935 
biplanar point and each tangent plane has 3 different lines in common 
with F*"). We found A to be double point in every plane not 
containing the line of intersection a of « and y. In every plane 
(fa or y) through a, A is cusp, for in such a plane a is the only 
line having no point but 4 in common with £*. 
Points as here described are excluded in what follows. 
Second case. The plane y bas no points in common with F'*, not 
situated on PQ. This case can be dealt with in almost entirely the 
same way as the first case’). Again A is biplanar point with a 
and y for tangent planes. We find 7 lines, namely 3 in «, PQ and 
3 more in the planes through PQ and the first 3. Of all these, 
except PQ, it can be easily shown that in no plane they count double 
or triple. For PQ remain the possibilities that this line counts single 
or triple in y. Accordingly the total number of lines on F* is 7 or 
9. In the last case PQ carries a second biplanar point of the same 
type as A. We leave undecided whether this last case can occur. 
Biplanar points as here described are excluded in what follows. 
They are a cross-type of those immediately preceding and those 
of p. 928. *) 
Third case. In y the curve consists of 2 lines through A, one of 
which counts double. The results of p. 931 show that through the 
double one passes a plane containing 3 different lines through A. 
Part of the reasoning given for the first case shows that this plane 
cannot contain one of the lines in «a. Neither can it be situated 
inside “ DAE or / EAF, for such planes always contain branches 
touching @. Hence the above mentioned plane would lie inside 
“4 FAH. Then however the double line is the intersection of 2 
planes inside “ FAH, neither of which contains a non-degenerated 
oval, and this is impossible according to the results of p. 934. 
Fourth case. Every plane y contains a non-degenerated oval with 
a for tangent plane at A. Obviously these ovals depart either all on 
[or all on //. Let us assume the latter, then these ovals contract 
1) A good drawing is to be found in table III of the above mentioned paper 
of KLEIN. 
4) That HD does not count double in the plane through PQ appears when we 
bear in mind that every plane (4 y) through PQ inside L AH contains an oval 
having « for tangent plane at A. In the planes on the side of y these ovals 
depart on J, and in those on the other side of y they start on JJ, for otherwise 
F3 could not be a twodimensional continuum. 
3) For cubic surfaces, compare Krein, loc. cit, p. 557. 
