72 ANALYTICAL TREATMENT OF THE POLYTOPES EEGULARLY 



so these repetitions must be arranged in the same manner round 

 every edge. But this decides that the arrangement of all the con- 

 stituents round every edge is the same, as there is only one other 

 constituent, viz g Q . So the degree of regularity of iV 18 , iV ig , iV 27 is |-. 



In all the nine cases of the second group there are two or more 

 different kinds of edge and the degree of regularity is y 3 ^. From 

 these cases Ave treat a couple of examples. 



Example e x M(C i6 ). All the repetitions of the groundform are repre- 

 sented by the system of the three symbols 



[Ga ± + 4, 6a 2 -\-2, 6% +0, 6« 4 +0], lLa L even) 

 -1 [6*!+ 3 + 3, 6û2+ 3-J-3, 6% + 3+1 , K + 3 + 1], Ha L odd . 

 — | [6^+3 + 3, 0a 2 4-3+3, 6^4-3-fl, 6a 4 4-3 + 1], Steven) 



So the repetitions r with 4, 2, 0, as vertex are: 



2, 0, 



-4, 0, 



3 — 1,-6 + 3 + 3, 

 3 — 1, 3 — 3,- 



3 — 1, 3 — 3, 



3 — 1,-6 + 3 + 3,- , , _, 



which may be indicated by the symbols r i9 r 2i . . , t 2 . Now the 

 adjacent vertex 2, 4, 0, is vertex of the six repetitions and 4, 0, 2, 

 of r i9 8 29 ti only. So there are two kinds of edges and the degree 

 of regularity is y 3 ^. 



Example e 3 JT(C7 16 ).If we telescope [pp i -\-Q l9 pp 1 + Q>,pp-à+ Q^PPnr\- QJ 

 into [<7 l5 ^ 2 , ^ 3 , ^ 4 ,] (ƒ;) pi,p 2 ,Ps,p!i the repetitions of the groundform 

 [2+ N/2, V/2, V/2, Y/2] can be represented by 



4 







4, 





re 



-2,6 



1 



2 



I 



Ï 



Ï 



'J 



[ 



[ 

 [ 

 [ 



3 + 1, 

 3 + 1, 

 3+1, 

 3 + 1, 



0]. 



. . u 



0]. 



■ ■ r 2 



3 — 3] . 



. . s t 



6 + 3 + 3] . 



. . s 2 



3 — 3] . 



. . 4 



+ 3 + 31 . 



. . t. 



[2- y '2, |/2 , |/2 , j/2 ], (1 + 21/2)2*! , 2« 2 , 2« 3 , 2a 4 , 2a; even, 



ï 



!"l_l_9i/2 1 1 11) 4 



LlT ^: 1 + l^i 1 + ^; ,_^j.(H-^)%+l, *.,+!, *«+l. 8..+1.2., odd, 



_}[! +TÏ !+; ,+ 3 ; J4 ,1+V2)JSi+l ' ** +1, 2as+1 - 2 " i+ + even - 



So the repetitions /• with 2 + Y2, V2 9 \/2, V% as vertex are only 



2 -1-1/2, >2 , j/2, 1X2 1, 



1 I- 2^/2 -f 1 - - K2, 1 -I- 2|/2 - 1 + \/%\ 1 + 2*/2 — (1 -f j/2), 1 + 2^/2 — (1 + |/2)1 



Now V 2, 2 + V 7 2, V2, V/2 is vertex of both, whilst on the 

 other hand 2+ \/2, \/2, V : l,—V% is vertex of the first only. 

 So two kinds of edges, degree of regularity A. 



The very last column of Table VII contains the results. 



