52 ANALYTICAL TREATMENT OF THE POLYTOPE3 KEGULAKLY 



Pr-l 



Pr— 2 



Pi 



r— 1 , r— 2,. ., 1 



Pr— 1 P r -2 H 



PO 



) 



'.-i 



( r t r _i, r— 8,.., 1, ) 





 — 1 



h 



h + r 



Po 

 r 



Po 



Pr—l 



P-2 



Pi 



r—l,.., 2, 1 ) 



Pr-l P-2 P,- 1 



(r+l f 



r , r 



1,.., 2, 1 ) 



— A> 



— GvH) 



^o + Po r 

 h + (A)+ 1)*' 



Pr— 2 



Pr-3 



Po 



Pr-l 



'Zr— 2 , 2r—3, . . , r , r- ) 



1 P r _ 2 P r — 3 PO 



P r" 1 



r—l 



{2r—l, 2r—2 , 2r—3,. ., r, r—l) 





P i~ 4 



r— 1 



Pr-2 



Pr— 3 



PO 



(2r— 1, 2r— 2 , 2r— 3, . . , r , r—l ) 



■# 



*o -+-(* + 1— Pr-i)r 



/• -f- ar 



For each of these groups of constituents have been indicated in 

 a second column the value of So,-, in a third column the value 



r— 1 



of /: = mipi. Moreover, in order to point out the regularity of 



i=i 



the process by means of the variation of these two sums, the use 

 of the zero symbol has been sacrificed for a moment, i. e. the 

 diminution of the digits by unity every time as the last zero is 

 replaced by r (exacted by the second of the two processes of the 

 preceding article) is not executed here, which implies that a digit 

 h jumping to the fore becomes h -\- r. Here at each step 2# 

 diminishes by a unit and k increases by r. *) 



But in the selection of a definite polytope (P) a of the list we 

 return to the zero symbol and suppose 



1) that the cyclical permutation of the partition cycle from which 

 (P) a has been derived begins by p t _ ± and winds up in p t , 



2) that p r of the p l zeros have been replaced by r, 



3) that the equation of the space S n _ i containing (1)^—1 is 



l ) This relation also holds when we pass from the last group of constituents to the 

 first, when we diminish all the a t by unity at the transition. From this point of view 

 we can introduce the notion of "cycle of constituents". 



