By the same reasoning it is proved that the roots common to B=0, R=0 are the same as those which satisfy the pairs R=0, R'=0; R'=0, R′′=0, &c. As the degree of the divisor is diminished at each operation, two equations must of necessity be obtained; the one in x, y, the other in x only or y only, and of which the pairs of common values are precisely the same, and in the same number as those of the proposed polynomials. The last, containing one unknown quantity, will, therefore, be the final equation required.

The last remainder but one being the common divisor of not only the polynomials A, B, but of all the preceding remainders also, it follows that the systems of roots which result from the equality to zero of the two last remainders satisfy all the other remainders as well as the proposed polynomials A, B. Whence it also follows that the same final equation and the same common divisor may belong to an indefinite number of equations; since with A, B others may be formed composed of A, B, as A, B are themselves by means of the remainders and quotients.

243. If this method of elimination is compared with that which is followed in the resolution of equations of the first degree, containing several unknown quantities, the analogy of the two is evident.

Let there be, for example, two equations of the first degree, these may be always put under the form x+p=0, x+q=0.

But substituting in the one the value of x taken from the other, p-q is obtained for final equation. Now this is the condition under which the two quantities +p and x+q admit a common divisor in x, as may be readily shown by division.

In general the different processes of elimination, which may be employed to effect the resolution of equations involving many unknown quantities, have this common object, to substitute for the proposed system of equations another system perfectly equivalent, and to decompose the difficulty which results from the simultaneous existence of many unknown quantities in all the equations compared together, by reducing successively their resolution to that of more simple equations, of which the last involves only one unknown quantity, and which may be resolved exactly, or at least by the known methods of approximation.

244. If the value of y, deduced from the final equation, causes the remainder of the first degree in x to vanish, the preceding remainder in r2 becomes the common measure of the proposed polynomials. Substituting the value of y in this remainder, and making it equal to zero, two values of x are obtained corresponding to the same value of y.

But if the value of y causes the remainder in r2 also to vanish, the remainder in 3 is the common measure of the proposed polynomials, and three values of x correspond to the same value of y.

And, in general, if the value of y causes the remainder in a to vanish, the remainder in " is the common measure required, and n values of x correspond to the same value of y.

If the last division is exact the two equations have a common measure, without any determination of y; they are therefore of the form Q×D=0; Q'xD=0; D being the common measure.

In this case both equations are satisfied by making D=0; and if D contains x and y the equation D=0 gives the means of determining one of the quantities x, y in terms of the other.

But if D contain only x or y, then z ory is determinate, and y or r altogether indeterminate.

By making, next, Q=0, Q'=0, two equations are obtained, which may give determinate solutions of the proposed equations. If, for example,

(ax+by—c)(mx+ny−d)=0; and (a'x+b'y-c')(mx+ny—d)=0, by making the common factor mx+ny-d=0, one equation only is given to

determine the two quantities x, y, and the question seems indeterminate ; but if the common factor is suppressed the equations

ax+by-c=0; a'x+b'y-c'=0

are obtained, which must give a determinate solution.

245. If the final equation is independent of y, or contains only known quantities, which do not destroy each other, the proposed equations involve contradictory conditions.

To render this more evident let two equations in x, y be,

yx3—(y3—3y—1)x+y=0; x2−y2+3=0.

Dividing the first by the second, and the divisor by the remainder as follows:

1st remainder

x2-y2+3)yx3—(y3—3y—1)x+y(xy
yx3 — (y3-3y)x

The final equation is +3, and the common measure x+y.

Denoting the first quotient xy by Q, and the second x-y by Q'; therefore, x2-y2+3=(x+y)(x−y)+3=(x+y)Q'+3.

yx3 — (y3—3y—1)x+y=[x2−y2+3]xy+x+y=[(x+y)Q ́+3]Q+x+y. Hence it appears that one of the proposed equations is formed from the other by multiplying that other by xy or Q, and adding x+y to the product, and consequently that the two equations involve contradictory conditions; also the final equation indicates that if the number 3 were not contained in the second equation, the sum x+y would be the common measure of the two polynomials.

Consequently, a system of two equations in x, y cannot give values of x, y, when the final equation is numerical.

246. It is necessary to observe that the quantities by which the partial dividends are multiplied to render the divisions possible often introduce into the last remainder factors which are foreign to the question, and which render this remainder not the true final equation.

