930 
not containing AC or AEF and this is impossible. This completes the 
demonstration that in no plane Ge) through BD or CE a further 
branch starts from A. 
Once more we consider a plane (# a@) through CE. If in this plane 
a further branch starts from a point (A A) of CE, then an oval 
touches CE at that point, as every point (# A) of CE is isolated 
on one side of «a. It can then be fairly easily shown that this 
point is uniplanar point of the second kind described in § 2. Such 
points however, we excluded, hence we conclude that in no plane 
(Ka) through CE a further branch starts from any point of CE. 
Let the planes 8,8, 2,.... all containing BD, converge towards 
a in such a way that the top parts move towards AC (fig. 2). These 
planes end up by containing ovals, converging towards CZ, inter- 
secting AB and AD, and facing at these points of intersection A 
with their convex sides. For if these ovals did not so face A, then 
they would have points in common with every plane through CZ, 
and in such a plane a further branch would depart from some 
point of CH. 
A point A as here described we shall call normal point of inter- 
section of a double and a single line. 
We now have to consider the case that a branch leaves A, for 
instance on J, not having @ for tangent plane. We distinguish 2 
possibilities : 
1. Through A passes no line of F*, not situated in a. 
2. Throagh A passes a line of F*, not situated in a. 
1. Planes through line AL of a (fig. 2) can be found, in 
which 2 branches depart on Z, and 1 branch on JJ and IV 
each, and this means that A is double point with AZ as one of 
the tangents. Hence a branch starts from A on 7 in the direction 
AL, and it follows that in every plane Feu) through the line AM 
of a (fig. 2) 2 branches leave from A on /. In each of these planes 
A is double point and we conclude that no line ( CE or BD) 
