DERIVED FROM THE REGULAR POLYTOPES. 89 



faces and take into consideration the expansions e k) (k=2,3,...n — 2), 

 of the M^ only. 



We now pass to the contraction. À motion of the limits of vertex 

 import of ± \a x a 2 . . . a n _j L a n ], i. e. of (# 4 a 2 . . . « n _ 4 a n ) towards the 

 centre gives (a x — A, a 2 — A, . . . ,a n i — A,<z n — A). So the only new 

 form we can get is (% — 1, a 2 — 1, . . . , a n __ 2 — 1,0,0), i. e. a 

 form deducible from M n ^ ] , etc. 



100. We conclude this part by proving the following theorems, 

 which will be useful in the next: 



Theorem LXIII. "The limits of truncation import of 



e,. e k . . e,. e,. HMA 2 ^ 2 ) are e,, _ ± e k _ 4 . . e k ± e k _ v S(n)%(y%)" 



A l A 2 k p-\ A p n A l 1 /l 2 1 >-l _1 *p 1 v J 



According to the preceding theorem we have 



k p~ k p-i k 2~ k \ A i 



p 



c, i 4 2 . . e kpx e kp EM n = \ \%p + 1 , 2p — 1 , . . . , 33 . . 3, 1 1 . . 1]. 



So the limits of truncation import are 



n ~ k P k p~ k p-i h~ k i k \~ x 



(2/>-+ 1, 2p — 1, . . ., 33. .3, 11. .1,- 1), 



i. e. 



n ~ k P k p~ k P -i h -i-h k \- 1 



(2^ + 2, 2p , . . .,44. .4, 22. .2, 0), 

 or reversed 



fcl _i /.^/^ *p-*p-i "■-'■■„ 



— (2/j + 2, 2jö , 2p - - 2, , 22 . . 2, 00 . . 0), 



i. e. — e,. i e,. * . . e k 1 <? A . , /Stw)( 2 J/2). 



A l 1 ''2 V h p—l~ l K p l K ' 



Theorem LXIV. "The number l\ of the units figuring in the 



A i 



symbol of coordinates \\a n a n _^ .... 11. .1] of an Jmipd. in S n 

 indicates how many limits of truncation import pass through any 

 vertex." 



n S k p~ k p-\ h~h k i 



The number of vertices of J [2p -\- 1, 2p — 1, . . . , 33 . . 3, 11 . . 1], 



n—k k —Ic . ka—k* /.-, -1 



p p p — 1 2 1 1 



respectively of its limits (2p -\- 1, 2p — 1, . . . , 33 . . 3, 11 . . 1, — • 1) 

 of truncation import is represented by 



2"- 1 . »! »! 



resp. 



(»— k p )\ {Jcp—k p -i)\,..{k 2 —h)\ h\ ' ""*" (n—k p )l{k p —kp-i)l..Jk,—h]Kki—iy. 



So the 2 n_1 limits of truncation import admit together a number 

 of vertices equal to k A times that of the /impel, itself. 



