560 HISTORY OF THE THEORY OP NUMBERS. [CHAP, xxi 



G. Osborn 87 gave Young's 55 identity and 



J. W. Nicholson, 88 using one solution of m 3 = w 3 +p 3 +r 3 , found that 



(my 6x) 3 = (ny bx} z + (py ax) 3 + (ry+ax} 3 



holds if x : y = m 2 bn 2 bp 2 a+r 2 a : mb 2 nb 2 pa 2 ra 2 . 



J. E. A. Steggall 89 , to solve x 3 u 3 = y 3 v 3 , took xu = p, x+u = q, 

 y v=s,y+v = r. Then (6) implies p 2 +3<? 2 = /iS, s 2 +3r 2 = jup, whence 



(3gr) 2 = ( M s -p 2 } ( M p -s 2 ) = (ps- 



= 2 = 



* q 



ps-k* ps-k 2 ' 



Since ps-k* = 3t 2 , we get p+?= (s 2 -|-p(3-&) }/(30, etc. Hence 



_L 2 +p 3 (3t-k) _ 



x=: a*~2 > y~ 



Qtp 2 Qtp 



^L 2 - 

 U ~ 



where L = /c 2 +3 2 , is the most general rational solution. 



R. D. Carmichael 90 obtained a rational solution, involving four para- 

 meters, of 



z 3 + y 3 + z 3 3xyz = w 3 + y 3 + w* 3 uvw , 



by employing the factor x+y+z of the left member. Taking z = iv = Q, 

 he deduced formulae (11) of Euler and Binet, which he proved to give the 

 general solution. 



T. Hayashi 91 noted that C. Shiraishi published in his book of 1826 the 

 solutions 910 (attributed to Gokai Ampon) of x 3 -\-y 3 +z 3 = u 3 : 



u = y+l, z = 3a 2 , rc = 6a 2 3a+l, ?/ = 9a 3 +6a 2 +3a or 9a 3 -6a 2 +3a-l. 

 Replacing a by a//3 and passing to the homogeneous form, we get 



and in like manner 



Further, S. Baba, Mathematics, vol. 2, 1830, gave the solution 

 z = (a 6 -4)a, ?/ = 6a 3 +a 6 -4, 2 = a 6 -6a 3 -4, -u = ( 



of (10); S. Kaneko, Mathematics, vol. 2, 1845, gave the first solution of 

 Frenicle. 43 Kawakita, in Algebraic Solutions, vol. 2, compiled from a 



87 Math. Gazette, 7, 1913^, 361. 



88 Amer. Math. Monthly, 22, 1915, 224-5. 

 "Proc. Edinburgh Math. Soc., 34, 1915-6, 11-17. 



90 Diophantine Analysis, New York, 1915, 63-65. 



91 T6hoku Math. Jour., 10, 1916, 15-27 (in Japanese). 



910 For a briefer account, see D. E. Smith and Y. Mikami, A History of Japanese Mathe- 

 matics, Chicago, 1914, 233-5. 



