102 ANALYTICAL TREATMENT OE THE POLYTOPES REGULARLY 



So A 3 , B 2i C 2 , D t remain uncovered, i. e. have still to be covered 

 by limits (/) 4 of E. We represent these limits (/) 4 of E by E a , 

 E b , E c , E (l , indicating by the subscripts the constituents with which 

 they are in (/) 4 contact and repeat these limits in the column with the 

 headings E a , E b , E c , E d . Finally from these limits we deduce the 

 constituent E itself, see the last column of the seventh part of the 

 table. We remark that this fifth constituent is a prism otope, the 

 two components of which are HM S (or e. 2 HM z ) and p k (orjp 8 ) ; it 

 presents itself if and only if either e B , or e 4 , or both operations 

 are present. 



In applying the second method to JS 5 we have to extend the 

 J/ 4 (2/)) of fig. 20 with the broken line O P ± P 2 P% P 4 of edges leading 

 from O to the opposite vertex P 4 into an MJ? v) with O P i P 2 P 3 P 4 P b 

 as corresponding broken line of edges from O to the opposite vertex 

 P. Tf we represent the midpoints of OP 5 , OP^, OP 3 respectively 

 by Q 5 , Q 4 , Q 3 we find for the new origins leading to the consti- 

 tuents C, A, D, E the points P i9 Q 5 , Q 4 , Q 3 with the coordinates 



2p, 



0, 



Q, 



o, 



.0 



P> 



?• 



p> 



p> 



p 



P> 



P> 



p> 



p, 







P> 



P> 



p> 



o, 







So in the case e.e.e.XM,, i.e. [3'2'Tl'l]V/2 with ^ = 5 + V" 2 

 the constituents A, E, E are obtained by the three processes 



6-f-V 7 2 , 



4 + V2 . 

 5-J-V/2 , 



2-f V2 



5 + V/2 



, 2+V/2 , 

 , 5+^2 , 



V/2 

 5 + V/2 



1 



6 + V/2 



5 + V/2 



— 1 , 



, 4 + V/2 



, 5+^2 : 



— 3 



, 2 + V2 

 5 + V/2 



, —3 , 



, 2 + V/2 , 

 , 5 + V/2 , 



— 5 



|V2] 

 



1 



6 + V2 



5 + ^2 



— 1 , 



, 4-f-V2 

 , 5+^2 



— 3 



, 2 + V/2 



, 5 + V/2 



, — 3 , 



,[2+V2, 

 , 



[V2] " U 



V2] 







I 



— 1 



— 3 



,^ + V2 , 



V/2] SU 



subtr. 



giving respectively \ [53311], | [3311] [l]\/2, l [311] [l'l]V/2. 

 The results obtained in this way are collected in Table XL To 

 this we have only to- add a few remarks. 



The processes used just now show clearly why the syllables 

 L[3311] and ^[311] of D and E must correspond in digits with 