To avoid the error arising from these factors the most obvious precaution is to substitute in the proposed equations each of the values given by the equation in then all the values which communicate a common measure to these equations necessarily belong to the question, and the others ought to be excluded.

y,

It is also evident that the final equation may be rendered incomplete by the suppression, during the process, of any factor in y.

The following method is free from these inconveniences.

Let equation A or x”+PxTM-1+QxTM-2 . . . . . +T+V be supposed

also let equation A be multiplied by "+A'x+B'a", &c. and equation B be multiplied by x-'+Ax"-2+BxTM-3, &c. it is evident that the products must be equal; therefore, (xTM+PxTM-1+QxTM-2+, &c.)(x"→1 + A ́x2¬2 + B ́x2¬3 +, &c.) = (x"+P′x”"~'+ 'Q'x2+, &c.) (x+Ax2+Bx+, &c.)

-2

E.

Performing the multiplications and making equal to each other, the coefficients of the same powers of x (Art. 174) m+n−1 equations are obtained

between the indeterminate quantities A, B, C, . . . A', B', C', . Now the number of indeterminate quantities in equation C is m-1, and in equation D, n-1; therefore the number in equation E is m+n−2. Of the m+n-1 equations m+n-2 suffice to determine A, B, C, . . . A', B′, C', ; and one equation remains between P, Q, R . . P', Q', R' ... which it is necessary to satisfy in such a manner that the equations C, D may have a common divisor x-a; this equation of condition is the final equation required.

Since none of the indeterminate quantities A, B, C... A', B', C'... is multiplied by itself, the equations by means of which these quantities are determined are of the first degree.

The final equation being resolved, and the values of y successively substituted in A, B, C, . A', B', C', the results are obtained from

the division of the polynomials C, D by the common divisor æ—a.

If the equations A, B are incomplete, the two products E cannot be complete polynomials of the degree m+n-1; but the terms which are deficient in the one are found in the other. For taking the least favourable case, viz., x"+P=0; x2+P'=0;

the identity which results from the equality of the two products is

(x+P)(x-'+A'x2+, &c.)=(x"+P′)(x"¬1+ÂxTM-2+, &c.)

Example. Let x2+Pr+Q=0;

x2+P'x+Q=0.

Denoting by x-a the factor which is to be rendered common to these equations by the suitable determination of y, the first equation may be considered the product of r-a by a factor x+A of the first degree; and the second the product of x-a, by a factor +A' also of the first degree.

and

••• x2+Px+Q=(x−a)(x+A),
x2+P'x+Q'=(x−a)(x+A'),
(x2+Pr+Q)(x+A')=(x2+Pr+Q)(x+A),

or x3+P x2+Q | x+QA'=x3 +P′ | x2+Q′ | x+AQ'.

+A+PA

+A+AP
+ AP

Making the coefficients of the same powers of x equal to each other,

P+A'=P'+A' or A-A'=P–P'

Q+PA'=Q+AP' or AP'-PA'=Q-Q'

QA'=AQ' or AQ'-QA'=0

1,

2,

3.

By means of these three equations of the first degree the two indeterminate quantities A, A' can be eliminated, and a single equation obtained in terms of the quantities P, Q, P', Q.

For, if from equation 1, multiplied by P, or AP-PA'=(P-P')P,
is subtracted equation 2, or AP-PA'=Q-Q',
the remainder is AP-AP'=(P-P')P―(Q-Q)
Whence A=(P―P')P—(Q—Q')

Similarly, A'=

P-P'

(P-P)P'-(Q-Q)

P-P'

If these values of A, A' are substituted in equation 3,

or (P-P')PQ'-(Q—Q')Q'—(P—P')QP'+(Q—Q')Q=0,

or (P-P')(PQ'—QP')+(Q—Q')(Q—Q')=0,

or (P-P)(PQ'—QP')+(Q—Q')2=0.

The quantities P, P, Q, Q, containing only y and known quantities, this is the final equation in y.

It has been already noticed that, if this equation is identical, the proposed equations have at least one common factor of the form r-a, whatever be the value of y; and that, if it contains only known quantities, these equations are contradictory.

When the final equation has the proper form, the factor x-a is obtained by dividing the first of the proposed equations by x+A, thus,

x+A)x2+Pr+Q(x+P-A
x2+Ax

(P-A)x+Q
(P-A)x+(P-A)A

Q-(P-A)A.

