406 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



of Legendre) is 



p 2 )m, Y = p <2 n 2cpq+cnq 2 , Z = p 2 2npq-}-cq 2 , 

 obtained from x = m, y = n, z = l. The complete integral solution of 



is found from x = y = z = l to be 



X= -bp 2 +cq 2 , Y= (b+d)p 2 +cq 2 -2cpq, Z= -bp 2 -cq 2 +(d+2b)pq. 



By changing the notation of the parameters, this becomes 



X = q 2 -bcp 2 , Y=(q+cp} 2 -acp 2 , Z=(q+bp} 2 +b(a+d)p 2 +dpq. 



J. Neuberg and G. B. Mathews 40 proved that the general rational 

 solution of x 2 -\-xy+y 2 = z 2 is x = p 2 q 2 , y = 2pq+q 2 , z = p 2 +pq+q 2 . A. 

 Cunningham 41 deduced %x+y = t 2 3u 2 , %x = 2tu from (|a;+?/) 2 +3(^c) 2 = z 2 . 



Ch. J. de la Vallee Poussin 42 proved that a necessary and sufficient 

 condition for integral solutions of ax 2 -{-2~bxy-\-cy 2 = mz 2 , where m is prime 

 to 2(b 2 ac), and the g. c. d. of a, 26, c is unity, is that m be representable 

 by a form of determinant b 2 ac and of the same genus as ax 2 +2bxy-\-cy 2 . 



E. S6s 43 found the complete solution of 



x 2 +bxy-\-y 2 = z 2 or y(bx-\-y) =z 1 x 2 



by setting y = \(z x), \(bx-\-y} =z-\-x. Eliminating y, we get 



. 7 X 2 -X&+1 p 



9 IV I - 



X 2 -l "g' 

 where p/q is a fraction in its lowest terms. Hence 



The same method applies to ax 1 -\-bxy +q/ 2 = 2 2 , a or c a square. 



A. Gerardin 44 found a general solution of aX 2 +6XF+cF 2 = /zZ 2 , given 

 one solut'on a, /5, 7, by setting X = a-\-mx, Y = p+my, Z = y. Then m is 

 determined rationally and 



X = cay*-2c(3xy- (aa+bftx 2 , Y = a(3x 2 -2aaxy- (ba+cp)y", 



Z = ayx^+byxy+cyy 2 . 



Gerardin 45 granted that ah?-\-bh+c = m 2 , replaced h by h+x, m by 

 m+fx, found x rationally, and hence obtained a solution of ay 2 -\-byz-\-cz 2 v 2 : 



y = hf 2 -\-ah-\-b 2mf, z=f 2 a, v = mf 2 (2ah-\-b)f-\-ma. 



A. Aubry 46 solved 2d 2 x 2 =F2dx+l d 2 = y 2 for d and made the radical 

 rational by means of a Pell equation. L. Valroff 47 made the substitution 



RS X 



_ X 2Y ' y S' __ 



40 Math. Quest. Educ. Times, 46, 1887, 97. See papers 36, 171. Cf. papers 68, 63 of Ch. IV. 

 "Ibid., 75, 1901,33-4. 



42 M6m. couronne's et autres me"m. acad. Belgique, 53, 1895-6, No. 3, 43-54. 



43 Zeitschrift Math. Naturw. Unterricht, 37, 1906, 186-190. 



44 Bull. Soc. Philomathique, (10), 3, 1911, 218. 

 46 Sphinx-Oedipe, 1907-8, 177-9. 



46 L'intermddiaire des math., 20, 1913, 144. 



47 Sphinx-Oedipe, 7, 1912, 74-6. 



