736 



apparently be a part of a degenerate 7', the r/' form a quadric. 

 This is cut by 0^ in 4 points ; among them are tiie 3 common points 

 of 0=* and G- ; through the fourth intersecting point passes a (f\ 

 which has four points in common with the figure (0' + 0'). 



From this we conchide that one conic «' can be drawn through H^ 

 and i/3. As each «' is coupled with a /5' (which passes in that case 

 through Hs and H,), [(>'] contains three jigures («' + ^'). 



6. Through a point S of the singular curve (/ pass 00^ curves ^\ 

 They cut the plane (f in the points H. To this system of f/ belongs, 

 however, also the figure consisting of the trisecant t passing through 

 S and a y'' lying in fp. From this ensues that the locus of the q^ 

 meeting in S, is a cubic surface 2\ passing through o' and the 

 points H, and consequently belongs to the net [<!>']. 



An arbitrary line passing through S, is a bisecant of 07ie q\ 

 and so intersects :S\ apart from 5 in one point. Consequently 2' 

 has a double point in S. Through 5 pass 6 straight lines of 2\ 

 one of them is of course the t mentioned before ; each of the remaining 

 5 is a bisecant p of oc' curves q\ so a singular bisecant 



All the q' intersecting p twice pass through S; so they determine 

 on ;; a parabolic involution, of which all pairs have the point Sin 

 common ; we shall call p a .singular bisecant of the first species. 



Through each point of q' pass therefore five singular bisecants of 

 the first species. 



Any line h passing through a cardinal point H is as well a 

 singular bisecant of the first species. 



The monoids -S' having two points of q' as double points, inter- 

 sect apart from q' in a q\ Through any two points S passes 

 therefore only one curve of the congruence. 



7. Let q be a bisecant of a (/, and at the same time a secant 

 of i)\ Tiie surface <f>' passing through q' and q' and a point of q 

 contains q, and belongs to the net [0'J. Consequently all <P' passing 

 through a point Q of q will cut this straight line moreover in a 

 second point Q' . Consequently q is a bisecant of oo' curves q\ and 

 the pairs of the intersections Q,Q' form an involution. We call q a 

 singular bisecant of the second species. 



In order to find the number of lines q that pass through a point 

 F, we consider the cubic cone k\ which out of F projects the q* 

 containing F, and the cone t which has F as vertex and q' as 

 curve of direction. To the 15 common generatrices belong the lines 

 drawn to the eight intersecting points of (/ and q\ The remaining 