The quotient is x+P-A, and the remainder is considered equal to zero, because it is reduced to zero by the substitution, for y, of a value deduced from the final equation.

Making the quotient x+P-A equal to zero, the value of x is x-A-P, and by substituting the value of A,

(P-PP-(Q-Q)

x=

-P.

This example is given as an illustration of the general method. From its particular form it admits of resolution by another and a much shorter process.

For if from a2+Px+Q=0

x2+P'x+Q'=0 is subtracted,

the remainder is (P-P')x+Q-Q'=0 ;

Q-Q'
P-P

OF THE DEGREE OF THE FINAL EQUATION.

247. The degree of the final equation cannot be depressed by the reduction of each of the coefficients P, Q, R . . . P', Q', R'. . . in the equations, 2+Px-1+Qx-2 x"+P'x"1+Q'x"-2

+Tx+V=0,

.. +Tx+V'=0,

to the term of the highest exponent in y which it contains; for the degree of each of the equations is not changed by the reduction. Therefore the reasoning may be applied to the equations,

x+ayxTM-1+by2xm-2

x"+a'yx"1+b'y2x2-2.

+ty"-x+vy"=0
+ty"1x+v'y"=0

which are of the same degree respectively as the preceding equations. The latter are reducible to

are numbers.

Denoting by a, B, y

...

the numerical roots of equation 3,
the numerical roots of equation 4,

and by a', B', y these equations may be decomposed into,

(-a) (—B) (~—y), &c.=0,

Substituting in equation 5 the roots of r from equation 6, viz. a′y, ß'y, &c., (a'y—ay) (a'y-ßy)(a'y—yy), &c.=0, (B'y-ay)(B'y-By) (3'y-yy), &c.=0, ('y-ay) (y'y-By) (y'y-yy), &c.=0.

Or, since the number of factors in equation 5 is m, and the number of roots in equation 6 is n,

y"(a'-a)(a'-B)(a'—y), &c.=0,
yTM (3′—a)(3′—3)(3′—y), &c.=0,
y" (y'—a)(y'—ẞ)(y'—y), &c.=0.

Consequently there are n equations, each of the degree m; these give all the solutions in y. The product of these roots (or solutions) of y is the final equation, since it becomes zero for all the values of y which render its factors zero, and only for these values. Now this product is evidently of the degree mn. Consequently the degree of the final equation (unless roots not belonging to the proposed equations are introduced by the process of elimination) cannot exceed the product of the degrees of the proposed equations.

It ought to be observed that the numerical values of the roots of y are changed by this process, but that their number remains undisturbed by it. OF THE EQUATIONS OF THE DIFFERENCES AND THE SQUARES OF THE DIFFERENCES OF THE ROOTS of an Equation.

248. Any equation containing one unknown quantity being proposed, it may be required to find another equation whose roots shall be the differences of the roots of the proposed equation combined two by two.

Also, let x, x be any two roots of equation A, and y the difference of these roots, so that x=x+y.

Equation A must be satisfied by the substitution of x+y for x'. Whence (x+y)+P(x+y)"¬1+Q(x+y)=-2+, &c.=0. Expanding the powers of x+y, and expressing by X, X', X", &c. the polynomial + Px”¬1 + QxTM¬2+, &c., and its successive derivatives (Art. 236).

-2

1

X+X'y+;X"y2+2.3X""y3+

a being a root of equation A, or X=0, the first term of the last equation may be suppressed; the other terms being divided by y,

Previous to the suppression of the common factor y, x, x were any two roots whatever of equation A. Suppose x'=x; the difference r'—x=y=0. Therefore the equation in y, before the suppression of the factor y, must have contained the root y=0; the equation being divisible by y, this is

manifest.

....

The division by y being made, the root y is suppressed, and the remaining m-1 values of y in equation B are the differences which would be obtained by subtracting the root denoted by x from all the other roots of equation A. For the sake of precision, let the m roots of equation A be denoted by a, b, c, d, . . . . and let it be supposed that in equation B each of these roots in succession is substituted for x, a number, m, of equations is the result of these substitutions. Of these m equations the first has for roots the differences between a and the other roots b, c, d, . . . . ; the second, the differences between b and the other roots a, c, d, .; the third, the differences between c and the other roots a, b, d, &c. Consequently if x is elimi

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