34 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



p — 2 . 1 



and a — — — ; as the true coordinates of O are — . — in both 



cases, the distance to is diminished by 



p—l 1 / p— 2 1 \ 



0+1 : — : I a — ; I = 



n + 1 % -f- 1 V » + 1 » + 1 / 



ƒ? - ■ 1 i J» 2 72 



# -|- 1- ^ -j- 1 # -4- 1 



Moreover it is evident that any two of these polytopes g of the 

 first form, e.g. those lying in the spaces œ ± = a-\-l, x 2 =a-\~\ 

 are separated by the right prism with the base polytopes 



00, = — CI ' 1 , 00 2 ~~~ CC , t2?g , 00^ , . . . === yO , C , . . . , U J , 

 X, <2 , <2?2 # "| J- ? <#?3> t#4> • • • (0 , 5, . . . , U) , 



while the corresponding tw r o g Q of the second form are in contact 

 with each other by the n — 2 -dimensional poly tope 



By combining the theorems IX and X we can find the symbol 

 in operators c and e k of any contraction form, i. e. of any form 

 the zero symbol of which contains two or more largest digits. To 

 that end w T e have. 



1°. to pass to the corresponding expansion form by adding one 

 to the first digit, 



2°. to treat the zero symbol of this expansion from according 

 to the rule deduced from theorem IX, 



3°. to put c before the obtained result. 



In the following we give some examples of the deduction of c 

 and e,. symbols from zero symbols, in connexion with the three 

 possibilities which may present themselves, if we consider the two 

 different zero symbols of a form without central symmetry, according 

 to the appearance of the contraction symbol; they are 



(4332210) = — (4322 110), i.e. e 2 e,e 5 £ (1) (7) = — e ± e z e h 8^{l\ 



(4432210) = — (4322100), „ ce i e 2 e^e 5 S^(7) = — e^e z e^\l) t 

 (3332100) = — (3321000), „ ce 2 e. à e, #W{7) = — ce, e 2 e 3 £ (1) (7). 



Remark. According to the developments of the preceding article 

 the contraction c k always cancels the expansion e k ; so we can deduce 

 from the theorems VI and IX that the operation c k can only be 

 applied to expansion forms in the zero symbol of which the h -j- 1 st 

 and the k -j- 2 nd digit are unequal and that the zero symbol of 

 the new form is found by subtraction of a unit from the first 



