260 
PEOFESSOE CAYLEY ON CUBIC SUEFACES. 
Q=FLr-j-B?/+F;z, 
R — Gx — Fy -f- Cz, 
t =fa+gy+hz, 
U = afyz + bgzx + clixy, 
V = 2 K xyz — a?yz — bQzx — cRxy 
— — aRy 2 z — hFz^x — cGx 2 y 
- aGyz 2 - bRzx 2 - cYxy 2 
+ ( — abc — af 2 —bg 2 — ch 2 -f- 4 fgh)xyz, 
W=(A, B, C, F, G, ET$ 'ayz, bzx , cxy) 2 , 
L =Jc 2 w 2 — 2Jdw— <b, 
M=M>U+V, 
N —% \dbc xyzw+^N : 
54. Then the invariants of the ternary cubic are 
S=L 2 -12£wM, 
T=L 3 — 18&WLM-54&VN ; 
and the required equation of the reciprocal surface is 
^{(F— 12 fe M ) 3 -( L3 - 18 m-54M) 2 }=:0, 
viz. this is 
0= L 3 N = (yfc V* - 2 to - <b)\Uabc xyzw + W) 
+L 2 M 2 +{Jc 2 w 2 -mw-®)\kivG+V) 2 
— 1 8&wLMN — 1 8 kw{k 2 w 2 — 2 ktw — <&)(kwVi + V )( 2 kabc xyzw + W) 
-16feM 3 —V$kw{kwTJ +V) 3 
— 27&WN 2 -27k 2 w 2 (2kabc xyzw+Wf, 
which, arranged in powers of lew, is as follows ; viz. we have 
Coeff. (lew) 7 = ‘Labe xyz, 
(, kwf= Zabc xyz{-§t)+Yf 
+ U 2 , 
(kw) 5 = 2abcxyz(-3®+12f)+W(-6t) 
+U 2 (-4^)+2UY 
— oGabcxyzTJ, 
