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

366

BITANGENTS TO A QUARTIC CURVE

[CH. XIX

so that the substitution induced by
/1ai 0a2 a3 a48
5 6 7 8/
transforms S = aP1234 into that induced by (15) (26) (37) (48)P1234,
which leaves unaltered only the symbols 15, 26, 37, 48. In
the former case, al,..., a4 is a permutation of 1, 2, 3, 4.
If this permutation is the identity, a permutes only 5, 6, 7, 8,
and, after applying a transformation not altering P1234, we
may take a=(78) or (56)(78), whence S leaves 8 or 4 symbols
unaltered. In the contrary case, we may assume that o- is the
product of (12) or (12)(34) by a substitution ao on 5, 6, 7, 8.
If oa is the identity, we transform by (18)(27)(36)(45) and are
led to the preceding case. There remain the cases
a=(12)(56), (12)(56)(78), (12)(34)(56)(78),
for which S leaves unaltered 4, 4 or 8 symbols, and the case
(12)(34)(56), which is equivalent to the second case.
THEOREM 8. There are exactly 4, 8, 16 or 28 real bitangents
to a real quartic curve without singular points.*
189. Real Lines on a Cubic Surface. If we adjoin to the
domain one root of the equation upon which depend the 28
bitangents to a quartic curve, the group reduces to a subgroup
simply isomorphic t with the group of the equation upon which
depend the 27 straight lines on the related cubic surface (~ 184).
The substitutions of period 2 of r which leave one symbol
fixed leave unaltered 3, 7 or 15 of the remaining 27 symbols.
THEOREM 9. There are exactly 3, 7, 15 or 27 real straight
lines on a general real cubic surface.:
* It is then not of the type excluded in the proof of Theorem 1. The group
discussion by Maillet (I.c., p. 323) is incomplete, as it fails to exclude the case of
no real bitangents. Weber, Algebra, 2d ed., vol. 2 (1899), devotes pages 458-465
to a proof that no other than these four cases can occur, and three pages to a proof
that all four cases actually occur. For geometrical treatments see Zeuthen,
Aathematicshe Annalen, vol. 7 (1874), p. 411; Salmon's Higher Plane Curves,
p. 220.
t Note that the quotient of the order 288 7! of r in ~ 187 by 28 is the order of
the group r in ~ 183.: That these four, but no other, cases actually occur was shown by SchlaiJi,
Quarterly Journal of Mathematics, vol. 2 (1858), p. 117.