making the denominators to disappear, we have

(Q — Q ) 2 — P (P — P') (Q — Q') + Q (P — P') 2 = 0, then developing the two last terms, and making the reduction

2

(Q — Q')2 + (P — P) ( P Q —Q P') = 0.

We have then only to substitute for P, Q, P' and Q, the particular values, which answer to the case under consideration.

189. Before proceeding further I shall show, how we may determine, whether the value of any one of the unknown quantities satisfies, at the same time, the two equations proposed. In order to make this more clear, I shall take a particular example; the reasoning employed will however be of a general nature. Let there be the equations

which we shall suppose furnished by a question, that gives y = 3. In order to verify this supposition, we must substitute 3 in the place of y, in the proposed equation; we have then

equations, which must present the same value of x, if that, which has been assigned to y, be correct. If the value of x be repre sented by a, the equation (a) and the equation (b) will, according to what has been proved in art. 179, both of them he divisible by x -a; they must therefore have a common divisor, of which x—« forms a part; and in fact, we find for this common divisor xnd ·2 (48); we have therefore « = 2. Thus the value y = 3 fulfils the conditions of the question, and corresponds to x = 2.

If there remained any doubt, whether or not the common divisor of the equations (a) and (b) must give the value of x, we might remove it by observing, that these equations reduce themselves to

from which it is evident, that they are verified by putting 2 in the place of x.

190. The method I have just explained, for finding the value of x, when that of y is known, may be employed immediately in the elimination of x

Indeed, if we take the equations (1) and (2), and go through the process necessary for determining, whether they have a common divisor involving x, instead of finding one, we arrive at a remainder, which contains only the unknown quantity y and numbers, that are given; and it is evident, that if we put in the place of y its value 3, this remainder will vanish, since by the same substitution, the equations (1) and (2), become the equations (a) and (b), which have a common divisor. Forming an equation therefore, by taking this remainder and zero for the two members, we express the condition, which the values of Y must fulfil, in order that the two given equations may admit, at the same time, of the same value for x.

The adjoining table presents the several steps of the operation relative to the equations,

x3 + 3x2 y + 3 x y2 980

x2+4xy-2 y2-10=0,

on which we have been employed in the preceding article. We find for the last divisor,

3

(9 y2 + 10) x — 2 y3 — 10 y — 98 ; and the remainder, being taken equal to zero, gives 43y+345 y 1960 y3 +750 ye y4.

- 2940 y - · 4302 = 0,

an equation, which admits, besides the value y = 3 given above, of all the other values of y, of which the question proposed is susceptible.

The remainder above mentioned being destroyed, that preceding the last becomes the common divisor of the equations proposed; and being put into an equation, gives the value of x when that of y is introduced. Knowing, for example, that y = 3, we substitute this value in the quantity

(9 y2+10) x2 y3 10y-98; then taking the result for one member, and zero for the other, we have the equation of the first degree

191. The operation, to which the above equations have been subjected, furnishes occasion for several important remarks. First, it may happen that the value of y reduces the remainder preceding the last to nothing; in this case, the next higher remainder, or that, which involves the second power of x, becomes the common divisor of the two proposed equations. Introducing then into this the value of y, and putting it equal to zero,

or rather (9y+10)x2+36xy3—18y4-110y3-100|(9y2+10)x-2y3—10y—98

we have an equation of the second degree, involving only x, the two values of which will correspond to the known value of y. If this value still reduce to nothing the remainder of the second degree, we must go back to the preceding, or that into which the third power of x enters, because this, in the case under consideration, becomes the common divisor of the two proposed equations; and the value of y will correspond to the three values of x. In general, we must go back until we arrive at a remainder, which is not destroyed by substituting the value of y.

It may sometimes happen, that there is no remainder, or that the remainder contains only known quantities.

In the first case, the two equations have a common divisor independently of any determination of y; they assume then the following form,

PXD=0, Q× D=0,

D being the common divisor. It is evident, that we satisfy both the equations at the same time, by making, in the first place D=0; and this equation will enable us to determine one of the unknown quantities by means of the other, when the factor D contains both; but if it contains only given quantities and x, this unknown quantity will be determinate, and the other will remain wholly indeterminate. With respect to the factors, which do not contain x, they are found by what is laid down in art. 50. Next, if we make at the same time

P=0, Q=0,

we have still two equations, which will furnish solutions of the question proposed.

Let there be, for example,

(ax+by—c) (mx+ny — d) = 0,
(a'x+b'y—c') (m x + n y — d) = 0 ;

-

by supposing, first, the second factor, common to the two equations, to be nothing, we have with respect to the unknown quan. tities x and y only the equation

mx + ny―d = 0,

ques ion will be indeterminate; but if we we are furnished with the equations

c = 0,

ax + by = c,

a' x + b'y — c' = 0,

a' x + b' y = c' ;

and in this case the question will be determinate, since we have as many equations as unknown quantities.

When the remainder contains only given quantities, the two proposed equations are contradictory; for the common divisor, by which it is shown, that they may both be true at the same time, cannot exist except by a condition, which can never be fulfilled.(D) This case corresponds to that mentioned in art. 68, relative to equations of the first degree.*

192. If then we have any two equations

x2 +Р xm¬1 + Q xm−2 + R xm—3 +Tx+U=0,

....

x2 + P'x2+ Q xn−2+ R′ xn−3....+Y'x+Z' = 0, where the second unknown quantity y is involved in the coefficients P, Q, &c. P', Q, &c. in seeking the greatest common divisor of their first members, we resolve them into other more simple expressions, or come to a remainder independent of x, which must be made equal to zero.

This remainder will form the final equation of the question proposed, if it does not contain factors foreign to this question; but it very often begins with polynomials involving y, by which the highest power of x in the several quantities, that have been successively employed as divisors, is multiplied, and we arrive at a result more complicated than that, which is sought, should be. In order to avoid being led into error with respect to the values of y arising from these factors, the idea, which first presents itself, is to substitute immediately in the equations proposed each of the values furnished by the equation involving y only; for all the values, which give a common divisor to these equations, necessarily belong to the question, and the others must be excluded. It will be perceived also, that the final equation will

* It will be readily perceived, by what precedes, that the problem for obtaining the final equation from two equations with two unknown quantities, is, in general, determinate; but the same final equation answers to an infinite variety of systems of equations with two unknown quantities. Reversing the process, by which the greatest common divisor of two quantities is obtained, we may form these systems at pleasure; but as this inquiry relates to what would be of little use in the elementary parts of mathematics, and would lead me into tedious details, I shall not pursue it here. Researches of this nature must be left to the sagacity of the intelligent reader, who will not fail, as occasion offers, of arriving at a satisfactory result.