REV. T. P, KIRKMAN ON THE K-PAHTITIONS OE THE R-GON AND R-ACE. 239 
The configurations producible about the loaded agonal axis through the nt\\ side of 
the w-gon, from any one solution of (B'.), are the product HA«^, of 
2A 
H=2^{2R”“^(4-l-2«o, g+Il^”‘“^''‘(4+2ao, So)}, 
and A„- 2 A=D( 2 +ffl„ ei).D( 2 H-a 2 , (I/n—2h<} € n—2l\ , 
^ \ 2A 2/i / 
and the sum of these products, one for every solution of (B'.), all values and orders of 
the quantities counting among the solutions, is the sum of the configurations bisected 
by the loaded agonal axes of all (l+^)-partitioned r-gons built on the ? 2 ,-gonal nucleus, 
to have loaded agonal axes, and clear agonal axes bisecting the equal intervals 
between the loaded ones. 
Now the whole number of these r-gons is that of these configurations, since no such 
r-gon has more than one ; and these r-gons are in number exactly 
2k 
for all these have (2^^-l)x■|^ loaded agonal axes, and among these are ^ equidistant 
ones, in the inteiwals of which lie an odd number (2^+l) of clear agonal axes. But 
]c)n is not in the number of these y-gons, because in the intervals of any 
equidistant loaded axes of these, lies an even number of clear axes. Wherefore, (^ > 0) 
XXXVIII. The highest value of h in (B'.) gives w— 2A=0; which reduces them to 
2r—2n 2k— n 
^ ^ ^ 0 ? ^0 1 » 
wherefore 
Tt ^ / \ / 
2r 2k— n 
n n 
We can next put =I in equations (B'.) and find R^^^ (r, and thus every number 
R^(r, k)„^ for any even value of A, so that 
w, r, and 
shall be multiples of 27i, being some number of diagonals on our list of agonally 
reversibles. 
Tit 
In XXXVI., equations (B.), the highest value of h gives —^^ = 1, or /?= 2 , when 
those equations become 
2r — 2n 
2k — n 
n 
-=«„ 
so that r is divisible by n\ D(24-«i, e,)=2A,. 
-agdi 
Hence 
