Mathematics. — " NuU-Si/stems in the Phne". By Prof. Jan 

 DE Vries. 



(Gommuuicaled in llie meeting of January 26, 1918). 



1. Ill a nnll-sjsteiu )ü{ct, i^) a group of <( straight, lines ?/, passing 

 tlirongli a point N is associated to tliat point; to a straight line ?i 

 belongs a gronp of j? points ^V lying on n. A point is called 

 singular, when it is nnll-point of oo null-rays; a straight line is 

 called si)i(/u/a)- if it has oo null-points. 



The null-systems, for which a or ^^ ïh e(\uii\ lo i {linear mtll-sj/^steins) 

 are characterized by the fact that they always have singular null- 

 points if i(=:l, always show singular null-rays if ,i = l. Considerations 

 concerning the case <( = 1 are to be found in my papers "Oji plane 

 Linear Null-Systems" (These Proceedings vol. XV, page 1165) and 

 "Lineare ebene NuUverwandtschaften" (Bull, de I'Acad. des Slaves 

 du Sud de Zagieb, July 1917, Auszug aiis der im Had. Bd. 215, 

 S. 122 veröffentlichten Abhaudlung). 



That a non-linear null-system does not necessarily possess singular 

 elements, appears among others from the consideration of the null-system 

 ^1{'S, 3?i — 6) formed by the points of inflection and their tangents 

 appealing in a general net of curves of order n '). Only for n = '6 

 we have in general a group of 21 singular null-rays, viz. the 

 straight parts of the binodal figures. 



2. Let us suppose that a ^'?(«, /?) possesses n singular points S, 

 which are singular null-points on each ray drawn through them, 

 and (J^ singular points S^, which replace two null-points on eacli 

 ray *). We further suppose that there are a singular rays .v and 

 r/^ singular rays .v^; the latter are characterized by the fact that 

 they represent two coinciding null-rays for each of their points. 



If the straight line Ji is caused to revolve round the point J^, 

 the ^ null-points iV describe a curve (/-*) of order {a -\- /^), which 

 has an a-fold point in F. 



Analogously the null-rays n, which have a null-point /V on the 



1) See my paper "Two null-systems determined by a net of ciibics'" (These 

 Proceedings vol. XIX, page 1124) 



3) In the linear null-system formed by Die tangents and their points of contact 

 of a pencil (c") the base-points are singular point.s ,S'^, the nodes singular points S. 



