DERIVED flllOM THE IIEGULA11 POLYTOPES. 1 5 



55. We apply the notion of end digits and middle digits of 

 the syllables, introduced in art. 12, to the syllables in round 

 brackets occurring in the symbols of the polytopes deduced from 

 the measure polytope, in order to be able to repeat theorem XXX, 

 in a version connected with the more practical unextended symbols, 

 in the following form : 



Theorem XXX 7 . "We obtain the unextended symbol of a poly- 

 tope (P) d the vertices of which are vertices of the given {P) n , if 

 we put the lowest k digits of the pattern vertex between square 

 brackets, where /' takes successively one of the values 0, 1, 2, . . ., d, 

 and place before it, of the n — k remaining digits, between 

 round brackets either one group of d — /; -|- 1 interchangeable 

 digits, or two groups containing together d — k ~\- 2 interchange- 

 able digits, or three groups containing together d — k -\~ 3 inter- 

 changeable digits, etc., this process winding up where the total 

 number of groups is n — d -\~ k for u < 2d — k -j- 1 and d for 

 n > 2d — k — 1". 



"This (P) d will be a limiting polytope of (P) n , if the syllables 

 between round brackets satisfy the two following conditions : 



1°. each syllable with middle digits exhausts these digits of the 

 symbol of (P) n , 



2°. no two syllables without middle digits have the same end 

 digits". 



The proof of this new version can be deduced fron the articles 

 10, 12 and 54. 



By means of theorem XXX' we deduce the limits (P) 6 of the 

 polytope (i J ) 10 represented by the symbol [5' 4' 4' 3' 3' 2' 2' 21' 1]/ 

 of which — as is easily shown 1 ) — the (P) 9 of art. 12 represen 

 ted by (5443322210) is the limit ^ of vertex import. If we put 

 together the different (P) 6 for which the k has the same value 

 we find for k = the 58 polytopes given in art. 12 and for 

 k= 1, 2,. . ., 6 successively groups of 33, 11 , 9, 6, 2 , 1 , i. e. 

 in toto 120 polytopes. If for brevity the last syllable — between square 

 brackets — is put at the head of each group, these are 



*) la rectangular coordinates the polytope g is (5' 4' 4' 3' 3' 2' 2' 2' 1' 1) which may 

 be simplified by passing to parallel axes with the point 1, 1, ..., 1 as origin, i.e. by 

 subtracting a unit from all the coordinates. If we then bear in mind that according to 

 art. 1 we have to divide the coordinate values by V2 if we pass to barycentric coor- 

 dinates on account of the new unit of length, we find (5443322210). 



From this relation between a polytope deduced from the measure polytope and its 

 polytope of vertex import can be deduced generally that the number of these polytopes 

 in Sn, the measure polytope itself included, is C + 2iV -f 1, where C and N represent 

 the numbers of central symmetric and of non central symmetric polytopes in Sn — 1 of 

 simplex extraction, the simplex itself included. 



