DERIVED FROM THE REGULAR POLYTORES. 33 



This difference in character implies a difference in the number of 

 different positions a constituent of definite form may admit. In the 

 case of a simplex net proper this number is two in general and 

 only one if the form is central symmetric. In the case of a measure 

 polytope net this number is one for the two extreme constituents, 

 whilst the intermediate form I, generally occurs in a number of 

 different positions indicated by half the number of limits J/ / . (2) of 

 M^ 2 \ i.e. in % r ~ k ~ ± (n) k different positions. 



In the case of the simplex net we have considered as kind of 

 constituent any polytope of the net with equipollent repetitions ; when 

 the partition cycle was a power cycle we have even been obliged to 

 split up a kind of constituent into several groups, in order to keep 

 the analytical treatment in contact with the geometrical facts. On 

 account of the extreme transparency of the measure polytope nets we 

 can allow ourselves to be less exacting and extend the notion of 

 constituent here by admitting that the 2" ~ u x {n) k different positions 

 of the intermediate form I, introduced above belong to the same 

 constituent. 



b). In order to be able to indicate the number of different con- 

 stituents according to the new point of view we fall back on the 

 different cases (e, c), (e, e), (c, c), (c, e) mentioned at the end of the 

 last theorem. By generalizing the results of the two examples given 

 above one finds immediately that the required number is in general 

 7i — p ~\- 1 , where p indicates the number of e's contained in the 

 symbol. But this general number n — p i-\- 1 is still to be considered 

 as a maximum, i. e. under circumstances the number of constituents 

 may become less. This decrease can be due to two different causes. 

 If in the first place in one of the two groups (e, c), (c, c) of a net 

 in S. n the expansion operator with the largest index is e,., where 

 h <Ç n — 1 , the constituents g,. ,g,. + i ,. . . , g n _ 2 are lacking together 

 with g n _i. If in the second place in one of the two groups {e, e), 

 (c, c) a net is semiperiodic the equal constituents of complementary 

 import may count for one constituent. 



c). Some of the intermediate constituents may become measure poly- 

 topes, this being even the case with all the intermediate constituents 

 of the net e n N(M n ). So by extending the notion of constituent 

 still more the number of the different kinds of constituent is lessened 

 in these cases, this number being unity for the net e n N(M n ). 



d). By comparing the cases g 2 under n = 4 we remark that the 

 prismotope (4 ; 4) which is the measure polytope C s of # 4 is indi- 

 cated by three different symbols; in the cases of the nets (e, c), of 

 the nets (e, e), of the nets (c,c) we get successively: 



Verh. Kon. Akad. v. Wetensch. I e Sectie Dl. XI No. 5. E 3 



