In making applications it is convenient that the inferior limit be the greatest possible; the nearest approximation possible should in consequence be made in determining l.

257. The limits of the negative roots are found by changing r into —x in the proposed equation. By this transformation the signs of the roots are changed, so that by seeking the limits of the positive roots of the transformed equation, and affecting them with the sign, the limits of the negative roots of the proposed equation are obtained.

When all the terms of an equation have the same sign, it is evident that no positive number can satisfy the equation. In such cases there is, therefore, no occasion to seek the limits of the positive roots.

In like manner there is no occasion to seek the limits of the negative roots of an equation, of which all the terms of an even degree have the same sign, those of an odd degree having the contrary sign; for it is evident that, any negative number whatever being substituted for x in an equation of this form, all the terms become of the same sign, and their sum cannot by consequence be equal to zero.

When the last term of an equation is wanting, one or more of its roots are equal to zero. Making abstraction of these roots, the preceding remarks are true alike for complete and incomplete equations.

258. Let X be a polynomial of the form

X=Mx"+Nx”+Px2+, &c.,

in which the exponents m, n, p... are integer or fractional, but positive and decreasing numbers, a positive value of x can be found such that, if any greater number is substituted for x in the proposed equation, the polynomial X will acquire values of the same sign as its first term, and which will continue to augment until they become infinite.

The proposed equation may be put under the form

Representing by Y the sum of those terms between the parentheses which have positive coefficients, and by Y' the sum of those terms which have negative coefficients,

Y

X=M(x+Y-Y)=M[xTM(1—2m)+Y].

Y'
X

Y'

-

x"

is a

By hypothesis " is a positive power of x. Now, whatever the value of the expression -Y', it is possible to find a positive value of x such that a shall be greater than Y'; therefore in this case <1. Therefore 1positive quantity. But a" is positive and Y is positive, therefore the whole quantity within the parentheses is positive; consequently, if M is positive the whole expression, or X, is positive, and if M is negative X is negative. Therefore X is of the same sign as M or Mr".

Y'

Denoting by A that value of x for which <1, and supposing that x is made to increase from A, the quotient of Y' by a is continually diminished while and Y are indefinitely augmented; therefore the quantity X will go on increasing to infinity.

The value of A may be obtained, as in the case of limits, by substituting successively increasing positive numbers for x until a exceeds Y',

OF THE INDICATIONS GIVEN BY THE SUBSTITUTION OF ANY TWO NUMBERS FOR THE UNKNOWN QUANTITY.

259. Let X be a polynomial of the form Ax+Bx-'+Cx*~2+, &c. (in which A, B, C, . . . are real quantities); if in this polynomial a is made to vary continuously the polynomial X will also vary continuously; in other words, if the difference of two values of x is less than any assigned quantity

the difference of the corresponding values of X shall also be less than any assigned quantity.

Let any value whatever be given to r; change r into x+h, denote by X, what X becomes when x becomes x+h; denote also by X', X".... the polynomials derived from X.

Then X, X+X'h + { X′′ h2 . . .. +AhTM (Art. 236).

The algebraic sum of the terms which follow X, and which contain the different powers of h, expresses the change produced in X by means of the change of x into r+h. Now, it is evident that by taking sufficiently small each of the terms of this sum may be rendered indefinitely small, and consequently also the sum of the terms.

260. If two numbers substituted successively instead of x in an equation X=0, of the form Ax+B+CxTM-2+, &c. =0,

give results of contrary signs, the equation has at least one real root comprehended between these two numbers. Let the two numbers which, being substituted for r in this equation, give results of contrary signs be represented by a and 3, of which a is less than 6, and let it be conceived that instead of x all the values from a to ẞ are substituted successively in X, the polynomial X must vary in a continuous manner. But by hypothesis the results obtained by making a=a, x=3 are of contrary signs; therefore, among the values which X obtains in consequence of the continuous increase of x from a to ẞ, the value 0 must be found at least once. Now the value of r for which X becomes zero is a root of the equation; therefore the theorem is demonstrated.

It is necessary to observe, that there might be an error in affirming that there can be only one root of the equation X=0 between a and 3; for the polynomial X, without ceasing to vary in a continuous manner, might have, for example, first, decreasing values which might terminate at a certain limit, after which it might increase to another limit, then decrease again, &c.; consequently there is no reason why r, in the interval between ra and x=ß, may not pass several times through zero.

261. If two quantities comprehend between them only one root of the equation X=0, and these quantities are successively substituted for x in the equation, they give two results of contrary signs; and in general, if two quantities comprehend between them an odd number of roots, the results obtained by substituting these quantities for r are of contrary signs. But if two quantities comprehend between them no root or an even number of roots, the results obtained by substituting these quantities for x are of the same sign.

