232 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-GOX AND E-ACE. 
different configmations, the numbers &c. here being those of a solution of equa- 
tions (A'.). And every one of these may be varied again by changing the posture of the 
(4-f-2ao)-gon, if it has a second configuration about an agonal axis, so that this axis 
shall come into a line with that of the n-gon, or by exchanging the (4-}-2ao)'goii for any 
other that is also (l-l-eo)‘P^i’titioned and has an agonal axis. The number of variations 
in our power by distm’bance of the (4-l-2ao)-gons alone, is 
2,„{2R-“^(4-f2ao, So)-}-E^-“^‘''(4+2a„, So)}=H, 
because each m-agonally reversible has two configurations about agonal axes and thus 
admits of two postures, while each ago-diagonally reversible has only one ; whatever be 
the number m of agonal axes (Theorem A.). 
Thus we see, that for every solution of equations (A'.) that we can write down, all values 
and orders of the numbers counting in the solution, we can make 
H . An— 2A 
'IF' 
different configurations about the axis through the nth. side of the ?i-gon. The proof 
that there are no two of them ahke need not be repeated here fi:om art. XVI. 
If then 2H . A^^ 
4A 
denote the sum of those products HA made from every solution of equations (A'.), this 
number is that of the r-gons (l-j-^)-partitioned, and having li agonal and h diagonal axes 
of reversion, the latter only being clear axes, which are constructible on the w-gonal 
nucleus. This is shown exactly as in art. XXVII. And by the reasoning of that 
article we obtain 
hi being any number h! such that 
n — 2h! r—n—2h'.{2-\-2ur) , k — n — 2h!sn 
VI “d — V/— 
are positive integers, for some values of and that are found in oiu register of 
(l + 2 o)'P^^titioned (2-(-2ao)-gons, having an agonal axis of reversion. 
Hence (oO), 
ih 
71 ““ 2^ 
XXX. The highest value of h in equations (A'.) gives ■ =0, or 2]i—n. In. this 
case Ao is the only imposed polygon, and the equations (A'.) become 
T — Zn ^ k—n 
71 
which =0 of course if the fractions are irreducible^ or if no such values are in our 
register of reversible polygons with an agonal axis. An;^= Ao is here to be considered 
