80 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



distinction of the different kinds of limits (l) p by what we have 

 called formerly "unextended" symbols. If we take care to exclude 

 always the petrified syllables we can formulate the method in: 



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

 tope (P) d the vertices of which are vertices of the given limpd. 

 of 8 n by applying to the n digits of the symbol of coordinates 

 i [ a i a 2 • - • a n—\ a n\ °f this poly tope one of the three following 

 processes : 



1°. Take the last digit a n , first with the positive and afterwards 

 with the negative sign, and place for both cases between pairs of 

 round brackets either one group of d -\~ 1 digits, or two groups 

 containing together d -\- 2 digits, or three groups containing together 

 d -\- 3 digits, etc., omitting the digits not included. 



2°. Place before -^-[11] of the remaining digits a ± ,a 2) . . . ,a n _ 2 

 between pairs of round brackets either one group of d digits, or 

 two groups containing together d -\- 1 digits, etc., omitting the 

 digits not included — and the syllable with one digit for d = 1. 



3°. Place before ^ \_a n _ fc + 1 a n _ k+2 . . .a n _ 4 a n ], w T here k = 3, 

 4,. . .,d successively, between pairs of round brackets either one 

 group of d — k -\~ 1 of the remaining n — k digits, or two groups 

 containing together d — k -\- 2 of these digits, etc., omitting the 

 digits not included — and the syllable with one digit for d = 7c." 



"In each of these cases the (P) d obtained will be a limiting poly- 

 tope of fanpd., if the- syllables between round brackets satisfy the 

 two following conditions: 



a) each syllable with middle digits exhausts these digits of the 

 symbol of the given hwpd. , 



b) no two syllables without middle digits have the same end digits/ 3 

 The proof of this theorem, forming an adaption of theorem XXX 



to the special character of the hwpd., embodied in the 1 before 

 the square brackets of their symbol, can be copied from that of 

 theorem XXX and theorem XXX'. 



We apply it to two definite examples, one in #(5), the other 

 in /SJj. 



Case J [5 5311]. — 



If we place before a vertical stroke the limits deduced from 

 55311 and after it the different ones furnished by 5531 — 1, 

 we get 



(/),... (53),, (31^, -J [11]» 



(/),. . .(553) l) (53l) i ,(3ll) 1 ,(53)J [11]. 2 (31- -1) 2 



(/) ;! . . .(5531).,, (5311),, (553)J[11] 1; J [311], (581 -l) t 



{l\. . .(5531 1),, i [5311], (5531 — 1\ 



