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X-X1Q=-X,

X1)X, (Q2
X2Q2

X=0, by its exponent, and then diminishing the exponent by unity. Divide X by X, until the remainder be of a lower degree than the divisor, and call the remainder -X2, or changing the signs of all the terms in the remainder, we shall have X2 = modified remainder. Proceed in the same manner with the functions X, and X2, and call the modified remainder X,, and so on, as in the marginal scheme, until the division terminates by leaving a final remainder independent of x; and let this remainder, having its sign changed, be called Xm+1. Then we have the series of functions.

X, X1, X2, X3, X. . . . . Xm+1;

X-X2Q2--X,

X1) X2 (Q3
X3 Q3

X2-X3Q3=-X,

which are of continually decreasing dimensions in x, and Xm+1 is altogether independent of x.

Now, if p and q be any two numbers of which p is less than q, and if these numbers be substituted for x in the above series of functions, we shall have two series of signs, the one resulting from the substitution of p for x, giving h variations of sign, and the other from the substitution of q for x, giving k variations of sign; then the exact number of real roots of the proposed equation between the limits Р and q will be h―k.

=

In order to simplify the demonstration of this beautiful theorem, we shall premise one or two Lemmas.

Lemma 1. Two consecutive functions cannot both vanish for the same value of x.

From the process above described for the determination of the successive functions, we have obviously these equations:

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Now, suppose X=0, and X,=0; then by eq. (3) we have X=0; hence, since X=0, and X=0; then by eq. (4) we have X=0; and proceeding in this manner we shall find that Xm+1=0; but as the equation X=0 is supposed not to have equal roots, the polynomials X and X, have no common measure (Prop. VII), and therefore there must be a final remainder, Xm+1, totally independent of x, and must therefore remain unchanged for every value of x.

Lemma 2. If one of the derived functions vanish for any particular value of x, the two adjacent functions have contrary signs for the same value of x.

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and if X=0; then X,--X1, and therefore it is obvious that X, and X, must have contrary signs.

Let p be nearer to

DEMONSTRATION OF THE THEOREM.

∞ than any of the real roots of the equatio. X=0, X1=0, X=0, X,=0, . . . . X=0;

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or from

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and conceive p to increase continuously until it becomes 0, and then to go on increasing until it becomes equal to q, which we may suppose to be nearer + ∞ than any of the real roots of the preceding equations. Now, while p is less than any of the roots of these equations, no change of signs will occur by the substitution of p for x, in any of these functions (Prop. VI. Cor. 4;) but whenever p in its continuous progress towards q, arrives at a root of any of the derived equations, that function becomes zero, and neither the preceding nor succeeding function can vanish for the same value of x (Lemma 1), and these two adjacent functions have contrary signs (Lemma 2); hence the entire number of variations of sign is not affected by the vanishing of any of the derived functions. While, therefore, p advances in the scale of numbers by minute additions, it will pass successively over the roots of the proposed equation, as well as over those of the derived equations; and in passing from a number very little smaller to a number very little greater than a root of the equation X=0, the sign of X will be changed from + to to (Prop. VI. Cor. 1); and the difference of these numbers may be made so small, that no change of signs can take place in the derived functions; hence the loss of a variation of sign arises from the change of sign of the function X. Again, when p becomes nearly equal to another root of X=0, the order of the signs of the derived functions may be changed, but the number of variations is not at all affected (Lemma 2); and, therefore, while p varies from a number very little smaller to a number very little greater than this root of X=0, there will be a loss of one variation of sign, arising from the change of the sign of X; and so on for the other roots of X=0. Whenever, then, the value of p passes over a root of the equation X=0, there is a loss of one variation of sign; and since a variation cannot be lost among the signs of the derived functions, nor can one be ever introduced, it is obvious that we are furnished with a simple and beautiful criterion for ascertaining the number of real roots between any two specified numbers, p and q. To illustrate this more fully, we shall suppose that the substitution of p and q for x in the series of functions, gives the two series of signs, viz.:—

X X, X, X3 X4

+

++

X X, X2 X3 X4
+++++

Now there are two variations of sign in the former row of signs, and no variation in the latter; hence one variation is lost in the signs of the derived functions, and the sign of X remains unchanged; but a variation cannot be lost in the signs of the derived functions, on the supposition that one root lies between p and q; besides, the sign of X is unchanged; hence there must be a number, m, between p and q, which, substituted for x in the series of functions, gives the sign of X negative, and hence there must be one root between and m, and another root between m and q. The loss of two variations of sign must, therefore, indicate the existence of two real roots between p and q; and, in like manner, the loss of three variations of sign indicates the existence of three roots in the interval, and so on. Hence, if the substitution of p for x gives h variations, and q for x gives h variations; then h―k— number of real roots between p and q.

p

Since all the real roots are comprehended between the extreme values - ∞ and we may readily ascertain the number of real roots by substituting and +∞ for x in the leading terms of the several functions, because the first term of each function must, for x=∞, be numerically greater than all the other terms in the function together; and hence the sign of the leading term will determine the sign of the whole function. Let h be the number of variations of sign arising from the substitution of - for x in the functions, and k the number for +∞; then h―k= the number of real roots in the equation, and n—(h—k)= the number of imaginary roots. To determine the initial figures of the roots, we may substitute the successive numbers of the series

0, −1, −2, −3, −4, . . . . .

till we have as many variations as numbers of the series

∞ produced; and if we substitute the

0, 1, 2, 3, 4, . . . . .

till we have as many variations as +∞ produced, then the numbers which first produce the known number of variations, will be the limits of the roots of the equations, and the situation of the roots will be indicated by the signs arising from the substitution of the intermediate numbers.

