EEY. T. P. KIEKMA]!^ ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 229 
disturbance of the solution, must differ from every C that we have before regarded about 
the axis through the wth side of the n-gon ; for it has no longer the same polygons in 
the same places of the interval between the axes. We have either changed the order of 
the imposed polygons, or we have in certain places put a (l+(9p)-partitioned (2-t-a^)-gon 
which was not employed in the construction of C, for a (l+^m)-pa'rtitioned (24-«ro)"gon, 
which was. Whatever be the solution we work from, we shall be able to produce from 
it, by mere disturbances of diagonals in the imposed polygons, distinct configura- 
tions, all different from the An- 2 h counted before ; and all the r-gons thus produced will 
ih 
have the property of being reversible about h agonal and h diagonal axes. And it is 
evident, that every configuration about an agonal axis of any ( 14 -^)-partitioned r-gon, 
ha\ing h agonal and h diagonal axes of reversion, will be produced from some solution 
of equations (A.), and some arrangement of diagonals in the imposed polygons. 
If, then, we denote by 2 . A»^ the sum of the products made from every solu- 
4A ih 
tion of equations (A.), every change, either in value or order, of See. being 
counted as a solution, we find that 2A»^, many terms of which Avill be equal numbers, 
ih 
is the exact number of configm'ations about an agonal axis which can be seen on any 
(l+/?:)-partitioned r-gon having h agonal and h diagonal clear axes of reversion. 
Now no ago-diagonally reversible has two configurations about agonal axes ; and 
every (l+^j-partitioned r-gon ha\ing ih agonal and ih diagonal axes, is reversible about 
h agonal and h diagonal axes, whatever positive number i may be; for it has h equi- 
distant agonal axes, and between every pair' of these (^—1) more agonal axes, and it has 
h diagonal axes, because ih diagonal axes bisect the angles between the ih agonal ones. 
Hence it follorvs that 
2A„_-^=2;Rf'^'''(r, 
ih 
T 7t ^xh 
where i is every whole number giving and — — positive integers ; con- 
ditions necessary to the existerree of equations (A.). 
XXVIII. Consequently (^>0), 
Rf ■“^‘"'■(r, A%=2A^--2iRf+‘^"-''^*'(r, /r)„, 
ih 
for all values of (z-l-l)/i=A', which make n — 2A', r—n, and all multiples of 44'. 
As r>w, n — 2A>0 in equations (A.) ; hence w<t;64, and if 24>|, Rf =0 ; e. 
i).=Er'V, i).=0. 
If w=6^, the equations (A.) become 
2n 
T—n 
j 2n 2n 
