2 ANALYTICAL TEE ATMEMT OP THE POLYTOPES REGULAELY 



(321 1) (0) (0) = (3211) = tl\ (321) (10) (0) = (321) (10) = P 6 , 

 (32)(110)(0) = (32)(I10) = P 3 ,(32)(1)(100)=(32)(100) = P 3 , 



(3) (2110) (0) = (2110) == CO, (3) (21) (100) = (21) (100) = P 3 , 



(3) (2) (1100) = (1100)= O, 



the numbers of which are respectively 



(6) 4 .2!:2 = 15 , (6) 3 .(3) 2 =60 , (6) 3 . (3), = GO , 

 (6) 3 • (3) 2 = 60 , (6) 4 . 2! = 30 , (G) 3 . (3) 2 = 60 , 



(6), .2! = 30 , 



i e. 



1 5 tT-\- 30 (O -f CO) + 60 P G + 180 ]\ = 315 limiting bodies. 



Limiting poly topes. There are four groups of limiting poly topes, viz. : 



(32110) (0) = (32110) = ^3 # 5 l ), (32 I) (100) = (6 ; 3), • 

 (32) (1100) = P , (3) (21100) = (21100) = e 2 <S 5 , 



the numbers of which are. 



(6) 5 =0 , (6)3 = 20 , (6) t =15 , (6), = 6. 



So we find 



6 e x e. s S 5 -\- 20 (6 ; 3) -f- 1 5 P -f- 6 e 2 8 S = 47 limiting polytopes 



and the characteristic numbers are 



(180, 630, 720, 315, 47), 



in accordance with the law of Euler. 



12. Though the introduction of the extended symbols has enabled 

 us to simplify the theoretical considerations it cannot be denied that 

 the unextended symbols are better fit for practical use. Therefore 

 we insert here a corresponding version of theorem III , but to that 

 end we have to enter first into a distinction of the digits of the 

 syllables of the unextended symbols. We will distinguish the digits 

 contained in any of these , syllables into end digits and middle digits, 

 the first and the last digits and the digits equal to these being the 

 end digits, the remaining ones — if there are some — the middle 

 digits. So in (3210) there are two middle digits 2 and 1, in (2110) 

 there are two equal middle digits 1, while in (2210), (21 00) there 

 is only one middle digit and in (1000), (1100) none. Now w r e can 

 repeat theorem JIT in the new form: 



Theorem III'. "We obtain a (P) (/ the vertices of which are vertices 



l ) Compare the small table unter art. 3. 



