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

till we arrive at a number which produces as many variations as+, then the numbers thus obtained will be the limits of the roots of the equation, and the situation of the roots will be indicated by the signs arising from the substitution of the intermediate numbers.

We shall now apply the theorem to a few

EXAMPLES.

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

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

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

Put + and

signs of the results are

; and,

for x in the leading terms of these functions, and the

*The process applied to the general cubic equation x3+ax2+bx+c=0, gives the fol lowing functions, viz.:

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 be found by substitution only. In order that all the roots of the equation x+bx+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 greater than

can be given.

(). When, therefore, b is negative and

all the roots are real, a criterion which has been long known, and as simple as

For

x=+∞,++++ no variation,

x=−∞, −+-+ three variations,

..3-03, 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.

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 V, 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.

Let

x= 1 signs -++ one variation,

x=·9....+++ two variations;

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

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

Let x+, signs of leading terms +++-+two variations

and all the roots of the equation are imaginary.

When, in seeking for the greatest common divisor of V and V1, we arrive at a polynomial V. (for example, at that of the second degree), which, put equal to zero, will only give imaginary values of x, it is not necessary to carry the divisions further, because this polynomial Vn will be constantly of the same sign as its first term for all real values of x; for if it gave a plus sign for one value, and a minus for another, there must be a real root between.* (3) Required the number and situation of the real roots of the equation 2x-11x2+8x-16=0.

* This consideration is of importance, as the calculations for determining the functions V2, V3 are long, especially toward the last, on account of the magnitude of their numerical coefficients.

and the roots of the quadratic 11x-12x+32=0 are imaginary, for 11 × 32 X4 is greater than 122; hence V2 must preserve the same sign for every value of x, and the subsequent functions can not change the number of variations, for a variation is only lost by the change of the sign of V. Hence,

For

x=+signs +++ no variation,

x=1∞. ++ two variations;

and the proposed equation has two real roots, the one positive and the other negative, since the last term is negative. (Prop. VIII., Cor. 5, p. 314.)

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

and the signs of the leading terms are all +; hence the substitution of must give three real roots.

and

To discover the situation of the roots, we make the substitutions

81

x=1.

+

hence the two positive roots are between 3 and 4, and we must, therefore, transform the several functions into others, in which r shall be diminished by 3. This is effected by Art. 251, p. 315; and we get

Make the following substitutions in these functions, viz.:

y=0 signs +--+ two variations,

hence the two positive roots are between 3-2 and 3.3, and we must, again, transform the last functions into others, in which y shall be diminished by 2. Effecting this transformation, we have

hence we have 3.21 and 3.22 for the positive roots, and the sum of the roots is 11; therefore, -11-3.21-3·22=-17-4 is the negative root.

When the equation has equal roots.

255. 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 measure of V and V1; and it has been shown (Art. 253, Scholium 3, p. 321) that if (x—a1)(x — a2) be the greatest common measure of V and V1, then V is divisible by (x—a1)o(r—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 (29); 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.

Find the number and situation of the real roots of the equation

x5—7x1+13x2+x2-16x+4=0.

Therefore we infer that there are four distinct and separate roots; one is -1, for V vanishes for this value of r; another between 0 and 1; a third is 2, and a fourth is between 3 and 4. The common measure - -2 indicates that the polynomial V is divisible by (x-2)2; and hence there are two roots equal to 2 (Art. 253, Cor. 1).

=

It may happen that one of the functions, V1, V... V-1, should be found zero either for x-A or r=B. In this case it is sufficient to count the variations which are found in the succession of signs of the functions V, V1, V2 ... V., omitting the function which is zero. This results from the demonstration in Art. 254, V, for the case where an intermediate function vanishes.

When the number of the auxiliary functions, V1, V2, &c., is equal to the degree of the equation, as is ordinarily the case, in consequence of each remainder in seeking for the common divisor being one degree less than the preceding, the number of imaginary roots in the equation may be found by the following rule: The equation V=0 will have as many pairs of imaginary roots as there are variations of sign in the succession of the signs of the first terms of the functions V1, V2, &c., to the sign of the constant Vm inclusive.

This follows from the fact that two consecutive functions, V-19 Vn, are the one of an even, the other of an odd degree. Then, if the two functions have the same sign for x=+, they will have contrary for x=-∞, and vice versa. So that if we write the succession of signs of V, V1, V2 .... Vm, for x=∞ and for x=+∞, each variation in the one succession will correspond to a permanence in the other. Thus, the number of permanences for x=—∞ is equal to the number of variations for x=+∞.

But for x= + the number of variations will be that of the first terms of the functions V, V1 ... Vm, which denote by i. Then there will be i permanences for x- and m-i variations. The excess of the number of variations m-i for x=-∞ over the number i for x= =+∞, is m-2i, which is therefore the number of real roots of the equation, and therefore 2i the number of imaginary roots, the whole number of roots being m.

HORNER'S Method of RESOLVING NUMERICAL EQUATIONS OF ALL ORDERS.

256. The method of approximating to the roots of numerical equations of all orders, discovered by W. G. Horner, Esq., of Bath, England, is a process of very remarkable simplicity and elegance, consisting simply in a succession of transformations of one equation to another, each transformed equation as it arises having its roots less or greater than those of the preceding by the corresponding figure in the root of the proposed equation. We have shown how to discover the initial figures of the roots by the theorem of STURM; and by making the penultimate coefficient in each transformation available as a trial divisor of the absolute term, we are enabled to discover the succeeding figure of the root; and thus proceeding from one transformation to another, we are enabled to evolve, one by one, the figures of the root of the given equation, and push it to any degree of accuracy required.

GENERAL RULES.

1. Find the number and situation of the roots by Sturm's theorem, and let the root required to be found be positive.

2. Transform the equation into another whose roots shall be less than those of the proposed equation by the initial figure of the root.

3. Divide the absolute term of the transformed equation by the trial divisor, or penultimate coefficient, and the next figure of the root will be obtained, by which diminish the root of the transformed equation as before, and proceed in this manner till the root be found to the required accuracy.