54 REV. T. P. KERKMAN ON THE REPRESENTATION 



also to use the duads At, ghy fg, ef^ ma ; hence the triplets 

 following are determined, 



mlii^ mgh, mfg, meft mae. 

 There are now 3 edges of the broken line in the face or, of 

 which ae is one; they will be in order «?, ad^ ac, which 

 must be combined with the duads de, cd; therefore 



ade, acd, 

 are determined. We have thus assigned all the 24 triedral 

 summits of the 14-edron on a 13-gonal base, which has 

 two triangular faces k and b, between which lie three faces 

 /, m, and a ; the values of X, ju, and a described above being 

 2, 5, and 3. These triplets are 



Aab Abe Acd Ade Aef Afg Agh Ahi 

 abc acd ade mef mfg mgh mhi 

 Aij Ajk Akl Aim Ama 

 lij jkl Imi mae. 



Of these summits 13 are in the face A, 7 in W2, 5 in /, 5 

 in c, 5 in a, 4 in A, 4 in z, 4 in J, 4 in c, 4 in of, 4 in ^ 4 in 

 ^, 3 in 6, and 3 in A. The faces of P are one 13-gon, one 

 heptagon, three pentagons, seven quadrilaterals, and two 

 triangles. 



8. The number of triangular faces is seen by inspection of 

 the triplets ; for it is equal to that of the triplets made with 

 three consecutive letters ; the middle letter, in alphabetical 

 order, always denoting a triangle. There can never be two 

 triplets made with four consecutive letters, as abc, bed", for 

 be, which occurs in the summits of the base A, cannot twice 

 occur again. Therefore b and c cannot both be triangles, nor 

 a and 6, nor any consecutive pair of faces. Hence follows 



Theo. VI. A ip-edron, having a (p — V)-gonal face, and 

 all its summits triedral, has not more than ^(p — 1) trian- 

 gular faces, when p is odd, nor more than |(p — 2), when p 

 is even, 



A q-peak, having a (q — \)-gonal summit, and all its 



