PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
where 
A=/+4(^+yw y V, 
^=^+^ z -ywy+^Ah+yw)+c[f-2(hz+yw)x]-dxy, 
V= ( — 8 ac + 9J 2 )(&z — yw) 2 
+(2fl*-W)[y-2(&+yfD>F] 
+ ( — 4a^ + Qbc)y(bz + yw) 
— 2 cdxy 
-)- d 2 x 2 
+ 4y^(2c.r 2 — 3 bxy -f ay 2 ), 
y=c-—bd, 
v =ad 2 —2bcd-\-2c 3 , 
and the equation is 
4S2?pW A! + 24UOT + 144 Kw?-(A>+36AXJ2«,+216VzV+86teV) J }=0. 
Or, expanding, this is 
A>-A 3 V+A 2 U 2 
+ 4 zw( -A 3 ^+12A 2 U i a-9AUY+8U 3 ) 
+ 3 6zV( 4A> 2 -4AU^+16U>-3V 2 ) 
+ 864z 3 w 3 ( 4U^ 2 -Y>) 
+ 1728*V( 4 j u, 3 -k 2 )=0, 
where observe that the value of 
-Kt>d-c 2 y-(ad 2 -Zbcd+2cy is = -d\a 2 d 2 +4,ac 3 +m-U 2 c 2 -8abcd). 
82. It is convenient to modify the form of the equation as follows ; write 
Ui=U-f Saybzw, Vj=V + ( — 8ac+9b 2 )yfow, 
so that 
A =?/ 2 +i {hz+yw)x, 
U, = - 2 y h? +2 a(h + yw) 2 + 3 by{lz + yw) + c[y 2 ~2(h + yiv)x] - dxy , 
V, =( — 8ac-\-9b 2 )(hz-\-yw) 2 
+ (2 c 2 —bd\y 2 — 20z + yw)x~] 
+( — 4 ad-]- 6bc)y(fo + yw) 
—2c dxy 
-f- d 2 x 2 
+ 4yo(2cA’ 2 — 3 bxy -f- ay 2 ), 
(*— c 2 —bd, 
v= ad 2 —2bcd+2c 3 , 
MDCCCLXIX. 
