388 A MEMOIR ON CUBIC SURFACES. [412 
Write as before (A, B, C, F, G, H) for the inverse coefficients (A = be — f 2 , &c.), and 
K = abc — a/ 2 — brf- — ch 2 + 2fgh ; and moreover 
= (A, B, C, F, G, H\x, y, zf, 
P = Ax + Hy + Gz, 
Q = Hx + By + Fz, 
jR = Gx + Fy + Gz, 
t =fe + gy + hz, 
U = afyz + bgzx + chxy, 
V = 2Kxyz — aPyz — bQzx — cRxy 
= — a Hy 2 z — b Fz 2 x — c Gx 2 y 
— a Gyz 2 — b Hzx 2 — c Fxy 2 
+ (— abc — af 2 — b(f — ch 2 + 4fgh) xyz, 
W= (A, B, G, F, G, R\ayz, bzx, cxy) 2 , 
L = k 2 w 2 — 2 ktw — <3>, 
M=kw U+V, 
N = 2 kabc xyzw + W: 
54. Then the invariants of the ternary cubic are 
S — L 2 — 12 kwM, 
T — L ?j — 18&W LM — 54 k 2 w 2 N; 
and the required equation of the reciprocal surface is 
IQg 2 {(L 2 — 12kwMf — (L 3 — 18kwLM — 54<k 2 w 2 R) 2 } = 0, 
viz. this is 
0 = L 3 N = (k 2 w 2 — 2ktw — C I>) 3 (2kabc xyzw + W) 
+ L 2 M 2 + (k 2 w 2 — 2 ktw — d>) 2 (kw U + V) 2 
— 18 kiuLMN — 18&«; (k 2 w 2 — 2ktw — <f>) (kw U+V) (2 kabc xyzw + W) 
— 16kwM 3 — 16kw (kwU + F) 3 
— 2*Ik 2 w 2 N 2 — 27 k 2 w 2 (2kabc xyzw + IF) 2 , 
which, arranged in powers of kw, is as follows; viz. we have 
CoefF. (kw) 7 = 2abc xyz, 
(kw) 6 = 2abc xyz (— 6i) 4- W 
+ U\ 
(kw) 5 = 2abc xyz (— 3<i> 4- 12i 2 ) + W (- 6t) 
+ U 2 (-4t) + 2UV 
— SQabcxyzU,