DEKIVED FEOM THE EEGULAR POLYTOPES. 51 



n + l 



to the next symbol we find by means of the second process (00 . . . 0), 

 which falls out. So we only get n kinds of constituents for r = 1. 



29. We come now to the general rule about simplex nets proper; 

 it can be stated in the following form: 



Theorem XV. "To every possible cyclical partition of n-\-\ 

 corresponds a definite simplex net proper of jS n ." 



In order to prove the theorem for the general case with n -f- 1 

 and the particular case with n groups of constituents we first of all 

 determine a list containing these different groups of constituents, to be 

 derived from the partition cycle. Then we select from this list a definite 

 polytope (P) a of a definite group and show that this ÇP) a is in contact 

 by any of its limits (l)\^ b) with one and only one other polytope (P) b 

 of the list, whilst the list contains no polytope overlapping (P) a . 



Case r > 1. We start from the partition cycle r (Pr— 1> Pr-i> • • *A> Po)n 

 and deduce from it the net symbol 



P r -\ Pr-2 H Cd 



(ra t -j-r — 1 , ra L -f- r — 2 , , ra t -\- 1 , ra h -\- 0) 



i. e. the symbol with /V-i digits congruent to r — 1 mod. r, p r _ 2 

 digits congruent to r — 2 mod. r, etc., the different quotients 

 a x , a 2 , . . ., a n + i of the division of these digits by r having a sum 

 2«. = , whilst the sum of the remainders r — 1, r — 2,...,0, 



r— 1 



i. e. 2 i Pi may be represented by Jc . 

 If we write this symbol in the form 



Pr-l Pr-2 Pi P0 



(ra t -f- r — 1 , ra 2 -f r — 2 , . . . . , ra L + 1 , ra L -f 0) 



(«! « ff a , an+l) 



and permutate only the remainders r — 1, r — 2,...,0, the net 

 is decomposed into the group of constituents to which the central 

 polytope belongs; but we can have this rather complicated symbol 

 in our mind quite as well if we simplify it by omission of the 

 immovable parts of the digits. So the first line of the following 

 list repeats the group of constituents to which the central polytope 

 belongs, while the other lines give all the other groups of consti- 

 tuents, deduced from the "central group" in the manner and order 

 of succession of the preceding article. 



4* 



