Mathematics. — "Representation of a Bilinear Congruence of 

 Twisted Cubics on a Bilinear Congruence of Rays." by Prof. 

 Jan de Vries. 



(Communicated at the meeting of February 25, 1922). 



In a communication entitled : Congruences of Twisted Cubics in 

 connection with a Cubic Transformation (these Proceedings Vol. XI, 

 p. 84, 19()8j I have shown that the congruence of the twisted 

 oubics (>' through five points (congruence of Reye) may be converted 

 b}' a simple transformation {x}cyk= 1» ^ = 1» 2, 3, 4) into a sheaf of 

 rays. Now I s/iall show how a different congruence \q^] likewise 

 bj' means of a cubic transformation, may be represented on a 

 bilinear congruence of rays. 



§ 1. The transformation in question arises in the following way. 

 Three crossing straight lines a^, a,, a^ are the axes of involutions 

 of planes with pairs «i, «'^ (/t = 1, 2, 3) ; to the point of intersection 

 P of the planes a^, «,, a^ the point of intersection P' of the 

 corresponding planes a\, a\, a\ is associated. 



For a point A^ of a^,a^ is indefinite; any point of the straight 

 line ^,3 which is the line of intersection of the planes «',, «', 

 corresponding to A^, may be considered as the image of A^. To the 

 points of the singular straight line a■^ the rays of a quadratic scroll 

 (^jj)' having a, and «, as directrices are therefore associated. 



Let t be a transversal of a^,a^ and a^, S the point of intersection 

 of the three planes «'a associated to the planes ajc ^ tajc. Evidently 

 S is associated to every point of t. The locus of the singular points 

 aS is a twisted cubic o*, each point of which is represented by a 

 ray of the quadratic scroll {ty having a^, a, and a^ as directrices. 



aS being especially associated to the points A^, A^, A^ where t 

 rests on a,, a,, a,, ö' is the partial intersection of the three scrolls 

 i^it)' AhiY > {inY 'y these have in pairs a straight line aic in common. 



When P describes (he straight line r, the pencils («/t) become 

 projective; also the pencils {a\) become projective and they produce 

 a twisted cubic 9' which is the image of the straight line r. As r 



