70 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



a P =(n) p 2"~» + {n) p+i 2 n ~\ p = Ç, 4,. . ., n — 1. 



Neither is it difficult to prove that the characteristic numbers 

 a' satisfy the law of Euler. To that end we go back to the 

 relations given above and transform the Eulerian expression 

 a — a\ -f- a 2 — . . . -f- (• — - 1)"' _1 d n . 4 into 



— [«2 — 4^3 + i «0 i«4 — «5 +-.. + (— I)" (»)»}], 



of which two sums between square brackets the first contains the 

 contributions of the first column (old elements) and the second of 

 the second (new elements). Now we add to each of these two sums 

 between square brackets -J- a — a f -f- a 2 . So we get 



0„ _ „, + «, _ . . . + (— i )»-*«„] 



— [i «,, — a t + 2« 2 - 4« 3 + J- a \{n\ — [n\ + ... + (- 1)» «„}]. 

 But as we have 



— ûfj -f 2a 2 — 4r/ 3 = -J- a ô j— (») t -f (») a — (rc) 3 | 



the second sum disappears, as it is equal to 



^o|l— W1 + W2 — Ws + .-. + C— l) n Wn|=i%(l — If- 



So we find that the Eulerian expression of ZO/„ is equal to 

 that of M n and has therefore the value 2 for n odd and the value 

 for n even , etc. 



We give here the results up to n = 8. They are 



»= 5 ... ( 16, 80, 160, 120, 26), 



n = 6 . . . ( 32, 240, 640, 640, 252, 44), 



» = 7 ... ( 64, 672, 2240, 2800, 1624, 532, 78), 



n=8 ...(128, 1792, 7168, 10752, S288, 4032, 1136, 144). 



In the outset we remarked that BJ\l b admits two kinds of limits 

 (/) 4 , viz. cells (7 16 and simplexes #(5). Here we remember that in 

 general for n > 4 the HM n is bounded by two kinds of limits 

 (/)„.,, viz. limits HM n _\ forming Avhat remains of the limits M n _ ± 

 of M n and limits S(u) replacing the vanished vertices of M n . It will 

 be useful to call the EM ' n _ i the "original", the S(n) the "trun- 

 cation" limits. 



93. In the cases of the offspring of simplex, measure polytope, 

 and cross polytope we have used two different methods for the 

 determination of the characteristic numbers, one fulfilling the exi- 

 gencies for n < 6 as far as these numbers only are concerned, an 



