( 261 ) 



e. Further wo must (ietermine how many screws of F- belong 

 to pencils of screws, whose planes are parallel to d, having with 

 (.], u) a screw in common. Therefore it is necessary to determine 

 the points common to screws of P^ and the screws of {A, a) lying 

 at the same time on a generatrix of C^. Let us first consider the 

 section of plane « with C^; it consists of the line a' and a conic 

 A ". In the second place we must notice that the feet of the per- 

 pendiculars through P on the generatrices of C^ form a conic JB^, 

 lying in a plane having moreover with C^ a right line b in common. 

 Both degenerated cubic curves A'^ + a', B'^ -{- b, intersect in three 

 points. As the lines a' and ö cannot intersect, those points lie in 

 such a manner that a' meets once B'^, h once A~ and A^ once B'^. 

 Now the last point L is the only point of intersection satisfying 

 the above condition; so P^- intersects K-d in one point. 



7. Tlie curves corresponding in — i with the screws, enveloping 

 the parabolas in the planes ti, are determined in the same way. 

 We shall show this briefly. 



a. Out of the pole of n with regard to the zero system through 

 {A, a) two tangents can be drawn to the parabola n~. So the cor- 

 responding curve n{~ is a conic. 



b. One screw in tti^ meets <' and a'; so n-^^ intersects /j in one 

 point. 



c. One screw in n lies at infinity; so /rr meets the plane i\ 

 in one point, not situated on K'^^. 



d. One screw of pencil {A, a) is parallel to the plane ji ; both 

 belong to the same pencil of parallel screws; so jiy' has one point 

 in common with K'^^ • 



e. The line « tt common to « and ?i meets C^ in its point of inter- 

 section with a' and moreover in two points # and 2V. Through J/ pass 

 two tangents of n'^. One of these tangents contains a point lying 

 on the generatrix m' of C^, conjugate to that on which M is situated; 

 the other one is perpendicular to m itself. The latter tangent deter- 

 mines with the screw A M a plane perpendicular to m ; consequently 

 in that plane there is a pencil having a screw in common with 

 (.J,«); this pencil belongs to those, whose plane is parallel to d. 

 The same reasoning can be applied to N. Consequently the corres- 

 ponding conic jr," intersects the conic K'^j, in two points. 



8. So the conies Pj^ and .ij^, having four points in common 

 with the curve K^^. + K-d + /], their number in .Zj would amount 

 to CO •■ ; there being however in .^ only x ^ cones and curves of the 



18 



Proceedings Royal Acad. Amsterdam. Vol. I. 



