58 EEV. T. P. KIRKMAN ON THE REPEESENTATION 



Kab Abe kcd Ade Aef Afg Agh Ahi Aij Ajk 

 nab bed g-de efg d'gh d'hi d'ij b'jk 



Akl Abn Amn Ana b'hn 

 tvkl Imn b'd'j. 



The first has the fifteen faces, Aabc . . . Imn, in order, of 

 14, 3, 7, 3, 5, 7, 3, 4, 4, 4, 6, 5, 4, 3, 6 angles, and may be 

 represented by 



A a b c d e fg h i j k Imn: 



14 3735 7 344 4 064 36 



the second differs only from this in the faces d, e, g, and j, 

 and is represented by 



A a b c d efg h i j k I m n» 



14 373843544554 36 



There are then two 15-edrons which have only triedral 

 summits, stand on a 14-gonal base, have four triangles, a, e,f, 

 and 7W, between which intervene 1,2, 6, and 1 faces, and have 

 the same two ridge-summits b'd-j and b'k'n. And, of course, 

 there are two 15-peaks which have twenty-six triangular 

 faces, have one l4-edral summit, have four triedral summits 

 a, c, y, and m, between which intervene, counting the edges 

 in order round the 14-edral summit, 1, 2, 6, and 1 edges, and 

 have the same two faces b'd'j and b'k'?i, no two of whose 

 angles, as b and d or b and n, are on a consecutive pair of the 

 edges about the 14-edral summit. 



These 15-peaks are accurately represented by the above 

 systems of triplets, in which abe . . . n stand for summits, 

 not faces, namely the summits which in order lie on the 14 

 edges of the 14-edral summit A. 



12. The species of a w-edron on a (n — l)-gonal base, 

 having k triangles, may be said to be determined, when the 

 number and position of the k triangular faces are assigned, 

 with the k — 2 ridge-summits. Of this species there may be 

 no varieties or many. The varieties depend on the manner 