Let it be supposed that a is a real root comprehended between a and 3, and the only root between them. The polynomial X is divisible by x-a; denoting the quotient by Y, X may be put under the form X=Y(x—a). Making successively ra, x=ß, the values of Y must have the same sign, otherwise there must be between a and 3 a root of the equation Y=0, and which would be a new root of X=0, comprehended between a and ß, which is contrary to the hypothesis. But if x is made successively equal to a and, the factor x-a must change its sign; for the differences a-a, ß-a are of contrary signs, because a is between a and 6; therefore the two results which are obtained by substituting a and 3 in X are of contrary signs.

Taking the more general case, that of an uneven number of roots, a, b, c, ... between a, ß; since a, b, c ... are roots of the equation X, X is divisible by the product (x-a)(x—b)(x-c)... Let the quotient be Y; then X=(x−a)(x—b)(x−c) ... ×Y.

In this case, as in the preceding, if x is made successively equal to a, ß, the two corresponding values of Y will have the same sign, while the sign of each of the factors x-a, x-b, x-c... is changed, and the number of these factors is odd; whence the results of the substitutions of a and ẞ in X ought to have contrary signs.

It is, in like manner, evident that if the number of factors x-a, x-b'... is even, the sign of the product of these factors is +. Whence the preceding demonstration also proves that if there are no roots between a and 6, or if there is an even number of roots, the results of the two substitutions will have the same sign.

When an equation has no real roots, if any real quantities whatever are substituted in it instead of x, the results will always be of the same sign. For if two results of contrary signs are obtained, it follows that the equation must have at least one real root comprehended between two of the quantities substituted for x, which is contrary to the hypothesis.

When an equation has been divided by all the factors corresponding to the real roots, the resulting equation ought to be in the case of this theorem; but an equation containing real roots may still give results of the same sign if each root enters an even number of times into the equation. For, if these roots are denoted by a, b, &c. the equation may be written thus,

(x-a)" (x-b) 2n

.xY=0.

Now, whatever real values are substituted for x, the factors (x-a)2", (x-b) cannot become negative (since (-a)2 as well as (+a2)=+a2); and Y cannot change its sign, because the equation Y=0 contains only imaginary

roots.

262. Every equation of an odd degree has at least one real root of a sign contrary to its last term.

The first term being always supposed positive, let, in the first instance, the last term U be negative, and denote by l a superior limit of the positive roots. If, in the proposed equation, x is made equal to +1, the result is positive, and if x is made zero the result is -U. Therefore between zero and + the equation has at least one real root which must be positive. Next, let the last term of the proposed equation be positive. If substituted for x in the equation, the first term (its exponent being odd) becomes negative. It is rendered positive by changing the signs of all the terms; by this change the last term is made negative; then, by the preceding case, the transformed equation has one positive root; therefore the proposed equation has one negative root.

-x is

263. Every equation of an even degree, whose last term is negative, has at least two real roots, the one positive and the other negative.

For, making successively r=0 and x=+1, two results of contrary signs are obtained; therefore the equation has at least one real root.

Then, changing x into - in the proposed equation, the first term of the transformed equation is still positive, but its last term is negative; it therefore has a positive, and by consequence the proposed equation a negative,

root.

When the proposed equation is of an even degree and its last term positive the preceding method does not give the means of discovering whether the equation has a real root, because then the substitutions do not give results of contrary signs. It is thus that equations which have only imaginary roots are necessarily of this form, for if they were of an odd degree they would have at least one real root; and if, being of an even degree, the last term were negative, they would have at least two real roots.

264. In an equation of an odd degree the real roots of a sign contrary to the last term are odd in number, and the roots, if any, of the same sign are even in number. When the last term is negative, the substitutions x=0 and x=+1 giving results of different signs, it is inferred from Articles 261, 262, that the positive roots are odd in number.

Changing x into -r, the equation is transformed into another, of which the first and last terms are negative, but which are rendered positive by changing the signs of all the terms. Now, whether in this equation z is made equal to zero or equal to +1, the result is positive. Therefore the positive

roots of the transformed equation, and by consequence the negative roots of the proposed equation, can only be of an even number.

When the last term of the proposed equation is positive the last term of the transformed equation must be negative. Whence, by the last case, the proposed equation has an odd number of negative roots and an even number of positive roots.

265. In an equation of an even degree, if the last term is negative, the positive roots are odd in number and the negative roots are also odd in number. But if the last term is positive the roots, if any, both positive and negative, are even in number.

