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

374

BITANGENTS TO A QUARTIC CURVE

[CH. XIX

This lemma implies that the sums of corresponding elements
of Co, Do, Cj, Dj, are all even. Thus the set of six pairs determined by A and B is identical with the set of six pairs determined by Co, Do. For such sets of six pairs, properties (A)
and (B) of ~ 186 therefore hold. The fact that also property
(C) holds and hence also Theorem 11, may be stated as
LEMMA 4. The sets AB, CD,... and AC, BD,...
have no further symbol in common.
In the proof of these two lemmas, it suffices to consider
symbols A, B having b3 d3 (mod 2), and hence (after interchanging A and B if necessary) with b3 0, d3 1. For, if
b3=d3, but bliad, the symbols A'=(a3b3 a2b2 albi) and B',
derived from A and B by interchanging the first and third pairs
of elements, lead by the proof below to just five pairs C', D',
from which we derive the required Cj, Dj, by interchanging
the first and third pairs of elements. Next, if each b -di,
then ai ci, for example, and we proceed as before with
A*= (bia b2a2 b3a3), B*=(dici d2C2 d3c3).
To prove Lemma 3, we may therefore assume that b3=0,
d3=1. If C and D are symbols for which congruences (27)
hold, then y3+W3+1 =0 (mod 2), so that either y3=0 or
W3-=0. Since the mutual order of C and D is immaterial, we
may set y3-0, whence W3=l (mod 2). The conditions that
C and D shall satisfy the condition (25) for a symbol are (26)
and Z1Wl+2W2+3 —1. By (27), the latter becomes
2
(28) (ai+c+X)(b+di+y()+a3+c3+x3- 1 (mod 2),
i=l
which determines X3 in terms of xl, yl, x2, y2. There are six
sets of values of the latter which satisfy (26). One of these sets
is xi-ai, y -bi (i=1, 2), whence x3-a3, C=A (and hence
D==B), since
(29) ca3 1+cild+c2d2 (mod 2).
'Since this set is to be excluded, Lemma 3 is proved. For use
in the proof of Lemma 4, we shall exhibit the five pairs Cj, Dp.
In view of alb-i+a2b21 —, b and b2 are not both even.