DERIVED FKOM THE KEGULAK POLTTOPES. 6 1 



Let us take the polytope (221000) of 8 5 . This polytope can belong- 

 to a net the period r of which is either 2 or 3. In the first case 

 we find 2 (221000) which can be reduced to 2 (1 00000) by going two 

 steps backward; in the second case we have ;i (221000) which passes 

 into 3 (2iil00) also by going two steps backward. 



Or, let us go back to the constituent 



Pr Pl- 1 Pq Pr—\ Pl+1 Pl~Pr 



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



used in art. 29 in order to make the proof as general as possible. 

 This constituent can belong to two nets, one with the period r, an 

 other with the period r -{- 1; the two partition cycles of these nets are 



r\Pl-\> Pl— 2> -'-jPo'Pr — li •••)/ J | + l) Pa ) 



r+iKpi'i Pl—l9 Pl—2> ' ' ' 9 Pof Pr — lf ' • • ? Pl + 4 > Pl pr) ' 



and may be reduced to 



r\Pr — \i Pr — 2>'''>Pl + l> Pl > Pl — 1 • • • > A' po) 

 r+l(pr> Pr— 1? Pr-2> ■ ■ • > Pl + l> Pl Pr> Pl — l'-'fplfpo) ! 



In the case of an asymmetric constituent it may happen as we 

 have seen that a definitely orientated one occurs in two different nets, 

 if we consider as different two nets as 5 a , 5 which are each 

 others reversions. So under this point of view the two positions of 

 3(221 000) occur together in three different nets. But the statement 

 of the theorem about each of the two positions of an asymmetric 

 constituent holds under any point of view. 



A second remark refers to the expansion symbols used in the table. 

 In order to bring the two different orientations of the asymmetric 

 constituents into evidence we have introduced the expansion symbols 

 provided with the negative sign. But the law of succession of the 

 different constituents of each net proceeding in the list from column 

 to column would have been much more evident if we had stuck to 

 expansion symbols without sign. Then the order of succession in the 

 case of net 5 7 would have been e , e x , e 2 , e Zi <? 4 leading to the sup- 

 position that in general at each step the index of each e increases 

 by unity, an illusion which is already destroyed by the series 



g >^i>¥2»¥3>¥4»?4' ^ anv ra ^ e this second remark places us 

 before the question by which rule the expansion symbols of the 

 constituents of a net can be deduced from the partition cycle. The 

 answer may be given in the form of: 



Theorem XXI. "The constituents of the net of S n corresponding 

 to the partition cycle r (p r _ i , p r _ 2 , . . . , p ) are found by applying 

 to the system of digits consisting of the series 



