224 REV. T. P. KIRKMAN ON THE PARTITIONS 
edges of the central quadrilateral, and the rest symmetri- 
cally about the two axes on the four half-edges of it, and 
on 4¢+4 other edges of HE’. That is, }(v-—2) points are 
to be laid in one or more groups in 7+2 positions in the 
fourth part of the circuit of E’, which can be done in 
e+o\n0 r—2k-2 ae. 
Kcapdion lstihior aa? 
- {1 a 
ways. 
When x#=4y, there are no points at the mid-edges of the 
central quadrilateral, but 42 points are deposited im the 
i+2 positions before mentioned, in some one of 
m2 pee 
L+4y 4 r—Qky 4 
zt sa, 
- = ——____—_—- ways. 
eh pep 
When & is even, ”=4é+4, there is no central quadri- 
lateral, and v=4j. In this case 7 points are deposited on 
one-fourth part of E’ on i+ 1=37" positions, in some one 
of 
meee ie r—2Qky2" 
——_ = ——__———__ ways. 
eee 
The number R? of r-gons E that can be thus constructed 
on the 7’-gon E’, to have two axes of reversion, may be 
thus expressed, 
k+1|4 k+1 1) Ki, 
Gate) 2 \ gas 2 (ie 
2 4 h 4 
R= - : Seaatie ae = > 
—— J2!? 
if we define that every irreducible fraction containing 
r or k shall be accounted zero. Of these three terms in 
R? either two or all will always thus vanish. 
7. We take up next an 7-gon of the number R(r, +1), 
having one axis of reversion, two marginal faces, and k+1 
