PARTITIONS OF THE X-ACE AND THE X-GON. 55 
The (22-+4+m-+1)-edra thus produced have no axial edge 
kK’, but they have all an axial point 7 and line 7, which 
neither of the previous sub-classes of this article have. We 
may write therefore, 
m1 
RRQe+m+1,7)=R(2Qr+1,2)- Pan 
jax=1 kax=0 
the subindex yar =1 denoting that these have all a summit 
z in the axial plane. This then is a portion of the number 
R(2Qve+m-+1,z2) of reversible (2e+m-+1)-edra having » 
triangles, and a the base odd-angled, (2#+)-gonal, 
as must necessarily be the case when the axis of reversion 
passes through one summit of the base. 
14. Although no reversible can be obtained from an 
irreversible subject of operation, the converse is not true ; 
and we proceed next to 
Prosiem g. To determine the number of irreversible 
(2a@+m-+1)-edra reducible to singly reversible (2v+1)- 
edra having 2 triangles. 
These will all be singly irreversible, because the subject 
of operation stands alike in no two positions, read in the 
same direction. We have 27-3 edges k, on which to 
place m points, giving us 
(+1)! 9 
]24 }1 
results, among which all those of the preceding problem 
will occur once and once only, while each of the remaining 
irreversibles will occur twice, in one result as the reflected 
image of another result, because of the reversible character 
of the subject. We obtain, therefore, dividing this re- 
mainder by 2, 
2vu—4 | 1 
IRQe+m+1,2)=IRQ0 41,2). tI. 2m 
m «a3 |1 
m (z+1) 
— tR(2¢+1,x)-2?-———- — 4 R(274+1,2) 
kax=1 
r—3 | 1 
ae kaw=-0 
