J 249 



ill Ivvo brandies. Now a priori, two possibiiilies exist, (liat P„ and P,' 

 end u)) b}' being situated on different brandies, which then face 

 eacii other with their convex sides, or that thej lie on the same 

 branch. We proceed to show that the last case is excluded. The 

 position in «„ is denoted bj the dotted line in fig. 2. These curves 



have for lintiiting set the lines b and c in n. The lower branch has 

 for limiting set the lower semilines of b and c, hence the upper 

 brancli must take the upper semilines for its portion. Now if we 

 choose through line PR a plane, for instance x (^y then in this 

 plane, P must be a cusp, where the branches meet from the same 

 side of tlie tangent and this is excluded for a curve of the third order. 



We are now in a position to prove theorem 2. Let the section 

 in a plane a consist of three unes forming a triangle A^ A^ A^ 

 (fig. 3). Above it a|)peared that in the neighbourhood of .4, the 

 connection is, for instance, as follows: A^B is connected with 

 At A^ and A^ A^ with .4, 6' boih above <(, and .1, /i with A^ 6^ and 

 A^ .1, with A^ A^ below u. Hence if a plane through A^ A, con- 

 verges, ill the neighbourhood of A, from above, towards (f, then 

 the oval is divided by the line at infinity into two branches having 

 resi)ectively for limiting sets the semilines A^ B -\- A^ A^ and the 

 semilines A. C -{- A^ A^. If on the other hand, the plane converges 

 in the neighbourhood of A^ from below towards cr, these limiting 

 sets become .4, B -}- A^ (J and .4, ^4, -f- -'^s ^i- 



From this we conclude: If a plane turning round a line a of 

 F" passes through a position in which the restcurve degenerates 



