856] NOTE ON A CUBIC EQUATION. 423
The remaining functions of a, /3, 7 are of course expressible rationally in terms
of the coefficients: we have
S/3 2 7 2 = — (— 6bd + 9c 2 ),
X/3 3 7 = — 3 (— 3 abd — 18ac 2 + 27 6 2 c),
Xu 3 = — (— 3 a 2 d 4- 27 abc — 27 b s ),
a 3
2/S 2 7 = — 2 (3ac£ — 9be),
CL
2a 2 = — 2 (96 2 — 6ac),
CL
and the final result is
3 (to — со 2 ) V(A) {2a (3/ + x) + 36} = — abd + 4ac 2 — 36 2 c
+ x (— a 2 d + 7abc — 6b 3 )
+ x 2 ( 2a 2 c — 2a6 2 );
viz. we have thus an automorphic transformation of the equation
ax 3 + 36ж 2 + Зсж + d = 0.