DERIVED FROM THE REGULAR POLYTOPES. 83 



number of the limits, the second that of the limits concurring in 

 any vertex. But in the sixth part, making its appearance for n = 5, 

 the arrangement is an other one, the character of the limiting poly- 

 topes and their total number having interchanged places; so in any 

 case the total number appears at the head of the column and the 

 character at the first of the two horizontal places in the column. 

 So the polytope e^HM 5 =^[31111] with the characteristic num- 

 bers 80, 400, 720, 480, 82 is limited by 480^ 3 and 240/? 4 of 

 which 18 and 12 respectively meet in any vertex, by 2407 7 and 

 240^3 of which 12 and 18 respectively meet in any vertex, and 

 by !0C iG , 40P r , 16/Y(5) and 16e 3 /S[5) of which 1, 4, 1 and 4 

 respectively meet in any vertex. 



Table IX, concerned with the hmpd. in S 6 , has been divided in 

 the same way into seven main parts. It will be clear without farther 

 explanation; only we are bound to add that in the first column of 

 the sixth part 2 „ means that 2(7 16 is to be taken 60 times and 

 that in this part and the next the numbers of limits concurring 

 at any vertex have been omitted. 



96. We insert a few remarks about the character of the limits. 

 Faces. We find only j» 3 , j» 4 , p e . 



Limiting bodies. The set of limiting bodies obtained for n = 5 

 is completed by the addition of C for n = 6. 



Limiting poly topes. In general the limiting poly topes are 

 1°. Simplex forms, deduced from S(?i), S{n — 1), . . ., #(3), 

 2°. Half measure polytope forms, deduced from HM n _ u HM n __ 2 , 



3°. Prismotopes the constituents of which are simplex forms, deduced 

 from S(n — 1), . . .,£(3), and at most one half measure polytope 

 form, deduced from BM n _ 2 , . . . , HM$. 



This general result shows that the list of limiting bodies is 

 complete for n = 6. Moreover that the list of fourdimensional limits 

 will be complete for n = 8 , as the case n = 8 brings C s for the 

 first time, etc. 



In order to show how theorem LVII w T orks we give the list of 

 the limits (P) 6 of the tendimensional form ^[9 775533311]: 



(9775533), — (9775) (5333), (9775) (533) (31), — (977) (55333), 

 (977)(5533)(31),(977)(553)(331),— (97)(755333),(97)(?5533)(31), 

 (97)(7553X331),(97)(755)(3331),(97)(75)(53331),(97X75X533K311), 

 (97)(75)(533)(31-1),(97)(75)(53)(3311),(97)(75)(53)(331— 1),— 

 (7755333), — (775533) (31), — (77553) (331), — (7755) (3331),— 

 (775)(53331),(775)(533)(311),(775)(533)(31— 1),(775)(53)(3311), 



