DERIVED FKOM THE REGULAR POLYTOPES. 5 7 



In the cases 2 m , S v , 4 7// , 5 V// , where the partition cycle consists 

 f n _|_ i units, we find back the self space fillers of simplex 

 extraction, p 6 = (210), tO = (3210), etc. The net symbol of these 

 self space fillers is characterized by the property that its n -\- 1 

 digits, when divided by n •-(- 1, leave all possible remainders 

 n,n — 1 , ...,1,0, each remainder once. 



In the case of the partition 2,2 of the net S IH an other particu- 

 larity presents itself: in the process of formation of new zero symbols 

 we fall back at the second step on the original symbol 



(1100), (2110), (2211) = (1100). 



This is due to the fact that the partition cycle consists of (two) 



equal parts. So this particularity repeats itself in the cases b IV , b VII 



and 5 y with the partition cycles (3, 3), (2, 2, 2) and (1, 2, 1, 2), 



in general if we have n -j- 1 = uv and the partition cycle consists 



of the v digits a ± , a 2 , . . . a v , this set of v digits being repeated in 



the same order of succession so as to have u sets. In the latter 



case where the partition cycle is said to be "a cycle of powers", 



n + 1 

 we find onlv = u constituents of different form ; it includes 



J v 



the self space fillers, which present themselves for v = n -\~\ ,u=\ 1 ). 



We point out two other particularities occurring for the first time 

 in jS 5 . The two partition cycles 1, 2, 3 and 1, 3, 2 of which the 

 second written in the form 3,2,1 is the inversion of the first, have been 

 inscribed both as b VI , as these two nets, differing only in orientation 

 with respect to the simplex of coordinates, are essentially the same. 

 On the other hand the two nets 5 /v and 5 x are essentially, different, 

 though the four digits of the partition cycle are two times 2 and 

 two times 1 for both. 



The fifth column forms the principal part of the table. As to the 

 number of different constituents of a net in S n this column is sub- 

 divided into n -I- 1 small ones. In the first of these n -f- 1 small 

 columns is placed the central polytope; on each horizontal line the 

 polytope mentioned in a following small column is deduced by the 

 two processes of art. 27 from that in the immediately preceding one. 

 For brevity we have only inscribed the geometrically different forms, 

 using from n = 4 upward the symbol e explained at the end of 

 art. 21 and indicating the orientation by means of the signs. 



l ) If we wish to indicate the number of constituents of different form and orientation 



n 4- 1 



we can complete theorem XIV by saying that this number is n for r = 1 and — — 



v 



if the partition cycle is a cycle of power v, 



