133 



Through i^,* there pass two straight lines </j and A, of H, through 

 F,* two straight lines «7, and A,. These four lines form a skew 

 quadrilateral; g^ and </, cut each other in S*, It, and h, in S,* ; 

 (7, and Aj rest on b,, </, and /<, on èj. The projections g,, h^, g,, A, 

 of liiese lines are evidently singular nail rays and form a quadri- 

 lateral which has the singular null points Si,S,: F,,F,; Bi,B, as 

 angular points. For B, ^ g, /i„ B,^gi A,; F^^g^ h„ F,^g, h^; 



•5, =gig,. 5, = /i, A, . 



The plane (>(/' cuts H along a conic, the tangents of which belong 

 to [)•] ; hence the straight line q, (the projection of ƒ ) is a singtdar 

 nidi ray. On q lie the singular null points F^,F^ and S. But .S' is 

 the intersection of a tangent 0, therefore also a point of the .9/»</?//a)' 

 null ray h^B^B,. Accordingly the singular elements of ^V(J,2) 

 form the figure of the angular points, the diagonal points and the 

 sides of a complete quadrangle. This null system is therefore of 

 the same kind as the -A^(l,2) which arises if to each straight line 

 there are conjugated as null points its intersections with the conic 

 in which it is transformed by an involutory quadratic correspondence.') 



7. Five tangents /■ define a /»/eörr C07??/)/ex ^ ; this has a congruence 

 (2,2) in common with the complex of the tangents of i/. The represen- 

 tation on « is again a iiull system ^V(l,2); for a point P defines 

 a point R and in q there lies one ray of the plane pencil which 

 in A has the null point of q as vertex; and a line / defines on B 

 a conic of which two tangents belong to the linear complex. 



This complex has two straight lines in common with each of the 

 scrolls of H, they form a skew quadrilateral ^, (7, A, /;,, the angular 

 points of which are singular points for the congruence (2,2). For 

 in A the point g^g, is the null point of the plane q defined bij g^ 

 and g,, so that any tangent at that point belongs to both complexes. 

 Consequently the points gi g„ g, hi, h, h,, and h^g^ are singular nidi 

 points of the null system (1,2) in «. 



As (/, and /(, rest on b,, //, and A, pass through B^; hence B, 

 and B, are singidar null points. Also here the six null points are 

 the angular points of a complete quadrilateral the sides of which 

 are si)igidar null rays. The plane pencil [0,uj'j contains one ray of 

 A which therefore also belongs to the congruence ; its intersection S 

 is the seventh singular point of j.Y(l,2). As aS lies on è and 5, and 5, 

 are singular, also 6 is a singular null ray. 



1) The general null system (1,2) has no singular null rays (l.c p. 1167). 



