140 
NOTE ON THE THEORY OF CUBIC SURFACES. 
[334 
The equation X + Y = x + y then gives 
IX + mv — 1, 
Ifi + mp — 1, 
which give the values of l and m, and thence the values of A and B; and collecting 
all the equations, we have 
where 
R = %(v + w), 
S = ^ (w + u ), 
T = ^ (u + v ), 
A 
1 /Xp — fiv\ 3 
VO \ p — v )’ 
b=-X ——Y, 
Vd V p-v J’ 
C = — 4c, 
Xx + py = {I (<ï> + VO Z7)} 4 , 
poc + py= (4> - VD C7)}3 
(i>, □ being respectively the cubicovariant and the discriminant of TJ= ax? + by 3 — c (x + y)% 
for the formulae of the transformation 
AX 3 + BY 3 + 6GRST = ax 3 + by 3 + c (u 3 + v 3 + w 3 ). 
X + Y+R + S+T=x + y + u + v + w. 
The equation ax? + by 3 + c (u 3 + v 3 + w 3 ) = 0, where 
x + y + u + v + w= 0, 
presents over the other form the advantage that it is included as a particular case 
under the equation ax? + by 3 4- ca 3 + dv 3 + ew 3 = 0 (where x + y + u + v + w— 0) employed 
by Dr Salmon as the canonical form of equation for the general cubic surface. 
5, Downing Terrace, Cambridge, April 29, 1864.