EEV. T. P. TTT-RKMAN ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 227 
2r.l(r, k)=V(r, R^, + ^:)| ; 
I 7/i 71ft X J 
which shows that, if we can find R”(r, Jc) and Jc) for all values of m>0, we can 
obtain the most numerous of all the classes, I(r, k)^ by a simple subtraction and 
division. 
XXIV. Our first step towards the actual solution of our problem of partitions, which, 
after all these tedious prolegomena, we may now think of taking, is to find the numbers 
in (XX.). We begin with 
Problem b. To find Ef •“^*'(r, k)„, the number of {\-\-'k)-jpartitioned r-gons, built on the 
\>g(mal nucleus, and havmg h agonal and h diagonal axes of reversion, all clear. 
There are, in every one of these, 2A equidistant terminations of diagonal axes, between 
7t 7t 
which intervene ^ sides of the w-gon ; the central side of these ^ is bisected by an 
agonal axis, wherefore ^ is an odd number. 
If we mark this central side as the wth of the ?^-gon, we see on either side of it 
1'^^— 1^ sides between it and a termination of a diagonal axis. This nih. side is one 
of 2h equidistant sides of the w-gon, bisected by terminations of agonal axes, namely the 
n 
Tfi 
2n 
. . nih. sides. 
which are all unloaded, being sides both of the r-gon and the ^^-gon. The | 
sides on either side of the nih are loaded, the first and (^^ — l)th with a certain -parti- 
tioned (2-f-a,)-gon, so as to form a configuration bisected by the agonal axis between 
them; the second and {n — 2)th by a gj-p^i’^tioned {^-k-a^y^on placed in like reversible 
manner, and so on ; in such a way as to satisfy the equations following. 
where am>0, and e^':ip>a^—l, since no (2-J-«,„)-gon can have more than 1 diagonals 
drawn none crossing another; wherefore if a„,—i), e„^=—l. 
71 ^Jt 
These — polygons «i« 2 , &c. are laid on in the same manner to correspond about the 
7t ^2t7t 
;^th ^th, &c. sides ; i. e. they fill up the 4/i intervals between agonal and diagonal axes, 
making a configuration reversible about any one of them. 
XXV. The equations (A.) must be satisfied, because the n summits of the M-gon, with 
the a, summits added 4/i times in the (2-f-ai)-gon, and the a^^ summits added 4A times in 
2 H 2 
