REV. T. P. KIRKMAN ON A CLASS OF POLYEURA. 
403 
is that of the singly irreversibles, where the appearance of a fraction in the function 
ii, reduces it, as it always must, to zero. That is, we obtain 
P(3x-i-l, Ti, as P.(3.r+3A:-f Z+1, 
ZZ(3a?+l, ^k, /) — k, 
as a portion of I(3a?+3A:+Z+ 1 , 3Z:+/). 
From an irreversible no reversible can be cut by this removing of summits; for as 
the summits of triangles of a reversible correspond in pairs, the arrangement of faces 
about the base will still be reversible, if all the triangles are supposed to vanish. 
Let us now operate on a reversible polyedron, whose faces about the base read 
backwards and forwards alike. 
There will be a certain period ahc klm reversed, in one of the three ways, 
ahc.,..klmlk....ch, 
ahc klmmlk ch, 
or ahc klmmlk cha. 
There is in any of these what may be called an axis of reversion, which in the first 
passes through the faces a and m, in the second through a and between two m’s, in 
the third between two a’s and between two m’s. 
It is evident that the number of triangles about the base of a reversible cannot be 
odd, unless the axis of reversion passes through a triangle ; as all faces recur in order 
reversed, through which that axis does not pass ; and the base must be (2a?+ l)-gonal, 
if the axis passes through one face only. The third case, of an axis of reversion pass- 
ing through no face, does not occur when all the summits are triedral. 
Fi rst let the base be even, and let the number of triangles be even also ; we have 
to consider the operations practicable upon a (2x+l)-edron R reversible, with 2k 
triangles. Some (2^+ 1 +2Z:+/)-edra can be cut from it reversible, and some irre- 
versible, by the removing oi 2k-\-l summits of the base, leaving none of the 2k tri- 
angles untouched. 
A reversible so cut will have on either side of its axis of reversion half the 2^-1-Z 
added triangles, unless it passes through an added triangle, in which case it will have 
on either side ^(2^-|-/— 1) of them. 
LetZ=2/'; then the number of possible operations on one side of the axis of rever- 
sion, which exhibits x summits, is ii{x, k, I') ; each of which gives by repeating it 
backwards one of our reversible (2^+ 1 +2Z’-l-2/')-edra. As the axis does nut pass 
through a summit, I cannot be odd, for the added triangles are all in pairs. 
If then R(2x+1, 2k) be the total number of reversible (2,r-|- l)-edra having 2k tri- 
angles, we obtain 
^{2x-\-\i2k).n(^,k,^, part of R(2.r-1- I H-2A:-[-Z, 2Z’-i-Z). 
and 
P(3j?+ 1, 3Zr)^ 
