58 REV. T. P. KIRKMAN ON THE 
+m-+1)-edra reducible to a doubly reversible (4¢+1)- 
edron having 22 triangles. 
We can produce singly reversible configurations bisected 
by either of the axial planes of the doubly reversible sub- 
ject. Taking first that plane which contains the axial 
edge k’, we see on one side of our plane 27-2 edges & 
(Art. 13) on which we can deposit 4m points 7, thus ob- 
taining 
a {23 | ie ery 2” 
results, which are to be repeated in order reversed on the 
other side of the bisecting plane. 
Among these will occur once every doubly reversible 
(4e¢-+m-+1)-edron with 2v triangles enumerated under 
problem 4 (Art. 15); for if the bisecting plane be drawn 
through the axial edge # of any of these, we see a configu- 
ration that we have been producing here; and evidently 
this configuration, and therefore this (4a+m-+1)-edron 
has been only once here obtained. And every singly 
reversible (4v-+m-+1)-edron among our present results 
will occur twice, because. of the reversible character of the 
semi-solid on which we have been operating, as pointed 
out in the preceding article. Moreover all these singly 
reversibles have an edge # in the axial plane, and all the 
singly reversible (47+ m-+1)-edra having an axial edge K’ 
which are reducible to a doubly reversible have been here 
constructed. Therefore, subtracting the results of Art. 15 
and dividing the remainder by two, we obtain 
2a—3 | 1 
zt) pk 
RR (424+ m+1,22)= jes 2 : 
kan=i 
a2|1 
zt) m= | 
oe es saan 2% (R(4v4+1,22); 
