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. 377
In the model the six lines, forming the sides of the triangles ABC, A'B'C', are
drawn on the surface of two brass plates which are carefully hinged together in such
a manner that the straight line XYZ, which passes through the intersections of the
pairs
BC, B'C'; CA, C'A'; AB, A'B',
is in the line of the hinges. Each of the remaining twenty-one straight lines is represented by a stretched string. On each plate the point at which any straight line cuts
the plate is marked by the Arabic number which denotes the line. In the explanation,
where it is necessary to distinguish between the points where any line, say 9, cuts the two
plates, the point where it cuts a side of the triangle ABC, in the left-hand figure, will
be denoted by 91, and the point where it cuts a side of the triangle A'B'C', in the
right-hand figure, will be denoted by 9,..
It will be observed that the lines 7112l151, 7,12,15,., in which the sides of the
triangle formed by the lines 7, 12, 15 cut the sides of the triangles ABC, A'B'C', meet
on the line XYZ.
We have now chosen three plane sections of the cubic surface, and we have one more
condition at our disposal. This is exhausted by the choice of the point 81, that is, the
point where the line 8, which cuts the three non-intersecting straight lines 2, 6, 7, cuts
the line 2. This determines the line 8, and therefore the point 8,..
As the lines 7, 8, 9 are complanar the straight line 7iZ8 cuts BC in 91 and cuts
the line XYZ in a point such that the straight line joining it to the point 7, gives the
points 8,., 9,..
In a similar way
41 and 91 give 13/
61, 8,, 101,,, 91,, 112,
2,,, 8,, 14,.
Since 10, 11, 12, and 13, 14, 15 form triangles,
101 and 121 give 111 11,. and 12,. give 10,
13,, 15l, 14 14,., 15,.,, 13,.
Lines 1 to 15 are now determined.
The remaining lines 16 to 27 form a double six.
Any triple tangent plane which passes through one of these twelve lines passes
through two of them, and also through one of the lines 1 to 15. We must, therefore, adopt
a different method to find one of the lines 16 to 27.
One of themii must be found by some quadratic method, and then all the rest can
be found as before. The line 17 was found by a method of trial and error from the facts
that 171 lies on BC and 17, on C'A', and that the pairs of Unes 7117E, 7,17,. and 141171,
VOL. XVIII. 48