EEV. T. P. EIBKMAJSr ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 237 
a sum of products easily obtained from our register of reversibles, a different product for 
every solution of the two equations just written in Kq, Sq, Sj. 
We next put or 2A=-, and find 
^ 4A ’ 4’ 
Hagdi —agdi 
E; (r, y^)„=SHMA,-E; (r, 
H, M and A, being properly determined from the numbers that satisfy 
^(2-{-2ao)-}- ai-l--(2a2-hl), 
In like manner we can obtain Ef k)^ for every value of Ih that gives, in equa- 
tions (A'".), 
w, T~Qh^ and k~~n—21i^^-\-z^^ 
each divisible by 4A ; Kq, Zq, and being numbers in our register of reversibles. 
4/i 4A 
It is easily seen that 
A:)6=0=Ef“^^‘(8, k\, 
whence by what precedes we obtain, for >1;>0, 
E"*“^*'(8, ^)=0, 
or the octagon has no partition ago-diagonally reversible, except itself. 
Thus we have determined the four numbers 
Ef ^)„+Ef k),,. 
XXXVI. The next step is to obtain 
E^-“^(r, A:)„=E^“^(r, /^;)„+Ef-^(r, ^)„+ES-^(r, k),. 
Problem f. To find Ee‘^^(r, k)„, the number of {\-\-V)-partitioned x-gons built on the 
n-gonal nucleus, to be reversible about h agonal axes, all clear. 
WTiether n be odd or even, 2h sides of the w-gon, viz. — 
the Tith, ^th, ^th.,.^^wth sides, 
k ii a 
and the ^th, ^th...^^^^%th sides, 
2A ’ 2A 2h 
which are equidistant from pairs of the preceding h, are unloaded, and the sides between 
71 
the nth and^th ai’e loaded, the wth with a (ld-6j„)-partitioned (2-j-fl5,„)-gon, &c., the 
9 Tt ““ 
configuration of this interval being reversed in the two adjoining intervals of sides 
each, so as to make a figure reversible about all the axes. And the equations following 
will be true : — 
