Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 12)

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.
	        
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