60 REV. T. P. KIRKMAN ON THE 
centre of the edge i’, in the axial plane, and from this 
draw either of the lines j, in every figure constructed. 
But none of these constructions will be doubly rever- 
sible, being not the same on both sides the axial plane 
containing #. If then we multiply the above by two, we 
shall have, as shown in the preceding article, double the 
number of singly irreversibles when m is odd, and no sub- 
traction of doubly reversible results has to be made. That 
is, 
m—-1 Qw—2} 1 
[eae 54 l) ml 
RR?*(4e+m+ Wee ee ies +1,22)- ea ae g2, 
a portion of R(4”+m-+1,27). In these the base is odd- 
jaz=1. 
angled, being (4a-+m)-gonal, and in these alone, having 
m odd, is there a line 7 in the axial plane, as denoted by 
the sub-index jaw=1. 
19. Prosrem &. To determine the number of singly 
irreversible (4¢-+-m-+1)-edra reducible to a doubly rever- 
sible (4¢+1)-edron having 22 triangles. 
We can distribute m points on the 4¢-3 edges & (Art. 
7). This gives us 
(m aE hee Jl o 
oe 
arrangements of lines 7. 
Among these results will occur every one of R@R? (Art. — 
15) once, and once only; for the lines 7 in any of these 
can be laid on the subject in one way only: every one of 
I°R2 (Art. 16) will occur twice, in one result appearing as 
the reflected image of the other: every singly irreversible 
(RR?2)’ will be constructed twice, viz. the same configura- 
tion being laid on both sides of that axial plane of the 
subject which is not the axial plane of (RR?)’. Every 
singly irreversible now to be enumerated will present itself 
four times; for any sequence read about the base of the 
subject of operation is read four times, beginning In every 
