656 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



"V. G. Tariste" 215 noted that (3) is derived from Bini's 204 formula by 

 equating to zero one of the six biquadrates. 

 0. Birck 216 stated that (3), viz., 



x=-y = p 2 +pq+q 2 , z = p*-q z , u = q z +2pq, v=-p*-2pq, 



gives the most general solution of x+y = z+u+v=Q with either (2) or 



He noted that 



= 3 4 +20 4 +26 4 , 51 4 +76 4 = 5 4 +42 4 +78 4 . 



A. S. Werebrusow 217 gave equal sums of three biquadrates involving 

 many parameters and derived Ge"rardin's 204 formulas by specialization. 

 He 218 gave 37 4 +38 4 = 26 4 +42 4 +25 4 and eight more such sets. 



E. Fauquembergue 219 gave the identity 



where v = 4a 4 -4a 3 /3+13a 2 /3 2 -36a/5 3 +24/3 4 , and found five sets making v= D, 

 all giving trivial solutions of (2). A. Tafelmacher 220 drew the same con- 

 clusion from a complete study of the identity derived by replacing a by 



0+7. 



L. Bastien 221 stated a solution of x\-\ ----- \-x* n = y\-{ ----- \-y* m , n^2, 



- 8r M 4 ), </< = 8*>pV rfr (i = 3, -, n), 



-; ----- a n . 



R. D. Carmichael 222 noted that x 4 +7/ 4 +42 4 = ^ 4 has the special solution 

 x, Z = p 4= F2cr 4 , y = 2p 3 <r, z = 2p<r 3 . Solutions involving two parameters are 

 given for z 4 +m/ 4 +az 4 = t 4 and x 4 +i/ 4 +a2 4 = at*, if a = 2 or 8. Also, 



the case p = k, q = 1 , of (3) . By Cunningham, 173 x*+y 4 4z 4 =1= f 4 . 



A. S. Werebrusow 223 tabulated all solutions, each ^50, of (4). 



E. Miot 224 gave a solution of (4) involving a parameter; likewise for 

 two equal sums of 4 or 5 biquadrates. 



Werebrusow 225 noted that 



for 



a = pv+(s+3t)U, 



215 L'intermediaire des math., 19, 1912, 183-4. 



216 Ibid., 255. 



217 Ibid., 20, 1913, 105-6. 



218 Ibid., 58; error in fourth set, p. 301. 

 2l <> Ibid., 245. 



22 Ibid., 21, 1914, 59-62. 



221 Sphinx-Oedipe, 8, 1913, 154-5. 



222 Amer. Math. Monthly, 20, 1913, 306-7. 



223 L'intermediaire des math., 21, 1914, 153-5. 



224 Ibid., 155-6. 



226 Ibid., 23, 1916, 223. Math. Sbornik. 



