Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

SHOWING THE 27 LINES ON A CUBIC SURFACE. 379
Let us consider the section of the surface made by a plane passing through one of
the lines; for instance, the line 1. We shall find five pairs of points, 2, 3; 6, 15; 9, 11;
16, 19; 17, 18, on this line and the other sixteen points will lie on a conic. In this
case there are 40 straight lines, each of which passes through three of the points.
Through each point on the conic 5 of the lines pass, and through each point on the
line 4 lines pass.
Next, let us consider a section of the surface not passing through a line.
It will be a cubic curve and the points on it where the 27 lines cut the plane
lie by threes on 45 straight lines, five straight lines passing through each point. From
these 45 straight lines a selection of 9 can be made, to pass through all the points. This
selection can be made in 200 ways. There are, also, 360 conics, each of which passes
through six of the points, 80 conics passing through each point. From these 360 conics
a selection of four can be made to pass through all the points except three lying on a
straight line. This selection can be made in 168 ways for each particular set of three
points, that is in 7560 ways altogether.
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