236 EEV. T. P. KIEOIAN ON THE K-PAETITIONS OP THE E-GOX AND E-ACE. 
An r-gon so constructed from any one solution has k diagonals, and is reversible about 
at least Ji agonal and h diagonal axes, all loaded. 
We can, as before, make the product 
D( 2 -{-( 2 i, ej)D( 2 -j-<? 2 ? 62 ) • • • •D/' 2 -j- Cln—Ak^ 4A\ — 4A 
\ Ah Ah } Ah 
with the numbers &c. in our solution of (A'".), which is the number of variations 
of diagonals that can be made in the (2+ai)-gon, (2-{-a2)-gon, &c., while the 4A rever- 
sible polygons upon the axes are undisturbed. We can also disturb these polygons alone, 
varying the 2h agonally reversibles, in 
different ways, as in XXIX., and the 2 A monogonally reversibles in 
\ Ah AhJ 
different ways, as in XXXII., wherefore the entfre number of r-gons (l+A’)-partitioned 
and having Ji equidistant agonal and as many diagonal axes of reversion all loaded, that 
can be formed by one solution of (A'".), is the product 
Ah 
all the numbers ^0 &c. appearing in the product being those formd in that one 
solution. The sum of these products, one for every possible solution of equation (A'".), 
all orders and values of the quantities as, ae, See. counting as solutions, is the total 
number of (I+A)-partitioned r-gons, built on the w-gonal nucleus, to have h agonal and 
h diagonal axes of reversion none of them clear, i. e. 
. agdi^^^ k)n= 2H . M . ; 
ih 
the number of products under 2 on the right being that of the solutions of (A'".). 
Wherefore A)„= 2H . M . A«^— S^Ilf k)„ {i > 0) ; 
Ah 
for whatever be the value of i, there must be h equidistant agonal and as many diagonal 
axes bisecting the intervals between them. 
XXXV. The highest value of h in (A'".) gives n — 2k=0, 2A= The equations 
become in this case 
2r — 5n 
2n =“o+an 
2k — 2n , 
which have of course no solution if the fractions are irreducible ; and 2Ao=I : wherefore 
A)„=0, 
R* (r, ^).=2{2.(2E-"<(4+2a., s.)+E«""s'«(4+2<.., s.))2.E'“ "«(3+«., s.)} = 2HMA., 
