Let d be the greatest common divisor of c and r, d, the greatest common proceed to prove that we can obtain all the solutions of the system A=0, B=0, without any foreign solution, by resolving the following systems: To establish this proposition, we shall first prove that the solutions of the systems (2) all agree with those of the system A=0, B=0; we shall afterward show that the solutions of the system A=0, B=0, are all comprised in those of the systems (2). [a] Dividing by d the two members of the first equation of system (1), it becomes d is entire, for c and r, by hypothesis, are divisible by d; hence, qB is divisible by d; but B, by hypothesis, is prime with respect to d; therefore, d divides q. Equation (3) shows that the values of x and y, which satisfy the equations C Τ =0, destroy also A; but and are prime with respect to each other. Consequently, 1°, all the solutions of the system B=0,=0, agree with those of the system A=0, B=0. [b] To obtain a relation between A, R, and di we multiply equation (3) by C1, and in the resulting equations place, instead of c1B, its value as found in the second member of the second equation of system (1); we thus obtain is entire, because r and q are divisible by d; more over, this quantity is divisible by d1; for d, divides and r1, and it is prime with respect to R. and taking, to abridge; d Dividing the two members of the above equation by d1, C equation of system (1) by which gives B=R+R. Since and r1 are, by hypothesis, divisible by d1, it follows that d, divides also CAR; but d d, is prime with respect to R; hence, d, divides. Dividing all the terms of c the equation by d1, and taking, to abridge, N, d cqi dd = N1, it becomes (5) Equations (4) and (5) prove that all the values of x and y, which reduce the Τι CC1 CCI polynomials R and to zero, destroy also A and -B; but and Τι dd d1 are prime with respect to each other; consequently, 2°, all the solutions of the system R=0, =0, agree with those of the given system, A=0, B=0. by multiplying equation (4) by c2, and placing, instead of c2R, its value found in the second member of the third equation of system (1); we thus find By hypothesis, de divides the first member of this equation, it also divides r2; it ought, then, to divide R. (M,q+Mc.); ; but R, and d, are prime with re spect to each other; d2 then divides the term by which R, in the above equation is multiplied. Designating the quotient by M, the equation becomes Multiplying equation (5) by c2, and then placing, instead of c2R, its value found in the second member of the third equation of system (1), it becomes We can demonstrate as before that the multiplier of R, is divisible by da, and, representing the quotient by N2, we find Equations (6) and (7) show that all the values of x and y, which reduce the to zero, destroy also the first members of these two T2 and are prime with respect to each other; conseFITS quently, 3°, all the solutions of the system R1=0, posed system, A=0, B=0. [d] The equation which gives a relation between A, R2, and can be ob tained by multiplying equation (6) by c, and placing, instead of caR1, its value as given in the second member of the fourth equation of system (1); we thus find Dividing the two members of this equation by ds, and designating by Ms the quo T2 tient obtained by dividing the entire polynomial M2q3+cMoy dз, there results T3 To obtain a relation between B, R2, and we multiply equation (7) by cз, and put in the place of c3R, the second member of the fourth equation of the system (1), which gives Dividing both members by ds, and designating by N, the quotient obtained 3 by dividing the entire polynomial Neqs+c3N by d3, it becomes da Equations (8) and (9) show that all the values of x and y, which reduce the to zero, destroy also the first members of those equa 73 ly, 4°, all the solutions of the system R=0,=0 concur_with_those_of_the proposed system, A=0, B=0. (II.) It remains still to be proved that any system whatsoever of values which satisfy the equations A=0, B=0, is a part of the systems of values which furnish equations (2). C To form the equations which demonstrate this second part of the theorem, let us first place in equation (3) N instead of and M instead of; it will become, transposing the term MB, Eliminate now R between equations (4) and (5). We can effect this elimination by subtracting one of these equations from the other, after we have multiplied the first by N1, the second by M1, remembering