Mathematics. — "On Elementary Surfaces of the Third Order." 

 (Fourth communication.) Bj Dr. B. P. Haalivieijer. (Com- 

 municated by Prof. L. E. J. Bkouweh.) 



(Communicated in the meeting of February 23, 1918). 



It has been proved that 7^* cannot exist if no plane section consists 

 of three lines ^). We now proceed to show that F^ cannot exist if 

 that surface does not contain 3, 7, 15, 27, or an infinite number 

 of lines. 



This note is divided into two parts, the first contains a few theorems 

 we shall want later on, in the second we show that except those 

 mentioned, no other number of lines is possible on F*. 



^ 1. Theorem 1. //' a plane section of F* consists of three lines 

 forming a triangle, then through none of the angles can pass a. 

 further line of F*. 



Let the plane be denoted by a, the lines by a,, a^, a, and their 

 points of intersection by A^, A,, and A, (A^ is the point where a, 

 and a, meet etc.). We choose on a, an arbitrary point B^ (not 

 coinciding with A, or A^). B^ is limiting point, both of sequences 

 of points of F* situated abo\ e « and below «. For suppose B^ were 

 limiting point only of sequences situated below n. Then choose a 

 plane ^ through B^, not containing the line a^ and not passing 

 through ^1. Let b be the line of intersection of a and /? and let b 

 intersect a^ at 7i, and a^ at B^. Let a parallel line in plane /? con- 

 verge towards b from above. The point B^ on the limiting line is 

 supposed not to be limiting point of points of F* situated on the 

 converging lines, hence B^ must count double on b as point of 

 intersection with the curve in (i. Besides b carries the points i>, and 

 j5, : a contradiction. 



From the above follows that in every segment of one of the 

 lines a■^, a^, and a^, containing no angle of the triangle, the sectors 

 of jP' meet from different sides of «. In plane « four branches meet at 

 Ay^-.At A^ and CA^^ on a^ and A^A^ and B A^ on «, (tig. 1). From 

 the above result, in connection with the assumption that F* is a 



^) Again line will be used for straight line. 



