228 REV. T. P, KIRKMAN ON.THE PARTITIONS 
sk+2=43(7 -4), or 
z=43(7 -—k-4). 
The number of figures H’ to which H” is thus reducible 
is 
(Gk)Yi yeia (4h)Yi—2 yh) | 4 
Wi yet 49 qe ee? 
2 
a sum which is shown by the reasoning of Art. 4 to be 
(4h) -** i: k—hr'+2 
=a) 11 “2 ; 
Any one of these S r'-gons H’ becomes an r-gon H, if 
4x points be placed in any of (37’—1) positions on half 
the 7’-gon, namely about the vertex of a marginal triangle, 
and on 3(7’—4) other edges of H’ on the same side of the 
axis, and if the operations be repeated in reverse order on 
the other side of it. That is, every figure H’ gives 
(doF1) 2 _ (Ar 42-2) eI 
Bein pen figures H. 
The product of this number into 8, summed for all 
values of 2=r-r', is the entire number of 7-gons having 
+1 diagonals and reversible about a drawn diagonal axis 
in every position about it. In other words, this product 
is, putting R“(r,4#+1)’ for those having one axis only, 
2R%(7,£+1)/+R(r,4+1), 
where for R? is to be put its value when & is even, and 
r—2k=4m; for each of R” has two configurations about 
the axis, and each of R?* has only one about the axis 
which does not pass through the marginal faces. Where- 
fore, putting 7’ =r- «2, we obtain 
R“(r, k+1)' 
(4(r— 2) 14 (3 Bisset 1 yt. 
12,4 - era) }21 ‘ pare 22 
El 
(F(r - 24)? 
ams ; é 
el 
where #<t0, w<tr—2k-4, vbr—-k-4, and irreducible 
