26 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



where g n is given, g is obtained by subtracting the digits of g n 

 from a and taking the differences in inverted order, while the two 

 syllables of g H _ k are got by taking the last n — k digits of g n and 

 the last Jc digits of g for £ = 1 , 2, . . . , n — 1. 



63. We 'now prove the following problem of positive tendency 

 completing the preceding one. 



Theorem XXXVIII. "In either of the two cases a = a l and 

 a = a x -\~ 1 the vertices of the co H polytopes (P) n of the preceding 

 theorem do form together the vertices of a net. The constituents 

 of this net are obtained by means of the algorithm developed at 

 the end of the preceding article." 



We march in the direction of the proof of this general theorem: 



1°. by deducing from the symbol of coordinates of the given 

 groundform (P) n the symbol representing all the repetitions of this 

 polytope and therefore all the vertices of the system, 



2°. by deriving from this new symbol the symbols of the polytopes 

 different from the groundform the vertices of which belong to the 

 system (which set of new constituents will prove to be equivalent to 

 that obtained above by the geometrical multiplication introduced above), 



3°. by showing that the system of polytopes obtained in this way 

 fills space, i. e. that there is neither overlapping, nor hole. 



Symbol of the total system of vertices. The symbol of a definite 

 repetition of the groundform is 



[20J a -f- a u 2b 2 a -f- a 2 , . . . , 2b n _ i a -f- (l n _ u 2b n a -f a n ], . . . T) 



where b l9 b 2 , ..., b n _ l} b a is a definite set of arbitrarily chosen 

 integers. So this symbol represents the total system of vertices, if 

 the bj denote all possible sets of integers. 



From the symbol T we deduce the frame symbol 



[2b. v a, 2b. 2 a,..., 2b n _ i a, 2b n a], F) 



representing the system of vertices of a net of measure polytopes 

 ~M n (2 ''\ one of which has the origin as vertex and the n spaces 

 x x -— 0, (i = 1, 2, . . . , n) as limiting spaces. 



Presumptive new constituents. The most general transformation by 

 which the total system of vertices T) passes into itself consists in a 

 transport of p<a units from the permutable to the unmovable part 

 of x Xi the n quantities p- t being integer. But this process is limited 

 by the restriction that in the case of a new constituent sought 

 the permutable parts placed within the same pair of square 

 brackets have to satisfy the conditions of theorem XXVIII, from 



