731 



shall indicate by ^," + '. Tlirongli a point P of /• i)as,ses only one 

 y", namely the curve lying in the plane {pr). 



The surface -S'/ + > belonging to an axis p has a node in P; for 

 a line / passing through P cuts the y" of the i)lane {pi) in (?z— 1) 

 points lying outside P. 



If r is made to i-evolve in a plane (p around a point then 

 -2"/' + ! describes a pencil. In order to determine the surface -S" which 

 passes through an arbitrary point P, we have only to find the ray 

 ?', which cuts the axis p of P. The base of this pencil consists of 

 the curve y" lying in (r and a twisted curve r/)'''+"+i, which cuts 

 y" in n{n-{-l) points. 



Any point P of this curve lies on oo^ curves /" ; its axes /; must 

 meet all the rays of the pencil {0, <p), consequently pass through 0. 



To the net of rays of the straight lines r, lying in (f, corresponds 

 a net of surfaces -^/ + '. Through two arbitrary points P, P' passes 

 the surface belonging to the straight line r, which cuts the axis p,^9'. 



7. Let us now consider the surfaces of this net belonging to 

 three straight lines, r, r' , r" of y, Avhich do not pass through one 

 point. The curve (>""+"+•, which two of these surfaces have in com- 

 mon, cuts the third surface in {n-{-l) {n--\-n-\-l) points. To these 

 points belong n (n-|-l) points of the y" lying in (p. 



Let H he, one of the remaining {n-\-l){n''-\-n-\-l) — {ii-\-\.)n = 

 {n-\-l){n^ -\-l) intersections. Through H pass the curves y" lying in 

 the three planes which connect // with r,r',r\; these planes do not 

 belong to a pencil, consequently H bears co^ curves y" and is therefore 

 a cardinal point (fundamental point) of the complex |y"|. Any straight 

 line through H is apparently an axis and determines by means of 

 its intersection with 7-, a pencil (2'"+^), consequently a curve rj"Hn+i. 



The complex |y"| has {n-\-'i){ii^ -\-l) cardinal points; they are at 

 the same time cardinal points of the complex of rays \p\ and of the 

 complex of curves |9"^+"+'|. 



The cardinal points are apparently base points of the net 1-2" +'1 

 belonging to the plane (p, or, more exactly expressed, ofall the nets 

 which are indicated by the planes <f in space. 



8. Let us now consider the curves of {y"| which send their planes 

 through an arbitrar} point F. Through a point P passes the y" of 

 the plane {Fp); through a straight line r passes the plane {Fr) and 

 this plane contains one y". So we have set apart out of the complex 

 a bilinear congruence [y"] which has F as pole. Its polar rays are 

 the axes p of the points P of the singular curve <>'•'+"+' ; they 



