REV. T. P. KIRKxMAN ON A CLASS OF POLYEDRA. 
405 
subtract from ii{n,k,l) all the possible reversible results, and divide the remainder 
by two. That is, by what precedes, we obtain from R, R' and R" 
l|n(2a7+ 1, 2h, l) — ii(^x, k, ^ j, 
2k-{-],l) — ii(x, k, j’ 
^\^i{2x, 2k, l) — ii(^x, k, 0 —ii(x, k, j* 
It is to be understood in these formulae that 
2k-\-l<2x in the first, 
2^d-l+/<2.r in the second, 
2k-\-l<2x — 1 in the third. 
If we multiply the first by R(2xd-1, 2k), the second by R(2a:'+1, 2^+1), and the 
third by R(2a?, 2k), we obtain the corresponding portions of I(2^+l -\-2k-{-l, 2k-\-l), 
I(2x+2+2^+Z, 2/C+1 + /), and l{2x-\-2k-^l, 2k+l). 
Let us next operate on a doubly reversible (4.r'|- l)-edron, with Ak triangles. All 
these are cut from the pentaedron 3434, and by the addition of an even number of 
triangles in each period. The operations by which doubly reversibles are cut from a 
doubly reversible are simply those whereby reversibles are cut from the reversible 
period of 2x summits, containing 2k triangles, being in number n(^x + 1, h by 
what has preceded. Or, if R^(4A’d-l, Ak) be the number of (4a7+l)-edra doubly 
reversible, with Ak triangles, 
R^(4a?+1, Ak) .a. ^^+1, k, =R^(4(a?+Z’)+/+l, Ak-\-l), 
there being no more ; for one of these latter can be cut from nothing but a doubly 
reversible with Ak-\-l faces fewer. 
From the same (4A^+l)-edra can be cut doubly irreversibles, namely so many 
as the irreversibles producible from sections of one period, or from a reversible 
(2A:+l)-edron with 2k triangles. This number is, as just proved, 
•||«7(^2a?+1, 2k, ^—ii(x-\-\,k, 
which multiplied byR^(4j';+l, 4Zr), constitutes a portion of V .{Ax-\-Ak-\-l-\-\, Ak-\-l). 
As a doubly reversible is also a reversible, reversibles can be cut from it. It is to 
be observed that, as a reversible, it has two axes of reversion, as indeed every 2;«-ly 
reversible has. Thus the enneaedron 35363536 has an axis through the two penta- 
gons and another through the hexagons. If we operate on one side of the axis for 
irreversible results, and revert our operations on tlie other side, we obtain reversibles. 
The number of such results in either position of the axis of reversion is that of the 
irreversibles producible by ^2A-+^^ sections of a reversible (2x-j-l) gon with 2k 
3 H 
MDCCCLVJ. 
