921 
First case. Let A be the point and a the corresponding plane. 
There exists a plane 3 with a section of the third order and this 
plane passes through A as a curve of the third order has at least 
1 point in common with every line of its plane. The curve in @ has 
only A in common with «, hence this curve passes at the point A 
from one side of a to the other. Then however the plane a divides 
the vicinity of A on £* into two parts with only one common limiting 
point (A), and this contradicts point 1 of the definition of F’.’). 
Second case. Let a be the plane of the oval, a a line intersecting 
the oval at two different points A and B and 8 a plane through a. 
Suppose the curve in 3 is of the third order, then one of the points 
A and B, for instance A, counts double as point of intersection 
with a. This is possible in three ways: 
The curve in 2 has a for tangent at A. 
A is cusp in 8, both branches coming from the same side of a. 
A is ordinary double point, two branches coming from each 
side of « °). 
The two first possibilities do not agree with theorem 1 p. 311 
comm. 2,*) for in « A is ordinary point of intersection of the oval 
and a, and in 8 we can find lines converging towards a without 
carrying points of 4” converging towards A. 
Remains the third possibility. The two branches departing from 
A in « are connected both above and below «a. The sector which 
forms the connection above « cannot have 8 for tangent plane along 
a branch in #, for in that case an infinite number of lines in 3 
would belong to F*- *) It follows that this sector crosses 8 twice and 
this is impossible as it connects two branches situated on different 
sides of 9. 
We conclude that the curve in 8 cannot be of the third order, 
hence it must obviously be of tbe second. The only condition imposed 
on @ was that this plane has two different points in common with 
the oval in a. On the other hand there exist lines carrying 3 different 
points of #*. Let y be a plane through such a line and intersecting 
the oval in « at two different points. In y the curve must be of the 
1) An analogous case was minutely dealt with in comm. 1 p. 104 and 105. 
2) In case the curve in @ consists of a double line through A and a single one 
through B, we can at once find another plane through a with a curve of the 
third order and without a double line. 
8) The demonstration there given made use of point 2 of the definition of F5 
only so far as sections of order higher than the third were excluded. 
4) This also holds when the curve in / has degenerated. 
60 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
