115 



Through onr transformation a congruence G passes into a system 

 G' that consists tirst of all the rays of y, eacli counted j^-fold, if 

 /? represents the class of G. For each line /' of y is associated to 

 the ii lines / of (t lying in the plane CI'. Further, if a is the order 

 of G, there are a rays of G which pass through C and are trans- 

 formed into as many sheaves of rays of G' . Another congruence 

 with the order a' and the class /j' has ««' -\- ,?/ rays in common 

 with G' . From this follows the well known theorem of Halphkn: 



Turn line congruences («, ^) and {a , ^') have tut' -\- ^?,i' Ihies in 

 common. 



§ 4. Before we give the general solution of our problem in an 

 Rn, we consider the special case that we have to do with the oo' 

 straight lines of an R^. By the aid of a point C and a space r in 

 /s*4, we arrive at the <x^ homographic representations that are each 

 characterised by a value of the anharmonic ratio ). :^ {CC' FP') if 

 C' is the point of intersection of CP en r. Again we consider 

 especially the representation belonging to / = 0. 



If we take a system S, of oc^ rays, this is transformed by the 

 latter representation into a ruled surface a^S'/ of the order q lying 

 in r, if Q represents the number of straight lines of S-^ cutting a 

 plane. If we consider further a system S ,, of oc° straight lines 

 of which an arbitrary plane pencil contains v., the rays that S.^ has 

 in common with F form a complex of the order x, so that *S'. con- 

 tains QK rays of *S'/. 



A system S^ of the order q has qv. rays in common with a system 

 «Sj of the order x. 



A system S, of oc* rays is represented on S^\ a congruence 

 («, /J) of r, if a is the number of rays of S, cutting an arbitrary 

 straight line (through C), and ji the number of straight lines of 

 S, lying in an arbitrary space (through C). A system S^ has a 

 congruence {q , V') in common with F, if fp and if' represent the 

 numbers of straight lines of S^ resp. belonging to a (three-dimen- 

 sional) sheaf of rays or lying in a plane. S^ has «7+^^*1' ^'^J-^ ^" 

 common with *S,'. 



A system S, («, li) has a (f + ^ l^' rays in common mith a system 



A system S, is transformed through our representation into a 

 system S/ consisting first of a complex of the order r lying in F^ 

 if r is the number of rays of *S, lying in a (three-dimensional) special 

 linear complex. Further, if S^ contains n rays through a given point, 

 to each of the /t straight lines / through C there are associated the 



8 

 Proceedings Royal Acad. Amsterdam. Vol. XXV. 



