real root comprehended between 0 and K + 1, which root is positive, and consequently affected with a sign contrary to that of the last term.

Suppose now, that the last term is positive.

Making x = 0, we obtain + U for the result; but by putting − (K + 1) in place of x, we shall obtain a negative result, since the first term becomes negative by this substitution; hence, the equation has at least one real root comprehended between 0 and (K1), which is negative, or affected with a sign contrary to that of the last term.

Fourth.

323. Every equation of an even degree, involving only real coefficients of which the last term is negative, has at least two real roots, one positive and the other negative.

For, let U be the last term; making x = 0, there results - U. Now substitute either K+ 1, or - (K + 1), K being the greatest co-efficient in the equation. As m is an even number, the first term am will remain positive; besides, by these substitutions, it becomes greater than the sum of all the others; therefore, the results obtained by these substitutions are both positive, or affected with a sign contrary to that given by the hypothesis a=0; hence, the equation has at least two real roots, one positive, and comprehended between 0 and K+1, the other negative, and comprehended between 0 and (K + 1).

Fifth.

324. If an equation, involving only real co-efficients, contains aginary roots, the number of such roots must be even.

For, conceive that the first member has been divided by all the simple factors corresponding to the real roots; the co-efficients of the quotient will be real (Art. 278); and the equation must also be of an even degree; for, if it was uneven, by placing it equal to zero, we should obtain an equation that would contain at least one real root; hence, the imaginary roots must enter by pairs.

REMARK.-325. There is a property of the above polynomial quotient which belongs exclusively to equations containing only imaginary roots; viz., every such equation always remains positive for any real value substituted for x.

For, by substituting for x, K+ 1, the greatest co-efficient plus unity, we could always obtain a positive result; hence, if the polynomial could become negative, it would follow that when placed equal to zero, there would be at least one real root comprehended between K+1 and the number which would give a negative result (Art. 311).

It also follows, that the last term of this polynomial must be positive, otherwise x = 0 would give a negative result.

Sixth.

326. When the last term of an equation is positive, the number of its real positive roots is even; and when it is negative, the number of such roots is uneven.

For, first suppose that the last term is + U, or positive. Since by making x=0, there will result + U, and by making x= K+ 1, the result will also be positive, it follows that 0 and K+ 1 give two results affected with the same sign, and consequently (Art. 313), the number of real roots, if any, comprehended between them, is even.

When the last term is U, then 0 and K+ 1, give two results affected with contrary signs, and consequently comprehend either a single root, or an odd number of them.

The reciprocal of this proposition is evident.

Descartes' Rule.

327. An equation of any degree whatever, cannot have a greater number of positive roots than there are variations in the signs of its terms, nor a greater number of negative roots than there are permanences of these signs.

A variation is a change of sign in passing along the terms, and a permanence is when two consecutive terms have the same sign. In the equation x — α = 0, there is one variation, and one positive root, x = a. And in the equation x + b = 0, there is one permanence, and one negative root, x = - b.

If these equations be multiplied together, there will result an equation of the second degree,

If a is less than b, the equation will be of the first form (Art. 144); and if a>b the equation will be of the second form; that is,

In the first case, there is one permanence, and one variation, and in the second. one variation and one permanence. Since in either form, one root is positive and one negative, it follows that there are as many positive roots as there are variations, and as many negative roots as there are permanences.

The proposition will evidently be demonstrated in a general manner, if it be shown that the multiplication of the first member by a factor-a, corresponding to a positive root, introduces at least one variation, and that the multiplication by a factor x + a, corresponding to a negative root, introduces at least one permanence. Take the equation

+ Tx+U =

0,

xmАxm-1 ± Вxm−2 ± Cxm−3 ± in which the signs succeed each other in any manner whatever.

The co-efficients which form the first horizontal line of this product, are those of the given equation, taken with the same sign; and the co-efficients of the second line are formed from those of the first, by multiplying by a, changing the signs, and advancing each one place to the right.

Now, so long as each co-efficient of the upper line is greater than the corresponding one in the lower, it will determine the sign of the total co-efficient; hence, in this case there will be, from the first term to that preceding the last, inclusively, the same variations and the same permanences as in the proposed equation ; but the last term Ua having a sign contrary to that which immediately precedes it, there must be one more variation than in the proposed equation.

When a co-efficient in the lower line is affected with a sign contrary to the one corresponding to it in the upper, and is also greater than this last, there is a change from a permanence of sign to a variation; for the sign of the term in which this happens,

being the same as that of the inferior co-efficient, must be contrary to that of the preceding term, which has been supposed to be the same as that of its superior co-efficient. Hence, each time we descend from the upper to the lower line, in order to determine the sign, there is a variation which is not found in the proposed equation; and if, after passing into the lower line, we continue in it throughout, we shall find for the remaining terms the same variations and the same permanences as in the given equation, since the co-efficients of this line are all affected with signs contrary to those of the primitive co-efficients. This supposition would therefore give us one variation for each positive root. But if we ascend from the lower to the upper line, there may be either a variation or a permanence. But even by supposing that this passage produces permanences in all cases, since the last term Ua forms a part of the lower line, it will be necessary to go once more from the upper line to the lower, than from the lower to the upper. Hence, the new equation must have at least one more variation than the proposed; and it will be the same for each positive root introduced into it.

It may be demonstrated, in an analogous manner, that the multiplication by a factor xa, corresponding to a negative root, would introduce one permanence more. Hence, in any equation, the number of positive roots cannot be greater than the number of VARIATIONS of signs, nor the number of negative roots greater than the number of PERMANENCES.

Consequence.

328. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots to the number of permanences.

For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences; we shall have m=n+p. Moreover, let n' denote the number of positive roots, and p' the number of negative roots, we shall have

whence

m = n' + p' ;

n + p = n' + p', or, n- n'p' - p.

Now, we have just seen that n' cannot be >n, nor can it be less, since p cannot be >p; therefore we must have n' =n, and p = p.

REMARK.-329. When an equation wants some of its terms, we can often discover the presence of imaginary roots, by means of the above rule.

For example, take the equation

x3 + px + q = 0,

p and q being essentially positive; introducing the term which is wanting, by affecting it with the co-efficient 0: it becomes x3 ± 0 . x2 + px + q = 0.

By considering only the superior sign, we should only obtain permanences, whereas the inferior sign gives two variations. This proves that the equation has some imaginary roots; for, if they were all three real, it would be necessary by virtue of the superior sign, that they should be all negative, and, by virtue of the inferior sign, that two of them should be positive and one negative, which are contradictory results.

We can conclude nothing from an equation of the form

for, introducing the term +0.x2, it becomes

which contains one permanence and two variations, whether we take the superior or inferior sign. Therefore, this equation may have its three roots real, viz., two positive and one negative; or, two of its roots may be imaginary and one negative, since its last term is positive (Art. 326).

Of the commensurable Roots of Numerical Equations.

330. Every equation in which the co-efficients are whole numbers, that of the first term being unity, will have whole numbers only for its commensurable roots.

For, let there be the equation

xm + Pxm-1+ Qxm-2 + + Tx+U = 0;

...

in which P, Q... T, U, are whole numbers, and suppose that it

were possible for one root to be a commensurable fraction

Substituting this fraction for x, the equation becomes

a

b