EEV. T. P. KIEK3IAN ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 243 
(C.) 
about at least one monogonal axis, which is an agonal axis of the %-gon (Theorem N), 
bisecting the side on which the polygon is placed. The intervals of sides being 
loaded as in the previous constructions, the equations of condition are 
T =%-hA(l-{-2ao)-j-2A^ai-|-«2H“ •• "h CLn—^h 
Il=^^-|-A.So “h • • “h n I 
\ 2A / 2A ^ 
^rri^^m I5 
and both a, s are chosen from our register of monogonally reversibles. Leaving undis- 
turbed the 2A polygons on the axes, we have in our power, from any one solution of 
equations (C.), 
I)(2-|-ttj, 6 j]D( 2 -1-<3525 62 ) . . . I)/2-1- Qt w— 2ft , 2ft \ Am— 2ft 
\ 2ft 2* / 2ft 
variations of arrangement of the ^i, ^? 2 ? diagonals in the intervals, producing as many 
configurations about the axis through the nih. side of the %-gon, and all reversible about 
at least h monogonal axes. With any one of these variations we can combine any poly- 
gon of the number 
M=S„E“(3+2«„, s„), 
and any one of 
M'=2„E’"-’”73+2a«, g„\ ; 
V 2A 2ft/ 
SO that we obtain MM'A^^ configurations from this one solution of (C.). 
2* 
If we form this product for every possible solution, in every order of values of a a, s e, 
&c., w^e shall produce about the axis through the nih. side of the ?z-gon, 
different configurations. 
Now two of these are seen in each of Eo''*(r, ^)„, 
one of them in each of E 4 *' *(r, ^)„, 
one in each of Ef'-°^*'(r, 
and one in each of k)n', 
for every one of the last three classes has hxi terminations of agonal axes of the w-gon 
loaded with monogonally reversible polygons. Wherefore 
2M.M'.A^=2:d2Ef ^)„-f Ef Jc\], 
2ft 
and (^>0), 
RrXr, ^)„+E|f ‘(r, k^y 
This result is useless until E^f (r, k)„ is known. 
XLII. Problem k. To find E“(r, k)„, the number of {l-\-h)-partitioned x-gons built on 
an xi-gonal nucleus, to have -^h loaded and ^h clear axes only, all diagonal axes. 
A clear diagonal axis of the r-gon is a diagonal of the nucleus, evidently. These 
2 K 2 
