776] 
389 
776. 
ON THE JACOBIAN SEXTIC EQUATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xviii. (1882), 
pp. 52—65.] 
The Jacobian sextic equation has been discussed under the form 
(z — a) 6 — 4a(z — a) 5 + 105 (z — af — 4 c(z — a) + 5 b 2 — 4 ac = 0, 
(see references at end of paper), but the connexion of this form with the general 
sextic equation has not, so far as I am aware, been considered. And although this 
is probably known, I do not find it to have been explicitly stated that the group 
of the equation is the positive half-group, or group of the 60 positive substitutions 
out of the 120 substitutions, which leave unaltered Serret’s 6-valued function of six 
letters. 
Tnvariantive Property of the Jacobian Sextic. 
Taking z — a as the variable, and comparing the equation with the general sextic 
equation 
(a, b, c, d, e, f, g\z - a, 1) 6 = 0, 
we have 
a, b, c, d, e, f , g 
= 1, — |a, 0, !&, 0, — §c, 5b 2 — 4ac; 
the Jacobian equation is thus an equation 
(a, b, c, d, e, f, gjoc, y) 6 = 0, 
for which c = 0, e = 0, ag + 9bf - 20d 2 = 0; but of course any equation, which can be by 
a linear transformation upon the variables brought into this form, may be regarded 
as a Jacobian equation. 
Hence, using henceforward the small italic in place of the small roman letters, 
the Jacobian sextic may be regarded as an equation 
(a, b, c, d, e, f gjx, y) e = 0, 
linearly transformable into the form 
(a, b, 0, d, 0, f gjx, yf = 0,