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.