1638 
sponds in the representation on a plane to the curve of Jacopr of the 
net of curves c,; the latter is of order 6 with u nodes in the u 
base-points of the net (the genus is consequently 10—,) and is more- 
over intersected by an arbitrary c, in 18—2u or (as u =9— n) in 
2n points. The intersection of two surfaces of S, has therefore 2n 
free intersections with ®7; a straight line in >’ touches the limiting 
surface P'y in as many points, so that the limiting surface is of 
order 2n. 
The genus of the curve of Jacopi of the c,-net corresponds to that 
of the intersection of a surface of |S, with Dj, and this curve cor- 
responds again point for point to the plane intersection of dj. The 
latter too has therefore the genus 
10 —u=n thi. 
The rank of ®'7 too is easy to determine. A surface of S, with 
a node D corresponds in =" to a tangent plane at 47 in the corre- 
sponding point D’. The curve formed by the points of contact of 
the tangent planes drawn at #'7 out of a point P’, corresponds to 
the loci of the nodes of the net of surfaces of S,, determined by 
the points P,,...,P, corresponding to P’, i.e. to the curve of 
Jacos 9 of this net of surfaces. 
g is intersected by a surface d> of S, in as many points as there 
are curves with a node in a pencil of the c,-net, consequently in 
12, ie. the rank of Dy is equal to 12. 
4. From what has been mentioned above it ensues that the plane 
intersection of @'7 is of order 2n, of genus n-+1 and of class 12, 
by which the remaining numbers of PLicker are known; it appears 
inter alia that this intersection must possess 30—n bitangents. 
A bitangent of ® happens to be the representation of the inter- 
section of two surfaces of S,, which possesses two nodes, in other 
words is degenerate. 
In order to determine the class and the order of the congruence 
of bitangents of '7 we investigate how often the curve of inter- 
section of a surface ® with the remaining surfaces of S, is degenerate, 
and how often such a degeneration will pass through one or more 
of n coupled points P,,..., P, (which determine a net out of Sy): 
5. From the representation of ® in a plane it appears that the 
degenerate curves of intersection are represented in different ways, 
viz. by: 
a. a c‚ possessing a node in ove of the u base-points of the c‚-system. 
Number u = 9—n. 
