981 



Uimiigli <I,,<J.„(I., in>d (i- aiul luive a node in (1. Tlie luisc ol' this 

 nel consists of the 6 lines ;/„;/,, iJ,J',,J',,Jh,; tiiey form a aege- 

 nerate twisted curve of the 6"' order with 7 a|)parent nodes. Every 

 two V have moreover in common a twisted cubic, wliich passes 

 through G and (r* and meets each of the lines ^i twice ; these curves 

 (f^ consequently form the above mentioned congruence. 



Through an arbitrary point passes a pencil (¥^), hence ona (/'. 

 On an arbiti-ary line / the net determines a cubic involution of 

 the second rank ; through the neutral points of this /% passes a 

 curve (/\ which has / as bisecant. The congruence [7'] is therefore 

 bilinear. 



Through a point >S' (A' </, pass oo^ curves r/^ they lie on the hyper- 

 boloid H\ which is determined by S, G, (J^, y„ g,. All the curves 

 (f^ lying on H\ pass moreover through the point aS''> in which H' 

 again cuts the line y^. 



To f^'] belongs the tigure formed by h^^ and a conic of the 

 pencil which is determined in the plane [G^'t)^) by the intersections 

 of y„y,, h,,, and the point G"". There are apparently 5 analogous 

 pencils of conies besides. 



Let us now consider the surface A formed by the ff\ which 

 meet the line /. Through each of the two points of intersection of 

 / and H' passes a (f\ cutting //, in S. From this ensues that the 

 three lines c/k are double lines of A. The lines hkiJi^^ki he on A, 

 for / for instance meets a conic of the pencil indicated in the 

 plane {G^'g^), and this pencil forms with h^.. a (f\ 



We determine the order of A by seeking for its section with the 

 plane (%j. To it belong J) the line </,. which counts twice, 2) 

 the conic in that plane, which rests on / and is completed by A^.,* 

 into a (j.\ 3) the lines h,., and h^„ which are component parts of 

 two degenerate (/\ of wliich the conic rests on /• From this ensues 

 that A is of the sid-t/t order. 



10. If the congruence [^7"] is made to intersect with a plane </, 

 a cubic involution {P') arises, wliich has the intersections of the 

 lines (/A-, hu, and hi,r as singular points. With the intersection Jh 

 oi' gk correspond viz. the intersections of the 7 ^ wliich cut </ already 

 in Ihy, they lie as we saw on the intersection (7i/,)' of the hyperbo- 

 loid H belonging to B/,. To the iiitei-section A, of A„ corresponds 

 the 7- on the intersection a, of the plane ((V*</,), originating from 

 the pencil of conies in that plane, etc. 



On {B,y lie the intersections oï <j„ (/,, g„ k,, and h^^, viz. the 

 points B„B„B„ A, and A,"" ; on the intersection a, of the plane 



