431 



Mathematics. — "Bilinear congruences of twisted curves, which 

 are determined by nets of cid/ic surfaces^ By Pi'of. Jan de Vries. 



(Communicated in the meeting of May 2it, 1915 ) 



1. The base-curves (/ of the pencils belonging to a general net 

 [0'] of cubic surfaces, form a bilinear congruence. For through an 

 arbitrary point passes 0)ie curve o', and .the involution of the second 

 rank, which the net determines on an arbitrary straight line, has 

 one neutral pair, so that there is one o' for which that straight line 

 is bisecant. 



The 27 base-points of the net are fundamental points of the 

 congruence. Any straight line ƒ passing through one of those points 

 F is singular bisecant; for through any point of ƒ passes a <>', at 

 the same time containing F. As the points of support of the curves 

 resting on / form a parabolic involution, ƒ may be called a parabolic 

 bisecant. 



Let t be a trisecant of a ^*" : through an arbitrary point of t 

 passes one 't*', and this surface contains all the points of t. By the 

 reiuaining surfaces of the net, t is intersected in the triplets of an 

 involution ; consequently ^ is a singular trisecant. The singular trise- 

 cants therefore form a congruence of rays; it is at the same time 

 the congruence of the straight lines lying on the surfaces of the net. 



A curve q^ has 18 apparent nodes, is therefore of genus 10. The 

 cone of order eight 5*, projecting it out of one of the points F has 

 therefore 11 double edges t '). 



Any point F is a singular point for tiie congruence [t], conse- 

 quently vertex of a cone .J formed by trisecants t. With A" this cone 

 has, besides the 26 straight lines FI^' to the remaining fundamental 

 points, the 11 double edges of Ts' in common. Consequently i is a 

 cone 'of order six ; the congruence \_t] has therefore 27 singular 

 points of order six. 



The trisecants of a (?' form a ruled surface, on which p" is an 

 elevenfold curve. With an arbitrary surface 0' this ruled surface 

 has moreover the 27 straight lines of '/>' in common; the complete 

 section is consequently a figure of order 126, and the ruled surface 

 in question has the order 42. 



Let us now consider the axial ruled surface -I, formed by the 

 rays of the congruence \f\ resting on a straight line a. With an 

 arbitrary (>' it has first in common the 27 sextuple points F, the 



1) A curve f" with /* apparent nodes is intersected in each of its points by 

 h — (n~% trisecants. 



