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 a 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,

PX D= 0, Q X 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.

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

mx + ny d = 0,

and in this view the question will be indeterminate; but if we suppress this factor, we are furnished with the equations

or

ax + by — c = 0,

ax + by = c,

ax + 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, xm + Pxm-1 + Qxm-2 + R xm−3 x2 + P'xn−1 + Q x2-2 + R'1⁄2"―3 where the second unknown quantity, y, is involved in the coeffi cients, 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.

.....

.....

+ Tx+U=0, + Y' x + Z' = 0,

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.

and this value of x will be known, or at least will be expressed by means of y, if we substitute for p its value deduced from the equations of the first degree, formed above.

This expression assumes, in general, a fractional form, so that

M

we have x = or NxM 0; and it may be seen in this

N'

case, that the values of y, which would cause M and N to vanish at the same time, would verify the preceding equation independently of x; this takes place in consequence of the fact, that by means of these values, the proposed equations would acquire a common factor of a degree above the first. It would not be difficult to go back to the immediate conditions in which this circumstance is implied; but the limits I have prescribed to myself in the present treatise do not permit me to enter into details of this kind.

194. Now let there be the equations

Q'

x2 + Px + Q = 0, x2 + P' x + Q′ = 0; the factors, by which xa is multiplied, will be here of the first degree, or x + p and x + p' simply; in this case,

=

R=0, R' 0, S'= 0, q = 0, q′ = 0, r = 0,

From the first two equations we obtain

(P — P') P — ( Q — Q);
—

P. P

P=

(P— P')

P— (Q — Q') Q = (P — P') P' Q — (Q—Q') Q or (P — P') (P Q — Q P') + (Q — Q')2 = 0.

Now if in the equation

x = p- P;

we put, in the place of p, its value found above, we have

195. In order to aid the learner, I shall indicate the operations necessary for eliminating x in the two equations

x2 + P x2 + Qx+ R = 0, x3 + P' x2 + Q′ x + R' = 0.

In this case, we have

S' = 0, r = 0 (193),

and are furnished with these five equations;

P+ P = P'

Q+Pp' + q = Q

+ P,

+ P' p + q,

R+QP+Pq = R' + Qp + P'q,
Rp+QqR'p + Q'q,
Rq' = R'q,

We may, by the rules given in art. 88, obtain immediately from any four of these equations, the values of the unknown quantities p, p', q and q'; but the simple form, under which the first and the last of the equations are presented, enables us to arrive at the result, by a more expeditious method. In order to abridge the expressions, we make

P-Pe, Q — Q = e', R-R'e";

and proceed to deduce from the first and last of the proposed equations,

then substituting these values in the three others, and making the denominator R to disappear, we have

(P' — P) Rp + (R R') q = R(e' — Pe). . . (a), (Q — Q) Rp + (RP' — PR') q = R (e" — Qe) . . . (b), (RR) Rp + (RQ' — QR') q = — R2 e . . . . . . (c).

If now we obtain, from the equations (a) and (b), the values of p and q (88), and suppress the factor R, which will be common to the numerators and the denominator, we have

P=

9=

(e' —

Pe) (RP'PR') — (R — R') (e'' — Qe) (P P) (RP PR') — (R— R') (Q' — Q)

(P' — P) (e" —

(P' — P) (RP'

putting these values in the

equation (c), we obtain a final equation, divisible by R, and which may be reduced to

-

(RR) [(e-P c) (RP' — PR') — (R— R') (e" — Qe)] + (RQ — QR') [(P' — P) (e“ — Q e) — (e′ — P e) (Q′ — Q)] =— Re[(P' —P) (RP' — PR') — (R — R') (✅′ — Q)] ; it only remains then to substitute for the letters e, e', e'', the quantities they represent.

==

196. If we have the three unknown quantities x, y, and z, and are furnished with an equal number of equations, distinguished by (1), (2) and (3); in order to determine these unknown quantities, we may combine, for example, the equation (1) with (2) and with (3), to eliminate x, and then exterminate y from the two results, which are obtained. But it must be observed, that by this successive elimination, the three proposed equations do not concur, in the same manner, to form the final equation; the equation (1) is employed twice, while (2) and (3) are employed only once; hence the result, to which we arrive, contains a factor foreign to the question (84). Bézout, in his Théorie des Equations, has made use of a method, which is not subject to this inconvenience, and by which he proves, that the degree of the final equation, resulting from the elimination among any number whatever of complete equations, containing an equal number of unknown quantities, and quantities of any degrees whatever, is equal to the product of the exponents, which denote the degree of these equations. M. Poisson, has given a demonstration of the same proposition more direct and shorter than that of Bézout; but the preliminary information, which it requires, will not permit me to explain it here; it will be found in the Supplement. At present, I shall observe simply, that it is easy to verify this proposition in the case of the final equations presented in articles 194 and 195. If we suppose the proposed equations given in those articles to be complete, the unknown quantity y enters of the first degree into P and P', of the second degree into Q and Q', of the third into R and R'; hence it follows, that e will be of the first degree, e' of the second, and e" of the third, and that the terms of the highest degree found in the products indicated in the final equation given in art. 194, will have 4, or 2. 2, for an exponent, and those of the final equation, art. 195, will have 9 or 3.3.