440 



degeneration of it) in coninion, determines a congruence of' twisted 

 curves o", of genus three, intersecting the singular curve o' eight 

 times '). The congruence possesses accordingly ten ftindamentnl 

 poinlx F. 



As q' has seven apparent nodes, <?" is intersected in each of its 

 points -S' by three singidar trisecaiils t. Using the image of the 

 monoid 2£' belonging to S, we find that the remaining three straight 

 lines of -S' meeting in S are shujuhir hisccatits b*. 



Through an arbitrary point P pass seven singular bisecants h. 

 Each point of a' is vertex of a cone of order four formed by 

 straight lines h. From this it ensues that the singular bisec/ints form 

 a congruence (7,12). 



The singular trisecants form n congruence (3,6) with ten singular 

 points, F, of order three. 



The characteristic numbers, connected with the surface A\ have 

 tlie same values as witli the congruence [. "] already dealt with. 



11. The surfaces of a net [*"], which have a plane curve ö' in 

 common, determine a congruence of tujisted curves q° oï g&nws four , 

 which possesses twelve fundamental points F. 



As (i" has now six apparent nodes, each point S of the singular 

 curve ö' bears two singular trisecants. 



To the surfaces '/'' passing through a figure (o', y") belongs a 

 figure consisting of the plane a of ö' and a hyperboloid ; q" is there- 

 fore the complete section of a hyperboloid with a cubic surface. 

 In connection with this the curves (<" intersecting o' in a point S, 

 form a lu/})erholoid 2-, passing through the points F. The surfaces 

 2' form a pencil") with base-curve [V, which determines in a a 

 pencil of conies q\ Any point of the plane a bears therefore a 

 figure consisting of a q^ and the curve [i". 



The section of o with the surface J belonging to the straight 

 line / consists of the nodal curve o» and the conies q' intersected 

 by / ; hence yJ is of order eight. 



Two surfaces yf have the singular curve 0', the curve ,?', and 

 eight curves q^ in common. 



1) If r'^ is replaced by a conic -2 and a siraight line s intersecting it, we under- 

 stand easily that any 'f has five points in common with a", and three points 

 with s. 



2) The net ('I''''] may be represented by the equation 



a,' + f. (a.' + V -^J ^ ft W + <-■/ ■fj = 0. 

 Through a point of x^ = passes the pencil for which 1 + ^ + /.t = 0. It 

 consists therefore of the plane ^4 = 0, and the pencil xihj.- — o-) — c,- = 0, 

 with base-curve bi° = 0, Cx- 0. 



