( 500 ) 



Because the regular Cj, leads us to find the regular 6,4, the 

 number of pairs of new cells is not six but five. 



I. General observations. 



1. If we understand for the regular cells bj e,k,f,r successivel}' 

 the number of the vertices, edges, faces, bounding bodies, by p, q 

 the number of bounding bodies through an edge, through a point, 

 by e' , k\ f the number of vertices, edges, faces of the bounding 

 bodies, then besides the relations 



of EuLER the equations hold 



qe = re' , j)k =: rlz , 2/ zir: rf , 

 out of which number of five we can easily deduce the relation 



(5-2). = (p-l0^ (1) 



The following table furnishes these quantities for the six regular 

 cells of S])^. 



2. We shall now endeavour to express the characteristic numbers 

 E, K, F, R of the first of the two new cells — and what is also 

 possible for these P, Q — in the characteristic numbers 6^, ^,/, ?•, p, gf 

 of the regular cell. 



"Tlie number of vertices of the new cell is k, i.e. E = k." 

 If we project the regular cell (see the diagram) on the plane through 

 one of the edges EJ\' and the centre ^>, the two new bounding spaces 

 |)assing through the centre A'^ project themselves according to the 

 perpendiculars /, /' let down out of K^ on the axes OEg, OE/ . The 

 section of the regular cell with a plane normal to the plane of 

 j)rojectiun in a point lying close to i^^pA'/ being an equilateral triangle, 



