenever then, in the course of the division, it is found that by placing any of the remainders equal to 0, an equation is obtained with imaginary roots only (Art. 291), it will be useless to obtain any of the succeeding remainders. This principle will be found very useful in the solution of numerical examples.