DERIVED FKOM THE REGULAR POLYTOPES. 77 



elements (l\, (/) 2 , . . . , (On-i- So ^is research is closely related with 

 theorem III of art. 10 which enables us to find the subgroups 

 of the same system of limiting elements l d characterized by different 

 symbols, the more so as we have the theorem: 



Theorem XXVI. "Any two limiting elements of the same group 

 (l) d belong to the same subgroup or to different subgroups , in the 

 sense of the scale of regularity, according to their zero symbols 

 being equal or different, if we consider two different zero symbols 

 of a central symmetric poly tope as being equal when they pass 

 into each other by inversion." 



This theorem is nearly self evident. A rigid proof of it can be 

 based on the consideration of the limits (/) n _i passing through the 

 (l) d . So in the case of the form (321100) treated in art. 11 the 

 different unextended edge symbols (32), (21), (10) correspond to 

 subgroups of edges with different positions in relation to the sur- 

 roundings. For, if we consider the four groups (32110), (321) (100), 

 (32) (1100), (21100) of limiting polytopes it is immediately evident 

 that the second group distinguishes (10) from the others, that the 

 fourth group distinguishes (32) from the others, whilst the third 

 group alone shows already that no two of the three subgroups of 

 edges can be equal. 



We remarked above that we count the contributions to the 

 numerator of the regularity fraction beginning at the vertices and 

 taking in only successive contributions. But the case may present 

 itself that a poly tope derived from the simplex shows also some 

 regularity at the side of the limiting elements (l) )l _ i of the highest 

 number of dimensions. We then indicate two fractions of regularity, 

 one for each side, as will be shown in an example in the next article. 



The fifth column of Table I contains the regularity fraction of 

 the different forms obtained in the cases n = 3, 4, 5, only counted 

 from the vertex side. In the fourth column the subscripts indicate 

 the numbers of the different subgroups of each limiting element (l) d . 



43. We elucidate the theory by applying it to several examples: 



a). Example (321100). Here we find three different groups of 



edges. So the vertices contribute 1, the edges contribute -| to the 



1 + - 1 - 3 



numerator and the fraction is — — = — . 



5 10 



b). Example (110000). This form has only one kind of edge (10) 



but two subgroups (110) and (100) of triangular faces. So we find 



l+i+ l i 



5 2' 



