CHAP. XXII] BlQUADRATES AND SQUARES. 659 



Take d~ = 2v, 3a 2 +& 2 =(+6) 2 . Thus 6, d, v are found rationally in terms 

 of a, t, whence 



(1) (4a0 4 +(3a 2 +2ai-^) 4 +(3a 2 -2ai- 



For o = l, t = 2, we get 3 4 +5 4 +8 4 -fl4 4 = 2 



A. Cunningham 240 , to solve x*-\-y*+z* = 2u? n , took as u any number of 

 the form 2 +3/3 2 , whence u 2n is of the form A 2 +35 2 and a solution is 

 x = B-A, y = B+A, z = 2B. 



A. Ge"rardin 241 noted that (l+raz) 4 +(?m/) 4 +(mz) 4 = (l+2ra:r) 2 if 



ra 2 4 +2/ 4 -hz 4 ) +4raz 3 +2z 2 = 0. 



Its discriminant must be a square, say (2Sx) 2 , whence x* y* z 4 = 2$ 2 . Set 

 S = zU, y 2 +kz 2 = x 2 . Then ky 2 +%(k 2 -l)z 2 =U 2 . Hence the problem re- 

 duces to a " double equation," that of making the two binary quadratics 

 squares. 



E. N. Barisien 242 noted the identity 



Mehmed-Nadir 243 gave two special sets of solutions of 



A. Cunningham and E. Miot 244 obtained solutions by use of the identity 



x*+y*+ (x+yY = 2(x~+xy+y 2 ) 2 . 

 A. G6rarflin 245 solved Z 4 +F 4 +Z 4 = A 2 + 2 by use of the identity 



setting q of 2 , p = 2bg 2 . It remains to solve ab( / 2 -f2gr 2 ) = X 2 . For a = b = 1, 

 we may take /= ra 2 2n 2 , X = ra 2 +2n 2 , g = 2mn. He noted (ibid., p. 90) that 



(o! 2 +/3 2 ) 4 - (a 2 -/3 2 ) 4 - (2a^Y = 2 {2a j S(a 2 -,S 2 ) } 2 . 

 R. Norrie, 199 pp. 90-92, would derive a second solution of 



from one solution ofH ----- [-a 4 n = a? by setting X = ra; t -+o,-, X = r 2 y+rx+a, 

 and making the coefficients of r and r 2 zero by choice of y, x. To obtain an 

 explicit solution when n>4, take t = x 2 +xy+y 2 in ( 2 +z 4 ) 2 ^ 4 +(z 2 ) 4 +2 2 z 4 , 

 whence 2 2 = z 4 +2/ 4 +(o;+?/) 4 . But # 4 +?/ 4 can be expressed as a sum of r 

 biquadrates P if r>2 [Norrie, 199 end]. Hence 



E. N. Barisien 2450 wrote Froth's 227 identity in the form 



240 Messenger Math., 38, 1908-9, 101, 103. 



241 Bull. Soc. Philomathique, (10), 3, 1911, 239-240. 

 242 Nouv. Ann. Math., (4), 11, 1911, 280-2. 



243 L'intermediaire des math., 18, 1911, 217. 



244 Ibid., 19, 1912, 70-71. 



246 Sphinx-Oedipe, 6, 1911, 21-22. 

 2460 Mathesis, (4), 4, 1914, 13. 



