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

~ 181]

DOUBLE-SIX CONFIGURATION

345

two of the lines A, B, C, whereas the triangle ABC is the complete intersection of its plane with the surface. We readily
exclude the case in which L passes through the intersection
of A and B. For, if so, we may take those three lines as concurrent edges of a tetrahedron of reference. Then Xi = 2=0,
Xl=x3=0, x2=x3=O are lines on the surface, whose equation
f =0 therefore has no terms in X3 and x4 only, none in x2 and X4
only, and none in xi and x4 only. Thus X4 occurs only in the
terms X1X2x4, X1X3X4, X2X3X4. Hence the first partial derivatives of 0 with respect to each xi vanish at (0, 0, 0, 1), which
is therefore a singular point. But not every cubic surface has
a singular point. Hence there are exactly 27 distinct straight
lines on a general cubic surface.
181. Double-six Configuration. Consider a line bl on the
cubic surface and the five pairs at, co (i=2,..., 6) of lines
1\/ \/3 j3 V oa XX
b 2 64 66
FIG. 16.
on the surface which meet bl. The five planes blaicz are distinct and three lines on the surface do not concur (~ 180).
Hence no two of the lines c2,..., co intersect. The locus of
a line L intersecting C2, C3, c4 is a surface of the second order *
* By choice of the axes, the equations of c2 become y=mx, z=c, and those
of C3 become y= —mx, z= -c. The line L joining the general point (a, ma, c)
of c2 with the general point (b, -mb, -c) of c3 is
a-b a-+b
x=dz+ce, y=mez+mcd, d — X e-.
2c ' 2c
This meets Cs: x=lz+t, y=rz+s, if
(d-I)z+ce-t= O, (me-r)z+mcd-s=O.
In the determinant of the coefficients of z and the constants, replace d and e by
the values obtained by solving the equations of L. We get
mxz- cy —mZ, cyz-mc2x —mtZ
yz- mcx- rZ, mcxz- c2y- sZ