CHAP, xxii] EQUATIONS OF DEGREE FOUR. 671 



JV=a 2 +(a-fl) 2 , whence 



JV 4 = A 2 + 2 , A= -4a 4 -8a 3 +4a+l, B= -8a 3 -12a 2 -4a. 



Then 1 = A B gives a(a-t-l) 2 (a2)=Q. The only answer, given by a = 2, 

 isN=13, 13 4 = 119 2 +120 2 . 



L. Bastien 315 solved the system y 2 +z 2 +t~ = 2x 2 , y*+z*+t* = 2x* by 

 eliminating x. Thus y 2 +z- t- = 2yz, y^Fz=t. Let y = z+t. Sub- 

 stitute this value of y in the first equation. We get zi (z-\-t-\-x)(z-\-t x). 

 Hence set z = ab, t = cd, z+t+x = ac, z-\-tx = bd, 2b c = hd, b 2c = ha. 

 The solution is now evident. 



A. Ge"rardin 316 gave special cases in which s 4 x, s*y, s 4 z are all 

 squares or all cubes, where s = x+y+z. 



L. Aubry 317 proved the impossibility of the system 



<7 4 +9/ 3 <7=D, 9/ 4 +3/0 3 =D. 



GeVardin 318 solved the system x*+x z y* = a z , y*+x 2 y 2 = tf, x+y = c*. M. 

 Rignaux 319 noted that a = ax, b = j3y, whence the system reduces to 



and is easily solved. 



E. Fauquembergue 320 discussed the system x* hy*= D, x 4 +% 4 = D. 



A. Gerardin 321 discussed the system SP 4 = 2C/ 4 , PQR= UVW. 



A. Cunningham 322 solved X 4 -Z = A 2 , X*+Z = B 2 by taking any odd 

 integer a and any even integer /3 and setting X = a z 



Euler, 254 and Euler 81 of Ch. XVI, made z 

 squares. Petrus 12 of Ch. XV made p 2 +s 2 , P+q 2 , pstq squares. Woepcke 48 

 of Ch. XVI treated a*+4><i'* = a\+<}>al= D. Gerardin 185 of Ch. XXII 

 treated x i +mx' 2 y 2 -\-y' i = a 2 with other quartics. 



316 Sphinx-Oedipe, 8, 1913, 173. 



316 L'intermediaire des math., 23, 1916, 150, 169. R. Goormaghtigh and A. Colucci gave 



solutions, 24, 1917, 134-5. 



317 Ibid., 23, 1916, 129-131. 



318 Ibid., 122-3. 



319 Ibid., 24, 1917, 65-6. 



320 Ibid., 39. 



321 Ibid., 100-1. 



322 Math. Quest, and Sol., 4, 1917, 4-5. 



