1247 



twodimensional continuum, we coiu'liide that each branch is con- 

 nected with the neighbouring ones alternately above and below «. 



Fig. 1. 



Let A^A^ be connected with BA^ above a by I, B A^ with CA^ 

 below a by II, CA^ with A^A^ above « by 111 and lastly A^A^ 

 with A^A^ below « by IV. 



Now suppose a third line L// of F^ passed thi-ough ^,. Of course 

 LH cannot be situated in a. Let the seniiline A^H depart on 1 

 and .!,/> on 11. We consider the plane [3 through AiH and A^D 

 (fig. 1). In ,i the curve consists of LH and an oval passing through 

 D. This oval also passes through A^ because in /? a branch must 

 depart from A^ on IV. Now let ^ turn slightly round LH, then the 

 same conclusions hold for these new planes and this means that in 

 an infinite number of planes through LH ovals would pass thi-ough 

 A^. This however contradicts the results obtained in the third com- 

 munication (p. 744 at the top). The only way to escape immediate 

 contradiction is to assume the ovals to be degenerated in all planes 

 f?. Then however F^ would contain an infinite number of lines, a 

 possibility we do not consider while looking for the possible finite 

 numbers. 



Theorem 2. //' F' contains a finite number of lines, this number 

 is equal to 3 -|- /i . 4 {n beiiuj a positive integer). This (Jivision can 

 be mad'; in such a way that the lines of which the first group of 3 

 coiisists, lie in one plane and every nem group of 4 intersects one of 

 these 3. Besides every group of 4 can be divided into ttuo groups. of 2, 

 each lying {with one of the first 3) /// one plane. 



Before proving thi.s we shall considei" some auxiliary (juestions. 



86* 



