521 



eighteeiifold straight lines of i2", ^-'^ has eighteenfold points in the 



points of intei'section of ^p with these directrices and each of these 



18 V 17 

 points contains— — =zl53 out of tiie nnniber of double points. 



1 X ^ 



The points of intersection of (p with the 70 double straight lines of 

 52'% i.e. the transversals d of fonr of the directrices, each of which 

 forms a pair of two degenerate conies of 52" together with the trans- 

 versals of (/ and the three remaining directrices, are double points 

 of 52", just as the 112 points of intersection of'/ with the 7 ■ 8 = 56 

 double conies of k" the planes of which pass through one of the 

 directi-ices. There remain accordingly 2715 double [)oints. 



Tlie surface formed h>/ the conies intersecting seven given straight 

 lines, has there/ore also a double curve of the order 2715. 



The intersection of Si" with a plane y tlirougii a,, consists besides 

 of (ij of a curve of tiie order 74, L'\ If we associate to a point 

 of k"* the conic of 52'^ passing through it, there arises a (l,l)-corre- 

 spondence between the conies of 52" and the points of F*. The 

 genus of k^* is accordingly 35 and the number of double points 

 73 X 36—35 = 2593. The points of intersection of ff and the six 

 directrices of ii" outside a, are eighteenfold points of /;", and each 

 of them is therefore contained 153 times in the said number of 

 double points. Also each intersection of fp with one of the thirty 

 double straight lines of 52" that do not cut (/,, and each point of 

 intersection outside a, of <p and one of the 48 double conies of 

 52" that cut a, only once, is a double point of k''\ There remain 

 therefore 1597 double points. Hence: 



The double curve of 11^° ctits each of the directrices of this sur- 

 face ill 1118 points. These are points through which there pass two 

 conies of our system that have there a common tangent plane through 

 the directri.x. 



Analogously it is possible to examine the double curves of the 

 surface 52'" formed by the conies intersecting six given straight 

 of the touching a given plane, and of the surface 52'" consisting 

 lines and conies intersecting five given straight lines and touching 

 two given planes. 



