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

376

BITANGENTS TO A QUARTIC CURVE

[CH. XIX

Let Cn be a real plane curve of order n having no double point.
Set
p=~(n-1)(n-2), Rp=22p-_-2P-1.
There are Rp curves of order -3 having simple contact with
Cn at n(n-3)/2 points. The determination of these curves
depends upon an algebraic equation E of degree Rp whose roots
are designated by (xy x2y2... xpyp), where xl,..., yp
form any set of integral solutions of
xlyl +x2y2+... +xpyp1 (mod 2).
The simplest case is n=4; then p=3 and the problem is
that of the R3= 28 bitangents to a quartic curve. For n _4,
Clebsch proved that, if u is any positive integer _ Rp for which
jL(n-3)/2 is an integer, the points of contact of Ct with the
A curves corresponding to the roots
(X'ly'l... x py'p),..., (XI(yl(()... Xp ()yp(G))
lie on a curve of order u(n-3)/2 if the congruences
x'p+"lp+... +xp)=0, y'p+ +... +yP)-0 (mod 2)
(p-l,..., p)
hold simultaneously. For n=4, the first case j= 2 is evidently
trivial, while the next case u = 4 is the one treated in Theorem 11.
The group * of equation E can be represented as a subgroup
of the abelian t linear homogeneous group on 2p variables
with integral coefficients taken modulo 2. Its substitutions of
period 2 are conjugate to certain simple types, from which
fact in connection with the Corollary in ~ 188 we find that the
number of real roots of equation E is one of the numbers
22p-i-l(k =..., p), 22p-2j-l _2P-1 (j0, 1,..., Tr),
where 7r=p/2 or (p-1)/2, according as p is even or odd. For
n=4, we again get the number of real bitangents to a quartic
curve (~ 188). For n=5, we have p=6 and see that, of the
2016 conics tangent at 5 points to a quintic curve without
* Dickson, Annals of Math., ser. 2, vol. 6 (1905), p. 146.
t Leaving invariant a certain bilinear function of two sets of cogredient variables, It is not a commutative group,