394 
A MEMOIR ON CUBIC SURFACES. 
[412 
which shows that there is at B 3 an eightfold point, the tangents being given by 
(X + E+ Z) (IX mY -h nZ) = 0, 
(X 2 , ft 2 , v 2 , — fiv, — v\ — AftJ YZ, ZX, X Y) 2 — 0. 
Each of the facultative lines is a double tangent of the spinode curve; whence /3'=18. 
Reciprocal Surface. 
66. The equation may be deduced from that for 11=12 — C 2 , viz. writing 
(a, b, c,f g, h^X, Y, Z) 2 = 2(X + Y + Z)(IX + mY + nZ), 
that is 
we have 
Writing also 
(a, b, c, f, g, h) = (21, 2m, 2n, m+ n, n + l, l + m), 
(A, B, C, F, G, H) = — (X 2 , ft 2 , v 2 , fiv , v\ , Xft ); K = 0. 
X, ft, v = m — n, n — l, l — m as before, 
\x + fiy + vz = cr, 
Iran xyz = 6, 
l (m + n) yz + m (n + l) zx + n (l + m) xy = v, 
l\ yz + m/i zx + nv xy = yjr, 
(m + n)x+ (n+l)y+ (l + m)z = t, 
we have 
U = 2v, V = 2 ayfr, W = — 4 yfr 2 , 
and then 
L = khu' 2 — 2ktw + er 2 , M = 2 (kiuv + o-yjr), N = 4 (4linnkxyzio — yjr 2 ); 
so that the equation is 
0 = DN = 
+ DM 2 
— ISkiuLMN 
— 16kw M 3 
— 27k 2 w 2 N 2 
4 (krw 2 — 2kiut + <r 2 ) 3 (4lewd — yjr 2 ) 
4- 4 (khu 2 — 2kwt + cr' 2 ) 2 (kwv + ayf) 2 
— 144 kiu (khu 2 — 2kiut + a 2 ) (kiuv + cryjr) (Jew6 — \fr 2 ) 
— 128 kw (kiuv + o-yjr) 3 
— 432 khu 2 (kwd — yfr 2 ) ; 
or reducing the first two terms so as to throw out from the whole equation the 
factor kw, the equation is 
4£ 2 {0L + (v 2 - yjr 2 ) kw + 2yjr (tyjr + ucr)} - 1SLMN - 16di 3 - 27kwN 2 = 0