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Y and Y" representing what Y becomes, when we replace in succession, by p and q. These two quantities Y' and Y", are affected with the same sign; for, if they were not, by the first principle there would be at least one other real root comprised between p and q, which is contrary to the hypothesis. above results more easily,

To determine the signs of the divide the first by the second, and we obtain

(p − a) (p − b) (p − c) . . . × Y'

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Now, since the root a is comprised between p

is, is greater than one and less than the other,

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a must have contrary signs; also, p-b and have contrary signs, and so on.

Hence, the quotients

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is essentially positive, since Y' and Y" are

affected with the same sign; therefore, the product

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will be negative, when the number of roots, a, b, c

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prehended between p and q, is uneven, and positive when the number is even.

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will have contrary signs when the number of roots comprised between p and q is uneven, and the same sign when the number is even.

Third Principle.

280. If the signs of the alternate terms of an equation be changed, the signs of the roots will be changed.

Take the equation,

xm+Pxm-1+ Qxm-2 + U = 0 · · (1);

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and by changing the signs of the alternate terms, we have

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But equations (2) and (3) are the same, since the sum of the positive terms of the one is equal to the sum of the negative terms of the other, whatever be the value of x.

Suppose a to be a root of equation (1); then, the substitution of a for x will verify that equation. But the substitution of a for x, in either equations (2) or (3), will give the same result as the substitution of +a, in equation (1): hence -α, is a root of equation (2), or of equation (3).

We may also conclude, that if the signs of all the terms be changed, the signs of the roots will not be altered.

Limits of Real Roots.

281. The different methods for resolving numerical equations, consist, generally, in substituting particular numbers in the proposed equation, in order to discover if these numbers verify it, or whether there are roots comprised between them. But by reflecting a little on the composition of the first member of the general equation,

xm+Рxm-1 + Qxm-2. + Tx+U=0,

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we become sensible, that there are certain numbers, above which it would be useless to substitute, because all numbers above a certain limit would give positive results.

282. It is now required to determine a number, which being substituted for x in the general equation, will render the first term xm greater than the arithmetical sum of all the other terms; that is, it is required to find a number for x which will render

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Let k denote the greatest numerical co-efficient, and substitute it in place of each of the co-efficients; the inequality will then become

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It is evident that every number substituted for x which will satisfy this condition, will satisfy the preceding one. Now, dividing both members of this inequality by ", it becomes

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Making x = k, the second member reduces to 1 plus the sum of several fractions.

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The number k will not therefore

satisfy the inequality; but if we make xk+1, we obtain for the second member the expression,

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This is a geometrical progression, the first term of which is

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Now, any number > (k + 1), put in place of x, will render

k k

the sum of the fractions + + still less therefore,

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The greatest co-efficient plus 1, or any greater number, being substituted for x, will render the first term xm greater than the arithmetical sum of all the other terms.

283. Every number which exceeds the greatest of the positive roots of an equation, is called a superior limit of the positive roots.

From this definition, it follows, that this limit is susceptible of an infinite number of values. For, when a number is found to exceed the greatest positive root, every number greater than this, is also a superior limit. The term, however, is generally applied to that value nearest the value of the root.

Since the greatest of the positive roots will, when substituted for x, merely reduce the first member to zero, it follows, that we shall be sure of obtaining a superior limit of the positive roots by finding a number, which substituted in place of x, renders the first member positive, and which at the same time is such, that every greater number will also give a positive result; hence, The greatest co-efficient of x plus 1, is a superior limit of the positive roots.

Ordinary Limit of the Positive Roots.

284. The limit of the positive roots obtained in the last article, is commonly much too great, because, in general, the equation contains several positive terms. We will, therefore, seek for a limit suitable to all equations.

Let xm-n denote that power of x that enters the first negative term which follows am, and let us consider the most unfavorable case, viz., that in which all the succeeding terms are negative, and the co-efficient of each is equal to the greatest of the negative co-efficients in the equation.

Let S denote this co-efficient. What conditions will render

xm > Sxm-n + Säm−n−1 + ... Sx + S ?

Dividing both members of this inequality by xm, we have

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x=/S+1, or for simplicity, making SS',

which gives,

SS'", and S'+1,

x =

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Moreover, every number > S′+1 or "/S+1, will, when substituted for x, render the sum of the fractions

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still smaller, since the numerators remain the same, while the denominators are increased. Hence, this sum will also be less.

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Hence, S+1, and every greater number, being substituted for x, will render the first term am greater than the arithmetical sum of all the negative terms of the equation, and will consequently give a positive result for the first member. Therefore,

That root of the numerical value of the greatest negative co-efficient whose index is equal to the number of terms which precede the first negative term, increased by 1, is a superior limit of the positive roots of the equation. If the co-efficient of a term is 0, the term must still be counted.

Make n = 1, in which case the first negative term is the second term of the equation; the limit becomes

1 √√S+1=S+ 1;

that is, the greatest negative co-efficient plus 1.

Let n = 2; then, the limit is 2/S+ 1. limit is 3/S +1.

When n = 3, the

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