66 REV. T. P. KIRKMAN ON THE REPRESENTATION 



Let C next appear with three duads of non-consecutive 

 letters. We write the scheme with three triangles 6, d, f\ 

 Aab Abe Acd Ade Aef Afa 



abc cde efa ac ce ea. 



We have three triplets to add containing C, which can be only 

 Cae Cce Cea. These faces are 



ACabcdej: (8) 



6 3636363 



There are thus found eight octaedra on a hexagonal base, 

 having all their summits triedral. 



18. We shall now examine the octaedra which have no 

 face of more than five angles. There must then be two 

 crowns. 



Two crowns may in general be adnate by a common side, 

 or connected by an intervening edge or broken ridge-line. If 

 an intervening edge connects them, it is the side of a face at 

 least hexagonal, if the summits are triedral; for that face 

 must have two angles common with either crown besides two 

 in the base. Our crowns, therefore, must be adnate at two 

 summits ; and the sum of their other summits cannot exceed 

 five, because only one edge can pass to the pentagonal base 

 from each of them. 



Let the crowns be a pentagon C and a quadrilateral D. It 

 is evident that between the two faces at the common summits 

 of C and D there can intervene neither more nor less than 

 one of the five faces about the base. Let a and c be the two. 

 Then the triplets CDa CDc determine all the rest, for D 

 must appear only twice more, and with every letter twice ; 

 thus — 



Aab Abc Acd Ade Aea CDa CDc 

 Dab Dbc Qcd Cde Cea; 



A C D a b c d e. (9) 



5 5 4 5 4 5 4 4 



