250 EEV. T. P. KIEKMAJ^ ON THE K-PAETITIONS OF THE E-G-ON AND E-ACE. 
(which is to be considered unity when cin = 0, the load being then simply the edge of 
2h 
the w-gon), 
So), 
— D(2 62 ) •D(2-|-®25 ^ 2 ) X • .D(2-f~C tn-2A , 6 n-ih \^ 
2h \ 2h 2h ) 
is the number of configurations that we can produce, about the axis through the nth 
side of the %-gon, read from that side, of (l+^)-partitioned r-gons built on this nucleus 
from this one solution, to have h loaded monogonal axes. And the sum of these pro- 
ducts belonging to every solution, all orders and values counting as solutions, is the 
exact number of all such configurations possible. Now one, and only one, of these is 
seen in each r-gon of the number ^)„, for n even. That is, 
2/H" . M . 
\ 2A / 
where n is even, and i>0 ; and H"=l whenever a^=0. 
2h 
LI. In equation (D'.), the highest value of h is h=n. Then those equations become 
r — n 
— 2-|-2o5o 
k—n 
■ = £n 
H=2(HA„)=2„ 
since Ao=I, where n is odd ; but Ej A:)„=0 of course, E is even. 
From this we can proceed to find E*'”'®(r, A:)„, for all odd values of A>1, which make, 
in equations (D'.), 
n—Ji, r— w— A(2+2ao) and k—n—Jua, 
divisible by 2A; and £„ being selected from our register of agonaUy and ago-diagonally 
reversibles. 
In equations (D".), the greatest value of h gives — 2/i=0, or h—^i. Those equations 
become 
2r— 3w 
o„ —ao+ai. 
2k — 2n 
=£o + s., 
Ao=SAo=l, 
whence, if ^ be an odd number (for h is always odd), 
— . mo 
E; (r,A:)„=2:H".M, 
^but this is =0 if 2 be even^, H" and M being defined in the preceding article, and the 
number of products H". M being equal to the number of solutions of the above reduced 
