56 REV. T. P. KIRKMAN ON THE REPRESENTATION 



our way by a line of edges to some point in the base A, which 

 lies between a and 7, or between y and a, as the case may be, 

 looking along the series of the faces abed ... w, of which 

 two are a and y ; and this point is an angle in a triangular 

 face, which lies between a and 7, or between y and a. 



And if between a and y there be more than one triangle, 

 we shall arrive by proceeding from aye along ay to one or 

 more such summits as aye, the unions of branches of the ridge 

 line leading to such triangles, and this only along edges of 

 faces between a and y (a and y being included among them) : 

 so that no face which does not lie between a and y, can be 

 one at such a summit. And if there be several triangular 

 faces lying between y and a, we shall, by proceeding along ay 

 through the point aye, arrive at one or more of such (summits 

 or) triplets made with faces that lie between y and a (these 

 two being included) : i, e., no face j3 which lies betwen a and 

 y can form part of such a triplet. 



10. Generally, if there be k triangular faces, there are to 

 be added to the triplets denoting their vertices and the n — 1 

 angles of the base 



2n—4—ni-l—.k=n—3—k 

 triplets; of which there cannot be either more less than 

 n — 1 — 2k, which contain each a duad of consecutive letters, 

 this being the number of such duads remaining to be repeated. 

 There must be, therefore, 



n^3—k-^(n-.\—2k)=k—2 

 triplets containing no duad of consecutive letters. 



If aye and ^A^ be two of these, there must be at least one 

 triangle lying between a and y, and at least one between 

 y and a ; and the same must be true of 0\, (j)e, &c. : nor can 

 X lie between a and y, unless X and (jt all lie between them, 

 a and y being counted as so lying. Here a and might be 

 one and the same. 



