DERIVED PROM THE REGULAR POLYTOPES. 77 



other giving for n > 5 not only the characteristic numbers but 

 also the numbers of any limit of any kind; here we will do likewise. 



So in the case of the polytopes connected with HM h in the 

 manner indicated in theorem LVI we have to determine: 



1°. the number of vertices according to general principles, 



2°. the number of edges concurring in any vertex and thereby 

 the total number of edges, 



3°. the limiting polytopes (/) 4 , which limits reveal at the same 

 time the limiting bodies (/) 3 , 



4°. the number of faces (by means of Eider's rule). 



But before applying this method to a definite example we give 

 some further explanation with respect to the equations of the four- 

 dimensional spaces containing the limits (/)n-i of the hwpd. deduced 

 from HM n in S n , as this will save us trouble in the exposition 

 of the direct method. 



If -J- [#i a 2 - • • a n~] i s the symbol of coordinates , where the digits 

 have been arranged in diminishing order, we consider the vertices 

 represented by 



\Ct\ u>2 • • • &y) ~ö~ \~~p4-i **"p -J- 2 * * * ^"n\ 

 O/j, cv 2 , • • • j cV p «^,_|_4> tf/ p + 2» • • • 5 °°n 



lying in the space S n _ i represented by the equation 



#i + œ % + • .• • + œ v = a± •+ a i + • • • + a p' 



Evidently these vertices will determine a limit (/) n _i of the po- 

 ly tope, if (a 1 a 2 ...a p ) and l>-\_a p + i a p + 2 . . .a lt ~] represent polytopes 

 (P) p ..\ and (P) n _ p respectively, this {l) n _^ being then a prismotope 

 which may be denoted by (P p _ ± ; P n . p ). Now (a ± a 2 . . . a p ) always 

 represents a (P) p _ i} unless all the digits a x a i9 . . . , a p are equal, 

 in which case {a x a 2 . . . ci p ) is a petrified syllable. On the other hand 

 2"[ ö p+i a P +-2 • ' a n\ always represents a {P) n _ pi unless we have 

 either p = n — 2, or p = n — 1 ; for, as we remarked already 

 p = n — 2 gives the syllable -J- [11], i.e. a line segment instead 

 of a square , and p = n — 1 gives a vertex only instead of an edge. 



To this we have to add a few words only about the extreme 

 cases p = I and p = n. For p = 1 we find the polytope with 

 the coordinate symbol |- [# 2 <% . . . ar n ] lying in a space S n _ i repre- 

 sented by + gj. = g A ; it can be deduced from HM n _ ± . For p = n 

 the result is different for n even and n odd, the polytope having 

 as HM n itself a centre of symmetry in the first case and two 

 different limits, either a vertex and an (/)„_! or two differently 

 shaped (/) u _i, opposite to each other in the second. Or otherwise, 



