60 REV. T. P. KIRKMAN ON THE REPRESENTATION 



rene in order 3, 6, and 2 faces, and a ridge-summit made by 

 the faces which are separated from those triangles the first 

 from the first by one face, the second from the second by 

 three, the third from the third by no faces, counting in the 

 order of those triangles — has four varieties, which are the fol- 

 lowing, each of 26 summits. 



Aab Abe Acd Ade Aef Afg Agh Ahi Aij 

 ahc a'cd j'de efg cgh e'hi e'ij 



Ajk Akl Aim Amn Ana 

 n'jk n'kl Imn d'na dyn, 



Aab Abe Aed Ade Aef Afg Agh Ahi Aij 

 abe n'ed j'de efg e'gh e'hi e'ij 



Ajk Akl Aim Amn Ana 

 n'jk n'kl Imn cna d'j'n. 



Aab Abe Acd Ade Aef Afg Agh Ahi Aij 

 nbe a'cd g'de efg d'gh d'hi d'ij 



Ajk Akl Aim Amn Ana 

 n'jk n'kl Imn d'na dj'n. 



Aab Abe Acd Ade Aef Afg Agh Ahi Aij 

 abc n'cd g'de efg d'gh d'hi d'ij 



Ajk Akl Aim Amn Ana 

 n'jk n'kl Imn cna d'j'n. 



The faces are described thus : 



Aabcdefghij k I m n 

 14 53467344464437 

 14 43557344464438 

 14 53494354454437 

 14 4358435445443 8. 



To give an example of the maximum number of triangles, 

 let «=14, A:=6 : let the triangles be 6, of, f h, ;, /, and the 

 ridge summits be aeg, ace^ akg, and gik. 



