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

~ 184 CUBIC SURFACES AND QUARTIC CURVES

351

184. Relation between Cubic Surfaces and Plane Quartic
Curves.* Using homogeneous coordinates, let
f(x)-f(X1, X2, X3, X4)=0
be the equation of a cubic surface without a singular point,
so that the first partial derivatives of f with respect to x,...
X4 are not all zero for a set of x's not all zero. The point (y+Xz)
on the line joining the points (y)- (y, y4) and (z) is
on the surface if
f(y+Xz) -f(y) +XL +x 2Q +3f(z) =0,
where, by Taylor's theorem,
L=zli+... + Z4, Q =2af+2zl1 2 af+
ayi ay4 yli2 aylay2
Take (y) to be a fixed point P on the surface f=0. Then
the line joining P and (z) meets the surface in two further
points which coincide if
(3) (Q)2 =Lf(z).
Hence this equation of degree four in Zl,..., Z4 represents
the tangent cone T4 to the surface f=0 with the vertex P on
f=0.
Any plane through a straight line I on the cubic surface
meets the surface in I and a conic c. Let I and I' be the two
intersections of I and c. The tangent to c at I is the limiting position of a line meeting the surface in three points and
hence meets the surface at three coincident points. Thus
I and this tangent line are principal tangent lines to the cubic
surface and their plane is a tangent plane. Hence any plane
through I is tangent to the surface at two points I, I', and thus
is a bitangent plane.
In particular, the plane through the fixed point P and any
line I on the cubic surface is a bitangent plane to the surface
and hence also to the tangent cone T4.
The section of T4 by an arbitrary plane E is a quartic curve
C4. The intersections of E with the above 27 bitangent planes
to T4 give 27 bitangent lines to C4.
* Geiser, Mathematische Annalen, vol. 1 (1869), p. 129.