If any of the numbers b, c, d, &c., is an exact root of the corresponding transformed equation, the process terminates, and we find the exact value of x. Also, if one of the transformed equations be identical with a preceding one, the continued fraction expressing the root is periodical; for, after that, the same quotients will recur in the same order; in this case a finite value, in the form of a surd, may be obtained for the root (see Continued Fractions) by solving a quadratic whose coefficients are rational, both of whose roots will be roots of the proposed, since the coefficients of the latter are supposed rational; consequently, the first member of this quadratic will be a factor of the first member of the proposed equation, which may, therefore, be depressed two dimensions.

EXAMPLE.

To find the positive root of x3-2x-5-0 under the form of a continued fraction.

Comparing this with 23-qx+r=0, we find that

1

therefore (Art. 258) the equation has two impossible roots; and since its last term is negative, its third root is positive. Substituting 2 and 3, the results are -1 and 16; therefore the root lies between 2 and 3. and the transformed equation is

y3-10y2 — 6y—1=0,

Assume x=2+:

in which 10 and 11 being substituted, give -61, +54. Assume y=10+ and we obtain

613-94z2-20x-1=0,

whose root lies between 1 and 2. Proceeding in this manner, we find

the value of the root in a continued fraction; the method of reducing which to a common fraction will be hereafter given.

This method may be combined with Sturm's theorem.

Here finishes our recapitulation of the older methods. What follows belongs to the present more improved state of algebraic science.*

"We shall here point out a method of finding the equal roots of an equation, which avoids the laborious process of seeking the common divisor, and which may be employed when any other than Sturm's process for discovering the roots of an equation is used.

1. If an equation whose coefficients are commensurable have a pair of equal roots and no greater number, these roots must be commensurable; for the common measure of the first member of this equation, and the function derived from it, will be a binomial expression of the first degree with finite coefficients, and which, when equated to zero, will furnish one of the equal roots; these roots, therefore, must be commensurable, that is, either integers or fractions.

2. If the leading coefficient in the supposed equation be unity, and the others integral, the equal roots must be integral, because no fractional root can exist under these conditions (Art. 246).

3. If an equation with commensurable coefficients have three equal roots, and no more, these also must be commensurable; for, in this case, the common measure will be of the second degree, and, when equated to zero, will give two of the equal rorts: these roots, as just remarked, must be commensurable; hence all the three roots must be commensurable.

BINOMIAL EQUATIONS.

298. Binomial equations are those which can be reduced to the form

mA or xm—A=0.

A being any known quantity whatsoever.

(1)

And, as before, if the leading coefficient be unity, and the others integral, the equal roots will be integral.

4. By the same reasoning, if an equation with commensurable coefficients have m equal roots, and no other groups of equal roots, these m roots must be commensurable; and they will be integral if the leading coefficient be unity and the other coefficients integers.

5. When the leading coefficient is unity, and the other coefficients whole numbers, and m equal integral roots enter, we may infer, from the formation of the coefficients (245), that the absolute number, and the coefficient of the immediately preceding term, that is, the coefficient of x, will admit of a common measure involving m-1 of these roots; that the coefficients of x and x2 will have a common measure involving m-2 of them; and so on till we come to the coefficients of am-2 and 2-1, which will have a common measure involving the multiple root once. For, if the depressed equation containing only the unequal roots be considered, it will involve none but integral coefficients, since its last term is formed from the penult coefficient of the proposed divided by one root; so that if the equal roots be now introduced, they can combine with none but integral factors. Hence, if the root occur twice, it will be found among the integral factors of the common measure of the coefficients An (the final coefficient) and An-1; if it occur three times, it will be found among the factors of the common measure of An, An-1, and An-2, and so on. And, therefore, by trying several factors of the common measure in question, by actually substituting them for x in the proposed equation, when from any circumstance multiple roots are suspected to exist, we may remove all doubt on the subject. In analyzing an equation, the doubts that may arise as to the entrance of equal roots are confined to certain definite intervals, or within determinate numerical limits; so that, of the factors adverted to above, only those falling within these limits need be regarded.

