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

368

BITANGENTS TO A QUARTIC CURVE [CH. XIX

and hence be a singular point of f=O by (12i). Hence, after
changing the sign of q if necessary, we may assume that
u -q = X21X2X12,
where Xi is a constant not zero. Then
u+q = 2(X3X13 -X4X14)/X1.
Adding and replacing ut by its expression (13), we get
(14) X4X14 = X3X13 +X12X2X12 -XI( -X1X23 +X2X13 +X3X12).
In the same manner (or by permuting 1, 2, 3 cyclically),
we get
(14') { X4X24 = X1X12 + X22X3X23 -X2(X1X23 -X2X13+X3X12),
X4X34 = X2x23 +X32XIX13 -X3(XiX23+X2Xl3 -X3X12),
where X2 and X3 are new constants not zero.
To determine the X's, divide equations (14') by X2 and X3,
respectively, and then add. We get the identity
(I~;~(~X24 X34X X12
(15) X4 X (- +X3X13-2X23 +X231,
X4(X2 X3 / X2
where I =X2X3 +X2/X3.- Since X4, Xi, X23 occur in the Steiner set
(8) with g =8, /1=3, and no two are paired, they are not concurrent (proof of Theorem 3); similarly, no three of the xi concur. Hence I, X4, Xi are concurrent, so that X4 is a linear function of I and xi. But
(16) X4= alx, +a2x2~a3x3,
where the a's are known constants each not zero. This sum
must vanish for I=0, xi =0, whence a3= a2X2X3. Thus
X2= h1a3, ' = hia~,
where hi is a new constant. Then, by (15) and (16),
X24 / 13 X
X2X X3 a2 a3