Mathematics. — "A Null System (1,2, 3)." By Prof. Jan dk Vries. 

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



1. We consider as given a congrneiice [(>"] of twisted cubios with 

 the base poiiils C^, C,, (J„ C\, (J^') and the crossing straight lines 

 a and 6. 



Through a point N lliere passes one cnrve (j' ; let /■ he the 

 tangent at N and t tiie transversa! of a and 6 through JV. We 

 conjugate i^^rt to iV as a }iull plane. 



The curves q' touching a plane r have their points of contact 

 in a conic p'. The transversal t lying in r, cuts (}* in the null points 

 JSF, and jV, of v. 



if V revolves round the straight line /, t describes a scroll (/)' and 

 p' a cubic surface through /. The locus of A^ is accordingly a 

 twisted curve ?.\ which has evidently /, hence also a and b, as 

 trisecants. 



We have therefore a 7iull system, with the characteristic numbers 

 a — 1, ,i=:2, y = 3. 



2. The points Ck are singuhir; for C'^ carries one straight line / 

 but od' straight lines r. The null planes of Ck form a pencil of 

 planes round t as axis. 



Also the points ^4 of a and B of h are singular. For each of 

 them carries oo' straight lines / which are combined to a plane 

 pencil. The null planes of each of these points form a pencil of 

 which the axis lies in the tangent r. These axes form two cubic 

 scrolls ()•)'■ 



Other singular points S can only arise through coincidence of the 

 straight lines / and r. Now the tangents of the curves p' form a 

 complex of the 6''^ order and this complex has a scroll (n)'' in 

 common with the bilinear congruence \f\. On each straight line n 

 there lies a point »S to which any plane through n corresponds as 

 null plane. 



As / is intersected by i2 straight lines n, the corresponding curve 

 A' contains 12 points S. 



1) The principal properties of this congruence are to be found for instance in 

 R. Stuem : Die Lehre von den geometrischen Verwandtschaften, Part IV, p. 470. 



