DERIVED FROM THE REGULAR POLYTOPES. 47 



So by this motion the coordinates œ it œ 2 ,. . . ,#&+i of any vertex 



(A — l)V /2 

 yfc-f-f 

 . . a? n of this point remain zero. So (/)'/, is represented by 



^ of (/) /c increase by — - — » , whilst the coordinates a? /t . +2 , ^+3, 



^'/c+2 == c %+3 ::::::z: • • • == X n ==z s 



from which it ensues that the symbol of coordinates of the new poly- 

 tope becomes 



A— 1 A — 1 



n— k— 1 



L ' /* — 1 # — J /• — 1 J 



So the new polytope satisfies the law of the equality of all the edges 

 expressed in theorem XXVIII if, and only if, we have either A = I 

 or A = l\ As A = 1 corresponds to the cross polytope itself, we have 



n — k-l 



to take A = h in which case we find [2 1 1 . . 1 00. . 0] as the 

 theorem requires it. 



Case k = n — 1. We consider the limit (/) n _i = S[n) repre- 

 sented by 



n — 1 



x, v , œ 2 , . . . , x n = (1 00 . . . 0) V 2 

 with the centre M, the coordinates of which are 



Vi 



00 1 00-y . . Où ft 



n 



and move this (l) n _± parallel to itself in the direction O M to a 

 position the centre M ! of which is determined by the relation 

 OM' = X.OM. Then we find in the way indicated above for the 

 symbol of coordinates of the new polytope 



n — 1 



L n n n J 



So, if we discard immediately the supposition À = 1 leading 

 back to the original cross polytope, the new polytope the symbol 

 of which contains no zero satisfies the law of theorem XXVIII, 

 if — and only if — we have 



l+ A ^iV A ^^ = (l+V/2):l 

 n ) n 



giving À — 1 = \ n V%. So we find the polytope with the symbol 



