CHAP, xxii] EQUAL SUMS OF BIQUADRATES. 655 



R. Norrie 211 gave several methods to solve 



(4) 



First, take x = rxi+a, y = rx z -}-b, z rxz+c, u rxi a, v = rx 2 -\-c, w = rx 3 -}-b. 

 We obtain a cubic in r whose constant term is zero. The coefficient of r 

 will be zero if X 3 = x 2 -i-2xia 3 /(b 3 c 3 ). Then r is the ratio of the coefficient 

 of r 2 to that of r 3 . Second, he noted that 



equals identically the sum derived by interchanging the subscripts 1, 2. 

 Replacing Xi, y\, x 2 , y 2 by their reciprocals and multiplying each root 

 by (xiyiXtf/z) 4 , we obtain a new integral function which is added to the 

 former. Hence 



is unaltered by the interchange of the subscripts 1, 2. Multiplying 



by the identity derived by interchanging the subscripts, we get two equal 

 sums of five bi quadrates. The third method is really Haldeman's 201 

 remark that Q v , x = Qv, n if 3i/ 2 +z 2 = 3v 2 -|-w 2 . The general solution of the 

 latter is stated to be 



x, u= {(3X 2 l)t>-K3X 2 =Fl)i/}/(2X), 



where X is arbitrary. Again, x 4 -\-y 4 -\- (x+y} 4 is unaltered when x is replaced 

 by (3x5y)/7 and y by (5x-\-8y)/7. Changing the sign of y and subtracting 

 the new identity from the former, we get 



Finally there is given the identity, in which r = ju 2 c 8 X 2 6 8 , 



If we replace vx 4 by SJi^t Sjzjic^t, we get a solution of 



In the last, Norrie made the restrictions that s = r, Ki = vi, whence Xi = ju t -. 

 A. Gerardin 212 noted the identity 



(re 4 -2y 4 ) 4 + (2x 3 y) 4 + (3^ 3 ) 4 = (x 4 +2y 4 } 4 + (2xy*Y+ (xy*) 4 . 



E. N. Barisien 213 noted the identity (1). 



Gerardin 214 quoted his 208 solutions of (2) involving two parameters with 

 = z+wand noted that (3) is simpler than Ferrari's 204 formula, which fol- 

 lows by taking a+c = p, 6+c= q. 



211 University of St. Andrews 500th Anniversary, Edinburgh, 1911, 62-75. 



212 Bull. Soc. Philomathique, (10), 3, 1911, 236. 

 J13 Nouv. Ann. Math., (4), 11, 1911, 280-2. 



814 L'interme'diaire des math., 18, 1911, 200-1, 287-8. 



