640 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



From p', q', we get p", q", etc. Any two consecutive terms of p, q, p', q f , 

 p", yield a solution with y = l. Proceeding in the reverse order, we 

 obtain a sequence q, p, q\, pi, g 2 , , any two consecutive terms of which 

 yield a solution. 



To obtain an initial pair of solutions, set y = l and let the quartic be 

 the square of ax 2 +bx d or of ax 2 bx d; then 



4bd b*-2ad-c 



X ~b*-2ad-c r X ~ 4ab ' 



To treat aC 4 j8=D, where rb/3 is a square a 2 , set C=(l+z)/(l x). 

 Then 



a 2 x 4 +4(a : F/3)^ 3 +6aV+4(a=F J 8)x+a 2 = D, 



which is of the above type. Euler treated in detail the cases 



Euler 144 treated V=A+Bx+Cx 2 +Dx 3 +Ex* = D. If V can be given 

 the form P 2 +QR, where 



P = a+bx+cx 2 , Q = d+ex+fx 2 , R = g+hx+ix*, 



then V=(P+Qy} 2 , where 2Py+Qy 2 -R = 0. The latter is also quadratic 

 in x, viz., Sx 2 +Tx+U = Q. From initial solutions x, y, we obtain 145 

 x' x T/S; then from x' we get y'= y2P'/Q f , etc. As in the pre- 

 ceding paper, we thus obtain two series of solutions of V = D . 

 If, for E 0, V =f 2 for x = a, we may take 



p=f, Q = x-a, R = B+C(x+a)+D(x 2 +ax+a 2 '). 



For a general V, let y=/ 2 for 2 = a. When x is replaced by a+t, let 

 7 become / 2 +a+/3 2 H-7 3 +S 4 . Then V = P 2 +QR for 



Euler 146 gave ten values of x for which 



a-+2abx+ (b 2 

 including 



x=( df)/e, z = a+bx; x=al(bf), z = x(ex+d). 



G. Libri 147 treated a?x*-\-bx 3 -}-cx 2 -}-dx-\-e = z 2 with all coefficients positive 

 (since we may replace x by Xi+h). Multiply by 4a 2 and set 



2az = 2a 2 x 2 +bx+v. 

 Thus 

 (3) (4a 2 



144 M6m. Acad. Se. St. Petersb., 11, 1830 [1780], 69; Comm. Arith., II, 474. Cf. Pepin. 88 



146 This method of solving any equation quadratic in x and in y was given by Euler also in 



M<5m. Acad. Sc. St. Petersb., 11, 1830, 59; Comm. Arith., II, 467. For applications to 

 rational quadrilaterals, see Kummer, 133 and Schwering 160 of Ch. V. Cf. papers 55, 143, 

 148, 155; also Pepin 140 of Ch. IV, Guntsche 91 - 162 of Ch. V. On the relation of elliptic 

 functions to an equation quadratic in x and in y, see G. Frobcnius, Jour, fiir Math., 106, 

 1890, 125-188. 

 148 Opera postuma, 1, 1862, 266 (about 1782). 



147 Jour, fiir Math., 9, 1832, 282. 



