positive value of y in the new equation corresponds to the least positive value of x in the given equation, it follows, that

If we determine the superior limit of the positive roots of the equation Y = 0, its reciprocal will be the inferior limit of the positive roots of the given equation.

Hence, if we designate the superior limit of the positive roots of the equation Y=0 by L', we shall have for the in

1

ferior limit of the positive roots of the given equation, I Second, If in the equation.

which gives the transformed equation Y0, it is clear that the positive roots of this new equation, taken with the sign -, will give the negative roots of the given equation; therefore, determining by known methods, the superior limit of the positive roots of the new equation = 0, and designating this limit by L", we shall have -L" for the superior limit, (numerically), of the negative roots of the given equation.

Third, If in the equation

Y

we shall have the derived equation Y" = 0. The greatest posi tive value of y in this equation will correspond to the least negative value (numerically) of x in the given equation. If, then, we find the superior limit of the positive roots of the equation Y" 0, and designate it by I", we shall have the

=

inferior limit of the negative roots (numerically) equal to

Consequences deduced from the preceding Principles.

First.

1

L

287. Every equation in which there are no variations in the signs, that is, in which all the terms are positive, must have all of its real roots negative; for, every positive number substituted for x, will render the first member essentially positive.

Second.

288. Every complete equation, having its terms alternately post tive and negative, must have its real roots all positive; for, every negative number substituted for x in the proposed equation, would render all the terms positive, if the equation be of an even de gree, and all of them negative, if it be of an odd degree. Hence, their sum could not be equal to zero in either case.

This principle is also true for every incomplete equation, in which there results, by substituting -y for x, an equation having all its terms affected with the same sign.

Third.

289. Every equation of an odd degree, the co-efficients of which are real, has at least one real root affected with a sign contrary to that of its last term.

be the proposed equation; and first consider the case in which the last term is negative.

By making = : 0, the first member becomes U. But by giving a value to x equal to the greatest co-efficient plus 1, or (K+1), the first term am will become greater than the arithmetical sum of all the others (Art. 282), the result of this substitution will therefore be positive; hence, there is at least one real root comprehended between 0 and K+ 1, which root is posi tive, and consequently affected with a sign contrary to that of the last term (277).

Suppose now, that the last term is positive.

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

Fourth.

290. Every equation of an even degree, which involves only real co-efficients, and 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 substi tutions, it becomes greater than the sum of all the others; there fore, the results obtained by these substitutions are both positive, or affected with a sign contrary to that given by the hypothesis x= 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) (277).

Fifth.

291. If an equation, involving only real co-efficients, contains imagr nary 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. 246); and the quotient must alsc 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 (289); hence, the imaginary roots must enter by pairs.

REMARK.-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 1, 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 com

prehended between K+ 1 and the number which would give a negative result (Art. 277).

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

Sixth.

292. 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. 279), 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, they comprehend either a single root, or an odd number of them.

The converse of this proposition is evidently true.

Descartes' Rule.

293. 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. A permanence is when two consecutive terms have the same sign. In the equation

X- απ

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, member by member, there will result an equation of the second degree,

= 0.

If a is less than b, the equation will be of the first form (Art. 117); 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 perma

nences.

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

in which the signs succeed each other in any manner whatever. By multiplying by xa, we have

xm+1Axm±B xm-1±C xm−2+ ..±U

The co-efficients which form the first horizontal line of this product, are those of the given equation, taken with the same signs; 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 in 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 varia tion than in the proposed equation.