PARTITIONS OF THE X-ACE AND THE X-GON. 538 
(m+ 1)*+! mae . 
jes 
results. Among these every result in the preceding 
problem will occur once, and once only, and every singly 
irreversible will occur three times, because the same se- 
quence will have been begun and carried round from three 
similar starting points, say three A-gonal faces, of our sub- 
ject. We have then 
(m+ 7 gah 
Sian 11 
ID(6x+m+1,3x) =1°(6x +1,8x) +4 Qm 
_G? Ps 4 : 
a 3 
12 [hed -2 
as a portion of the number I(6z+m+1,8z). 
13. Prosrum f. To determine the number of singly 
reversible (2v-+m-+1)-edra reducible to singly reversible 
(2+ 1)-edra with 2 triangles (Art. 3). 
In a singly reversible one vertical plane and one only 
can be drawn bisecting the base, such that the configura- 
tion on one side of it is a repetition in reversed order of 
that on the other. The number of edges k is 27-3, an 
odd number. This bisecting plane, which we may call the 
axial plane, or plane of reversion, will therefore either 
contain one edge 4, or bisect one. 
First let the axial plane contain an edge #’: this edge 
can receive no point 7; for a line 7 being drawn from it on 
oue side the plane would require, in the reversal of opera- 
tions on this other side, another line 7 to be drawn from the 
same point on the other side, which would make? a 4-ace. 
We have then 4m points 7 to plant on 4(227-3-1)=x%-2 
edges & on one side of the plane. This gives us 
a—3 [1 
(S41 ) 3 
17311 
results, so that mis of necessity even. 
