226 REV. T. P. KIRKMAN ON THE PARTITIONS 
side of the axis, and by the repetition of the operation in 
order reversed on the other side of it. 
The number of figures G thus obtained from G’ is 
(jet 12 _ Ge’ +a)-)eO4 
Ler—pn pean 
2 
and the product of this number into S just found is the 
entire number of 7-gons G that have an agonal axis of 
reversion placed in every position about that axis. Among 
them will be ouce constructed every doubly reversible R? 
which has an agonal axis perpendicular to the axis through 
its marginal faces, and every singly reversible R” twice, 
once with either extremity of its axis upwards. This pro- 
duct then, summed for all values of v=r-—-7’, is 
(R?+2R)(r,k+1), 
wherefore putting for R? its value from Art. 6, when & is 
odd, and 7 - 2k=4m, 
Rr, kK+1) 
a! (ar As lj |—1 (4 (k 7) 1) )#e—#-#-8) ia Poe 
fi mall ee) a Rome) 11 +o ben are 
k+1 
(a(r-2h)) ? 
a ja 
1 ee 
where <0, wr<tr—-2k-4, wpr—k—-5. 
The fractions here, as before, are to be reckoned zeros, 
if they are not integers. This requires that & shall be odd, 
and 7 and & both even. 
9. The number of R’”*(r,£+1) of r-gons K having two 
marginal faces and a monogonal axis is next to be enu- 
merated. If between the two edges forming the angle 
through which the axis passes we introduce an edge, the 
r-gon K becomes an (7+ 1)-gon K’ having an agonal axis, 
and the figure is either one of R*(r+1,4+1), where & is 
odd and r+1=24+4m, or one of R“(r+1,4+1). And 
each of R*(r+1,4+1) will give one K, by the vanish- 
ing of an edge carrying the agonal axis which does not 
