412] 
A MEMOIR ON CUBIC SURFACES. 
401 
where 
81. Forming the invariants, these are 
= A 2 + 24Uzw 4-144yz 2 w 2 , 
— J = A 3 + 36 A Uzw + 216 Vzhv 2 + 864vz 3 w 3 , 
A = y 2 + 4 (Sz + 7w) x, 
TJ = 27S# 2 + 2a (Sz — 7iv) 2 + 3by (Sz + yw) + c [y 2 — 2 (Sz + 7w) x] — dxy, 
V = (— Sac + 9b 2 ) (Sz — yiu) 2 
+ (2c 2 — bd) [;y 2 — 2 (Sz + 7w) x\ 
+ (— 4ad + 66c) y (Sz + 7w) 
— ‘led xy 
+ d 2 x 2 
+ 47S (2cx 2 — 3bxy + ay 2 ), 
y = c 2 — bd, 
v = ad 2 — Ibcd + 2c 3 , 
and the equation is 
43ST~ r ~ 7 K A2 "h Uzw + 144yz 2 w 2 ) 3 — (A 3 + 36AUzw + 216 Vzhu 2 + 864za2 3 w 3 ) 2 } = 0; 
or, expanding, this is 
A 4 y-A 3 V+A 2 U 2 
+ 4zw ( — A 3 v+12A 2 U/jl—9AUV+8U 3 ) 
+ 3 Gzhu 2 ( 4;A 2 y 2 — 4 A Uv +16 U 2 y — 3 F 2 ) 
+ 864^ 3 w 3 ( 4 Uy 2 — Vv) 
+ 1728z 4 w 4 ( 4y 3 —i> 2 ) = 0, 
where observe that the value of 
4y 3 — v 2 , =4 (bd — c 2 ) 3 — (ad 2 — 3bed + 2c 3 ) 2 is = — d 2 (a 2 d 2 + 4ac 3 + 4b 3 d — 3b 2 c 2 — Gabcd). 
82. It is convenient to modify the form of the equation as follows; write 
U 1 — U + SaySzw, V 1 =V+(— Sac + 9b 2 ) 7Szw, 
so that 
A = y 2 + 4 (Sz + 7w) x, 
U x = — 2ySx 2 + 2a (Sz + 7w) 2 + 3by (Sz + 7w) + c [y 2 - 2 (Sz + 7w) x] — dxy, 
V 1 = (— 8ac + 9b 2 ) (Sz + 7iv) 2 
+ (2c 2 — bd) [y 2 — 2 (Sz + 7w) x~\ 
+ (— 4ad + 66c) y (Sz + 7w) 
— 2cdxy 
+ d?x 2 
+ 478 (2c# 2 — Sbxy + ay 2 ), 
y= c 2 — bd, 
v = ad 2 — 2bcd + 2c 3 , 
C. VI. 
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