REV. T. P. KIRKMAN ON A CLASS OF POLYEDRA. 
407 
from a reversible period of 2x summits and 2k triangles by ^(6/c+3/) sections, and 
is, by what precedes, 
^\i.i{2x-\-\, 2k, /)— k, 
from each subject of operation. This, multiplied by R^(6xH- 1, 6Zr), is to be added to 
P(6 j:+1 + 6^-1-3/, 6A+3Z). 
The reversibles obtainable from a triply irreversible are found only about one axis 
of reversion. Thus 735373537353 has only one axis through a heptagon and a pen- 
tagon. We are to cut by \{Qk-\-2l) sections on one side of this axis all possible 
irreversibles from a reversible system of 3 j? summits and ^k triangles. These results 
reverted on the other side of the axis, will give all possible reversibles. Among these 
will be all triply reversible (6a;-l- 1 +6Z--l-2/)-edra, with Qk-\-2l triangles, for these are 
all reversible ; and none of these can occur more than once. We have these to sub- 
tract from our results, leaving 
ii{^x-\- 1 , ^k, 1) —ii 
reversibles from every subject ; which number, multiplied by R^(6a?-1-1, Qk), forms 
part of R(6a^-l-l+6Z:+2/, Qk-\-2l). 
If we treated a triply reversible as an irreversible by Qk-\-l sections, we should 
ii{Qx-^\, 6k, /) results. 
Among these every triply reversible is found once; every triply irreversible twice, 
one place showing the reflected image of the other ; every reversible three times, each 
time the same operations commencing at a different period; and every irreversible 
six times, being begun both backwards and forwards in three different periods. If 
then we subtract from u{6x-\-\, 6k, 1) every triply reversible, twice the triply irre- 
visibles, and thrice the reversibles that can be made by 6k-\-l sections of a triply 
reversible (6a; fl- l)-edron having 6k triangles, there will remain six times the number of 
irreversibles that can be so cut from the same. This remainder, after division by 6, is 
|^|z7(6a;-f- 1, 6k, k, ^ — ii(^x-\-\, 2k, k, 
— oii(^3x-{-l, 3k, 0 -|-3^7^a;-l- 1, k, 
=^|n(6a?-|- 1, 6k, l)-\-3U(^x-\-\, k, ^—ii(2x-\-\, 2k, ^ — 3/7^3a;-l- 1, 3k, 0|, 
which, multiplied by R^(6a;-f-l, 6Z-), forms part of I(6a;-1- 1 +6Z--1-Z, 6k-\-l). 
It is most convenient to treat the case of the triply reversible having only three 
triangles by itself. From this heptaedron can be cut one triply reversible by cut- 
ting every summit of the base. One triply irreversible only can be made, a deca- 
edron, by cutting each triangle once. Two reversibles can be cut, by four sections, 
3 H 2 
^x-f-l, k, ^ 
