Mathematics. — - " Representation of a Tetrahedral Complex on the 

 Pohits of Space." By Prof. Jan dic Vkies. 



(Communicated at the meeting of April 28, 1923). 



1. Let there be given a pencil of quadratic surfaces which has 

 a twisted curve (j' as base curve. The polar planes of a point P 

 with respect to these surfaces pass through a straight line p, which 

 we shall call the polar line of P. Through P there pass two 

 bisecants of q^ : the straight line p joins the points of these bisecants 

 which are harmonically separated from P by q*. If P lies in the 

 vertex of one of the four cones belonging to the pencil, tlie polar 

 line becomes indefinite; any straight line of the plane wa:^ Oi Om On 

 may be considered in this case as a polar line. 



The complex of rays T of the polar lines p is represented on 

 the space of points | P\. The side Ok 0/ is represented in any of 

 the points of the opposite side O,,, 0,,- If a straight line r is to 

 belong to T, its polar lines /•' and r" with respect to the surfaces 

 «' and ii' of the pencil, must cut each other. If the straight line r 

 describes a plane pencil, »•' and r" describe two projective plane 

 pencils; the plane pencil (»•) contains accordingly two rays for which 

 r' and 7-" cut each other. The complex T is therefore quadratic^) 

 and has four cardinal points Ok and four cardinal planes wj; hence 

 it is tetrahedral . 



A point P of (}* is the image of the straight line/* which touches 

 Q* at P. The scroll of the tangents of q* is therefore represented 

 in the points of q*. 



1. If P describes a straight line r, the polar planes of P with 

 respect to «' and t^' describe two projective pencils round the polar 

 lines r' and r". The polar line /) describes accordingly a quadratic 

 scroll (pY; Ihe conjugated scroll consists of the polar lines of r 

 with respect to the quadratic surfaces through o'. The points of 

 intersection of r with the cardinal planes w^. are the images of four 



1) If the pencil is defined by EukXi^ = and I^bad* = 0, the polar planes of 



4 4 



the point y have aayt and bii/t for coordinates. The coordinates of p are in this 

 case pi2 = (0364 — 04 03)2/3^4 etc. If we put a^a^b.,h^ + aoOibib^ = €1^,^, T is 

 represented by Ci2,34Pi2P34 + £33,14^33^14+ Csi,24P3i 2^24 = 0. 



