PARTITIONS OF THE X-ACE AND THE X-GON. 51 
having 2v triangles. The remaining portion we shall find 
presently. 
10. Observe that the number II’ above written is always 
to be considered zero, if 4m is not integer; and in all future 
: Ra; m — m-e 
expressions of this kind, the occurrence of = Oe not 
an integer in a factorial is to be considered equivalent to 
that of the factor zero before the factorial. We may thus 
dispense with the use of circulating constants. 
Proptem c. To determine the singly irreversible (4x 
+m-+1)-edra having 2 triangles reducible (Art. 3) to 
doubly irreversible (4v + 1)-edra having 2v triangles. 
If we distribute m points 7 in every possible way on the 
402-3 edges j of the subject P operated on in the pre- 
ceding article, and draw m lines 7 in every possible way, 
we shall obtain 
(m+1)"2 
1*#4+ {1 
results. Among these will evidently be found all those 
of the preceding article, once, and once only. The re- 
maining results will be simply irreversible, and each will 
be twice constructed; for the result P’ presenting a single 
sequence of configuration read round the base from a face 
h, will be identical with another result P”, having the same 
reading round the base beginning at the face of P opposite 
to h. 
If then from our total number of results we subtract 
those of the preceding article, the remainder divided by 
two will be the correct number of single irreversibles 
required. 
That is, if II’(4v-+m-+1,2z) be this correct number, 
44/1 
TP(4e +m 41,20) =3{ fan 
2a—2 | 1 
Gave ie 
pert 2} -T?(4e +1,22), 
