131 



are conjugated .t planes y. The planes 01 and tlie tangent planes 

 Q define on any stiaiglit line a oorrespondenoe with characteristic 

 nnml)ers P.rr and 2.t. Through eacli coincidence there passes one 

 image line ;•; accordingly the system Q-, ^) is represented hy a 

 scroll of the order (A -|- 2) .t. 



A system (A, .-t) contains 2/. straight lines / passing through i>, 

 or through B^. As each of them carries .i line elements, the scroll 

 contains 2A straight lines of the kyperholoid, each of wiiich is a 

 jT-fold straight line of the scroll. 



The system (1,.t) in which the points P form a curve (P) of the 

 order n- which has a x-fold point D and where all straight lines 

 / meet in D, has to be examined separately. For here a plane 01 

 contains only (.t — x) points P and defines therefore only (,7 — x) 

 planes q. The characteristic numbers of the correspondence between 

 the points of a straight line are in this case (.t — y^ and 2.t, so that 

 I he system (1,-t) is represented by a scroll of the order (3jr — «) on 

 which the straight line OD is evidently 2:7r-fold. 



A system (l,Jr) of the kind in question is found in a null system 

 xV(fi,i') which is the locus of the null points of the rays of a plane 

 pencil round a point D. For this null curve is a curve of the order 

 (fi -|- r) with a jj-fold point D, so that the line elements form a 

 system (1, ft -|- r). 



5. A null system N{n,v) is represented by a congruence of rays 

 [?•]. The straight line a^ is a null ray for r of its points Pand the 

 straight line r representing (P.a^), coincides with «,. Hence n, and 

 a, are r-fold rays of the congruence; the field-degree of \r] is 

 accordingly 2r. 



Let Q be the central projection of the point F. The null curve 

 of Q is projected by a cone of the order (fi -(- '') and this cone has 

 2(fi + >■) points R in common with the conic which is the inter- 

 section of H and the polar plane of F. From this follows that the 

 sheaf-degree of the congruence is 2(fi -|- v). The image of an N{[j,v) 

 IS therefore a congruence (2 f« -|- 2 i', 2 r). 



Accordingly a hilinenr null system Is{\-,\) is represented by a 

 congruence (4,2). The singular points «S,, aS,, 5, define three points 

 Pi, P,, P, on H\ these are the vertices of three plane pencils 

 (''i). (^'i). ('".). representing the plane pencils round the points S, hence 

 singular />oints of the congruence [r]. The line elements on the three 

 singular straight lines s^^ S,S„ s, and s, are represented by the 

 tangents of three conies Ok' througii 0. Their planes Oj. are singular 

 planes of the congruence. Also the plane <t^P, P, P, is singular 



