434 



F. The five straight lines t lie, iilve q, entirely on JS'' ; thej art 

 apparently singular trisecant.'i. Each i^traiglit line t is intersected by 

 the curves (>' in S, and in a pair of an involution. 



Two monoids have the straight line q and a y' in common. Conse- 

 quently in general a curve of the congruence is determined by two 

 of its intersections with q. The sets of four points of support form 

 therefore an involution of the second rank. So there are on q three 

 pairs of points, which each bear co' curves (>" ; in other words, the 

 net contains three hmoclal surfaces, of which the two nodes lie on 

 q. We may further observe that q is stationary tangent of six 9° 

 and bitangent of four q". 



Each trisecant t of a <>"' is singular (§ 1); the straight lines t 

 form a congruence of order 8, with 20 singular points F. The 

 cone 5", with vertex F, which projects a q", has 8 double edges 

 and contains 19 straight lines FF' ; from this it ensues that the 

 straigiit lines passing through F form a cone -S so that F appears 

 to be a singular point of order fve. 



In any plane passing through q lie 6 choi'ds of a (/, through any 

 point of q pass 8 chords. The straight lines resting on q and twice 

 on p% form therefore a ruled surface of order 14. As they belong 

 to the trisecants of the figvire {q, q"), the ti'isecants of q* must form 

 a ruled surface of order 28. 



Let us now again consider the axial ruled surface ^i, formed by 

 the trisecants resting on the straight line a. With a definite q' 3i 

 lias the 20 quintuple points F and 28 triplets of points of support 

 in common; from this it ensues that 1\ is of order 23. The singular 

 trisecants consequenthj form a congruence (8,15). 



5. The surface II is here of order 9; it contains q and has 20 

 nodes F (§ 1). Its section with the cone, it\ which projects the q^ 

 laid through P, consists of that curve, 8 singular trisecants (which 

 are nodal lines for both surfaces) the 20 singular bisecants Pi^ (each 

 with a |»araboiic involution of points of support) and moreover three 

 straight linos h, which apparently must also be singular bisecants. 

 These straight lines we find moreover by paying attention to the 

 intersections of -S' with q\ to them belong the four points, which 

 q has in common with the <,>" projected by that cone. If S is one 

 of the remaining three intersections the straight line PS belongs to 

 a '/'■' of the net, is consequently cut by that net in the pairs of an 

 involution and is therefore bisecant of oo^ curves o^ 



For a point .S' of q TI consists of the monoid ^^ and a cone of 

 order sit: For, a bisecant of a o" not laid through S is at the same 



