Mr. A. Cayley on the Theory of Cubic Surfaces. 495 

 and we then have 



U= JU {\v + w)*-{vx + py)*\ =AX 3 + BY 3 , 



which is satisfied by 



Y=m(vx + py), 

 if 



AP = -?t=, Bm 3 =---L.. 



Vu vu 



The equation X + Y=#-f 2/ then gives 



Vk + mv = \ } 



lfA-\-mp = l, 



which give the values of / and m, and thence the values of A 

 and B ; and collecting all the equations, we have 



'-mv ™" \/n\ p—v / 



xp-fiv K \/n\ p— y 



\p—fiv K rsf " Vu\p— yJ 



E= \(v + w), C=-4c, 



S = 



g(«0+«)j 



where 



T= J(«+»), 



(<l>, □ being respectively the cubicovariant and the discrimi- 

 nant of V = aa? -\- by 3 — c{x + y) 3 ), for the formulae of the trans- 

 formation 



AX 3 + BY 3 -f- 6CRST = ax 3 + by 3 + c{u 3 + v 3 + ufi), 

 X + Y + R + S-f T =x + y + u + v + w. 

 The equation ax 3 + by 3 + c (u 3 + v 3 + w 3 ) = 0, where 

 0G + y-\-u-\-v + w=zQ t 



