737 



7 are bisccants of <^)' iiitersecting o*, therefore lines r/. Coiise(|iieiilly 

 the lines q form a coiu/nience of order seven. 



We can also arrive at this result in another way. A straight line 

 passing through P is generally speaking, a bisecant of one </ ; we 

 call /?, R' its intersections with r/ and consider the surface .t, which 

 is the locus uf the pairs R, R' . Oji any generatrix of the cone /;' 

 one of those points lies in P, hence .t has in P a triple [)oint with 

 k^ as tangent cone; .t is consequently a surface of order 5. It passes 

 through q\ and has nodes in the four cardinal points. For an arbi- 

 trary ()' has in common with .-r the intersections with the bisecants 

 which it sends through P, and in 8 poinis of ()\ so twice in each 

 point H. 



Now ji^ and k^ have in common the q' which ))asses through P, 

 further they can, by reason of the definition of rr, only have lines 

 in common which contain oo^ pairs R, R' each. Therefore eleven 

 singular bisecants pass through P. To these the four straight lines 

 hk^ PHi, belong; for through aiiy point of PHt passes •aq\ which 

 meets this straight line again in the cardinal point Hi^, so that PH^ is 

 a singular straight line of the first species (which, however, does 

 not rest on ^% and consequently may not be interchanged with a 

 straight line /;). The remaining 7 singular bisecants passing through 

 P are therefore straight lines q. 



¥ov a point S of q^ the surface n^ degenerates, and consists of 

 the monoid 0^ with node S and a quadratic cone, formed by the 

 straight lines q, which intersect q^ in S. 



In an arbitrary plane lie five points of 9^ consequently 10 straight 

 lines q ; they belong therefore to a congruence of rags of class ten. 



The singular bisecants of the second species form a congruence 

 (7, 10), lüldch lias q^ as a singular curve. 



The section of n^ with a plane passing through P is a curve 

 with a triple point, consequently of class 14, of its tangents 8 pass 

 through P. Therefore the tangents of the curves q^ form a complex 

 of order eight. 



8. The 9^ which [intersect a given line /, form a surface A, of 

 which we intend to determine the order x. Any monoid 0' contains 

 three q\ which intersect /, and rest in the vertex aS on ^* ; conse- 

 quently ^^ is a ti'iple curve of A. 



The surfaces J, A' belonging to two lines /, /' have, besides the 

 threefold curve o'' only the x curves o'' in common, i-es(ing on / 

 and /'. So we have the i-elalion x'-^ z= -ix -\- 3'-.5, hence x = 9. 



On J" lies one trisecant t\ for the curve y'', which intersects /, 



