483 



backwards and forwards the same. If P^. be the number of ,r-edra 

 having an (x l)-gonal base, and all their summits triedral, 



the symbols on the right denoting the numbers of <r-edra of the six 

 varieties that make up P x . 



Each variety is again subdivided according to the number of tri- 

 angular faces. Thus, if P(x, k) denote the number of ,r-edra on an 

 (x l)-gonal base, having k triangular faces, and all their summits 

 triedral, 



P(#, k) = l(x, k) 4- l\x, A) + l\x, k) + R(x, k) + R 2 (>, k) + R\x, k\ 



x _ \ 

 The number k is not <2, nor > , and P x =2P(x, k), for all 



values of k. 



It is necessary to solve the following 



Problem. To determine the number of (,r + A: + /)-edra, none of 

 which shall be the reflected .image of another, that can be made 

 from any .r-edron having k triangular faces, by removing k+ I of its 

 base-summits, thus adding k + I triangular faces, so that none of its 

 k triangular faces shall remain uncut. 



Thea?-edron is supposed to have an (x l)-gonal face, and all its 

 summits triedral; no edge is to be removed, and k-\-l not ># 1. 



When the ,r-edron, the subject of operation, is irreversible, all 

 the resulting (* + A + /)-edra will be irreversible. If it is reversible, 

 some of them will be reversible and others irreversible ; if it is mul- 

 tiple, some of them will be, and others will not be, multiple. 



If the subject of operation is irreversible, the number required by 

 the problem is 



r _i_j.'|-i -l '-"- 1 



iitrk n 2* -^ _ - _ X (~> a \^ 2* 



[7TT a( 



taken for all values of a not greater than the least of k arid /; i. e. 

 ka not <0, not >/ a. 



The complete answer to the problem is expressed by the follow- 

 ing equations, in which, of the capitals on the left, the first ex- 

 presses the result, and the second the subject of operation. That is, 

 IR 2 (.r, k, /) denotes the number of irreversible (.r + A: + /)-edra having 

 k + 1 triangular faces about the (x + k+l l)-gonal base, that can 