And further, if the repeated root occur but twice, the square of it must be a factor of 20 or An; if it occur three times, the cube of it must be a factor of An, and the square of it a factor of An-1; if it occur four times, the fourth power of it must be a factor of An, the cube of it a factor of An-1, and the square of it a factor of An-2, and so on. And thus, of the factors of An to be tested, those only need be used whose powers also are factors, entering, as here described, according to the multiplicity of the roots.

6. These inferences may be easily generalized: they apply, whatever be the integral value of the leading coefficient, and whether the repeated root be integral or fractional.

a

Thus, let the repeated root be x=-, a and b having no common factor; then, if the root enΤ ter m times, the original polynomial will be divisible by (bx-a)m, giving a quotient involving the remaining roots, and into which none but integral coefficients enter (253). Let us now return to the original polynomial by multiplying this quotient by bx-a m times: the first multiplication by bx-a will evidently give a product, into the first term of which b must enter as a factor, and into the last of which a must enter; the next multiplication must, therefore, give a product, into the first term of which 62 must enter, into the second b, into the last a2, and into the last but one a; the third multiplication, therefore, must give a product whose first three terms involve b3, b2, b respectively, and last three a3, a2, a, reckoning these last in reverse order, and so on. Hence the coefficients A1, A2, A3, &c., will be divisible by bm, bm-1, bm-2, &c., respectively, down to b; and the coefficients A., An-1, An-2, &c., by am, am-1, am-2, &c., down to a. In other words, the coefficients, taken in order, reckoning from the beginning, will be divisible by the corresponding decreasing powers of the denominator of the repeated root; and the coefficients, reckoning from the end, will be divisible by the like powers of the numerator.

7. The inferences still have place, whatever be the degree of the multiple factor entering the proposed polynomial, so long as this factor, as well as the original polynomial, have none but integral coefficients. This is plain, from the reasoning in the preceding case, which remains the same, as respects the entrance of the factors b, a, whether the repeated multiplier be bx-a or bxm+....+a.

We perceive immediately that the m roots of this equation are different from one another; for the first member rm-A has no common factor with its derived function mam, and hence the proposed equation (Art. 253, Schol.) can not have equal roots. The roots, if we raise them to the power m, ought each to produce A, since they are the same as the values embraced in the expression r=A. We know, then, that this radical has m different values; but we shall recur to this subject again, and more at length.

299. When m is any composite number, the solution of equation (1) reduces itself to the solution of several binomial equations, the degrees of which are the factors of m.

Suppose m=pqr, instead of the equation P=0, we can take the equations x2=x', x'=x'', x'''=A,

in which x', x' are new unknowns.

It is evident that, after we have solved the equation "=A, the preceding equation x'=x" will make known the values of x', and that then the equation x=x' will give all the roots of the proposed equation. This agrees with the formula demonstrated in the theory of radicals (Art. 63), viz.,

VA="VA.

300. Designate by a a quantity whose mth power is A, and take x=ay. The equation A becomes amymam; dividing by aTM,

=

ym=1;

hence y=V1, and, consequently, r=a√1.

=

We conclude, therefore, that the roots of the equation A can be obtained by multiplying one of them by the roots of the equation yTM=1; or, in general, that the different mth roots of a quantity can be obtained by multiplying one of them by the mth roots of unity.

301. Let us consider more particularly the case in which A is a real quantity; and, to distinguish the hypothesis of A being positive or negative, write the binomial equation in this form:

These conclusions will greatly simplify the research after equal roots, and will either enable us wholly to dispense with the laborious process for the common measure, or will, at least, render the more tedious steps of it unnecessary

The first of these can have no fractional repeated roots, because the leading coefficient 2 has no factor a perfect power; the equal roots, if any, must, therefore, be integral. Unity, which always has claims to be tried, does not succeed; and from the factors of 9 and 6, it is plain that +3 and -3 are the only other numbers to be tested; and as no higher power of 3 than the square enters 9, we infer that more than two equal roots can not have place in the equation. By testing 3, we find this to be one of a pair of equal roots. Equal quadratic factors could not possibly enter the equation, since, as the first coefficient shows, the polynomial is not a complete square. In the second of the above equations no fractional roots can enter. Applying, therefore, +1 and -1, we discover that +1 is wice a root, and -1 three times. The remaining equal roots -2 and -2 are found from the resulting quadratic obtained by suppressing from the given equation the five factors of the first degree.

