78 ANALYTICAL TREATMENT OF THE POLYTOPES EEGULARLY 



as follows. For n even the diagonals of M n split up into two 

 groups of non adjacent ones, of those bearing vertices belonging also 

 to HM n and of those bearing vertices cut off by the alternate 

 truncation leading from M n to HM n \ the 2 n ~ 2 diagonals of the 

 first group are normal to two limits of vertex import *) in the 

 considered polytope, whilst the 2 W_2 diagonals of the second are 

 normal to two limits which may be called of truncation import 

 as they are derived from truncation limits of HM n in passing to 

 the polytope under consideration. For n odd there is only one 

 group of diagonals of M n , each of which bears only one vertex 

 of HM n ; so each of these diagonals is normal to two differently 

 shaped limits of the polytope, to one limit of vertex and to one 

 limit of truncation import. But in the two cases, of n even and 

 n odd , we have to deal with the two equations 2 + œ i = ^ a i 

 and S + %j = Sflj — 2 , the last digit a n = 1 having to be taken 

 with the positive sign for limits of vertex import, with the negative 

 sign for limits of truncation import. 



After this introduction we treat a definite example. 



94. Case | [53 3 11]. 



The number of vertices is 2 4 times 5! divided by 2 2 , i. e. 480. 



The vertices adjacent to the pattern vertex 5 3 3 11 are 



51331 



3 3 5 11 

 35311 



5313 1 

 5 13 13 

 53113 



533 — 1-1 



which may be indicated by the brackets and the negative sign after 

 the two units in the symbol 



i (-) 



So seven edges concur in any vertex, i. e. the total number of 

 edges is half the product of 480 and 7 , i. e. 1680. 



Now we have to pass to the limiting polytopes. 



The spaces # 4 represented by + x i = 5 give 2.(5^ = 10 limits 

 •£[3 311] of polytope import. 



') Also the import of the different limits (0 n _i of HM n will be considered in relation 

 with the limits (0„_i of M n . So the equations ± x { = a t will give limits of (On— 1 

 import, the equations ± x i ± x- = a x + a 2 will give limits of (l) n _ 2 im P ort î etc -i 

 this series ending in general in limits of body and limits of vertex import, as no edge 

 or face of M n partakes in the limitation of HM n . 



