( 499 ) 

 BN _ 1 



So the llicoreiii liolds : 



"If we describe iji tlie spaces Sp,> bearing the bounding'simplexes 



S^,j^\ ( — 1/2 J of a regular simplex ;S,, f — - — l/2 j simplexes 



S^j^\ ( - l/2 J concentric and oppositely orientated to the original ones 



f n \ 2« -f- 1 _ 



we find the (/> + !) , , vertices of a -^ (J/,,)." 



VP + ly 2?t 



Fov odd n = 2n' -\- 1 we have in particular : 



"If we describe in the spaces Spn' , bearing the bounding simplexes 



f2n' -f 1 \ , o /'^n' + 1 \ 

 Sn'+\ I ^ — 1/2 of a regular simplex b-2n'+\ I — ^ V^ I sim- 

 plexes >S„'+i ( 77 1/2 ] concentric and oppositely orientated to the ori- 



/2n' + n 

 ginal ones we find the (n'-f-'l)l I vertices of a Don--" 



1 

 In connection with the results tound above the length — 1/2 



til 



appearing here for the edges of the new simplexes contains a con- 

 firmation. 



Mathematics. — "On five pairs of f oar-dimensional cells derived 

 from one and the same source" By Mrs. A. Boole Stott 

 and Prof. P. H. Schoute. 



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

 Introduction. 



As this paper must be regarded as a short completion of the 

 handbook of the "Mehrdimensionale Geometrie" included in the 

 Sammlung Schubert we keep the notation used there. 



We regard in succession each of the six regular cells C^ , (\ , 

 6*1,, 634, C,3o, Cg„o of the space Sp^ and derive from these two 

 new four-dimensional cells. The first, which has the centres K^ of 

 the edges of the regular cell as vertices is formed by a regular 

 truncation at the vertices as far as the centres of the edges; the 

 second is the reciprocal polar of the first with resi)ect to the sphe- 

 rical space of the points K^ . 



34 



Proceedings Royal Acad. Amsterdam. Vol. X. 



