Rev. T. P. Kii-kmau un x-edra kaviiiy an {x — l)-gonul Face. 73 



543543, whose six faces about the base are a repetition of an irre- 

 versible period of three ; or triply irreversible , as the decaedron 

 643643643, whose faces exhibit a thrice-repeated irreversible period; 

 or they are reversible, dcublr/ reversible, or triply reversible, as the 

 hexaedron 53443, the eimeaedron 63536353, or the heptaedron 

 535353, exhibiting a single, double, or triple period, all reading 

 backwards and forwards the same. If ¥x be the number of cr-edra 

 having an (>r— l)-gonal base, and all their summits triedral, 



P^= I,, + 1^. + 1?, -f R,, + R' + rI 



the symbols on the right denoting the numbers of A'-edra of the six 



varieties that make up P.^. 



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

 angular faces. Thus, if V{x, k) denote the number of .z'-edra on an 

 (j_l).gonal base, having k triangular faces, and all their summits 

 triedral, 

 V{w, k) = \{x, k) + \\x, k) + l\x,k) + R(a-, k) + K\j:, k) + ^\x, k) . 



x—\ 

 The number k is not <2, nor >'-2~' ^^^ P*=SP(.r, k), for all 



values of k. 



It is necessary to solve the following 



Problem. — To determine the number of (a'-t-A:4-Z)-edra, none of 

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

 from any ^--edron having k triangular faces, by removing /c + / of its 

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

 k triangular faces shall remain uncut. 



Thei?-edron is supposed to have an (j?— l)-gonal face, and all its 

 summits triedral; no edge is to be removed, and k-]rl not >x—\. 



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

 the resulting (a' + 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— 1— //!-' ;i"l~' cr— 1 — 2A'~"'"^ 

 iiix k l) = 2" — -S ('•2«-l).2*-«.A— . r . 



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

 k—a not <0, not >l—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 th^subject of operation. That is, 

 lK-(x, k, I) denotes the number of irreversible {x + k + l)-edra. having 

 A-f/ triangular faces about the (a' + A:-^-/— l)-gonal base, that can 

 be cut from any doubly reversible ^-edron having k triangles about 

 its (x—l)-gonal base. 



Whenever A or / in the function ii(x, k, I) is not integer, the func- 

 tion, by a geometrical necessity, is to be considered =0. 



