406 
REV. T. P. KIRKMAN ON A CLASS OF POLYEDRA. 
triangles, or 
2k, ^^—n(x+\, ; 
whence we obtain, from both }30sitions of the axis, 
R".(4j?+1, Ak).^ii(^.v+\, 2k, , k, ^y| 
as part of 11(4.^+ 1 +4A: + /, 4A’+/). 
If we forget for a moment the character of one of these R^(4w+1, 4A), and treat it 
as an irreversible, we obtain by 4A-f-/ sections Ak, 1) results. Of these the 
doubly reversibles can occur only once, every doubly irreversible will occur twice, in 
one result as the reflexion of the other; every reversible will occur twice, tbe opera- 
tions in the first period in the second result being those of the second period in the 
first ; and every irreversible will occur four times, twice by the exchange of the 
operations on the first period for those on the second, and twice again by the rever- 
sion of all the operations, producing reflected images of two preceding results. 
That is, if we subtract from ii{4x-\-\, 4k, 1) all the doubly reversibles, twice the 
reversibles, and twice the doubly reversibles that can be cut from a doubly reversible 
(4.r-l- l)-gon having 4k triangles, by 4A-1-/ sections, there remains four times the num- 
ber of irreversibles that can be cut from it, by 4k-\-l sections. 
This remainder, divided by 4, is 
n{4x-\-\, 4k, l) — ii[x-\-\, k 
— 1 , 2k, gj— k, 
-|/y2.r-{-l, 2k, /?yx-[-], k, 
=^1 ii{4x-\-\, Ak, l) + 2ii{x-\-\, k, j)— 3/7(2^+ 1, 2k, 
which, multiplied by R'(4x-1- 1, 4k), is to be added to I(4a;-i- 1 +4A-1-/, 4A + /). 
It remains that we handle now trebly reversible (6.r+ l)-edra, having 6A triangles. 
If j?>l, the number of triangles in a triply reversible cannot be less than Qk, as 
they are all cut from the heptaedron h35353, by an even numbei’ of sections in every 
period. By operating on one reversible period of 2x summits and 2k triangles for 
reversibles by ^(6A-|-3/) sections, we obtain all the triply reversibles. The number 
of these so found is n.{^-\-\, k,^, giving so many (6x-l- 1 -|-6A TSO-cdra triply 
reversible with Qk-\-3l triangles, for each subject; and in all 
IT(6x+l, 6A)(/7(:r-}-l, k, =:IT(6A-f 1 +6A+3/, GA-fS/). 
1 he number of triply irreversibles is equal to that of the irreversibles producible 
