Mathematics. — "A Congruence (1,0) of Twisted Cubics". By 

 Prof. Jan de Vries. 



(Communicated at the meeting of February 24, 1923). 



1. The twisted cubics throiigii four points C\, C„ C„ C^ cutting 

 the straight line b twice, form a linear congruence [p*] ; for through 

 any point there passes one y'. The base points C are the cardinal 

 points, 6 is a cardimil chord. 



If (7 is a chord of one of the p', d (C, C, C, C,) = b{C, C, C. C,). 

 The chords (/ form therefore a tetrahedral coviplex ; a ray / not 

 belonging to this complex, is not cut twice by any p' : the class of 

 the congruence is zero. 



Togetiier with Ck and b a chord (/ detines a iiyperboloid ; on this 

 there lie oo' curves p' and these define on (/ an involution ; d is 

 consequently a tangent to tivo curves. 



The tangents meeting at a point P, lie on the complex cone of 

 P, their points of contact form a twisted curve of the 5''' order, 

 p', passing through P. 



2. Let B^ be the point of intersection of 6 with the plane 

 y,,, ^ C, C, C',. Each conic (j' through the points C^,C\,C\,B, is 

 a component part of a degenerate <>' ; the transversal t^ tiirough 

 C^ resting on b and p' is the second component part. The straight 

 lines t^ form the pencil of rays through C\ in theplane C\ b. There 

 are therefore four pencils of rays formed by singular straight lines. 



The pairs of lines of the pencil (9') produce three figures each 

 consisting of three straight lines, e.g. the combination of C, C\, C\ B^ 

 and the straight line t^ resting on C, C,. There are evidently 

 twelve figures consisting ot three straight lines. 



3. With a view to finding the order of the surface A formed 

 by the y' cutting a straight line /, we determine the intersection 

 of A with the plane y,„. It consists of two conies of the pencil 

 ((»'); the former cuts /, the latter is a component part of the p" 

 which is defined by the transversal through 6', of 6 and /. Hence yi 

 is a surface of the 4"' order; tiie cardinal points C are apparently 

 double points of A*. A 9' not lying on ./\ can only cut this surface 

 in the points C and on the cardinal chord b ; from this there follows 

 that 6 is a double straight line. 



