46 REV. J. P. KIRKMAN ON THE 
(2f+1)-edra, the number of (2/+m-+1)-edra on (2f+m)- 
gonal base having f triangles is determined. And thus 
the entire number of (#+1)-edra on an «-gonal base 
can be found, and classified according to the number f 
of their triangular faces. 
It is to be understood that in this enumeration no 
(#+1)-edron shall be either the repetition in a different 
position about the base, or the reflected image of any 
other. The avoiding of these repetitions is the great 
difficulty of our problem. 
5. Polyedra, whatever be the rank of their faces and 
summits, as well as partitions either of the polygon or the 
polyace, fall into two leading classes, the reversible and. 
the reversible. 
An irreversible is in no position identical with its re- 
flected image. A reversible is in certain positions identi- 
cal with its reflection. 
Again, an irreversible regarded at one point of its base 
may be identical with itself regarded at another or at two 
other points of the base. If at one other only, it is called 
a doubly irreversible ; if at two others, it is called a triply 
irreversible. In the former, the sequence of faces about 
the base repeats itself once in the circuit; in the latter, it 
repeats itself twice: that is, it is three times read in the 
circuit. A singly irreversible is in no two positions iden- 
tical with itself. 
In like manner, a reversible may be singly, doubly, or 
triply reversible, exhibiting a configuration about the base 
that occupies the whole circuit, half the circuit, or one 
third of it. 
A singly reversible can be placed in two positions before 
a mirror so as to be identical with its image, in which 
position a bisecting axial plane is perpendicular to the 
mirror. A doubly reversible has four such positions in 
two axial planes at right angles to each other. A triply 
