4 ANALYTICAL TREATMENT OF THE POLYTOPE3 REGULARLY 



and dividing the sums by 3. It is clear, that , in order to get a 

 polyhedron similar to ce 2 1 , the division by 3 may be omitted. 

 The auxiliary polyhedron so defined will be specified in the follow- 

 ing computation by ce 2 I'. 



First we compute the vertices of ce 2 I' (two types): 



1°. I 1 -f- e , , 2 



/ , 2 , 1 +e 



J , -2 , 1-f-e 



ce 2 T 1 -f e , , 4 -f - 2<? ; 



2°. ƒ 2 , 1 + <? , 



I , 2 , 1-fe 



/ 1 + * , , 2 



cr 2 /' 3 -f e , 3 -f-, e , 3 '-f- e . 



Since the faces of / now considered have two vertices in common, 

 the vertices of ce 2 1' deduced from them are adjacent. Their dis- 

 tance will be found to be 2 -}- 2e. 



Hence to obtain the coordinates of the vertices of ce 2 I , those 

 of re 2 I' must be multiplied by 



A 



2 I 2 e ■ 



2 + 2e 



Two coordinate-symbols are found : 



1°. ce 2 I' 1 -f e , , 4 -f le 



ee 2 I [ 2 , , 3 -j- e ] : 2 12 vertices 



2°. ce 2 I' 3 -f e , 3 + e , 3 -f e 



ce 2 I fl-f-e , 1+ S -, l+e ] 8 „ 



Total . . . 20 vertices. 



119. To deduce e l J by composition of ce l I and / we must 

 add to the coordinates of each of the vertices of ce l I those of 

 each of the adjacent vertices of I, i. e. of each of the vertices of 

 ƒ from which it is obtained in art. 117. We omit however thoee 

 additions which would produce a vertex included in a coordinate- 

 symbol already obtained. 



