f c A =B q+ Rr ^B =R?i+R,r, c2R =R|</j-|-R5r3 Let d be the greatest common divisor of c and r, di the greatest common divisor of -r and r„ rf2 that of -rr and ra, rf3 that of Tj—t and r,. We shall 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: \i=° \i=° ]j3=° « (B=0, (R = 0, /.Rl==0, (R.J=0 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 5A=5B+5R(3> ■ J is entire, for c and r, by hypothesis, are divisible by d; hence, qlH 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 r err B=0, ^=0, destroy also ^A; but -g and g are prime with respect to each r other. Consequently, 1°, all the solutions of the system B = 0, ^=0, agree with those of the system A=0, B=0. [6] To obtain a relation between A, R, and ^, we multiply equation (3) by clt and in the resulting equations place, instead of etB, its value as found in the second member of the second equation of system (1); we thus obtain Cit-\- qql The quantity -g— is entire, because r and q are divisible by d: more CCi over, this quantity is divisible by di; for d, divides -j and r„ and it is prime with respect to R. Dividing the two members of the above equation by dlt q Ci'+Wi and taking, to abridge; g = M, ■—^—=Mi, it becomes H;A=M1R+MR1J (4) To obtain a relation between B, R, and -r, we first multiply the second c ccj eq, c cci equation of system (1) by -g, which gives -^-Bur-^-R-l-^Riri. Since -g and Ti are, by hypothesis, divisible by dlt it follows that d, divides also i DUt d\ is prime with respect to R; hence, ds divides Dividing all the terms of the equation by du and taking, to abridge, ^=N, ^ =Ni, it becomes gB=NlR+NR1J (5) Equations (4) and (5) prove that all the values of x and y, which reduce the polynomials R and j| to zero, destroy also A and J^fB; but and jp are prime with respect to each other; consequently, 2°, a// <A« solutions of the system R=0, j^=0, agree taith those of the given system, A=0, B — 0. [c] We obtain a relation between A, Ri, and g-, by multiplying equation (4) by c2, and placing, instead of c,R, its value found in the second member of the third equation of system (1); we thus find ■^-A=R,^M1?J+Mc35ij +1^^,. By hypothesis, <£s divides the first member of this equation, it also divides ra; it ought, then, to divide R, (m,?,-}- Mc2^; but Ri and d* are prime with respect to each other; di then divides the term by which R, in the above equation is multiplied. Designating the quotient by M2 the equation becomes g|A = MiR, + M1R1J (6) Multiplying equation (5) by Cj, and then placing, instead of c2R, its value found in the second member of the third equation of system (1), it becomes ^B=R1(N1„+Nc8J)+N1Rsr,. We can demonstrate as before that the multiplier of R! is divisible by d}, and, representing the quotient by Ng, we find ig-B=N3R1+N1RJla (7, Equations (6) and (7) show that all the values of x and y, which reduce the polynomials Ri and -r to zero, destroy also the first members of these two cciC, r2 ... , equations; but ^ g- and ^- are prime with respect to each other; conse quently, 3°, all the solutions of the system R^O, ^=0, SUH those of the proposed system, A=0, B=0. [d] The equation which gives a relation between A, Rj, and -j-, can be ob tained by multiplying equation (6) by c„ and placing, instead of CjR,, 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 and designating by M3 the quotient obtained by dividing the entire polynomial M2f/3+c3M|-7- oy dz, there results ^A=M3R,+ M3£ (8) r3 To obtain a relation between B, Rj, and -r, we multiply equation (7) by c3, <*» and put in the place of c3Rj the second member of the fourth equation of the system (1), which gives ^B=R,^N,?,+C3Nl5ij +Nar3. Dividing both members by and designating by N3 the quotient obtained by dividing the entire polynomial N3g3+c3Ni^- by d3l it becomes sraB=NA+N«s! (9) Equations (8) and (9) show that all the values of x and y, which reduce the polynomials and -j- to zero, destroy also the first members of those equals tions; but ^ ^ ^ and -j are prime with respect to each other; consequently, 4°, all the solutions of the system Ri=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). To form the equations which demonstrate this second part of the theorem, c q let us first place in equation (3) N instead of ^, and M instead of ^; it will become, transposing the term MB, NA-MB=R2 (10) 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 Ni, the second by Mi, remembering the values previ ously given to Ni and M,; 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 (M,B-N1A)R+(MB-NA)R,5-=0. r Placing instead of MB—NA its value previously determined, —R^, and suppressing the factor R|, this equation becomes N,A-M1B = -R,^ .... (11) Finally, we eliminate Ri 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 (MaB-NjAJR^M.B-N.AJRa—O. Placing in this equation, instead of MiB—NiA, its value, determined in (11), Rj^-, and suppressing the factor R^ it becomes NJA-MsB=R2jg- (12) In the same manner we obtain the equation N^_M,B=_5ro-. Equation 13 shows that every system of values of x and y which gives A=0, B=0, ought also to satisfy the equation T_r±r±r±_0 an equation which requires that one of its factors equal zero, whence it follows that the equations r Tt rt r3 give all the correct values of y. This being established, let x=o, y=j3 be a system of correct values of the equations A=0, B=0. T If the value y=P is a root of the equation ^=0. it is clear that the system x=a, y=P will be a solution of the system B=0, ^=0. If the value y=p does not verify the equation ^=0, and if it is a root of the equation -j-=0, we perceive, by equation (10), that the system x=a, y=P will give R=0 ; consequently, it will be a solution of the system R=0, I If the value y=/? verifies neither the equation ^=0 n°r the equation ~r=0, and is a root of the equation ^=0> we see, by equation (11), that the system x=o, y=8 will give Ri=0; consequently, it will be a solution of the system R,=o, T4=o. a, r T\ If the value y=/? does not verify any one of the equations ^=0. ar==0i ^=0, and is a root of the equation ^"=0, we see by equation (12) that the system x=za, y=P, will give R, = 0; consequently, it will be a solution of the system Rj=0, ^=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). The equation j. . g?. =0i which gives all the correct values of y, is called the final equation in y. REMARKS ON THE PRECEDING METHOD. It niay chance that in one of die. equations of system (2), for example, -j =0, R=0, a value of y, derived from the first equation, destroys some of the coefficients of the powers of x 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 x in R, the equation R=0 would not give any value of x. In Qict, 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 Sx°-|-Hx°-1-fKx°-' -f- =0, we suppose that the quantities which enter into the coefficients S, H, K, Arc, are of such a nature that wo 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=U, 15=0, suppose that y=fl, 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 make y=/3, the term MR^' will be reduced to zero, and the degree of the term M,R will be lowered with respect to x one or more units. Again, we can not suppose that the terms of M,R, which are reduced to zero, have been destroyed, until we have made y=/3 in the terms of MR,j, because the degrees of A, B, R, Ri, 6rc, are decreasing, and we see without difficulty, from the relations which exi9t between M, M,, M„, 6cc, that the degrees of these quantities with respect to x go on increasing. It will be CCi necessary, then, in order that y may have the value (3, that the degree of grrA with respect to x be lowered as many units as the degree of R is lowered. We can prove, in the same manner, that the value y=p 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=? one or more infinite values of x. As to the reciprocal proposition, that the solutions of the equations A=0, B=0, in which x is infinite, ought to be found among the solutions of systems (2), it is not the fact, as will be seen in the second examplo following. EXAMPLE I. |