the values previ ously given to N, and M1; but the calculations will be simpler if we multiply equation (4) by B and equation (5) by A. Subtracting the two resulting equations the one from the other, we find Placing instead of MB—NA its value previously determined, —R1⁄2, and suppressing the factor R1, this equation becomes Finally, we eliminate R1 between equations (6) and (7). To do this, multiply equation (6) by B and equation (7) by A; then subtract the one of the resulting equations from the other, we thus obtain T3 (M ̧B—N ̧A)R1+(M,B-N,A)R=0. Placing in this equation, instead of M,B—-N1A, its value, determined in (11), Equation 13 shows that every system of values of x and y which gives A=0, B=0, ought also to satisfy the equation an equation which requires that one of its factors equal zero, whence it follows that the equations 2=0,0,0,0, give all the correct values of y. This being established, let x=a, y=ß be a system of correct values of the equations A=0, B=0. If the value y=ẞ is a root of the equation 0, it is clear that the system If the value y=ß does not verify the equation=0, and if it is a root of the equation=0, :0, we perceive, by equation (10), that the system r=a, y=ẞ will give R=0; consequently, it will be a solution of the system R=0, =0. Τι If the value y=ß verifies neither the equation=0 nor the equation=0, Το and is a root of the equation =0, we see, by equation (11), that the system da x=a, y=ß will give R1=0; consequently, it will be a solution of the system :0, If the value yß does not verify any one of the equations =0, = di 0, we see by equation (12) that the T2 T3 =0, and is a root of the equation = de system x=a, y=ß, will give R2=0; consequently, it will be a solution of the system R=0, 7=0. C c Hence, all the systems of values which satisfy the equations A=0, B=0, form part of the values which furnish equations (2). It may chance that in one of the equations of system (2), for example, di =0, R=0, a value of y, derived from the first equation, destroys some of the coefficients of the powers of r in the second equation, after the highest power of x; in this case we only obtain a number of values of x inferior to the degree of the equation R=0; and if the substitution of the value of y should destroy all the multipliers of the powers of r in R, the equation R=0 would not give any value of x. In fact, it can be proved, by a method similar to that which we have employed with reference to the general equation of the second degree (Art. 191), that if in an equation of the form S2"+Hxa¬1+ Kx"→2 +...=0, we suppose that the quantities which enter into the coefficients S, H, K, &c., are of such a nature that we have S=0, H=0, &c., the equation has infinite roots equal in number to the consecutive coefficients which are reduced to zero. But it should be remarked that the theory by which we have proved that the solutions of systems (2) are the same with those of the system A=0, B=0, only applies to solutions expressed by finite values of x and y. To prove that the solutions of systems (2), in which the value of x is infinity, also suit the proposed equations A=0, B=0, suppose that y=ß, verifying the equation =0, causes one or more of the multipliers of the higher powers of x in R to vanish. If, in the two members of the equality (4) we Τι di Τι 'd make yẞ, the term MR, will be reduced to zero, and the degree of the term MR will be lowered with respect to x one or more units. Again, we can not suppose that the terms of MR, which are reduced to Τι zero, have been destroyed, until we have made y=ß in the terms of MR because the degrees of A, B, R, R1, &c., are decreasing, and we see without difficulty, from the relations which exist between M, M,, M2, &c., that the degrees of these quantities with respect to r go on increasing. It will be CC1 necessary, then, in order that y may have the value ẞ, that the degree of A dd with respect to r be lowered as many units as the degree of R is lowered. We can prove, in the same manner, that the value y=ß ought also to cause one or more of the coefficients of the higher powers of x in B to vanish. The equations A=0, B=0 will give then for y=3 one or more infinite values of x. As to the reciprocal proposition, that the solutions of the equations A=0, B=0, in which r is infinite, ought to be found among the solutions of systems (2), it is not the fact, as will be seen in the second example following. EXAMPLE 1. (y-1)+(y+1)x2+(3y2+y-2)x+2y=0, |