DEBITED FÜOM THE REGULAR POLYTOPES. 1 5 



on the division of the limiting elements (l) p of p dimensions accor- 

 ding to their symbol into different groups; as introduction we explain 

 what this means by considering the edges of the polytope (432110) 

 treated in the preceding article. 



The edges are obtained by joining two points which pass into 

 each other by interchanging in the symbol of coordinates two digits 

 with difference unity. So we have edges with four different symbols, 

 viz. (43), (32), (21), (10), if by the symbol (p, q) we indicate any 

 edge the end points of which pass into each other by interchanging 

 two coordinates with the numerical values p and q. 



It is easy to find their numbers. To that end we calculate first 

 the numbers of edges of different symbol passing through a deter- 

 minate vertex, e.g. through the pattern vertex 4, 3, 2, 1, 1, 0, which 

 point will be indicated by P. Through P passes only one edge 

 (43) and one edge (32), as the symbol of coordinates contains only 

 one 4, one 3 and one 2; but two edges (21) and two edges (10) 

 concur in P, as the symbol of coordinates contains two digits 1. 

 So we have 



1 edge (43) -f 1 edge (32) 4- 2 edges (21) -f 2 edges (10) 



through P. Now the number of vertices of the polytope being 300, 

 the numbers indicating how many times each of the four edges of 

 different symbol passes through any vertex would be 360, 300, 

 720, 720; as each edge bears two vertices we find 



180 edges (43) -f- 180 edges (32) + 360 edges (21) -f 360 edes(10), 



i.e. once more altogether 1080 edges. 



In order to be able to extend the method of deduction of edges 

 with different symbols to limits (/) /; of p dimensions we are obliged 

 to introduce some new terms. So we call (43), (32), (21), (10) the 

 unextended symbols of the four groups of edges found above and 

 designate as the extended symbols of these groups respectively 



(43)(2)(1)(1)(0), (4)(32)(1)(1)(0), (4) (3) (21) (1) (0) , 



(4) (3) (2) (1) (10), 



containing also the remaining four digits, each placed in brackets. 

 Moreover we call the constituents (43), (2), (1), (1), (0) of the 

 extended symbol (43) (2) (1) (1) (0) the syllables of that symbol 

 and exclude from our considerations the "petrified" syllables with 

 two or more equal digits as (22), (111), etc., where the influence 

 of the interchange of the digits is nihil. 



