404 
REV. T. P. KIRKMAN ON A CLASS OF POLYEDRA. 
Now take the reversible (2^+ l)-edron R' having 2^+1 triangles; the axis of rever- 
sion passes through one of them. As this triangle cannot be untouched, it must be 
twice touched, so that we have only other seetions to make, one half 
of these on the x—\ summits on one side of the axis, neglecting the summit of the 
central triangle. The possible operations are 
or, if R(2^-l-l, 2k-\-\) be the total number of reversible (2a?-l- l)-edra having 2A:-1-1 
triangles, we obtain 
R(2x-1- I, 2k -\- 1 )ii(^x, h, part of K{2x-\-2-\-2k-\-l, 2k -\- 1 -p/). 
We take now a reversible R" having a (2x — l)-gonal base, and 2k triangles. 
The axis of reversion passes through a summit and through a face, which we shall 
suppose to be not a triangle. It is not difficult to prove that it can be no triangle if 
2^:— 1 >3, all the summits being triedral. 
We have on either side of the axis ^{2x — 2) summits, that in the axis being 
neglected, and half the 2k triangles. We have to distribute sections on these 
X— ] summits. The number of ways to do it is 
which requires I to be even ; or if I be odd, we may cut the summit in the axis, and 
distribute sections on the ,r— 1 summits, giving u(^x, A, These opera- 
tions reversed on the other side of the axis give us all the possible results. 
If then R(2x, 2k) be the total number of reversible 2a?-edra having 2k triangles, 
we can cut from all these 
R(2^, 2k)^^i(^x, A, ^-[■iii^x, A, ^)| 
(2x-i-2A-l-/)-edra reversible, having 2A+/ triangles. One of the terms in the second 
factor is always zero. The polyedra so cut are a portion of R(2:r-1-2A-1-/, 2A-|-/). 
When k-\-lz=x—-i, all the angles about the base of the reversible with A triangles 
are cut, and the result is of necessity reversible. 
But if A-l-/<.r~-l, some of the results of k-\-l sections will be irreversible. The 
whole number of results, if we treated the w-edron R or R' or R" as irreversible, would 
be ii{n, A, /), but these are not all different. 
They will all, except the reversible ones, have a different order on the two sides of 
the axis of reversion ; and each irreversible will occur twice, the second time reverted 
by exchanging the arrangements of the two sides of the axis, so as to make a poly- 
edron and its reflected image. As we are not to count these reflexions, we have to 
