435 



time bisecant of a <,/" belonging to ^', consequently a singular 

 bisecant b. The locus of the straight lines b drawn through S 

 forms therefore with S' the surface 77. An arbitrary plane 

 consequently contains six straight lines b, and the singular bisecants 

 form a congruence (3,6). The three rays b out of a point P lie in 

 the plane {Pq); the six rays in a plane .-t meet in the point {tq). 



The curves q^ meeting a straight line / form again a surface vi'. 

 On it q is triple straight line, for each monoid ^' contains three 

 curves resting on / and meeting in <S'. Two surfaces A have besides 

 q the 9 curves q^ in common, resting in the two straight lines /. 

 The points F appear this time again to be triple. 



In a plane <i the congruence [(>*] determines an octuple involution, 

 which possesses a singular point of order three (intersection 6^ o{ q). 

 The curve of coincidence r/'" (^ 3) has now a node S. 



As A^ and (/'' have now, outside S, 48 points in common, the 

 curves y", which touch the plane ff, form a surface </>". On it q 

 is a J 6-fold straight line; for the monoid that has an arbitrary point 

 of q as vertex, cuts ff', outside q, in 16 points. The plane rp cuts 

 «f>" moreover along a curve r/->" with 12-fold point S. The curves 

 <f" and '/" have 24 intersections in S; as their remaining common 

 points must coincide in pairs, there are 96 curves q" osculating </ . 



The curve f/" has with the surface V" (belonging to a plane 

 If') 6 X -48 — 2 X 16 =: 256 points in common outside (/ ; there are 

 consequently 256 curves i>'',ujhich touch lino given planes. 



6. If the surfaces '/'' of a net have two non-intersecting straight 

 lines q and <[' in common, they determine a bilinear congiiience of 

 twisted curves q\ of genus four, for whicii q and q' are singular 

 quadrisecants ; it has 13 fundamental points F. The curves o" have 

 11 apparent nodes. 



If the monoid 2' containing the curves q' , which intersect j in »S, 

 is represented in the usual way, the system of those curves passes 

 into a pencil of curves cf', which has a triple base-point on q and 

 double base-points in the intersections of three other straight lines t 

 of the monoid ; the remaining .base-points are the images of the 

 points P^, and the intersections of the two straight lines b*, which 

 may moreover be drawn on JS'' through S (and which apparently 

 rest on q'). The straight lines 6* are singular bisecants (parabolic 

 bisecants), the straight lines / are .lingular trisecants. The locus of 

 the singular bisecants b* is a ruled surface of order four with nodal 

 lines q and q'. 



Through an arbitrary point P pass six singular trisecants; they 



