14 ANALYTICAL TREATMENT OF THE POLYTOPES EEGULAKLY 



a). The poh/tope obtained is a {P) d . 



Bv the exclusion of petrified syllables we are sure here too that 

 any syllable ( . . ) k with k digits allows the vertex , the coordinates 

 of which are the n digits of the symbol of (P) n , to coincide successively 

 with all the vertices of a definite /• — 1 -dimensional polytope (P) fc _ i 

 situated in a space S k _ i determining equal segments on /• of the n 

 axes OX r Moreover the unique syllable [. .] /v ' with k' digits allows 

 that vertex to coincide successively with all the vertices of a definite 

 ^'-dimensional polytope (P) r situated in a space 8 k . parallel to the 

 space of coordinates S' L . containing the k' axes OX it where i is 

 successively n — k' -\- 1 , n — // ~\~ 2 , . . . , n. The spaces bearing 

 these n — d -\- 1 poly topes (P) k , [k =■ k lt k 2 , . . . k n __ d ), and (P),,. 

 are by two normal to each other. For (P) k lies in the space S k . = 

 0(X, X 2 . , .X k ), {P\ lies in the space S k% = 0(X kt + i X ki +2 . .X,^^), 

 etc. and now the spaces S k , /S^,. . ., ^/ Cn _ d> &k' form a se ^ of 

 coordinate spaces containing together all the axes OX once, i. e. 

 they are by two perfectly normal to each other. So, as two spaces 

 lying in spaces perfectly normal to each other are themselves perfectly 

 normal to each other, the spaces bearing the n — c I -p 1 polytopes 

 found above partake by two of that property. So the polytope under 

 consideration is a prismotope with n - — ■ d -\~ 1 constituents and 

 this prismotope is a (P) ( ; for its number of dimensions is the 

 sum of the numbers k y — 1, k 2 — I, . . ., k n _ d — 1 , k' of the 

 dimensions of the constituents, i. e. the sum of the numbers 

 k i} k 2 , ... k n _ d diminished by n — d, i. e. n diminished by 

 n — d, i. e. d. 



b). The (P) d obtained is a limit of (P) n . 



According to the manner in which (P) d is obtained the coordi- 

 nates of its vertices satisfy the n — d mutually independent equations 



if p ± is the sum of the first k x digits of the pattern vertex, p 2 the 

 sum of the next k 1 digits, etc. As in art. 10 these equations 

 can be written in the form 



Ar t frj+frj fei -f K + . . . + k n _ d 



2 x i =p ± ,JÏ-œ i = p [ -f-'ft , . . . , E x { =p x -\-pir\- .... -\-p n _ d , 



representing n — d limiting spaces 8 n __ i of (P) n , as each of the 

 right hand members is a maximum. For the rest of this part we 

 refer to art. 10. 



c) By weans of the theorem ice obtain all the limits {P) d of (P) n . 



For this part compare also art. 10. 



