523 



plane pencils of (his complex containing a, correspond resp. (o the 

 vertex T of K and the generatrices of this hypercone. A straight 

 line of K in a plane -Vj, represents a plane pencil of 6' the vertex 

 of which lies on a, and a plane pencil of C of which the plane 

 passes Ihrongh n, is associated to a straight line of a plane V,,. 



Now we assnnie on K a point 6' and in the fonr-diinensional 

 space R, a tliree-dimensional space /?,. The representation mentioned 

 in ^ 1 arises, when we associate to each straight line / the projec- 

 tion L of P out of S on Hf, if /-* is the point on A" corresponding 

 to /. ') 



^ 3. The straight line *■ of C of which S is the image point on 

 K, is a singular straight Hue of the second order for the correspon- 

 dence (/ — L). For all the points of the plane q that the three- 

 dimensional tangent space R of A' at S, lying in /?,, has in common 

 with ft,, are associated in R, to this straight line. 



In R there lie the two planes T'^^ and Y ^ of K of whi('h the 

 intersection is the generatrix 6, of K through aS. To these planes 

 there correspond resp. the star of C, that has its vertex in the point 

 of intersection A of s and a, and the field of 6' consisting of the 

 rays of the plane a that pas.ses Ihrongh .v and a. Tlie star A and 

 the tield « have in common the plane pencil [A, a) to which the 

 straight line è, on K is associated. 



The planes V^ and V] cut q resp. along the straight lines p, 

 and v^ each consisting of points that are singular for the corre- 

 spondence (/ — L). For to each point L of p, there corresponds on 

 K a straight line of V^ through 5, hence in C a plane pencil 

 containing ,v, with vertex in A. Likewise a plane pencil in « con- 

 taining .V, is associated to each point L of i?,. The point of inter- 

 section B^ of /), and v, is the image point L for all rays / of the 

 plane pencil (^1, ((]. In this way the go' straight lines of the star A 

 correspond to the oo' points of p,, the oo' rays of the field a to 

 the oo' points of i\. 



To a plane pencil with vertex on a a stiaight line on K in a 

 plane Vfj, which accordingly intersects T^„,, is associated ; consequently 

 to such a plane pencil in R, corresponds a straight line cutting i\. 

 Inversely the plane through *S and a straight line of R, cutting v,, 

 intersects the hypercone K along a straight line in V,-^ through S, 

 to which there corresponds the plane pencil of C that is associated 



1) The method applied here, has been indicated for the rays of space by 

 Felix Klein. Gf. Mathem. Annalen, Bd. 5, p. 257. 