For x=0 and x=+1, the results are of contrary signs; therefore the positive roots are odd; and making x=-x, the first term of the transformed equation is positive and the last negative; therefore the positive roots of the transformed, and by consequence the negative roots of the proposed, equation are odd.

When the last term of the proposed equation is positive the last term of the transformed equation is also positive, and so is the first; therefore the real roots, if any, whether positive or negative, must be even in number.

In all cases, therefore, the total number of real roots is odd if the degree of the equation is odd, and even if the degree of the equation is even; hence, if imaginary roots are found in an equation the number of such roots must be even.

266. If an equation is composed of a series of positive terms after which there are only negative terms, it has one positive root, and only one. Since the last term of the proposed equation is negative, the equation has certainly one positive root (Art. 262, 263).

To demonstrate that it cannot have more than one,

denote by Y the sun of the positive terms,
denote by Y' the sum of the negative terms,

and by the lowest power of a contained in Y;

The proposed equation may, consequently, be put under the form

Y Y

Let x=a be the positive value which satisfies the equation: for this value Y Y'

=

Y

x

If in this expression x is augmented, is augmented, or (in the case of

Y'

Y composed of one term) remains constant, while is diminished; but if x

Y'

Y

is diminished is augmented, and is diminished. In either case the equation is not satisfied by values of r greater or less than aa is the only positive root of the proposed equation.

a; therefore

SEPARATION OF THE ROOTS, BY THE METHOD OF LAGRANGE. 267. When an equation is proposed for resolution it is convenient, first, to find the commensurable roots, equal and unequal, then the other equal roots, if the equation contains any, and to suppress them by division. As this can be done by the methods of Article 252 and Article 251, it is assumed, in the following investigation, that the equation to be resolved has neither commensurable nor equal roots. Consequently, when the expression Real roots is employed, the meaning is real roots that are incommensurable and unequal.

The investigation of these roots is divided into two distinct parts. The first treats of the determination for each root of two numbers between which one root only of the equation is comprehended. This part is named

the Separation of the roots. The second part treats of the evaluation of the roots with that degree of approximation which is required.

If in an equation the unknown quantity a is changed into -x, and the positive roots of the transformed equation are taken with the sign the negative roots of the proposed equation are obtained. It will therefore be sufficient to show in what manner the positive roots of equations are determined.

268. The first discovered rigorous method of effecting the separation of the roots of an equation is named the method of Lagrange. By it the separation of the roots is effected in the following manner :

Let X=0 be any equation which contains no equal roots; and let it be conceived that, instead of the unknown quantity x, there are substituted suc. cessively in X the positive numbers p, q, r, s, forming an increasing

series which commences at the inferior limit of the positive roots, ends at the superior limit of these roots, and is, besides, chosen in such a manner that between each number and the following there cannot fall more than a single root of the equation. From the theorem of Article 260 it is known with certainty, by the signs of the results of the substitutions, which of these numbers (p, q, r, &c.) comprehend between them a root, and which of them do not. Then the separation of the roots will be complete.

The essential condition to be fulfilled by the numbers p, q, r, s, is, that no two of them, taken consecutively, shall include between them more than one root of the proposed equation. Now this condition would be fulfilled if for p, q, r, s,... were chosen the terms of an arithmetical progression, of which the ratio & is less than the least difference which exists between the positive roots of the proposed equation.

Hence the difficulty is reduced to that of discovering the ratio d. The method of Lagrange gives by means of the equation of the squares of the differences of the roots of the proposed equation. Thus,

Let X=0 be the proposed equation;

Let Z=0 be the equation of the squares of the differences of the roots of the proposed equation;

Let A be the inferior limit of the roots of the equation Z=0.

Then it is evident that A is less than the square of the least difference between any two roots of the equation X=0; and, consequently, that √ is less than the least difference between any two roots of the equation X=0. & may therefore be assumed equal to √.

As is generally incommensurable, a rational number less than A is usually taken in preference.

When for A is found an incommensurable number greater than 1, the whole number immediately greater than is taken. Then, to effect the separation of the roots it is only necessary to substitute certain whole numbers in the equation X=0. When it has been discovered by this means that there is a root between two of these numbers, the intermediate whole numbers may then be substituted in the equation X=0, and it can be determined by the signs of the results between which of the consecutive whole numbers the root is comprehended.

Most frequently is a fraction less than 1; consequently & is also a fraction less than 1; hence it becomes requisite to substitute fractional numbers in the equation X=0. But such substitutions can be avoided by a simple transformation, namely, by making a=r'. Then, in order that two consecutive values of x may have between them a difference greater than è, it is evident that the values of a ought to differ from each other by more than 1. Consequently it is possible, by the substitution of whole numbers, to find the two numbers which comprehend between them each real root of the transformed equation in x'.