( 508 ) 



The following table slio\vs the results which are obtained by 

 substituting the values of e,k,f,r,p,q for the five different cases. 

 For the sake of completeness those quantities are also included which 

 indicate how many vertices are situated in face and bounding space. 

 We must here notice that, the first cell having two kinds of faces 

 and bounding bodies, we are obliged to take four new quantities, 

 namely the nnmbers of vertices >S'i and 1\ in face and bounding 

 body of one, the numbers of vertices S^ and T', in face and bounding 

 body of the second kind. Here S.^ and 7^ will relate to the truncating 

 bodies with faces of the same kind and >S\ and T.^ refer to the trun- 

 cated bodies, where we must then consider as far as S^ is concerned 

 those faces which the truncated bodies keep in common. We mnst 

 likewise for the second cell, with two kinds of vertices and edges, 

 introduce the four new quantities F^ , P,', Q/, Q^, 



As is evident J\^^ Q^ =. h', whilst T^=Q^' is the number of 

 vertices of the regular polyhedron with q faces. 



In a second part we shall submit each of these five pairs of new 

 cells to a separate investigation. 



Mathematics. — "The analogon of the Cf. of Kummer in seven- 

 dimensional space" . By Dr. J. A. Barrau. (Communicated by 

 Prof. D. J. Korteweg). 



(Communicated in the meeting of December 28, 1907). 



§ 1. In a preceding communication a method was indicated to 

 generate the Cff. in spaces of (2/' — 1) dimensions," which can be 

 regarded as analoga of the Cf. of Kummer ^). 



1) A quotation in Hudson's Kummefs Quartk Surface (p. 187) drew my 

 attention to the fact, that these Cff. have already been obtained by an altogether 

 different method by VV. WmTiNGER (Göttinger Nackrichten, 1889, page 474 ; 

 Monatshefte fur Mathematik mid Physik \, page 113; Mathem. Annalen 40, 

 page 74). In these papers tlie varieties are also investigated, for which the elements 

 of such Cff. are singular in the same way as those of the Cf. (log) for Kummer's 

 quartic surface. 



