290 
NOTE ON THE EXPRESSION FOR 
[346 
then the Resultant is 
that is, the Resultant is 
= - 2 2l 3 + 27 2153 + 27 g, 
= — 2 (ad' — a'd — 3bc' + 3b'cf 
+ 27 (ad’ — a'd — She' + 3Vc) x 
{(ac — b 2 ) (Ve' — d! 2 ) — I (ad — be) (a'd' — Ve') + (bd — c 2 ) (a'c' — ò' 2 )} 
+ 27 { (a 2 d — 3abc + 26 3 ) (— a'd' 2 + 3b'cd' — 2c' 3 ) 
— 3 (dbd — 2ac 2 + b 2 c ) (— a'c'd' + 2b' 2 d' — Ve' 2 ) 
+ 3 (— acd + 2b 2 d — be 2 ) ( a'b'd' — 2a'c' 2 + b' 2 c ) 
— (— ad 2 + 3 bed — 2c 3 ) ( a' 2 d' — Sa'b'c' + 26' 3 )}. 
In particular assume 
(a[, V, c', d'\x, y) 2 = a? + y\ 
so that 
. (a', b', c!\x, y) 2 = xy, 
B', C, D'^x, y) 3 = x 3 — y s , 
and thus 
a' =d' = 1, V = c' = 0, 
a ' = c' =0, b' = i, 
A' = -D' = 1, B' = G' = 0. 
21 = a — d, 
53 = — b — be —ad, 
(S = A +D= a 2 d — ad 2 — 3 abc + 3bcd + 2Ò 3 — 2c 3 , 
or, putting for shortness, 
we have 
and Resultant is 
a — d= 9, and therefore a = d + 9, 
21 = 0, 
53 = be — d9 — d 2 , 
($ = 2 (b 3 - c 3 ) - 3bc9 + d 2 9 + d6 2 , 
- 29 s 
+ 27 9(bc-d 2 -d9) 
+ 27 {2 (V - c 3 ) - 3bc9 + d 2 9 + d9 2 }, 
Avhich is 
= — 20 3 + 54Ò 3 — 54c 3 — 546c0, 
or rejecting the factor — 2, it is 
= 0 3 — 276 3 + 27c 3 + 27 cb9.