934 
PQ passes through A, not situated in «, leaving on / (AP) and 
on lI (AQ). 
We note that no line through A carries 2 other points of F? 
without belonging entirely to that surface. 
Let y denote an arbitrary plane through PQ inside “FAH. The 
part of the section in y which is not situated on PQ, can consist of: 
1. Two different lines through A. 
2. Nothing. 
3. One line through A (possibly to be counted double). 
4. An oval touching « at A. 
No two planes y can contain a section falling sub 1, 2 or 3, for 
then an infinite number of planes through A inside / EAD could 
be found, each containing 2 lines not situated in @ and having only 
A in common with #* (namely the lines of intersection with the 
2 planes y) and this is impossible considering that in all those 
planes A is double point with a tangent in a. 
Hence it suffices to consider 4 cases. In the first 3 all planes y 
except one, contain a non-degenerated oval and the excepted plane y 
shows a section falling sub 1, 2 and 3 respectively. In the fourth 
case every plane y contains a non-degenerated oval. 
First case. In a plane 7 through PQ inside “FAH (fig. 4) the 
curve consists of 3 different lines through A. In plane 8 through 
PQ and BE no third line passes through A, for then sector / would 
ercss this plane twice and this cannot be as sector / comes from 
the one side of 8 (AF) and ends up at the other side (AC). Neither 
PQ nor BE can count double in @, hence the curve in 8 consists 
of 3 lines forming a triangle. 
Almost the same reasoning holds for the plane through PQ and 
DH) and substituting / for // and vice versa, also for the plane 
through PQ and CF. Besides for PQ can be substituted either of 
the remaining lines in y. Hence in all 15 lines on #* have been 
found, namely 6 through A and 9 others. 
From the foregoing follows at once that none of the 3 lines in 
« can be intersected by other lines as those mentioned. Also it can 
be easily shown that none of the 15 lines counts double or triple 
in any plane. Hence the total number is 15. 
Every line through A in a or y is tangent at that point and on 
the other hand every tangent is situated in «a or y. Hence A is 
‘) DH cannot count double in the plane through PQ, for then sector JJ could 
not cross y three times, without passing more than once through the plane o 
PQ and DH. 
