30 ANALYTICAL TREATMENT OF THE POLYTOPES1REGULARLY 



G a , G b/ So the proof of the lemma is given. So neither of the two 

 systems of constituents can admit holes. 



In order to show that no two polytopes of any of the two 

 systems can overlap we remark that by means of the symbols T) , 

 T'), T") any polytope of the chosen system can be promoted to 

 central polytope , which shows that not a single vertex can lie 

 inside that polytope. 



So we have proved completely now the theorem under consi- 

 deration. 



64. We now formulate the manner of deduction of all the 

 measure polytope nets as follows : 



Theorem XXXIX. "Let G = [a ± , a 2 , . . . , a a _ i} a H ] be the symbol 

 of coordinates of the "groundform" of the net. Deduce from it the 

 symbol O = [a — a n , a — a n _ 4 . . . . , a — a 2 , a — <%] of the "opposite 

 form", where a is either a f or a ± -f- 1. Derive from these two 

 symbols G, O the mixed symbol I k of the "intermediate forms" 

 represented by 



XP'n — ft + 1 J ^n - ft + 2 ' • • » a n — 1 ? ^nj 



\a a n _ ]. , ct ct n _ k _!,... , ci a 2 , ct a^j , 



of the two syllables of which the first contains the last k digits of 

 G, the second the last n — Jc digits of 0. Then G, the forms I k , 

 (Jc = n — 1 , n — 2 , . . . , 2,1), O are respectively the constituents 



"The frame of the constituent g n _ k is 



where we have (i i = 2ö i and /3' £ = 2 b t -\- 1 , the b { being integer 

 and the digits of the first syllable being related to the odd, those 

 of the second syllable being related to the even multiples of a". 

 "If (e, c), etc. indicates a net with an expansion groundform and 

 a contraction opposite form, the theorem includes the four cases: 



a n = 1 , a = % (e, c) , 



a n = I, a = ^ -\- I (e,e), 



a n = , a = % (c, c), 



a n = 9 a = a x -\- 1 (c, e)." 



In this theorem the deduction of the intermediate constituents 

 differs slightly from that given in the preceding article, the two me- 

 thods passing into each other by interchanging h and n — k, and 

 the two syllables. In the new form the succession of the different 

 constituents is a more regular one, as the following examples prove. 



