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of the case. In every plane through a (Fe) lies an oval touching a 
at A. Degeneration being excluded, the branches departing from A 
on the oval, cannot pass from one side of a to the other, unless 
their plane passes ¢. Let the semiplane in- which the oval touches 
a, first turning one way and then the other, converge towards 
e. If in both cases the ovals contracted towards A, then they 
would suffice to give to A the character of a point of a two- 
dimensional continuum and a sequence of points on a with A for 
limiting point could not be fitted in anymore. 
Considering ¢ has only « in common with /* there remains as 
only possibility that when their plane turns one way, the ovals 
contract towards A and in the other case towards the entire line a 
(from both sides with branches connected via the line at infinity). 
This means that A is cusp in every plane not containing a and 
that every other point of a is point of inflexion in every plane 
not containing a, with all tangents situated in ¢. Hence A is unt- 
planar point and the vicinity on #* forms a twodimensional 
continuum. According to our definitions a counts triple in e. Evidently 
F* contains no further lines and as a counts single in every plane 
apart from ¢, the total number of lines on F* is 3. The numbers of 
lines being the principal object of the present investigation, we 
exclude this case in what follows. 
Above we excluded degenerations of the oval in planes through a 
(Ge) on ground of the assumption that no second line of /? intersects 
a. If we only assume that no second line passes through 4, then it 
is possible for the oval to degenerate in two lines of which one 
coincides with a and the other does not pass through A. Then in 
a plane turning round a from the starting position e the following 
changes take place: At first ovals have a for tangent at A, these 
ovals extend and in a plane a occurs the generation in a and a 
line not containing A. After that we find ovals again touching a 
at A (but now from the other side) and when their plane converges 
towards e‚ these ovals once more contract towards A. It can easily 
be shown that no further degeneration takes place. A is another 
kind of wniplanar point, with twodimensional vicinity on #”. In plane 
x line a counts double. In « line a counts single and A can be con- 
sidered as a point-oval. From the foregoing follows directly that #? con- 
tains no further lines. The second line obviously does not count 
double in any plane. It can also be easily shown that it does not 
count triple in more than one plane, but we leave undecided if this 
can happen in one plane. Hence the total number of lines on F” is 
