Mathematics. — ^^Nn/l-Si/siems determined by tivo linear con- 

 gruences of rays'. By Professoi- .)an dk Vrik,s. 



(Communicated in the meeting of April 26, 1918). 



1. A twisted curve, n'' intersected by a straight line a in (/> — 1) 

 points, determines a linear congruence (!,/>), of wliicli each ray n 

 rests on a and on (J'. Analogously a curve /^9 intersected by the 

 straight line b in [q — J) j^oints determines a congruence (1 ,</), of 

 which the rays v rest on b and (i'l. 



Thi-ongh the point zV pass in general one ray u aiuJ one ray v. 

 If the plane r=^uv is associated as null-plane to JV a null-system 

 arises in which a plane r has in general /> (/ nnll-points, viz. the 

 intersections of the p rays of u with the q ra^ s of v. 



If JSl describes a straight line /, (he rays n. and v describe two 

 ruled surfaces, which are successively of order (/> -|- 1) and order 

 ((/ -j- 1), and intersect along a curve (/) of ordei' {p q -\- p -\- q)- An 

 arbitrary plane v passing through / has with (/) the p q null-points 

 of V in common, and moreover (/> -\- q) points lying on /, which 

 belong each as null-point to a definite plane v. In other words, the 

 straight line / is ( p -\- q) times null-ray. In R. Stukm's notation the 

 null-system has therefore the characteristic nnmbeis a=z{,^=ipq, 

 Y = p -\- q, may consequently be indicated by '^^{\,pq, p -{-(])■. 



2. If V coincides with //, any point of that straight line has any 

 plane passing through that straight line as null-plane. Now, the 

 congruences (!,/') and (!,(/) have iji general ipq-\-'^) rays in 

 common. There are consequently {pq-\-^) singular straight lines s. 



The curves ar and ji^ are also loci of singular points. Thi-ough 

 a point -A* of «i' passes a ray v* and a plane pencil of rays u. In 

 any plane passing thi-ough y* lies one ray v; so A* is null-point to 

 any plane of a [lencil that has v as axis. The straight lines v* form 

 a ruled surface of order p{q-\-\}; for a plane passing through b 

 contains p rays v* and a point of b bears pq rays ?;*. Finally the 

 points of a. and h too are singular iitill-points. A point A^ of a 

 bears one ray z;* and go ^ rays u, which form a cone of order /; 

 with {p — l)-fold generatrix. Any plane passing through v^ contains 

 p rays u, so that ^l^,. is to be considered as /^fold null-point. The 

 rays v^ form a ruled surface of order {g-\-^). A straight line u 



