( 502 ) 



tangential spaces to tliat spherical space in those poinls, i. e. by 

 the spaces in the points K^ normal to the axes OK^). By polar 

 inversion of the result found above we arrive with respect to this 

 second new cell at the following results : 



"The number of bounding spaces of the new cell is h, i. e. E' = /;." 



"The bounding bodies have y> + 2 vertices and are double pyramids 

 with a regular polygon with 2;-sides as base lying in a plane bisecting 

 the connecting line of the vertices at right angles." 



"The faces are isosceles triangles." 



"In a bounding space lie 2p faces, so ph is the number of faces, 

 i. e. F' = pL" 



"The system of the vertices consists of two groups, namely of e 

 regular vertices and r semi-regular vertices, i.e. E' = e -\- r'. 



"The number of edges K' is i {qe -f- }>^ + >"•" 



So the result is : 



"The second of the two cells, {L' , K' , F' , W), deduced out of the 

 regular cell {e, k,f, r, p, q) is bounded by double pyramids with a 

 regular polygon with />-sides as base and has the characteristic 

 numbers 



E' = e -]- r, K' =r i (<7^ + pk) + r, F' = pk, R = A-." 



It might appear as if it were possible to deduce more pairs of 

 new cells out of the regular cells by doing for the ends F^ of the 

 axes OFo the same as has been done above for the points K^- This is, 

 however, not the case, because for each regular cell the centres 7^,, of 

 the faces form the centres A'^ of the edges of another regular cell 

 which is for the cells C^, C^^ dualistically related to themselves a cell 

 of the same kind, for the cells related in pairs to one another {C\, Cu) 

 (C'ljo, C'goo) ^ cell dualistically related. And as is immediately evident, 

 the pointgroups E^ and R^ can neither lead to new results. 



We conclude these general observations with the remark that the 

 two cells deduced from the regular cell {e, k,f, r) show much regularity ; 

 of the former the vertices and the edges, of the latter the faces and 

 the bounding bodies ate mutually equivalent groups of elements, whilst 

 the faces and the bounding bodies of the former and the vertices and edges 

 of the latter form groups of elements consisting of two subgroups. Do 

 these new cells furnish the maximum amount of regularity for 

 polytopes not entirely regular? We do not intend to go into further 

 details here, as the Mathematical Society at Amsterdam is proposing 

 a prizequestion about what is to be understood by "semi-regular 

 polytopes". 



1) The handbook quoted above contains in Vol II pages 256—261 some com- 

 munications about the corresponding polytopes in the space SjJn- 



