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

372

BITANGENTS TO A QUARTIC CURVE

[CH. XIX

form an Aronhold set, the remaining 21 roots j= jt (i, j=1,.., 7; i5j) are rational functions of ~1,..., 47 with
coefficients in R (Lemma 1). If cl,..., c4 are the ratios
of the coefficients of f, any rational relation between the roots
with coefficients in R is of the type
4( (1,.., 47, 12,..., 67, C,..., C14)=0,
where 4 is a rational function of its arguments with rational
coefficients. First, we replace the ~j by their rational expressions in terms of the c, ct. Next, we replace each ck by its
rational expression (Lemma 2) in terms of the coefficients C,
mt (i=l,..., 7) of the seven bitangents of our Aronhold
set. But these 14 quantities can be chosen at will. Hence
after our replacements, relation 4=0 becomes an identity
in the A, mi. Thus )=0 remains true if we substitute for
t1,..., 47 the seven roots in any order of any Aronhold set,
provided of course we replace each ~v by the root which arises
from it by our substitution. But r is the group of all such substitutions. Hence G = r.
COROLLARY. The adjunction of one root of a certain
equation of degree 36 reduces the group of the equation for the
28 bitangents to the group of the general equation of degree 8.
In fact, the subgroup E(~ 187) of r is simply isomorphic
with the symmetric group on 8 letters and is of index 36 under r.
Since we know the generators of r and the representation of
each substitution of r in terms of the generators (end of ~ 187),
we can prove by a straightforward argument that r is a simple
group (Weber, I.c., pp. 454-6).
191. Symmetrical Notation for the Bitangents to a Quartic
Curve. The separation of the Steiner sets into two types and
likewise for the Aronhold sets was due to the lack of symmetry
in the notation of Hesse and Cayley and not to a geometrical
difference. A perfectly symmetrical notation was discovered *
* Riemann, Werke, 1876, p. 471. Weber, Theorie der Abelschen Functionen
vom Geschlecht 3, Berlin, 1876, p. 82. Clebsch, "Ueber die Anwendung der
Abelschen Functionen in der Geometrie," Crelle, vol. 63 (1864), p. 211, who
used the notation (X1, x2, X3;,Y, y, y3). Appell and Goursat, Theorie des Fonctions
4lgebriques, 1895, p. 511,