PARTITIONS OF THE X-ACE AND THE X-GON. 63 
versible (6z-+m-+ 1)-edra having 3z triangles that are redu- 
cible to triply reversible (67+ 1)-edra having 3~ triangles. 
It is of no consequence about which of the three axial 
bisecting planes in our subject we operate, for all of them 
contain an axial edge /, and reduce by the vanishing of 
triangles to the tetraedron. We have 67-4 edges & on 
which to plant m points 7, that is, 37—2 on one side of 
the axial plane, on which to lay 3m points 7, for singly re- 
versible results ; operations which are to be reversed on 
the other side of the plane. Among the 
results obtained from our subject will occur every triply re- 
versible of the preceding article, for they are all of course 
reversible configurations about our bisecting plane, and 
will occur only once, as will also every singly reversible. 
For as there is no axial plane at right angles to the one 
we operate on, the configuration of the subject reads dif- 
ferently from the two extremities of the axis of reversion 
in the base; i.e. we can only produce a given reversible 
result by one position of our m lines 7 on the subject. 
Wherefore, subtracting the triply reversibles, 
RR*(62+m+1,32) 
3a—3 | 1 ea 22) 
Ey Gey") 
=R'(6r +1,32) (2 a 8 ea 
which is part of the number R(6v+m+1,3z.) 
kax=1. 
22. Prosirem n. To determine the number of triply 
irreversible (67-+m-+1)-edra having 32 triangles, which 
are reducible (Art. 8) to (6v+1)-edra triply reversible 
with 3z triangles. 
We have (67-3) edges k on which to lay m points, or 
22-1 in the third part of the circuit on which to lay $m 
