DERIVED FROM THE REGULAR POLYTOPES. 27 



which it ensues that the extent of the restriction depends on the number 

 of syllables which the symbol of any constituent may contain. This 

 number is evidently two at most. For the process can only afford 

 besides the original minimum digit a n one new minimum digit, 

 viz. zero in the case a = a i and unity in the case a = a ± -f~ 1. 

 So we have to hunt up only new constituents the symbols of which 

 are either monosyllabic or composed of two syllables. 

 If we take all the p t equal to one we find 



[(2^,-f- 1)05+^!— a,{U 2 -\- \)a-\-a 2 — a,. . ., 



(2^_i + l) a + a^—a, (26 n -f l) a -f a n — a], 



or, if we replace negative permutable parts by the positive ones 

 of the same absolute value, rearrange these positive parts according 

 to decreasing order and substitute for brevity j2' for 25 -f— 1 , 



winding up in zero for # = a^ and in unity for a = a ± -\-\. So 

 we find the repetitions of the new constituent y of the last list 

 of the preceding article. This form </ and the given form g n we 

 started from are the only constituents of measure polytope descent 

 proper. 



If we transform the first k digits of T by the transport of a 

 units from the permutable parts to the immovable ones and put 

 each of the two sets of digits, the set of the h transformed ones 

 and the set of the n — h untransformed ones, between square 

 brackets, we get after rearranging, if j2\ still replaces 2b i -f- 1 and 

 /3, is substituted for 2ö i 



[fi\a-\-a— a„,p 2 a-\- a— a k . u . . . ,£'*£* + « — a A ~\ 



[fik+l a + ^/.+l > Afe+2 a + a k+2 , . . . , /3 n _ 4 a + dnr-U Pn a + «n]i ■ • T ') 



revealing the new constituent 



[a — a k , a — ct> k —\ > • • • > a — a \ \ [_ a k+\ > a k+2 > • • • > a n— i > a n\ > 



a prism otope (P k ; P n _ k ) with the constituents (P) A . and (P) n _ fe repre- 

 sented by each of the two syllables of the symbol taken separately; 

 if the digits of the second syllable correspond to the coordinates œ it 

 œ 2 , . . . , œ n _ k and those of the first syllable to £C n _ k+i , w n __ k+2 , . . . , so ni 

 this prismotope is the constituent g n _ k of the last list of the 

 preceding article. In the latter case the different positions of ÇP) n _ k are 

 parallel to 0(X ± X 2 . . -X n _ k ), those of (P) A .to 0(X n _,, +i X n _ k+i . . .X n ). 

 So we find again all the new constituents obtained formerly by 

 geometrical multiplication. 



