439 



therefore eiglU fundamental points F. Tlie curves q^ have eight 

 apparent nodes, they are consequently of genus tivo. 



The monoid ^' belonging to a point S of the singular quadri- 

 secant q contains a singular trisecant passing through -S'. From the 

 image of -S'' it appears tliat the remaining four straight lines of ^' 

 passing through S are singular hisecants b*. 



The curves 1/ intersecting the singular conic 0' in a point S* 

 also forui a monoid 2'. These curves are represented by a pencil 

 ((ƒ*), wliich has double base-points in the intersections of the four 

 singular trisecnnts t meeting in S*. The simple base-points are the 

 images of the 8 points i*" and the intersection of a 5//?^?<^/r iwctvni^ 6*. 



The sixth straight line passing through S* must be component 

 part of a compound {>". It must cut 0" and q, belongs thei-efore to 

 the plane pencil in the jilane of (f, which has the point Q of q 

 as vertex. 



Any ray (/ of that plane pencil is component part of a degenerate 

 0', for an arbitrary point of (/ determines a pencil (*') of which 

 all figures pass through d, consequently have a curve (t^ in common 

 besides, which intersects 0' four limes, q three times, consequently 

 possesses four apparent nodes. To the surfaces *' passing through 

 the figure {<f,q,d,ö^) belongs the figure composed of the plane and 

 the hyperboloid 25' passing through q and the points F ; this dege- 

 nerate figure ap|)arently replaces the monoid belonging to Q. The 

 hyperboloid D' is the locus of the curves tl' ; its intersection (f on 

 a contains the points D^{d, ö^) ; all curves d' pass through the 

 four intersections of d^ with a'. 



From the consideration of the surfaces 77" and S^'. which are 

 determined by a point P it follows readily that P bears five singular 

 bisecants b. Four of these straight lines rest on a', the fifth on q. 

 Any point of 0" or of q is the vertex of a cone of ordei' four, formed 

 by straight lines b. The singular hisecants consequently form two 

 congruences; a congruence (1,4) with directrix q, a congruence (4,8) 

 with singular curve 0". 



The singular trisecants t form a congruence possessing eight sin- 

 gular points, F, of order three. The trisecants of a p" form a ruled 

 surface "^t". In connection with this we find that the straight lines 

 t determine a congruence (4,6). 



As [(>"] again intersects a plane (f> along & sextujde involution wWh 

 three singular points of order three, we find for the characteristic 

 numbers connected with it the same values as in § 8. 



10. A net ['/'"J, of which the figures have a cubic 0' (or a 



