54, REV. J. P. KIRKMAN ON THE 
Let R(Qv+1, os ds dene the whole number of singly 
reversible (2041). “edra having # triangles, in which one 
edge x’ is axial or in the axial plane, (the subindex kaw=1 
signifying one is axial,) among which uumber is the sub- 
ject of our operations. Then we have 
(3 +1) 
RR(2e2+m-+1,z) =R(Qwr +1,2)- sae aes 
kax=1 kax=1 
for the number of our results, which form a portion of 
R(Qv+m+ aly pa (2e-+m-+1)-edra singly reversible 
having wv enh in which the axial plane meets none, 
i.e. contains one, of the edges not meeting the base. 
Next let the axial plane bisect an edge /’. 
We can now operate upon X’ on both sides the bisecting 
plane, without producing 4-aces. It is easily seen that our 
results are, if m is even, 
m 
(5+1) of 
Pen sae) 
and we obtain the equation He 2)1 
st 
RR(2Qve+m-+1, a EEL 2) +2 
kax=0 
as a portion of R(Qn m+ 1,v), the whole number of 
(27-+m-+1)-edra singly roveruible eivihe x triangles, in 
which no edge & is axial, as denoted by the subindex kar 
=0. The base in both these sub-classes is even angled 
(2a +m)-gonal. 
If however m is odd, we can produce as just pointed out, 
with the even number m-—1 operations, 
m— ] aE |" jae 
m—1 
results, and then plant one point more in the axial plane 
at the centre of the edge /’, from which of course two lines 
j can be drawn, doubling this number. 
