433 



(3,3), when eaoli two surfaces, intersecting on a straight line /, are 

 considered as homologous. It is found then that the locus of the 

 curves {>\ resting on /, is a surface / of order nine, which lias tiie 

 fundamental points as ti-iple points.') 



Two straight lines are therefore intersected by nitw curves o'. 



The curves q^, intersecting a straight line ƒ, form therefore a 

 surface of order six, with nodes F. 



A plane passing Ihrongh /, intersects ^' moreover along a curve /.*, 

 the latter has in common with / the points of support of the curve (>', 

 which has / as chord {nodal curve of J"). In each of the ren)aining 

 points of intersection of / with /.** the plane is touched by curves o\ 

 The points, in which a plane </ is touched by curves of the con- 

 gruence lie therefore in a curve 7^ The latter is the ciu-ve of coin- 

 cidence of the nontiple Involution which [n''~\ determines in </ -, this 

 involution possesses no exceptional points ; each point belongs to 

 one group. 



As each point of intersection of '/' with a surface J' belonging 

 to an arbitrary straight line / indicates a cur\e 9', which touches '/ 

 and rests on /, the curves y' touching (f form a surface *'\ This 

 surface has moreover in common with (f a curve ((" ; as the latter 

 can only touch the curve <i\ theie are 126 curves (/ , osculatiiu/ a 

 given plane. 



If the curve (/" is brought in connection with the surface •/■"'', 

 belonging to a plane <I', then it appears that two arliitrary planen 

 are touched by 324 curves (>'. 



X.' 



4. If the surfaces of a net ['/'"] have the straight line q in 



common, the base-curves {>" of the pencils form a bilinear congruence 

 with singular quadrisecant q. As a 1/ is cut by a surface 'l>, outside 

 q, in 20 points, the congruence has 20 fundamental points F. 



Each point -S' of (/ is singular; the oc' curves 9** passing through 

 5 form a monoid -2'' belonging to the net, with nodal point in S. 

 In order to confirm this more specially we consider two pencils of 

 the net, and make them projective by associating any two surfaces, 

 which touch in S. The figure which they produce then consists of 

 the common figure of the pencils and the monoid .2'^ 



If ^■'' is represented by central projection out of .S' on a plane 

 7-1, the images of the curves o" form a pencil of curves r/". The 

 image of the quadrisecant is triple base-point, the images of the five 

 trisecants t, which a (>* sends moreover through *S, are double base- 

 points. The remaining 20 base-points are the images of the points 



') A p' which does not intersect / will cut ■." only in the 27 points F. 



