402 
REV. T. P. KIRKMAN ON A CLASS OF FOLYEDRA. 
I(a?+A-+/, k-\-l), from other x-eclra having k' triangles, by removing k'-\-l' summits 
about the base, none of the k' triangles being untouched, if /r'+/' = A:+/. 
Next let our subject of operation be an irreversible containing more than one 
period of faces about the base. If it be a doubly irreversible, the base will be 2x-gona\, 
as every face will be opposite to a similar one. We have then two periods each of 
X faces, and if we operate on one of these so as to remove angles, leaving no triangle 
in the period untouched, and then repeat the operation exactly in order in the other 
period, we shall obtain a doubly irreversible for our result. 
Let P(2a’d-1, 2k) be the whole number of doubly irreversible (2x+l)-edra on a 
2x-gonal base, having 2k triangles. Then k, ^ is the number of ways in 
which we can remove k-\-^ summits from one period, and therefore 
P(2j;+1, 2k)ii(x-^l, k,^, part of V‘{2x-\- \ -\-2k-\-l, 2k-\-l), 
is the whole number of doubly irreversibles having 2k-\-l triangles, and a (2^7+2^+/)- 
gonal base, that can be cut from all the P(2 j?+1, 2k) polyedra before us. 
If/ is not an even number, n^x+1, k, ^ is to be considered nothing; for it is im- 
possible to retnove a fractional number of summits. 
We can cut also from these P. (2x-|-l, 2k) doubly irreversibles a number of singly 
irreversibles. If each of these doubly irreversibles were single, it would give rise to 
ii{2x-\-\, 2k, 1) singly irreversibles ; but the double character of the subject of opera- 
tion causes every method of removing 2^:+/ angles, which is not alike in both periods 
of the subject, to appear twice in the number just written. That is, we are to sub- 
tract from this number all the doubly irreversibles that can be made, and take half 
the remainder, which is, after division, 
^\ii{2x-\-\, 2k, l) — ii(^-\-l,k,^^- 
The second term of this is zero when I is odd. We obtain thus for the number of 
singly irreversibles that can be cut to have 2k-\-l triangles from all the V{2x-{-\, 2k) 
under consideration, ' 
P.(2x-\-l, 2k) .\^ii{2x-\-\ , 2k, /) — 
a part of \{2x-{-\-\-2k-\-l, 2k-\-l). 
Next let us consider the operations that can be effected on triply irreversible 
(3.r-l- ] )-edra having ^k triangles. Let their number be P(3.t7+1, ^k). It is easily 
proved by a repetition of the preceding argument, that k, ^ is the number 
of triply Irreversibles that can be cut from each of them, and that 
3 
//(3a?-j-l, ^k, 
k 
