DERIVED FEOM THE REGULAR POLYTOPESt. 41 



and' 4 3(2 1)0 belongs to only one. So there are two different kinds 



2 1 

 of edges and we find - = -. 



Remark. In S lt the degree of regularity is a minimum, i. e. 



for a poly tope and . — — for a net, 



2n r J r 2(0+1) 



1°. if the symbol of the polytope or that of the groundform of 

 the net contains no zero, 



2°. if the net admits a constituent 'ff n _ v 

 For in both cases there are at least two kinds of edges: in the 

 first case the edges [1], in the second case the erect edges of the 

 prisms g n _ x differ in character from the remaining ones. 



The results about regularity have been indicated in the Tables IV, 

 V, VI. In Table IV the regularity fraction is contained in column 

 5, whilst the subscripts in column 4 give the différent groups of 

 limits (/)„. In Tables V and VI in the cases n = 4 and n = 5 the 

 last column contains the regularity fraction, the last but one 1 ) the 

 different groups of limits (l) ki whilst the part n = 3 of Table V 

 contains two columns more, one indicating the number of the 

 Andreini diagram of the net, the other indicating the particularities 

 of the edges passing through a vertex (see Andreini's list, page 

 30 — 32 of the memoir quoted in art. 22). 



Section III : Polytopes and nets derived from the cross polytope. 



A. The symbol of coordinates. 



72. In this section which is so closely related to the immediately 

 preceding one that it may be considered as a mere supplement of 

 the latter we have to start from the cross polytope C 2 n (2) of S n repre- 



n — 2 



sented by the symbol [100 ... 0] \/2 and to remember that we 

 are to prove by and by that there is no difference whatever between 

 the offspring of this cross polytope and that of the measure polytope 



rn....i] of s n . 



For ^=3,4, 5 we have successively in the symbols of M 1S . Stott : 2 ) 



1 ) The numbers of the different groups of limits (/) /c for k >> 1 have been found in 

 the manner indicated for the simplex in Table III, but we have judged it of no impor- 

 tance to insert an analogous table for the measure polytope. 



2 ) For the deduction of the e and c symbols from the symbols of coordinates compare 

 part D of this section. 



In Table IV second column are inscribed the e and c symbols of the polytopes deduced 

 from the cross polytope corresponding to the symbols of coordinates of the third column. 



