524 



to tlie singular |)oiii( of intersection of llie dioseii straight line with v,, 

 and along a straigiit line cutting F,,,, which lies therefore in a plane 

 V/, and corresponds to a plane pencil of C the vertex of which lies 

 on (I. In the same way it is evident that tlie pencils of 6' in planes 

 through a, are represented on tlie straight lines of R, which cut y>i, 

 and that the plane pencils containing a are associated to the straight 

 lines through the point of intersection B^ of />, and r;, (for a plane 

 through SB^ cuts the hypercone /("outside 55, along a generatrix). 

 To a star of C, the vertex of which lies consequently on a, there 

 corresponds on K a plane V^j that cuts F",., along a straight line 

 and the projection of which on R^ passes accordingly through v^. 

 Hence a plane throtigh c, is associated to a star of C in R,. It is 

 easily seen that also the reverse holds good and that the fields of 

 C, the planes of which pass through n, are represented on the 

 planes of R, through /;,. 



^ 4. A congruence r{«, j3) of the order a and the class (i has in 

 common with C a scroll 52 of the order « -|- i^ that has a as an 

 a-fold directrix. If further F has the rank r, there are r plane 

 pencils through a containing two straight lines of 12. 



The curve y in R, on which £2 is represented, cuts /), in the a 

 points that are associated to the « generatrices of i2 which pass 

 through A, and i\ in the (^ points that correspond to the tS gene- 

 ratrices of a in the plane [a, s). A plane through y>,. cuts y outside 

 p, in the ff image points of the straight lines which the corresponding 

 field of C has in common with 11, and it appears in the same way 

 that a plane through i;, intersects the curve y outside v, in « points. 

 Hence the order of y is n -\- ^. 



To the )■ plane pencils through a that contain two straight lines 

 of a, there correspond in R, as many bisecants of the curve y 

 through Bj. Besides the lines /), and v^ which cut y resp. n and jJ 

 times pass through fi,. The number of apparent double points of y 

 is accordingly : 



,. + i„(«_l)-j-i^(^_1). 



We shall just mention an application that Sturm gives on p. 271 

 of his book quoted in ^ 1. The order of the focal surface of the 

 congruence F is equal to the number of sheaves with vertices on a 

 containing two straight lines of r, hence also of i^, that are infinitely 

 near to each other. These are represented on the planes through v, 

 touching y outside r,. Hence the order of the focal surface of P 

 is equal to the number of points of intersection outside y ofi'i with 



