# 
422 NOTE ON A CUBIC EQUATION. [856 
viz. the general cubic equation oc 3 + Sex 4- d = 0, adjoining thereto the radicals T lt T 2 
may be regarded as an Abelian equation. 
In particular, if c, d = — 1, 1, then we may write T 2 = 0, T. 2 = 1; the cubic equation 
is here 
x 3 + 3.« — 1 = 0, 
and the roots a, /3, y are such that /3 = a 2 — 2, y = ¡3? — 2, a ~ y 2 — 2 ; in fact, taking 6 
a primitive ninth root of unity, d 6 + 6 3 + 1 = 0; we have a, /3, y = 6 + 0 8 , 6 2 + 9 7 , 6 4 + d 3 ; 
values which satisfy x 3 + Sx — 1 = 0, and the relations in question. 
The same question may be considered from a different point of view. Take the 
transforming equation to be 
y = A 4- Bx + Gx\ 
then assuming that the values of y corresponding to the values x = a, /3, y are 
/3, y, a respectively, we have 
¡3 = A 4- Ba + (7a 2 , 
y = A + Bj3 + (7/3 2 , 
a = A. + By + Gy-, 
and the transforming equation thus is 
y > 
1, 
x, 
¿c 2 
/3, 
1, 
a, 
a 2 
7> 
1, 
/3, 
/3 2 
a, 
1, 
7» 
7 2 
This may also be written 
(/3- y)(y-*)(*-&) +/3+ y +x)} 
= /3 2 7 2 + 7 2 a 2 4- a 2 /3 2 — \ (¡3 s y + fty 3 + y 3 a 4- 7a 3 + a 3 /3 4- a/3 3 ) 
+ x { a 3 + {3 3 4- 7 3 — \ (¡3 2 y 4- ¡3y 2 4- 7 2 a 4- 7a 2 4- a 2 /3 4- a/3 2 )} 
4- x 2 {/3y + yot 4- a/3 — (a 2 4- /3 2 4- 7 2 )}. 
We have 
— 27 
(/3 — 7) 2 (7 — a) 2 (a — /3) 2 = —— (a 2 # 4- 4ac 3 4- 4<b 3 d — 3b 2 c 2 — Qabcd), 
-27 
a 4 
A, 
or, say 
(/3-7) (7 - a) (a - /3) = 3 ^ ~ M ^ V(A), 
tv 
if A be the discriminant, and w an imaginary cube root of unity, {(« — <y 2 ) 2 = — 3}. 
% .