494 Mr. A. Cayley on the Theory of Cubic Surfaces. 



may be identically satisfied. We have 

 ax 3 + hj 3 + c(u 3 + v 3 + w 3 ) 

 = ax 3 + by 3 + c\_(u-{-v +w) 3 — 3 (v -f w)(w + u)(u + v)~\ 

 = ax 3 -\-by s —c(x + y) 3 — Sc(v + iv) (w + u) (u + v), 



which is to be 



= AX 3 +BY 3 + 6CRST; 



and we may find X, Y, R, S, T, linear functions of x, y, u, v, w, 

 so as to satisfy these equations, and so that in virtue of 



x + y + u + v + w = 0, 



we shall have also X + Y + R+S + T = 0. For, assuming 



AX 3 + BY 3 = ax* + by 3 -c(x + y) 3 , 



X + Y = x+ y, 



R 



= g (*> + «>), C = -4c, 



S 



1/ 



T 



1/ 



we have identically 



ATL 3 + BY 3 + 6CR$T=ax 3 + by 3 --c{x + y) 3 -3c{v + w)(w + u)(u+v), 



X+Y + n + S + T!=x+y + u+v + w', 



and thus it only remains to show that we can find X, Y linear 

 functions of x } y, such that 



AX 3 + BY 3 = a* 3 + % 3 -c(tf + 2/) 3 , 

 X+ Y =x + y. 

 This is always possible ; in fact if 



U = ax 3 + by 3 — c(x + y) 3 , 

 then taking <3> for the cubicovariant, and D for the discrimi- 

 nant of U, we have ^{Q + VnU), g(«|)-\/nU) each a 

 perfect cube, say 



2(^ + V / DU) = (X^ + ^) 3 , 



