1250 



in such a way, that the tliree lines do not pass tliroiigh one point, 

 and if tlie branches throngh corresponding points of a first face 

 each other with their convex sides, then after passing through the 

 degeneration the brandies will face eacii other with their concave 

 sides (or vice veisa). 



Now let F' contain' 3 lines in one plane: ri^, ((^ and a\. If there 

 is a fourth line h^, it cannot lie in that plane, but it does inter- 

 sect one of Ihe first 3 lines, for instance a^. 



In the plane through a^ and h^ the curve consists of three lines, 

 hence another line 6, intersects a■^. In the third coniniunication we' 

 concluded that F" caunol exist without 3 lines situated in one plane, 

 as otherwise the convexity and concavity of the branches through 

 corresponding points of ^/, lead to a conti-adiction. Now the above 

 results show, that the arraiigeiuent of this matter, which was made 

 [)0ssible by a first degeneration of the oval in a plane through a^, 

 is upset again by a second degeneration, and a third is necessarj' 

 to put things straight once moi'e. 



This co)icludes the proof of theorem 2 (we remind the reader of 

 the fact that degeneration into 3 lines through one point is only 

 possible, when the branches in the converging planes face each other 

 with their convex sides. 



^ 2. If F^ contains moi'e than 3 lines their number is at least 

 7, of which at least f> intersect the same line (according to theorem 

 2). Now on this line can be situated, at the utmost, two meeting 

 points of corresponding points, and it follows, that in at least one 

 plane the curve consists of 3 lines foi'iuiing a triangle. Let a be this 

 plane, a, h and c the lines, and .4, B and C their points of inter- 

 section {A is situated on J> and c etc.). If F^ contains 7 lines then 

 one of the lines in d, for instance d, is intersected by 4 other lines, 

 not situated in a. We proceed to show, that if the number of lines 

 on F^ is greater than 7, it is at least 15. 



If F^ contains more than 7 lines, then at least two lines in 

 <( are each intei-sected by at least 4 lines, not situated in «, or 

 one of them is intersected by at least 8 lines not situated in a. Let 

 us consider the first 'case: a and b are each intersected by at least 

 4 lines of i^"\ not situated in a, in othei' words : through each of the 

 lines a and b pass at least 2 planes different from a, in which the 

 oval is degenerated. 



Let rtj be a line of F'^ not situated in ((, but intersecting a at a 

 point A^. According to theorem 1, A^ cannot coincide with either 

 B or 6'. Let /? be a plane through b in which the oval is degene- 



