1 C) ANALYTICAL TREATMENT OF THE POLYTOPES SECULARLY 



10. We now prove the general theorem: 



Theorem III. "We obtain all the groups of ^-dimensional limiting 

 polytopes (P) d with different symbols of any given ^-dimensional 

 polytope (P) n derived from the simplex JS>(n-\- 1 ) of 8 n , if we split 

 up the n -\- 1 digits of the pattern vertex in all possible ways into 

 n — d -f- 1 groups of adjacent digits and consider these groups, 

 each of them placed in brackets, as the syllables of the extended 

 symbol." 



We remark that the extended symbols of the four groups of edges 

 of the preceding article satisfy the conditions of the theorem. 



We represent the n — d -)- 1 different syllables by (. .) M , (. .) fe2 , 

 . . .,(. ,) k n—d + i f where k l9 k 2 , . . . 9 k n _ d + i indicate the numbers of 

 the digits, so that we have k± -\- h % -j- . . . -\~ /c n _ d + i = n -\- 1; 

 moreover in order to fix the ideas we suppose that the coordinate 

 values of (. . ) /l1 correspond to the coordinates x l9 x 2} . . . x k9 those 

 of (. .) />2 to the coordinates x k + ± , x ki + 2 , • • . os k + /fa , etc. 



Proof. If we put it short the three moments of the proof are: 



a) As petrified syllables have been excluded we obtain by proceeding 

 according to the indications of the theorem a ^/-dimensional polytope 

 P d , the vertices of which are vertices of P n . 



b). As the digits of the syllables are adjacent digits of the symbol 

 of (P) n , the {P) d obtained is a limiting polytope of (P) n . 



c) As the system of equations representing any limiting polytope 

 (P) d of (P) n occurs under all the systems of equations corresponding 

 to the limiting (P) d of (P) n furnished by the theorem, we obtain 

 by means of the theorem all the limiting polytopes (P) d of {P) IV 



We consider each of these three parts for itself. 



a) The polytope obtained is a (P) d . 



By the exclusion of petrified syllables we are sure that any syllable 

 (. .) /l with k digits allows the vertex, the coordinates of which are 

 the n -j- 1 digits of the symbol of (P) n , to coincide successively with 

 all the vertices of a determinate k — 1 -dimensional polytope {P) k _ i 

 situated in a space S k _ l parallel to a limiting space S k _ i of the 

 simplex 8{n -\~ 1) of coordinates. So in the case of the n — d -j- 1 

 syllables (, ,) w , (. .) />2 , . . .,(. .) fc n-d-t-i under consideration the poly- 

 tope obtained will be a prismotope, the constituents of which are 

 polytopes (P),._ i , where /• is successively & l9 fc 2 , . . . 9 /c n _ d+ . 1 , situated 

 in spaces parallel to the limiting spaces /S,, i _ 1 = (A t A 2 . . -,^-k), 

 S k i = C^x + i ^k y + 2- • - > d k + /,), etc. of the simplex of coordinates, 

 which spaces are by two normal to one another according to art. 5. 

 This prismotope which may be represented by the symbol (P H _ i ' 9 

 Pk2—i'> - • ; Pjc _d + i— i) ^ S a polytope {P) d . For its number of dimen- 



