DERIVED FROM THE REGULAR POLYTOPES. 23 



The (54) (43) (32) (2210) is a prism of the third rank on a tT 

 as base; it may also be considered as a prismotope [C ; t'T). 



If in the case of {P) n we deduce the limits (P) w _i we find them 

 in the order of succession g n _ i9 g n -2>- • -><j!o °* polytope import to 

 vertex import, when, in proceeding from left to right we take in 

 the first syllable as many digits as possible and keep in it the 

 first digit as long as possible. This principle has been followed 

 throughout in the enumeration of the limiting (P) 6 of the given 

 (P) 9 , as well as in the sixth columm of Table I, 



In the notation of art. 10 (page 17 in the middle) a limit (P) n _ 4 , 

 represented as to its import by g ki is a prismotope {P k ; P n _ k _ i ), 



14. It is worth noticing that in space S n the series of limiting 



elements may include the series of the measure polytope M, for 



n even up to the polytope M n of jS h , for n odd up to the polytope 



"2 Y 



M n + i of S n + i . So, for n = 2m -\- 1 the {P) 2m +i represented by 



2 2 



e ± e 2 e 3 . . . e m S {2m -f- 2) = {2m -j- 1, 2m, 2m — 1 , . . . , 3, 2, 1, 0) 

 admits as limiting element {P) )H + 1 the M llt + 1 with the symbol 

 {2m + 1, 2m) {2m — 1, 2m — 2). . .(3, 2) (1, 0). 



On the other hand, amongst the polytopes themselves, no measure 

 polytope occurs and of the cross polytopes only the octahedron 

 presents itself. We prove that this must be so, for each of the two 

 series separately. 



Measure polytopes. The number of vertices of (5443322210) is 



10! f . . , . ' e 10! 



which can be written m the tor m -— — — -^ — --i—, so as 



2! 2! 3!' 3! (2!) 2 (L!) :5 



to be able to generalize it for any (P) n as 



a\b\c\. ..M' 



where a,b,c,.. .k are arranged in decreasing order and their sum 

 is n -f- 1. Now this form is a product of binomial coefficients 



{n -f- l) a {n — a + \\ {n — a — b -f- l) c 



and there is only one possibility under which this product contains 

 no factors different from two and is therefore a power of two, 

 i. e. in the case n -\- 1 = 2 P , a = 2 P — 1, b=I, giving 



2'M 



. = 2 P 



{2 P — 1)! 1! 



