47 



and belongs to tlie net ['/''], again contains all the (/' intersecting 

 the singular curve o' in ,S'. In representing 2" on a plane </ the 

 system of those curves passes into a pencil of hyperelliptic onrves 

 (f'\ with a double base-point and 12 sim|)le base-points. The first is 

 the intersection of a singular trisecant /, consequently' of a straight 

 line |)assing thi'ough S, which is moreover twice intersected by all 

 the ^' lying on 2'. 



To the simple base-poinfs belong the central projections of the 7 

 fundamental points. The remaining five are itingular bisecants h, 

 consequently slniight lines, which have a second point in common 

 with any (^>^ passing through S. With tiie trisecant already men- 

 tioned they form the six straight lines of — " passing through ,S'. The 

 straight lines fi, are, as well as the straight lines ƒ passing through 

 the fundamental points, pai'aholic biseatnts. 



9. In the same way as above (§ 4j it is jiroved that an arbitrary 

 point bears eight singular bisceniits q. i.e. straight lines, which are 

 intersected by [</>"] in the pairs of an involution; they belong to 

 the complex of secants of o\ The straight lines q passing through 

 a point S of o^ again form a cubic cone, so that [c/] is a congruence 

 of rays (8, 12). 



The singular trisecants t form a congruence of order one, which 

 has the points F as singular points. The singidar cone Ï belonging to 

 i' is a quadric cone as it has in common with the cone o% which projects 

 an arbitrary o* out of F, six siraight lines FF' and a trisecant t, 

 which is nodal edge of T\ As (he tri.secants of rj' form a ruled 

 surface .'t\ tiie axial ruled suifucc ?l, belonging lo a siraight line 

 a, has in common wiih a {>^ the six points of support of two 

 ti-isecants and the seven nodes /'', consequently is of order four. 

 But in that case [/] is of cltas three, consequently the congruence 

 of the bisecants of a cubic t\ passing through the sexen points F. 



As in § 6 we find that two arbitrary straight lines are intersected 

 by nine curves <j'', that two arbitrary planes are touched by r/ /u<«(//'f(/ 

 curves, that there are tldrtii curves osculating a given plane. 



Here too, the fundamental points are triple on yi', decuple on 0". 



