437 



for both surfaces), the 15 parabolic bisecniits PF and nix suH/nlar 

 bisecants b. 



For a point S of rr TI" consists of the monoid ^' and a cone 

 of order five formed by straight Hues b. Hence the siruiular binecnnts 

 b form a congruence (6, 10). 



Let us now consider tiie straight lines which intersect the figure 

 (o', <>') tlirice, consequently form together a figure of order 42. Any 

 point of (f bears 10 chords of p' ; in the plane o of that conic 

 there ore 6 of them, viz. the straight lines connecting the 6 inter- 

 sections of 0^ and (>' with the point R, which 9' has moreover in 

 common with o. The chords of {t' meeting o^ consequently form a 

 i)i''. The chords of (f meeting 9', form the plane pencil {R, 0). 

 Consequently tlie trisecants of q^ form a v'\ 



In connection with this we easily find now that the .siui/ulur 

 triseca?it.s form a congruence (5,10) possessing 15 singular poirits F 

 of order four. 



The surface -i' has now a triple conic, o\ and 15 triple points 

 F. In a plane '/ [^'J determines again a septuple involution with 

 two singular points of order three. In connection with this we find 

 for this congruence [9'] the same characteristic number as for the 

 [9'J treated in § 6. 



8. Passing on to congruences of twisted curves (>", we suppo,se 

 ill the first place, that the surfaces of [*'] have three non-inter- 

 secting straight lines q, q', q" in common. They are then singular 

 qunch'isecants of the congruence [y"]; consequently the curves </ (genus 

 one) pass through six fundamental points F. 



The curves (, " intersecting q in S form again a monoid ^'. They 

 are represented by a pencil ('/'), having a triple base-point on q and 

 double base points in the intersections of two straight lines t. To 

 the base belong further the images of the points Fand the intersections 

 of two straight lines b* (singular bisecants). 



The sixth straight line of ^\ passing through S, is component 

 part of a degenerate curve ^\ It is the transversal (/ of q' and q" 

 passing through ,S; for through an arbitrary point of that transversal 

 pass go' surfaces </ ' having d in common and therefore intersecting 

 moreover along a cn.rve d^ (of genus one), which has q, q', q" as 

 trisecants. The planes {dq') and {dq") each intersect 2^" along one 

 of the straight lines b*. 



The ruled surface S>- with directrices q, q , q" contains all the 

 straight lines d forming the second system of straight lines. With a 

 monoid 2' J)" has three straight lines (/ in common of which one 



