1 



DERIVED FROM THE REGULAR POLYTOPES. 1 3 



1920 f^^SP 3 2P Q 2G2P 8 tT2tCO: 



= 160 CO + 960P 3 + 320P 6 +480<7+240i> 8 -f 160/7 7 -f 80 tCO 



. e. 2400 limiting polyhedra, 



According to the symbols the limiting polytopes split up into five 

 groups viz. (3'2'2'L'i), (3' 2' 2' l')[l], (3'2'2')[T 1], (3' 2') [2' 1' 1], 

 [2' 2' 1' 1], i. e., taken in the same order of succession, of e 2 e- 3 S(o), 

 P c0 , (3; 8), P tC0 , e ie . 2 C 8 . 



So we find through P 



e 2 e 3 tf(5) + F co + (3 ; 8) + 2 P tc0 + e, e, C 8 

 and therefore in toto 



1920 (^ — 60 — + ^4 + ~2^ + ~96~ + ~T92~ y 



= 32 e x e 3 S(b) + 80 2> co + 80 (3 ; 8) + 40 P tC0 + 10^ e 2 ft, 



i. e. the same 242 polytopes found in the preceding article. 



54. If we exclude once more the "petrified" syllables (11), (1 11), 

 etc. introduced in art. 9 we can state the : 



Theorem XXX. "We obtain the extended symbols of all the 

 groups of ^/-dimensional limits (P) d with different symbol of any 

 given ^-dimensional polytope (P) a derived from the measure 

 poly tope M n of space S ni if we split up the n digits of the pattern 

 vertex in all possible ways, either into n — d or into n — d -\- 1 

 groups of adjacent digits, place all these groups with exception of 

 the last one of the second case between round and this last one 

 between square brackets, and consider these bracketed groups as 

 the syllables of the extended symbol." 



Proof. As in art. 10 we represent the n - - d different syllables 

 in round brackets by (. .)*'»(■ . ) A \ ...,(. . ) '"~ d . So, in the first 

 case we have the relation l\ -f- k 2 . . . -f- k n _ d = n, whilst addition 

 of the syllable [. .]*' with k' digits leads in the second case to 

 the condition Jc x ~\~ k 2 -\- . . ~f- k n _ d -j- k' = n. In both cases we 

 suppose in order to fix the ideas that to ( . . ) h ' 1 correspond the coor- 

 dinates a? 4 , a? 2 , . . ., œ k , to ('. . ) k * the coordinates œ kl+i , ^ /ii+2 , . . ., 

 x k +/ . , etc. and in the second case to [ . . ~] k ' the coordinates x n _ A , + 1 , 



^n — k' + 2 1 • • • j %n • 



Here too the proof splits up into three parts. As the first case 

 can be deduced from the second by supposing k' = 0, we indicate 

 the alterations which the three parts of the proof of art. 10 have 

 to undergo for the second case only. 



