Kummer's quartic surface, by R.W.H.T. Hudson.

154

CURVES OF DIFFERENT ORDERS

[CH. XIV

As before, the product
(\/yz + X Vyz') (-/X + ~ \/zx') (V/Xy + v Vx ')
is rational on <>, and therefore the three quartics form the complete
intersection with a cubic surface S, and the three quadrics are
circumscribed about a quadric G in consequence of the identity
ABC- S2+ GP.
Since S contains the curves (A, 4>), (B, (<), it touches ( at
their points of intersection. Hence the common generator of A
and B is a bitangent of the cubic surface S and therefore lies
entirely on it. Since the three lines common to (AB), (BC), (CA)
lie on S, they must also lie on G which therefore consists of their
plane repeated.
Since through any point of 1' six bitangents can be drawn, and
there are thirty octads, it follows that five quartics from different
families cut in the same two points. It is easy to see that the
five characteristics together with the zero characteristic make up
a set representing six coplanar nodes.
~ 90. SEXTICS THROUGH SIX NODES.
There are thirty-two families of sextic curves on the surface,
two for each characteristic. They are of two kinds: sixteen are
cut out by surfaces of order - (n + 1), = 2, passing through a conic,
and the other sixteen are cut out by surfaces of order - (n + 3), = 3,
passing through three concurrent conies.
Taking a family of the first kind, the quadric is subjected to
five conditions in containing a given conic, and there remain five
arbitrary coefficients. Hence the family contains five linearly
independent curves.
The sextic meets one trope at six nodes and every one of
the fifteen others at two nodes and two points of contact. The
inscribed cubic surface cuts the one trope in a plane cubic passing
through the six nodes, and every other trope in a conic and the
line joining the two points of contact.
Two quadrics through the same conic cut again in another
conic; hence two sextics of the same family have six variable
intersections lying on a conic. The corresponding inscribed cubic
surfaces touch at these six coplanar points and therefore the plane
cuts them in the same cubic curve.
Among the quadrics cutting out the family there are fifteen
containing the four conies of a Gopel tetrad, for when one conic is