Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 1901 ACTUAL GROUP FOR THE BITANGENTS

371

For proof, we have only to reverse the argument made
for Lemma 1, now taking the as as independent variables.
Then x23, X13, X12 are determined rationally in terms of the aj
by means of (21)-(23), and then X4, X5, X6, X7, X4{, X5{, X6t, X7{
are determined rationally by means of (20), (19) and the analogous equations mentioned above.
Substituting the expressions for x23, X13, X12 into (121)
and (13), we obtain the equation f=0 of a quartic curve, whose
coefficients are rational functions of the aj. Its discriminant
is not identically zero, since we saw in the proof of Lemma 1
that we can deduce equations (21)-(23) from the equation of a
quartic curve without singular points.
For i=4, (20) and (21) give (16) and (18). From these and
(19) we get (19'). Substitute the left member of the latter
into (15). In the resulting term -X4X23/al, replace X4 by its
value (16); in the term -kaixlx4, replace kx4 by its value
(18). We obtain the right member of (15). Hence we have
(15) and the equations derived from it by permuting 1, 2, 3
cyclically. From the relation between (14') and (15), it follows
at once that (14) and (14') hold. Define q by the equation
preceding (14). Then u-q has the value indicated, so that
(12') follows from the equation written below it. The equations obtained from (12') by replacing 4 by 5, 6, 7 follow
similarly from (20), (21) and the equations of type (19). Hence
X4, X5, X6, X7 form with xi, X2, xa an Aronhold set.
THEOREM 10. If the ratios of the 15 coefficients of a ternary
quartic form f are independent variables and if R is the domain
of the rational functions of these 14 ratios with rational coefficients,
the group G for R of the equation E(~)=0 upon which depends
the determination of the 28 bitangents to f=0 is the group F
of ~ 187.
It was proved in ~ 187 that every substitution of G is in r.
To prove the converse, it is sufficient, in view of the Corollary
in ~ 149, to show that every rational relation with coefficients
in R between the roots of E(~)=0 is preserved by each substitution of r. To this end, let ~1,..., ~7 be the roots corresponding to the bitangents 18,..., 78. Since the latter