Mathematics. — "On the Plane Pencils Contaming Three Straight 

 Lines of a given Algehrnical Congruence of Rags". Bj Dr. 

 G. ScHAAKE. (Communicated by Prof. Henukik dk Vhiks). 



{Communicated at the meeting of June 30, 1923). 



^ J. In liis ,,K(ilku/ der Abzahlenden Geometrie", p. 33J , Schubekt 

 finds tiial tlie vertices of the plane pencils containing tliree siraiglit 

 lines of the congruence which two complexes of rays of the orders 

 m and m' have in common, form a surface of tlie order: 



\mm' {mm'— 2) (2mm'— 3m— 3/»' + 4), 

 and the planes of these pencils enxelop a surface of tlie same class. 

 In this paper we shall examine wiiat these results become for an 

 arbitrary algebraic congruence of rays. With a view to this we 

 make use of the representation of a special linear complex C on 

 a lineal' three-dimensional space R, which is described in Sturm: 

 „Liniengeometrie" , I, on p. 269. First, however, we shall give a 

 derivation of this representation which differs from the one 1. c. 



^ 2. If we associate to a straight line / with coordinates y>j, ... /j, 

 the point P in a linear five-dimensional space R of which the six 

 above mentioned quantities are the homogeneous coordinates, a 

 special linear complex 6' is represented on the intei'section of a 

 variety V with the equation 



Pi P, + P. Ih + />, P, = 

 and one of its four-dimensional tangent spaces R^. 



This intersection is a quadratic hypercone K that has its vertex 

 T m the point where R touches the variety V. As the generatrices 

 of K intersect an arbitrary three-dimensional space in the points of 

 a quadratic surface, K contains two systems of planes each of which 

 projects one of the scrolls of the surface in question out of T. Two 

 planes of the same system have only the vertex T in common, two 

 planes of different systems a generatrix of A'. The planes F^ of 

 one system are the representation of the stars of rays of the com- 

 plex C, which have therefore (heir vertices on the axis a of C', and 

 the fields of C the planes of which pass through a, are associated 

 to the planes Vv of the other system. The axis a of 6' and the 



