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

~ 191] SYMMETRICAL NOTATION FOR THE BITANGENTS 373

in connection with the theory of theta functions of odd characteristics
(24) (xlyl X2y2 x3y3),
where each xi and yj is 0 or 1, and
(25) xlyl -+2y2 +x3y3 1 (mod 2).
If X3 -1 (mod 2), the congruence determines y3 in terms
of Xl, yl, X2, y2, so that there are 24 such sets of solutions. If
X3-0 (mod 2),
(26) xlyl +x2y2 1 (mod 2)
has the four sets of solutions X2=1, y2-=1-xiyl, and the two
sets X2=-, y2-O or 1, xl=yi-1; since y3=0 or 1, we obtain
2X6 sets of solutions of (25) with X3=0. Hence there are 28
symbols (24).
THEOREM 11. The 28 bitangents to a general quartic curve
can be designated by the 28 symbols (24) in such a way that the
8 points of contact of the four bitangents
A=(albi a2b2 a3b3), B=(cidi c2d2 cad3),
C=(xlyl X2y2 x3y3), D = (Zlwl Z2W2 Z3W3)
are on a conic if and only if
(27) as+c,+x~+z0-, b++d~+yI+W-=-O (mod 2) (i=1,2,3).
This theorem, which leads to a symmetrical notation for the
bitangents and presents the problem of the bitangents in a form
suitable for extensive generalizations (~ 192), was deduced in
the papers last cited from the theory of abelian functions.
We shall here give a very elementary proof,* depending upon
two lemmas.
LEMMA 3. If A and B are any two distinct symbols (24),
there exist exactly five pairs of symbols Co, Do;...; C4, D4,
distinct from each other and from A and B, such that the sums of
corresponding elements of the symbols A, B, Cj, Dj are all even,
as in Theorem 11.
* Pickson, Wul, Awer, Math, Soc, vol, 20 (1914), pp, 463-4,