DERIVED FROM THE REGULAR POLYTOPES. 1 7 



sions is the sum of the numbers k A — 1 , k 2 — 1 , . . . , k u _ d + i — 1 

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

 />\, k 2 , . . . , k n _ d + i diminished by the number of the constituents, 

 i. e. n -j- 1 diminished by n — d -j- 1 , i. e. d. 



We pass from the extended symbol of a (P) d formed according 

 to the prescriptions of the theorem to the unextended symbol by 

 omitting the syllables containing only one digit. So the unextended 

 symbol contains only syllables with two and more than two digits. 

 If all the syllables of the unextended symbol bear two digits, the 

 polytope (P) d is a measure polytope; if this symbol contains only 

 one syllable with more than two digits, the polytope (P) d is a prism, 

 may be of higher rank; if the symbol contains at least two syllables 

 of more than two digits the polytope (P) d is a prismotope, may 

 be of higher rank, in the restricted sense of the word. This explains 

 how we have to interprète the result found above that all the 

 limits of (P) n are prismotopes. In the particular case of the limits 

 (/) n _ t of the highest number of dimensions, where we have to split 

 up the digits of the pattern vertex of (P) n into two groups, we 

 find, if n -\- 1 is split up into k and n — k -\~ I, the result 

 (Pk—il P n -k)> which is a non specialized 1 ) polytope (P) n _i for k = 1 

 and k = n, a prism on a non specialized polytope (P) n _ 2 as base for 

 k = 2 and k = n — 1 , and a prismotope in the narrower sense 

 for all the intermediate values; i. o. w. of the limits (/) n _ 4 represented 

 elsewhere {Proceedings of the Academy of Amsterdam, vol. XIII, 

 p. 484) in connexion with the notion of import by g , g \, . . . ,y„_ 4 

 the forms g and g n _ i are non specialized polytopes, the forms g x 

 and g n __ 2 are prisms and the forms g 2 , g 3 , . . . g n _ 3 are prismotopes. 



b). Tit e (P) d obtained is a limiting body of (P) n . 



A polytope (P) d , the vertices of which are vertices of (P) n , is 

 a limiting body of (P) a — and not a section of it — , if we can 

 indicate n — d limiting spaces /6 > >i _ 1 of (P) n containing it. Now, 

 according to the manner in which (P) d is obtained, the coordinates 

 of its vertices satisfy the n — d -\~ 1 equations 



Xi + a? 2 -f- . . . -f œ kx = Px, œ ki + ! + œ ki + 2 + • • • + ®k + k 2 = V* 



œ ky + k 2 + 1 " " 93 k l + A-, + 2T * ' • \ W k l + /,', + k 3 =z Pó-> e ^ c - 5 



iî % Pi is the sum of the first k x digits of the pattern vertex,^ the 

 sum of the next k 2 digits, p 3 of the then next k 3 digits, etc. These 

 n — d -\- 1 equations, only connected by the relation holding for 



') Here "non specialized" means: according to the mode of generation neither prism 

 nor prismotope. About this last form art. 13 will give more particulars. 



Verhand. Kon. Akad. v. Wetensch. (l ste Sectie) Dl. XI. G 2 



