1252 



the point of intersection of h^ and //j, and this is impossible according 

 to theorem J . 



Hence c is intersected by at least 8 lines not sitnated in a, and the 

 same can be proved for h. This brings the total np to at least 27. 



It remains to be proved that the nnmber of lines is intinite, if 

 greater than 27. 



Theorem.: 4 lines of F\ not intersecting each other and having one 

 common hitersecting line, have at least two inte7\secting lines in common. 



Let the lines be a,, a,, r/,, a^ and let a be a tirst line, intersecting 

 these fonr. Of conrse a belongs to F* also. The third lines in 

 the planes throngh a and a^, a^, a^, a, we denote respectively by 

 a\, a\, a\, a\. 



The oval ninst degenerate in a fifth plane a throngh a. Let the 

 lines is this plane be b and c. The foregoing resnlts show, that h 

 and 6' are each intersected by at least 8 lines of F\ not situated 

 in « M. Let us consider these 16 lines: j)i, }>^, ■ ■ ■ />ie- Each of these 

 intersects 4 crossing lines of the set: a,, a,, a^, a^, a\, a\, a\, a\. 



It is impossible that 2 lines p for instance p,, and />„„ intersect 

 the same 4 lines a,,, for in that case these 4 lines woidd have 3 

 different inteisecting lines in common (/>„, />,„ and a), hence an 

 infinite number, and F^ would contain an infinite number of lines. 



Now from the lines a^ . . . a^, a\ . . . a\ we can choose in 16 

 diffei-ent ways, 4 lines which cross each other (they are situated by 

 twos in the same plane), and considering there are also 16 lines /;, 

 each group of 4 lines a,, which cross each other, have a common 

 intersecting line, different from a. Hence this holds for the gr-oup 

 a,, a,, a^, a^ which we set out to prove. 



Now that this theorem aiid theorem 1 of § 1 have been proved, 

 refer the reader for the further proof that 7^^* contains an infinite 

 number of lines if that number exceeds 27, to the demonstration 

 given by Juel (Math. Ann. 76, p. 561 and 562). 



This finishes the examination of the numbers of lines, without 

 which F' cannot exist. 



We conclude with a more or less independent result, which happened 

 to present itself: 



1) The possibility tuiglit be pul forward, that the point of intersection oföandc 

 is situated on a, for which case tlie foregoing results have not been established. 

 This objection can be met as follows: We choose three lines a, an and a',,, 

 forming a triangle. The former results then show, that an and a'n are each inter- 

 sected by 8 lines, which do not intersect a. These lines can then be used instead 

 of the lines Pi, p^ . ■ ■ ■ Pu, introduced in the text. 



