22 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



is the group of limits (l) n __i of vertex import, sometimes denoted 

 by g , the vertices of which form exactly all the vertices of the 

 expansion form, each vertex taken once, and now the operation c 

 consists in this: all these limits undergo a translational motion, of 

 the same amount, towards the centre O of the expansion form, by 

 which any of these limits gets a vertex or some vertices in common 

 with some of the others. By this contraction the edges of the expansion 

 form parallel to the axes of coordinates are annihilated. 



We have now the general theorem : 



Theorem XXXVI. "By applying the contraction c to any expansion 

 form all the digits of the symbol of coordinates of this form are 

 diminished by a unit". 



This theorem, which shows that the preceding one still holds for 

 contraction forms deduced from M} 2 \ is almost self evident. So, 

 as the motion of the limit y lying in that part of S n where all 

 the coordinates are positive takes place in the direction of the line 

 making in that part of N n equal angles whith the n axes, all the 

 coordinates of the pattern vertex diminish by the same amount, and 

 this process has to go on untill the smallest of the digits disappears. 

 For then we once more obtain a poly tope the symbol of coordinates 

 of which satisfies the laws of the first part of theorem XXVIII 

 (art 47). 



Bemarfc. By combining the theorems XXXV and XXXVI we can 

 find the symbol in the operators c and e k of any form deduced from 

 M£ 2) . But this process can be simplified by introducing the opera- 

 tion e which transforms the centre O of M n (2) considered as an 

 infinitesimal measure poly tope M n {0) into M n (2 \ Then the contraction 

 symbol c can be shunted out by substituting e k e L . . . e m M n (0) . for 

 c e k e t . . . e m M£ 2 \ but this implies that we replace e k e t . . . e m M n (2) 

 by %e k e t . . .e IH M n (0) . This remark will be useful in part F of the 

 next section (compare theorem LIII). 



E. Nets of poty topes. 



61. The theory of the nets derived from M n (2) is based entirely 

 on the consideration of the most simple of these nets, the net 

 N (M n (2) ) of the measure polytope itself. So we begin by the analytical 

 representation of that net N (i/ n t2) ). 



By means of the symbol [2a ± -f- 1 , 2a, + 1 , . . . , 2a n -\- 1] the 

 net of M^ 2 is decomposed into its measure polytopes, if a u a 2 , . . , a n 

 are arbitrary integers and the heavy square brackets mean that in order 

 to obtain a definite M n {2) of the net we have to permutate and to 



