46 



of '31', a plane passing tliiough a contains three straight lines nnore 

 hence the singular trisecants form a coiignience (2, 3). 



6. A straight line / intersects three curves q'' of a monoid 2' ; 

 consequently a* is a tripk curve on the surface A foi'med by the 

 y', intersecting /. As two surfaces A^, outside a*, have but x cur- 

 ves p' in common, we have x'' = 5x i- 36, hence x := 9. An arbi- 

 trary curve (»' intersects A' on ö"" in 10 X 3 points, consequently 

 fifteen times in Fk; so A^ has five triple paints Fk. On A' lie (^ 3) 

 siv straight lines and six elliptic curves 9"; the (>', for which / is a 

 chord, is a nodal curve. 



In a plane K passing through /, the congruence [y'] determines 

 a quintuple-involution possessing four singular points 5 of order three. 

 It transforms a straight line / into a curve // with four triple 

 points, and has a curve of coincidence of order six, y', with four 

 nodes S. With an arbitrary surface A^ the curve y', has outside 

 &, 9 X 6 — 4 X 3 X 2 = 30 points in common. The curves y\ 

 touching a plane '/, consequently form a surface </>'" ; on it o* is a 

 decu2>le curve {2' intersects y', outside Sh, in 3 X 6 — 4x2 points) 

 while Fk are decuple pjinis (an arbitrary ^z' intersects *^°, out- 

 side a*, in 5 X 30 — 10 < 10 points). 



<P'° has in common with </ another curve f/,'", possessing four 

 sextuple points S ; it touches y" in 30 points; '/ is therefore 

 osculated by tliirl-f/ curves y\ 



Two surfaces */»'" ha\e, outside c/, 100 curves y' in common , 

 two planes are therefore touched by 100 curves (>'. 



7. When all the surfaces <I>' of a net have an elliptic twisted 

 curve Ü* in common, the variable base-curves q' of the pencils 

 comprised in the net form a bilinear congruence of hyperelUptic 

 curves. Each q^ rests iu eiyht points on 0' and has with an arbitrary 

 surface */>' moreover seven fundamental points Fk in common. As 

 the net is completely determined by 0^ and five points F, the points 

 F cannot be taken arbitrarily. 



The singular curve a^ may be replaced by the figuie composed 

 of a curve 0' and one of its chords, or by two conies having two 

 points in common. ') 



8. The monoid 2', which has the singular point S as node 



1) In both cases a *', containing 12 points of the base-figure, will contain it 

 entirely. This eluciilates the fact that *' needs only to be laid through 12 points 

 of the elliptic s* in order to contain it entirely. 



