DERIVED FROM THE REGULAR POLYTOPES. 47 



(rbi -f q l3 rb, -f q 2 , . . . , r3 #l _4 + gr„_ l5 ri fl +0, K*n+i — 1) +r), 2^=0 



by allowing r units to pass from the -immovable partr£ w + 1 of the 

 digit rb n + i -|- to the permutable part; for by that variation we 

 alter only the grouping of the vertices of the net to vertices of 

 polytopes but not the total system of vertices of the net. If now 

 we write b for b n + 1 — 1 and put the permutable digit r fore- 

 most we get 



(rb + V, rb, + q x , rb 2 -f q 2 , . . . , r£ n _< + q n _ x , ri M -J- 0) , 2^ = — 1 , 



bringing to the fore the constituent with the zero symbol (r, q u q 2 , . . . 



<7n-l> 0). , 



2°. In the case of the zero symbol (g ± , q 2i . t . q n _ i , 1, 0) con- 

 taining only one zero an application of the same process leads from 



0A + </i , rb 2 + <ƒ,,..., rb n _ x + g^ , ri n + 1 , rb n+i + 0) , Hb t = 



to 



(rfl -f r, r8, + f/i , rb 2 -f g 2 , ' . . . , r^ + # n _ l5 /•£„ -f- 1 ), 2i, = — 1 



and therefore to the constituent (r,q ±i q 2 , • •' •> ?n-i» 1)> ^he zero 

 symbol of which is (V — 1 , q x — 1 , q 2 — 1 , . . . , q n _ i — 1 , 0). 

 In order to obtain this zero symbol we can write the decomposing 

 symbol in the form 



(rb Q + 1 -f- r — 1, rb, + 1 +#! — 1, . . . , rb n _ x -f 1 



+ q n -i - 1, rb n + 1 + 0) , 22, = - 1 



and pass to an other sum ^Lq L — (n -j- 1) of all the digits by omitting 

 the unit of the immovable part of the digits. 



So, if we take notice only of the zero symbols of the constituents 

 deduced by means of the two processes, we can word these pro- 

 cesses as follows: 



1°. "If the zero symbol of the given constituent contains more 

 than one zero, we can replace one of these zeros by r". 



2°. "If the zero symbol contains only one zero, we can replace 

 this zero by r and diminish all the digits by unity afterwards". 



We now come back to the difficulty about the pure nets stated 

 above. To that end we have to ask under what circumstances one 

 of the two processes leads back to the original constituent; therefore 

 we repeat that: 



the first deduces (r,q u q 2 ,..., q n _ i ,0)ÎYom(q [ ,q 2 ,...q n _ A) 0,0), 



„ second „ (r— 1,^—1,^—1,...,^^— 1,0) „ (j^fti-fr-i»!» )- 



