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



F. Ferrari 204 noted the identity 



(a?+2ac-2bc-W) 4 +(W-2ba-2ac-c z y+(c' 2 +2ab+2bc-tf) 4 



= 2(a 2 +6 2 +c 2 -a6+ac+6c) 4 . 

 while U. Bini (ibid.) gave the identity 



[a(d+c)-b(c-3d)J+[2(bc-ad)J+[a(d-c)-b(c-3d)J 



= [a(d-c)b(c+3d)J+[2(bc+ad)J+[a(d+c)+b(c-3d')J, 



with the plus sign. A. Gerardin (ibid., 19, 1912, 254) stated that the sign 

 should be minus and gave other such identities. Welsch (ibid., 132, 184) 

 gave another method of correcting the signs : retain the plus sign, but change 

 the final term of the first member to b(c+3d). 



A. Cunningham 205 found numbers expressible in several ways in the 

 form 4 +2/ 4 +2 4 by use of x*-\-y*=2v? z*, u=x z -{-xy+y 2 , z = x-{-y, and 

 expressing this u in the form A 2 +35 2 in several ways. 



E. Miot 206 stated that [the case b = c of Ferrari's 204 identity] 



(1) (4pq)*+(3p*+2pq-q*y+(3p*-2pq-q^ = 2(3p*+q>)* 



and noted cases when a sum of three squares equals a sum of three bi- 

 quadrates and a sum of three eighth powers. Welsch 207 stated that Miot's 

 solution is erroneous and noted that 



always implies that 



A. Gerardin 208 noted cases of two equal sums of three biquadrates and 

 gave four methods of finding particular solutions of 



(2) x*+y 



the fourth leading to the solution 



[It is expressed by the next identity with h = l,l = q, and p replaced by 2p.~] 

 He gave 16 identities which follow by a change of variable from 



In conclusion, he gave 

 (3) (p 2 -g 2 



A. Martin 209 gave (1) and (3). 



E. Miot 210 noted the solution 37, 17; 35, 26, 3 of (2). 



204 L'intermediaire des math., 16, 1909, 83. 



206 Math. Quest. Educ. Times, (2), 14, 1908, 83-4. Same in Mess. Math., 38, 1908-9, 101-2. 

 206 L'intermediaire des math., 17, 1910, 214. 

 2 " Ibid., 18, 1911, 64. 



2 8 Assoc. frang., 39, 1910, I, 44-55. Same in Sphinx-Oedipe, 5, 1910, 180-6; 6, 1911, 3-6; 

 8, 1913, 119. 



209 Math. Magazine, 2, 1910, 351. 



210 L'interm6diaire des math., 18, 1911, 27-28. 



