6 2 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



1 , p r -i 1, /V-l + Pr + 2 — 1» > n — Po 



preceded by the repetition of all its terms after having subtracted 

 n -\~ 1 from each (of which repetition the negative terms with an 

 absolute value surpassing n may be omitted) a number of n times 

 the process of increasing all the digits by unity and throwing out 

 (as soon as they appear) digits surpassing n — 1 and (afterwards 

 when the process is finished) all the negative digits. Then the 

 n -f- 1 rows obtained represent the indices of the e operations to 



be applied to (00 ... 0) in order to obtain the expansion symbols 

 of the constituents". 



Before proving this general rule we elucidate its meaning by 

 applying it to an example, for which we choose the case 5 VI b . 

 Here the series is — 1,0,8 which has to be proceeded by — 3. 

 So, if we indicate in heavy type the figures which are to be kept, 

 the operation is 



— 3—10 3 



— 2 14 



— 112 

 2 3 

 13 4 

 2 4 



giving e e 3t e e i e 4 , e x e 2> e e 2 e 3 , e ± e 3 e 4 , e 2 e 4 for the six expansion 

 symbols of 5 VI b . As we have e i e 3 e^ = — e e ± e 3 and e 2 e 4 = — e e. 2 

 this series is the same as that inscribed in the table. 



The proof of this general theorem splits up into three parts. 

 In the first we show that the top row corresponds to the constituent 



P r — \ ? r — 2 ''l^O 



for which r (r — l,r — 2, ...,1,0) is the zero symbol. In the 

 second we explain that the addition of a unit to all the digits 

 corresponds to what happens to the digits in the processes of art. 

 27 but for the transplantation of the digit at the end to the begin- 

 ning. In the third we will be concerned with the influence of 

 that transplantation. 



The first and the second parts are mere consequences of theorem 



P r-l Pr—2 Pi Pd 



IX. In the case of the zero symbol r (r — 1, r — 2, . . ., 1, 0) 

 the unit intervals present themselves behind the digits of rank 



Pr_U Pr-l + Pr-2> , ^ + 1 A) 



and this proves in connection with theorem IX the first part. 

 Moreover the circular permutation over one digit to the right hap- 