We can determine, at least by approximation, a positive quantity a such that we have a=A. Take, again, x=ay, equation (2) will become

This is the equation to which I shall confine myself exclusively.

302. The following remarks may be made with regard to this equation: 1. When m is an odd number, and the equation is y"=1 or y"—1—0, it evidently has the root y=1; and it has no other real root, for every other positive value of y will give y">1 or ym <1, and a negative value will render y negative. To obtain the equation on which the m- -1 imaginary roots depend, we shall divide yTM—1 by y-1, and thus obtain the equation

ym1+ym2+ym¬3...+y+1=0,

which belongs to the class of equations called reciprocal.

2. When m is an odd number, and the equation is y=-1, it has evidently for a root y=-1. By a reasoning analogous to the preceding, it may be proved that the other roots are imaginary; and we obtain the equation on which they depend by dividing yTM+1=0 by y+1. But to obtain all the roots of the equation y=-1, it is well to remark that this equation can be derived from y=-1 by changing y into y. It will suffice, then, to take all the roots of y"=1 with contrary signs.

3. Suppose m is an even number, and let m=2n, the equation y2=1, or y2-1=0, has for its roots y=+1 and y=-1. The other roots are imaginary, and the equation which contains them can be obtained by dividing y2a—1 =0 by (y−1)(y+1), or y2-1; but it will be well to observe that y- 1 =(y"—1)(y"+1), and that, consequently, the equation ya—1=0 can be replaced by two others more simple,

y"-1=0, y"+1=0.

4. Finally, when the equation is y2"-1, or y2+1=0, we know that the even powers of real quantities will always give positive results; we hence conclude that all the roots are imaginary. Taking y=z, the equation reduces to the degree n, and becomes simply z"=-1.

303. I now proceed to determine the solutions of the equations y"-1=0, y+10, in some particular cases.

Let m=2; the equations to be resolved are

y2-10, whence y=±1;

y2+1=0, whence y=√1.

Let m=3; to resolve the equation y3-1=0, observe that it has for a root y=1; we divide it by y-1, and it becomes

If we take the equation y+1=0, we shall observe that its roots are the same, except as regards sign, with those of y3-1=0; consequently, they

Let m=4; the equation y1-10 may be decomposed into two others, y2—1=0, y2+1=0; and from these equations we derive the four roots y+1, y±√1.

The equation y'+1 will be resolved differently; by adding 2y to both members of the equation, we can write it thus:

we can then decompose it into two others,

y2+1=y√2, y2+1=−y√2;

and, finally, from these we derive the four values of y,

y = {√2 ± ¦ √ −2, y = −1 √2 ± { √ —2.

We could have treated the equation y+1=0 as a reciprocal equation. We might have observed, also, that it gives y±√ −1, and that, taking successively√−1, −√ −1, we have

We have then only to reduce these values to the form a+ß√1 by the process in Art. 104.

By raising the equation y1=0 successively to the 10° degree, we shall find that its resolution depends on that of the preceding cases, or on the resolution of reciprocal equations, which reduce it to a degree less than the 5o.

Let us examine, first, the odd degrees. If we have the equation y3—1=0, having observed that it has the root y=1, we divide it by y-1; it then be

comes

y'+y+y+y+1=0,

a reciprocal equation, which we shall reduce to the 2° degree. To do this, we first write it under the form

1

1

Then take y+=z, which gives y2+ =2 -2; and, consequently, the

y2

for

These values being known, those of y will be by the relation y+==

and we have only to substitute instead of z successively each of its two values,

in order to find the four imaginary values of y. We have then the five values