781] 
411 
781. 
ON THE AUTOMORPHIC TRAN SFORM ATION OF THE BINARY 
CUBIC FUNCTION. 
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883), 
pp. 103—108. Read Jan. 11, 1883.] 
I consider the cubic equation (a, b, c, dfgx, l) 3 = 0. It is shown (Serret, Cours 
d’Algèbre supérieure, 4th ed., Paris, 1879, t. n. pp. 466—471) how, given one root of 
the equation, the other two roots can be each of them expressed rationally in terms 
of this root and of the square root of the discriminant; viz. making the proper 
changes of notation, and writing 
A, B, C — ac — 6 2 , ad — be, bd — c 2 , X = V— -J, 
LI— B 2 — 4 A C, = a 2 d 2 + 4ac 3 + 4& 3 cZ — 3 b 2 c 2 — 6 abed, 
— \fLl + B 0 + 2(7 
a = —îw—TF\—» P = 
-2 A 
8 = 
-Xffl-B 
2 XfLl 
2X fil 
(values which give a + S = — 1, a8 — ¡3y = — 1, and therefore also 
a 2 + aS + 8 2 + ¡3y = 0, 
which is the condition in order that the function <£x, = , may be periodic of the 
third order, (f>*x = x), then, u being a root of the equation, say (a, b, c, d\u, l) 3 = 0, the 
other two roots are 
au + ¡3 
T y u + o 
and 
_ ( g2 + fiy) u + /3 (a + 8) _ 8u — ¡3 
(j) u, — 9 u, ry (« + 8) u + 8 2 + ¡3y ’ — yu + a ’ 
where observe that, by the change of \/fl into - \/Ll, a, /3, y, 8 become 8, - ¡3, -7» a 
so that the last-mentioned value is, in fact, the value obtained from $u by the 
mere change of sign of the radical. 
52—2