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.

378 MR TAYLOR, THE CONSTRUCTION OF A MODEL
14r17r meet on the line X YZ. All the other points were then obtained by drawing
straight lines in the following order, in which the suffixes are omitted because the
description applies equally both to the left-hand and to the right-hand figures.
7, 17 give 22
10, 17,, 20
9, 20,, 25
12, 25,, 18
13, 18,, 23
13, 22,, 19
12, 19, 24
9, 24, 21
15, 21,, 26
15, 20, 27
10, 21,, 16
All the lines on the surface are now fully determined.
The diagrams represent not only the lines used in finding the points, but each
diagram gives the 32 straight lines which represent the intersections with the plane of the
triangle of each of the 32 triple tangent planes that do not pass through a side of the
triangle. From these 32 straight lines a selection of 8 lines can be made to pass
through all the 24 points in the diagram. This selection of 8 lines can be made in
40 ways. The following numbers give such a set of eight straight lines for the lefthand figure (triangle ABC):6, 4, 5; 15, 12, 7; 9, 20, 25; 11, 23, 26; 16, 21, 10; 19, 22, 13;
17, 14, 24; 18, 8, 27.
It may be noticed that in the left-hand figure the points 4...15 lie by threes on
eight straight lines, two of the lines passing through each point.
From these eight lines two sets of four can be chosen passing through all the twelve
points.
We cannot draw a pair of conics through the twelve points.
Of the remaining twelve points, 16...27, no three lie on a straight line, but eight
conics can be drawn, each passing through six points, and there are four pairs of conics
passing through all the twelve points.
There are also 48 conics, each of which passes through 6 points in the diagram, and
12 of which pass through each of the 24 points, and from these 48 conics a selection of
4 can be made so as to pass through ail the 24 points; such a selection can be made in
168 ways. The following numbers give one such set of four conics for the left-hand figure:17, 4, 5, 18, 12, 7; 6, 20, 25, 15, 23, 26; 9, 21, 10, 11, 22, 13;
16, 14, 24, 19, 8, 27.