CHAP, xxil] ax*-\-by 4 +dx-y- MADE A SQUARE. 637 



V. A. Lebesgue 117 noted that if x 4 -\-ax 2 y 2 +by 4 = z 2 has the solution x = r, 

 y s, z = p, it has also the solution 



x = r 4 6s 4 , y = 2prs, z = p 4 (a 2 46) r 4 s 4 . 



A. Desboves 118 remarked that this generalization of the result by 

 Lagrange 68 for a = is insignificant since it is made by replacing his initial 

 identity (the following for d = 0) by 



(u 2 -bv 2 ) 2 +d(u 2 -bv 2 )(2uv+dv 2 )+b(2uv+dv 2 ) 2 =(u 2 +duv+bv 2 ) 2 , 



which Lagrange gave in his addition IX to Euler's algebra (French transl., 

 2, 1774, 640). 



A. Genocchi 119 proved by descent that x 4 +x 2 y 2 -\-y 4 ^= D. 



T. Pepin 120 treated x 4 +8x 2 +l = D. 



E. Lucas 121 deduced two solutions (X, Y, Z) from a given solution 

 (x, y, z) of 



X 4 ~2(a+2f 2 ')x 2 y 2 +(a 2 +b 2 )y 4 =z 2 . 



For brevity, set 



A = 4/ 4 +4a/ 2 - 6 2 , n = z 2 +4f 2 x 2 y 2 , 



m= -bxyzf[x 4 -(a 2 +b 2 }y 4 ^ a 



j8 = 4m 2 x 2 y 2 n 2 z 2 , 7 = 



Then 



A. Desboves 122 noted that if x, y, z satisfy ax 4 +by 4 +dx 2 y 2 = cz 2 , then 



X = ax* - by 4 , Y = 2xyz, Z = c 2 z 4 + (4a6 - d 2 )x 4 y* 

 satisfy X 4 + a6c 2 F 4 + cdX 2 Y 2 = Z 2 ; while 123 

 X = x(4bcy 4 z 2 - g 2 ), Y = y(4acx 4 z 2 -q 2 ), Z = z{ 4fxYq z - (cW -fxY) 2 1 



satisfy the initial equation if q = ax*by*, f=d 2 4ab. Cf. Desboves. 90 



T. Pepin 124 treated ax 4 -\-2bx 2 y 2 +cy' l = n 2 , a necessary condition for which 

 is that the quadratic form (a, 6, c) represent n 2 . If one such representation 

 is known, all are given by quadratic functions of two parameters. But in 

 returning to our quartic we are led again to the problem to make a quartic 

 a square. 



Moret-Blanc 126 found solutions of x 4 5x 2 y 2 +5y 4 = D and 



by Euler's method. 



S. Realis 126 proved that 2y*-2y z +l = D only for y = Q, 2. 



117 Jour, de Math., 18, 1853, 84; Nouv. Ann. Math., (2), 11, 1872, 83-6. 



118 Comptes Rendus Paris, 87, 1878, 925. 



119 Annali di Sc. Mat. e Pis., 6, 1855, 302. Cf. Adrain. 113 



120 Atti Accad. Pont. Nuovi Lincei, 30, 1876-7, 222-i. Cf. Euler, Algebra 2, Ch. 9, Art. 144. 



121 Nouv. Ann. Math., (2), 18, 1879, 73. 



122 Ibid., (2), 18, 1879, 384; for a=c = l, p. 437. Verification, (2), 19, 1880, 461-2. 

 12S Ibid., (2), 18, 1879, 440; implied, Compte3 Rendus Paris, 87, 1878, 522. 



124 Atti Accad. Pont. Nuovi Lincei, 32, 1878-9, 79-128. 



125 Nouv. Am. Math., (2), 20, 1881, 150-5. 



129 Bull. Bibl. Storia Sc. Mat. Pis., 16, 1883, 213; reproduced, Sphinx-Oedipe, 4, 1909, 175-6. 

 See papers 19-25 of Ch. XVII. 



