436 



are nodal lines of the surface /7" determined by /' and nodal edges 

 of the cone SV, which projects the curve p' laid through P. These 

 two surfaces have besides that q^ and the six straight lines t, nfiore- 

 over the 13 parabolic bisecants PF and four .s-iniju/ar binecants b 

 in common. The straight lines b are found back if '&' is brought to 

 intersection with q and q' ; on eacii of the singular quadrisecants 

 rest therefore two sti'aight lines h. 



Each point of q or q' bears a cone of order 5 (completing ^' 

 into a surface /7") formed by singular bi.secants. The .iingulnr 

 bisecants consequent!}- forin tmo couqruences (2,5). 



The locus of the trisecants of the figure [q, q' , 9') consists of four 

 ruled surfaces, together foiming a figure of order 42. The straight 

 lines intersecting q, q' and p' apparently form a ruled surface .'v°. 

 The bisecants of q'' resting on q or on q' lie on a i" with quin- 

 tuple straight line. Consequently the trisecants t of q' will form a 

 .^v" (with sextuple curve o'). 



According to the method followed above (§§ 1, 4) we find now 

 that the singular trisecants t form a, congruence (6, 10), possessing 

 in the 13 fundamental points F singular points of order six. 



On two arbitrary straight lines nine curves of the congruence rest 

 now too. The surface A^ has two triple straight lines, q and q' . In 

 a plane (( arises a septuple involution with a curve of coincidence 

 q' possessing Uoo nodes, where the involution hiii singular points of 

 order three. The curves y' touching (f , form a *" with 14-foId 

 straight lines q and q' . 



There are 70 curves q' osculating a plane 7, and 196 curves 

 touching tiuo given planes. 



7. If the surfaces 'I'' of a net have a conic 0^ in common they 

 determine a bilineai- congi'uence of tuiLfted curves q'' , of genus five. 

 Every q^ rests in six points on the singular conic <?'. The congruence 

 possesses consequently 15 fundamental points F. 



In representing the monoid ^', containing the curves n\ which 

 intersect 0' in a point S, the system of those curves passes into a 

 pencil of curves q'\ The}' have five nodes in the intersections of the 

 singular trisecants t, meeting in »S; the remaining base-points are 

 the images of the 15 points F, and the intersection of the straight 

 line b* of ^', which forms with the 5 straight lines t the set 

 of six straight lines passing through jS. Apparently b* is here also 

 a singular bisecant (parabolic bisecant). 



The surface /7* belonging to a point P and the corresponding 

 cone .f have in common a o', five singular trisecants (■ (nodal lines 



