EEV. T. P. KIEKMAJS' ON THE K-PAETITIONS OP THE E-GON AND E-ACE. 241 
which may be the same or not with the former, are any that we choose from our register 
of agonally reversibles. The intervals are loaded as before, so as to satisfy 
r—n 
-|~/i(2 -j“2oj0) \ 1 
\ 2A / \ 2A/ I 
Jc -|-^2 ~b • • ~\~€ n-2 h 
V 2/t 
)+&.; j 
(B”.) 
where 
Having chosen any solution of these equations, and posited the polygons, we can, 
without distm’biag the first-named 2A polygons on the axes, make 
— € n—2f\ — An— 2A 
\ 2/i 2/t / 2h 
variations of the e^, ... diagonals, and produce about the axis through the wth side as 
many configurations. And with each of these variations we can combine any (4-1- 2 Ko)- 
gon in our register which has gp diagonals and an agonal axis, and also any ^4-{-2o5^j-gon 
having diagonals and an agonal axis, and in any postm’e giving h agonal axes of the 
2A 
r-gon. That is, we can, without altering om* solution of (B".), produce about the axis 
through the Tith side H.H'.A^^ different configurations reversible about h agonal axes 
2/t 
all loaded as described, where 
H =2,{2.R-“^(4-l-2a„, g+R2.<'^‘^>(4-h2a„, go)} 
H'=2,J2.R’”"^/4-l-2an, g»\-f g„\l. 
If then SHH'A denote the sum of these products made from every solution of (B".), 
obtained by every change of value or order in the numbers aa g^, this will denote the 
entire number of such configm’ations. 
Nc^ two of these are found in each of (r, k )^ ; 
one of them, namely, about the loaded axes, in each of k\, 
and these are among the configurations obtained with c(,^=.(x,r^ and go=g^; 
2ft 2ft 
also one of them is in each of 
and one of them in each of E^f k\\ 
for both these have li equidistant agonal axes loaded, of course alike /ao=a^ and go=g^\. 
\ 2ft 2ft/ 
Nor is there anything to prevent the 2hi terminations of axes in some of Eo““^(r, k\ being 
loaded all alike ; for this cannot hinder the configurations about alternate axes from 
being different. Therefore 
SH.H'. A„^=2,^2EJ''“^(r, E|f or 
E^(r, /Y«=i{2:H.H'.A2^ft-5:,^2Eo^f'+‘’®^(r, E|f«’(r, ^)„-|-Ef k), 
2 K 
MDCCCLVII. 