175. When the equation has equal roots, one of the divisors will divide the preceding without a remainder, and the process will thus terminate without a remainder, independent of x. In this case, the last divisor is a common neasure of X and X1; and it has been shown (Prop. VI. Cor. 3, p. 292), that if (x—a,) (x—a2)2 be the greatest common measure of X and X,, then X is divisible by (x-a1)2(x—a,)3, and the depressed equation furnishes the distinct and separate roots of the equation; for Sturm's theorem takes no notice of the repetition of a root. The several functions may be divided by the greatest common measure so found, and the depressed functions employed for the determination of the distinct roots; but it is obvious that the original functions will furnish the separate roots just as well as the depressed ones, for the former differ only from the latter in being multiplied by a common factor; and whether the sign of this factor be + or —, the number of variations of sign must obviously remain unchanged, since multiplying or dividing by a positive quantity does not affect the signs of the functions; and if the factor or divisor be negative, all the signs of the functions will be changed, and the number of variations of sign will remain precisely as before.

We shall now apply the theorem to a few examples.

EXAMPLES.

(1.) Find the number and situation of the roots of the equation

x3-4x2-6x+8=0*

The process applied to the general cubic equation a3+ax2+bx+c=0, gives the following functions, viz.:

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These functions in (1) and (2) will frequently be found useful in the application of Sturm's theorem to equations of the third degree, since the derived functions in any particular example may

Here we have X3-4x2-6x+8

X1=3x2-8x-6;

then, multiplying the polynomial X by 3, in order to avoid fractions, 3x2-8x-6) 3x3—12x2-18x+24 (x−1

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It is now unnecessary to continue the division further, since it is very obvious that the sign of the remainder, which is independent of x, is —; and, therefore, the series of functions are

Put

X = x3- 4x2-6x+8

X1 = 3x2- 8x -6

X11 =17x-12

X1 =+

and ∞ for x in the leading terms of these functions, and the signs of the results are

For a∞, + + + + no variation

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.. k=0

+three variations .. h=3

.. h―k=3-0=3, the number of real roots in the proposed cubic equation.

Next, to find the situation of the roots we must employ narrower limits than +∞ and -∞. Commencing at zero, let us extend the limits both ways, and since the proposed equation has only one permanence of sign, one of the roots is negative, and the remaining roots are positive.

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be found by substitution only. In order that all the roots of the equation x3+b+c=0 may be real, the first terms of the functions must be positive; hence-2bx and -463-27c2 must be positive; and as -27c2 is always negative, b must be negative, in order that -463 and -26 may be positive; therefore, when all the roots are real, 463 must be greater than 27c2, or

(9) greater than (;-)' When, therefore, b is negative and )>(-)2, all the roots are real, a criterion which has been

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long known, and as simple as can be given.

We perceive, then, by the columns of variations, that the roots are between 0 and 1, 5 and 6, −1 and -2; hence the initial figures of the roots are -1, 0, and 5; and in order to narrow still further the limits of the root between 0 and 1, we shall resume the substitutions for x in the series of functions as before. But as the substitution of 1 for x, in the function X, gives a value nearly zero, we shall commence with 1, and descend in the scale of tenths, until we arrive at the first decimal figure of the root.

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hence the initial figures are -1, ·9, and 5.

(2.) Find the number and situation of the real roots of the equation

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+ two variations +−+++ two variations;

and all the roots of the equation are imaginary.

(3.) Required the number and situation of the real roots of the equation

2x1-11x2+8x-16=0.

The first three functions are

X= 2x-11x2+8x-16

X= 4x3-11x +4
X=11x2-12x +32;

and the roots of the quadratic 11x2-12x+32=0 are imaginary; for 11 × 32 × 4 is greater than 122; hence X, must preserve the same sign for every value of x, and the subsequent functions cannot change the number of variations, for a variation is only lost by the change of the sign of X. Hence, For signs +++ no variation

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and the proposed equation has two real roots, the one positive, and the other negative, since the last term is negative. (Prop. VI, Cor. 5, p. 279.)

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Hence the initial figures of the real roots are 2 and 2.

+

When two roots are nearly equal to each other.

(4.) Find the roots of the equation

x+11x2-102x+181=0.

++

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and the signs of the leading terms are all +; hence the substitution of

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