( 270 ) 



incident to: 

 SP, 



Sp! 



Sp, 



sp, 



sp< 



Sp, 



Sp 6 



By the method of intersecting and projecting triplets and doublets 

 of consecutive kinds of elements are to be transformed into elements 

 of Sp, or Sp, ; thus are formed e.g. a cf (21504 31 ) of points and 

 planes, with 80640 c/-lines; and a plane c/ (20I6„, 21504.) of 

 points and lines, or reciprocally. 



§ 7. If we represent the system of letters and that of signs of 8 

 resp. by Si and T and if we repeat the combination 



Si S, T T 



and 

 S, Si T -T, 



we obtain systems for 16 belonging to each other, etc., the operation 



allowing of indefinite continuation ; one always arrives at a regular 



Abel substitution group 6V and a suitable anallagmatical system 



for the signs. 



These always furnish in Rrf— i a cf, analogous to that of Kummer 



with the notation : 



Q 



( 2 C2''-D.aP- ] 



arising from an arbitrary starting element by an operation with a 



geometrical Abel group : 



the identity and 2 2 /> — 1 collineations on one hand 



and (2p — 1) . 2 0— ' focal systems mutually in involution with 



(2H-i — 2/' -|- 1) . 2?- 1 polarities on the other hand. 



The quadratic situation of the elements incident to one element 



can always be proved by reduction of the determinant according 



to the example of § 5 '). 



: ) A more extensive treatment also for spaces of other numbers of dimensions 

 will follow in the dissertation to be published: J. A. Barrau, Bijdragen tot de 

 theorie der cff. (Amsterdam 1907). 



