Mathematics. — ''Surfaces that may be represented in a (jlane by 

 a linear congrnence of ra.ys'\ By Prof. Jan de Vriks. 



(Communicated in the meeting of April 27, 1917). 



1. In order to obtain a representation (1/1) of a cubic surface 

 in a plane we may make use of the bilinear congruence which has 

 two straight lines lying on that surface as directrices. To this pur- 

 pose the linear congruence (1,3) may also be used, which is formed 

 by the bisecants of a twisted cubic lying on the surface ^). It would 

 also be possible to make use of the congruence (J, 3) of the rays 

 that intersect a twisted cubic and one of its bisecants, provided 

 that those two directrices are lying on the surface. 



We shall now consider, more in general, the surfaces that can 

 be represented by means of a linear congruence of rays (1,»)- 



The representation of a surface fp"+p+^ with an ?z-fold and a 

 />»-fold straight line by means of a bilinear congruence has been 

 amply treated by Guccia and Mineo '). 



2. Let now be given a surface <f»-"+i with an /i-fold twisted 

 cubic «'. Through a point P of *P passes a bisecant of a, which 

 intersects the plane of representation t in P' ; in general a poin^ 

 P has o?ie image P', and, inversely, a point P' of t is the image 

 of one point P. 



Let us now consider the ruled surface (^) formed by the bisecants 

 t projecting the points of a plane section y'^"+i. In the plane of that 

 curve lie three straight lines t, ^^ is consequently of order (2?z-|-4). 



That cone, which projects «' out of one of its points, determines 

 on y2"+i evidently {n-\-2) points P, consequently a is an (?2-)-2)-fold 

 curve on @2n+4 



The image of .y2"+i is therefore a curve of order (2n-f-'l) it^ith 

 three {n-\-2)-fold points A ; this curve will be indicated by the symbol 



The images of two plane sections have the (2?2-|-l) points P' in 

 common, which are the images of the points P lying on the inter- 

 section of the two planes. 



1) See e.g. R Stukm, Geometrische Verwandtschaften, IV, 288 

 2; Guccia, Sur une classe de surfaces, représentables point par point sur un 

 plan (^Ass. francaise pour I'avancement des sciences, 1880). Mineo, Sopra una 

 classe di superficie unicursali (Le matematiche pure ed applicate, volume 1, p. 220). 



