238 EEV. T. P. KIEKMAN OX THE K-PAETITIOXS OF THE E-GOX AXH E-ACE. 
r—n-{-2h^ 
2A / 
1 
(B.) 
The above construction from every solution of this equation gives an r-gon ^’-partitioned 
end reversible about at least h clear agonal axes bisecting 2h equidistant sides, both of 
the r-gon and %-gon (Theorems B, N). And with this solution, i. e. -without changing the 
Aveight or order of the imposed loads, we can, by disturbing the arrangement of the 
diagonals of the (2-f-«m)-gon in nny way whatever, produce a new configuration about 
the axis bisecting the n\h. side ; and the enthe number of configurations so brought into 
view by such disturbances will be 
D(2+a„ eJD(2-p«2, ^2)...D/2-t- Cin—^h^ 2a\ — ^ n~2h \ 
\ 2A 2A / 2n 
and a like expression for the number of configurations generable from eA ery solution of 
equation (B.), i. e. for every change in the weight or order of imposed loads, being added 
to the above, we obtain different configurations about the axis through the ??th 
2A 
side, the number of terms under 2 being that of the solutions. Among these -will be 
found once, and once only, every aspect of any -partitioned r-gon about a clear 
agonal axis, which has 2 A equidistant sides in common -with the M-gon, bisected by agonal 
axes of reversion. Now two of these configurations are found on every one of Ji)„, 
for each has ^ X 2A such equidistant sides of the ?^-gon ; one Avill be found on each of 
k)^ for the same reason, and one Avill be found on each of R^''^*'(r, k)„ and on 
each of same reason (Theorem C) ; that is {i > 0), 
^)„+Rf-^'^‘(r, k)„], and 
2A 
R'‘-^(r, ^)„=i|2A^A-2,(2RJ('+*>^(r, A’)„+Rf -''^*(r, j. 
XXXVII. Before we proceed with the discussion of this formula, it is necessary. 
Problem g. To find R 4 c®(i'? k)„, the number of {l-]r^)’^artitioned r-go^is having ^ clear 
and ^ loaded agonal axes^ and built on the n-gonal nucleus. 
These r-gons have all on h sides of the %-gon, the 
, n 2ra , h — 1 , . , 
mh, ^th, -^th, . . . — ^ wth sides, 
a (l+Soj^pai’titioned (4 + 2o£o)-gon ha-ving an agonal axis, and nothing on the 
^th, ^th . . . sides. 
2h ’ 2h 2h 
The equations to be satisfied are 
r=w+A(2-{-2o!o)+2/?^«,d-«2+ •• ~l~ 
k~n — h-\-hzQ -\-2h^i •• ~\~en—' 
J— 2a\ 
ih / 
(B'.) 
«3 and So being any number in our list of reversibles about an agonal axis. 
