DERIVED FROM THE REGULAR POLYTOPES. 6 3 



peiling at each step of the two processes of art. 27 (see the 

 example following theorem XIII) changes the ranks k -\- 1 and 

 k -\- 2 of two adjacent digits into k -j- 2 and k -\~ 3, i. e. — 

 according to theorem IX — the operation e k+i is still to be applied 

 or has already been performed on the new constituent according to 

 the operation e k being still to be applied or having already been 

 performed on the original constituent; i. e. if e k occurs in the 

 ^-symbol of the original constituent, e k+i must occur in the ^-symbol 

 of the new one, what proves the second part. 



In the third part we have to consider all the possible cases of 

 the transplantation of a digit from the end to the beginning; these 

 cases, four in number, are the following: 



(r — 1, .., I, 0) becomes (r — 1, r — 2, .., 0) .. loss of e n _ i} gain of e , 



(r — 1, .., 0, 0) „ (r, r — 1, , 0) .. gain of <? , 



if, , I, 0) „ (r — \,t — 2,..,0) .. loss of e n _ u 



(r, , 0, 0) „ (r, r, , 0) .. neither loss nor gain. 



So we find the two rules : 



1°. If e n _ i appears in the symbol of the original constituent it 

 falls out in the next one , though an other e n _ l may be introduced 

 (if e n _ 2 was contained also in the original symbol). 



2°. If the number of the operation factors e k is r — 1 the symbol 

 e appears in the next constituent. 



But this is also the effect of the operation indicated in the 

 theorem, the first rule being a consequence of the omission of the 

 digits surpassing n — 1, the second being deducable from the 

 repetition of the series — I , p r _ A — I , p r _ x -\- p r _ 2 — 1 , etc. If, 

 in order to add still one word about the second rule, G p _ ii G p , 

 G p+i indicate three constituents, consecutive in the sense of the 

 theorem, and the ^-symbol of G p bears only r — 1 expansion 

 factors, then the ^-symbol of G p _ ± contains n — 1 and therefore 

 also — (n -\~ 1) -\~ (ii — I) = — 2, before the negative digits have 

 been omitted; this — 2 becomes — 1 for G p and for G p+i . 



33. The theorem XXI enables us to show how the "principal" 

 net of S n , i. e. the net with the period r =* 1 always inscribed 

 first , can be transformed successively into all the other ones. 



The result for S 3 is given in the following table, in the left half 

 in the symbols to be applied to the different constituents of N(T, O), 

 in the right half by the results of this application. 



