1187 



therefore 14 nodes. Of these 5 lie on (>'', fi in the (|uadrnple point 

 Ijing on b^; tiie remaining 3 are represented by a triple point 

 originating from a trisecant restijig on b^. As />„ in each of its points 

 of intersection with q^ meets two trisecants, {t) is consequently a 

 ruled surface of order jive. ^) 



3. A ^* cutting [,^ in ;S forms with it the base of a pencil (0') 

 the surfaces of which touch in S. We shall now consider two pencils 

 (V) and (52) in the net ['/^^], and associate to each surface V the 

 surface 52', by which it is touched in S. The pencils having become 

 projective in consequence, produce a figure of order 6, which is 

 composed of the surface <f»' common to both pencils and a surface 

 2^. On a line / passing through S a correspondence (2,2) is deter- 

 mined by {W^) and (i2) ; one of the coincidences lies in S, because 

 / is touched in S by two' corresponding surfaces. The remaining 

 three are intersections of / with the figure of order 6, mentioned 

 above; the latter has consequently a triple point in S, from which 

 it ensues that *S is a node of ^'. The curves q\ which meet q^ in 

 S, form therefore a cubic surface passing through q\ which possesses 

 a node in aS; q^ is therefore a singular curve of order three for the 

 congruence [()^J. 



Through S pass 6 lines of ^^ ; to them belong the two trisecants 

 t, meeting in S\ the remaining four are singular bisecants of the 

 congruence. Such a line p is cut by oo^ curves q^ in two points, 

 of which one coincides with S, (singular bisecant of the first kind). 



The 00* rays h, which may be drawn through the cardinal points 

 i/i, H^ possess the same property. 



4. x4n arbitrary line /■ passing through a point P is cut by 07ie 

 Q* in a pair of points R,R' , the locus of those points is a surface 

 n of order 5 with triple point P. 



If P lies on c)\ then W consists of the surface ^""^ belonging to 

 S^P and a quadratic cone, of which the generatrices eive singular 

 bisecants q. Each line q is bisecant of x* curves of the [q*]. 



If, on the other hand, q is bisecant of a (/ and at the same time 

 secant of </% then the cubic surface passing through (>% r>^ and q 

 belongs to [*''] ; consequently q is cut by the surfaces of this net 

 in the pairs of points of an /-, is tlierefore bisecant of oo^ curves (>^ 

 (singular bisecant of the s-econd kind). 



The lines (/ meeting in a point P, belong to the common gene- 



^) Other properties of the ij^ of genus 1 are to be found in my communication 

 "On twisted quintics of genus unity" (volume II, p. 374 of these Proceedings). 



