76]
SURFACES OF THE THIRD ORDER.
455
of the existence of the lines aj)^, a 2 b 2 c 2 , a s b 3 c 3 , a x a 2 a 3 , bj) 2 b 3 , c-fifa is an immediate
consequence of the theorem quoted above with respect to curves of the third order—
a theorem from which the entire system of relations between the twenty-seven points
on the curve might have been deduced a priori. But returning to the system of
fifteen points, suppose the lines aj)^, a 2 b. 2 c 2 , a 3 b 3 c 3 , and bj) 2 b 3 , and also the point
a 6 to be given arbitrarily. The point a 7 lies on the line <\a 3 , suppose its position
upon this line to be arbitrarily assumed (in which case, since the ten points a 1} a 2 , a 3 ,
bi, b 2 , b 3 , c x , c 2 , c 3 , are sufficient to determine a curve of the third order, there is no
curve of the third order through these points and the point a 7 ). If the points
b 6 , c 6 , b 7 , c 7 can be so determined that the sides of the quadrilateral b 6 b 7 c 6 c 7 , viz.
b 6 b 7 , b 7 c 6 , c 6 c 7 , c 7 b 6 pass through the points b x , a 3 , c x , a 2 respectively, while the angles
b 6 , b 7 , c 6 , c 7 lie upon the lines a 7 c 3 , a 6 c 2 , a 7 b 2 and a 6 b 3 respectively, the required condi
tions will be satisfied by the fifteen points in question; and the solution of this
problem is known. I have not ascertained whether in the case of an arbitrary position
as above of the point a 7 , it is possible to determine a complete system of twenty-
seven points lying three and three upon forty-five lines in the same manner as the
twenty-seven points upon the curve of the third order; but it appears probable that
this is the case, and to determine whether it be so or not, presents itself as an
interesting problem for investigation.
Suppose that the intersecting plane coincides with one of the triple tangent planes.
Here we have a system of twenty-four points, lying eight and eight in three lines;
the twenty-four points lie also three and three in thirty-two lines, which last-mentioned
lines therefore pass four and four through the twenty-four points. If we represent by
a, b, c, d, a', b', c', dl and a, b, c, d, a', b', c', d', the eight points, and eight points
