48 REV. J. P. KIRKMAN ON THE 
2E=88, also 
F4+S=E+2, | whence 
3F+2E=3E+6, 
E=38F -6=6z2 -3, 
of which 62-38 edges 2 are in the base, and 2 are sides 
of triangles, leaving 27-3 edges not meeting the base and 
none of them in a triangle. 
8. If we draw in any non-triangular face h collateral 
with the base a line 7, from a point of the base to a point @ 
of any edge of h which does not meet the base, and con- 
sider h fractured along that line into two faces, the result- 
ing (2@+2)-edron (Art. 4) is still singly irreversible ; for 
the new summit of the base is the only one not in a tri- 
angle. If we draw another such line / either in 4 or some 
other non-triangular face, the result is still singly irrever- 
sible; for, by definition (Art 5), the new summit of the 
base is part of a configuration which nowhere else is read 
about the base: and in like manner, any number of such 
operations upon a singly irreversible subject will give a 
result singly irreversible, and no two of these results can 
be alike, because in our singly irreversible subject of 
operation the configuration read round the base from any 
face is different from that read round from any other. 
Our problem is to determine in how many different 
ways such operations can be performed on our subject 
PR, 
There will be added m summits 7 out of the base, which 
will be planted on the 2v-3 edges & not in triangles. 
The number of ways in which these m points can be dis- 
tributed on those edges, w on any edge, (wo) is, putting 
a 'i=a-a+1-a+2...(6 factors), 
(m+1)*4!1 
[eit » 
And as the joining line 7 through any of these points can 
be drawn to the base through either of the faces meeting 
