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

~ 185]

STEINER SETS OF BITANGENTS

355

there are five distinct values of X for which the quadratic form
is a product of two linear functions. We thus have *
THEOREM 1. Given any two bitangents x=0, y==0 to a general quartic curve, we can determine in exactly five ways a pair
of lines = 0, 7 = 0, such that the quartic becomes xyv = Q2. Then
the eight points of contact of the four bitangents x, y, i, X lie on
the conic Q = O.
Such a set of six pairs of bitangents is called a Steiner set
(the older term was Steiner group).
Let a, b; c, d; e, f be three pairs of bitangents of the Steiner
set determined by a, b. Then the quartic is abcd=Q2 or
ab(cd+2XQ +X2ab) = (Q+ Xab)2.
As above, we can determine X (X 0) so that
cd + 2XQ+X2ab =ef.
Eliminating Q between this and abcd=Q2, we get
4abcd = (ab +cd e )2
X X
Replacing a, c, e by a/X, Xc, Xe, we get
(6) 4abcd = (ab+cd-ef)2,
which may be written in the symmetrical forms
(7) a2b2 +c2d2 +e2f2 = 2abcd +2abef+ 2cdef,
(7') ab+ V/cd + ef =.
Transposing the first radical, we derive
4cdef = (ef+cd-ab)2.
Hence the points of contact of c, d, e, f are on a conic.
THEOREM 2. The eight points of contact of any two pairs of
bitangents of a Steiner set are on a conic. Thus the same Steiner
set is determined by any one of its six pairs.
Since the ~28 27 pairs of bitangents lie by sixes in a Steiner
set, there are 63 Steiner sets.
* Also by ~ 180 and Exs. 2, 3 of ~ 184, with r and l'1 as the given bitangents.