DERIVED' FROM THE REGULAR POLYTOPES. 107 



The proof of this theorem is to be based on the remark that the 

 order of the group must be half of that of theorem XL1IT on 

 account of the alternate truncation. 



113. As to the application of Elte's scale of regularity we have 

 to use theorem XLIV. We illustrate this , sticking to the original 

 scale, by the following examples. 



a). Example -^-[11111]. Here we find one kind of edge, one kind 

 of face, but two kinds of limiting tetrabedra, viz. tetrahedra of 

 body import and tetrahedra of truncation import. So the contribu- 

 tions to the numerator are 1 from each of the three groups of 

 vertices, edges, faces, and 4 from the limiting bodies. So the frac- 



tion is — - — - 



5 10* 



b). Example -^[553111]. Here we find three different groups of 



1 + - 

 edges (5, 3) , (3, 1), ![!, 1]. So the fraction is — ~— — = \ 



c). Example IV C*or 5 , CV» 5 > This simple net admits one kind of 

 edge, one kind of face, but two kinds of limiting tetrahedra, as a 

 tetrahedron of body import of HM h is common to four HM 5 , a 

 tetrahedron of truncation import to two HM 5 and one Cr 5 . So we 



3 + i 1 



find 6 12- 



d). Example e l «? 3 3TJf 6 . Here we have to deal with three groups 

 of constituents represented with their frames in the table 



B . . [321000] 2. . . (2 Pl , 2je 3 , 2p s , 2 P4> , 2p- , 2 H ) 5, Eleven, 

 C . . [221000] 2 ... (2 Pl , 2 n , 2 Ps , 2p 4 , 2 Ph , 2 Pg ) 5,2/; odd, 

 A . .1[555311] . . . . (2 Pl + 1, 2 lh + 1, 2^3 + 1, 2/> 4 -f- 1, * A -f- 1, 2p 6 -f 1) 5. 



So through the vertex 6, 4, 2, 0, 0, pass 



6, 4, 2, 0, 0, 0] B i 



10+4,10+6, 2, 0, 0, 0] B 2 



10+4, 4, 2, 0, 0, 



5 + 1, 5—1, 5—3, 5—5, 5—5, 5—5 



i[ 5+1, 5—1, 5— 3, (—5 + 5, 5—5, 5—5) 



c 



A 



. . A 2 , A- 3 , A k 

 5 + 1, 5- -1,5— 3, (--5+5,- -5+5, 5— 5)].. J 5 , J G ,A 

 5+1, 5— 1,5— 3, —5+5, —5+5,— 5+5] J 8 . 



Now the edge (64)2000 belongs to all these poly topes with 

 exception of C, 6(42)000 belongs to all with exception of B 2 , 

 whilst 64(20)00 belongs to seven only. So we find three kinds of 

 edges and the fraction is A. 



