ee 
PARTITIONS OF THE X-ACE AND THE X-GON. 59 
which is a portion of BGe pm 1,22). = m is of ne- 
ran= 
cessity even, and the expression vanishes for x=1, if 
m> QO. 
18. We consider next the bisecting axial plane of our 
subject which cuts the axial edge #’. On either side of it 
we see 27-1 edges f, including half of #’, on which, m be- 
ing still supposed even, we have to deposit $m points 7. 
The number of results is 
2Qa—2 | 1 
m 
(y +1) 2 
yet 2. 
By reversing these operations on the other side of the 
axial plane, we shall again produce, and once only, every 
result of Art. 15; for if in any of these the bisecting plane 
be drawn to cut the axial edge #, we shall see one of the 
configurations here constructed. And evidently this has 
been constructed only once by the process of this article. 
Also by the reasoning of the preceding article it -appears 
that every singly reversible (4¢+m-+1)-edron here ob- 
tained has been twice obtained, and has no axial edge. 
Subtracting then and dividing by two as before, we get 
2v—2 | 1 
m 
(5 +1) m2 
BR (Got m+ 1,20)— [= eS 2 
m a—2|1 
G +1) m—t 
— ea -2 7 [R(4e+1,22), 
which is a part of the number R(4v+m-+1,2z2). 
kax=0. 
Suppose next that m is odd, the axial plane still cutting 
the edge #’. We can with the even number m-1 produce 
as before 
Qv—2 | 1 
a mes 
ee 
Pan « 
results. We can then deposit one more point 2 at the 
