62 REV. T. P. KIRKMAN ON THE REPRESENTATION 



It is evident that there must be, in such an w-edron P, e 

 faces which do not meet the {n — e — l)-gonal base A: and it 

 will be convenient to say that P has e crown-faces, or e 

 crowns. 



First, let c=l ; and let w=8 ; the base or most-angled face 

 is a hexagon, and the crown cannot have more angles than 

 the base. Suppose it also hexagonal. We shall have of 

 course a system of triplets symmetrical with regard to the 

 base A and the crown C ; viz. 



Aah Abe Acd Ade Aef Afa 

 Cab Cbc Qcd Cde Cef Cfa, or 



AC abcdef (I) 



6 6444444 



Next, let the crown be pentagonal. The system 

 Aab Abe Acd Ade Aef Afa, 

 Cab Cbc Ccd Cde Cca, ef fa, ea, 



must be completed; which can only be by the triplet efa, 



giving us 



ACabcdef (2) 



6 5544453 



It is to be observed that, while in the former system there 

 are none but duads of consecutive letters, neglecting A and 

 C, there is of necessity in the latter one duad ea in Cea of 

 non-consecutive letters combined with C. Nor can there be 

 more ; for C must combine with exactly five letters, twice with 

 each, so that two duads used with A, namely, those containing 

 the sixth letter, must remain unemployed with C. But as 

 there can only be (2w — 4=) 12 triplets, not more than one 

 duad besides those two can remain to be repeated : if now C 

 had two duads of non-consecutive letters, two used with A 

 and two used with C would remain to be repeated ; which 

 is absurd. 



15. When C is 4-lateral, it may combine with one duad 

 of non-consecutive letters, or with more. If with one only, 

 the system to complete by two additional triplets is 



