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

~ 1921 FURTHER PROBLEMS OF CONTACTS OF CURVES 375

After interchanging the first and second pairs of elements
in all of our symbols, if necessary, we may set b2 —1. Then
we may set
A=(a b ab+l 1 e 0), B=(cldi c2d2 c31),
in which, as well as below, C3 is given by (29). The sets of solutions of (26), other than the above excluded set, are evidently
the five sets of the first four elements in Co,..., C4 below.
After determining x3 for each by use of (28), we see that the
five pairs of symbols specified in Lemma 3 are
Ck=(l 1 k 0 Zk 0), Dk=(a+ci+l, b+dl+1, ab+l+c2+k, d2+1, e+c3+zk, 1)
[zk= a+b-c1+++dc2+d2+-adl+bcl+abd2+k+d2k+e, k=0, 1];
C2=(a+l, b, ab+b+1, 1, e+di+bd2, 0) D2= (cl+l, di, c2+b, d2, C3+di+bd2, 1);
C3=(a, b+1, ab+a+1, 1, e+cl+ad2, 0), D3= (c, di+1, c2+a, d2, c3+cl+ad2, 1);
C4= (a+l, b+1, ab+a+b, 1, e+a, 0), D4= (cl+1, di+l, c2+a+b+l, d2, c3+a, 1)
[a= Cl+dl+d2+1 +ad2+bd2].
To prove Lemma 4, we have to show that if C is one of
these 10 symbols and E is one of the 8 not paired with C and
not identical with C, the new set AC, BD,... does not contain E. If it did, there would be a symbol paired with E whose
elements are the sums of corresponding elements of A, C, E.
But we readily verify that condition (25) is never satisfied for
this symbol paired with E. After treating the cases in which
C= C, we need not consider the cases C=Di, since if P and Q
form any pair of our ten symbols, A+B-P+Q-C++D2
imply A+D,+PA +C,+Q. Hence, treating Ck(k=0, 1)
together, we need consider only six cases with C=C2, four
with C=C3, two with C=C4, and C=Co, E=D1. For example,
A+C2+Ck=(0 1 b+k 0 s 0) is not a symbol satisfying (25).
192. Further Problems of Contacts of Curves. The preceding symmetrical notation for the bitangents to a quartic
curve is in accord with that used by Steiner * and Clebsch t
in their treatment of a series of problems on contacts of curves.
* Journal fir Math., vol. 49 (1855).
t Ibid., vol. 63 (1864).