228 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OE THE E-GON AND E-ACE. 
the (2-f a„)-gon, must make up the r summits of the r-gon. And, supposing that 
everywhere >0, the n—2h loaded sides of the w-gon, with the diagonals of every 
(2+«^)-gon 4A times laid on, must make up the k diagonals of the r-gon. And if 
in which case the mth and (w— w)th sides of the n-^on are imloaded, whereby 4A 
sides besides the 2h above-mentioned remain sides common to the r-gon and the w-gon, 
the appearance of e^=—l corrects the error made by counting those 4A sides for 
diagonals of the r-gon, and the second equation still remaius true, however many of 
the numbers a^, &c. =0. But these equations are impossible, and Ef'^*(r, A')„=0, 
unless r—n, n—lh^ and w-|-2A are all multiples of 4A. 
Consider any single solution of the equation (A.), and let N be an r-gon bearing, on the 
mth side everywhere reckoned in both directions from the one bisected by an agonal 
axis, a (2-j-«,„)-gon having diagonals. Whatever be the posture of those diago- 
nals, and whatever be the character of the (2+«„J-gon, the figure is reversible at least 
about 1i agonal and h diagonal axes. And let C be the configuration read in X about 
the agonal axis thrpugh the wth side of the w-gon. 
XXVI. Let, next, any disturbance be made in the arrangement of the diagonals in 
one or more of these imposed polygons, so that the reversible character about the 
2h axes be preserved, and the diagonals of the (2+a„)-gon on both sides of any axis 
shall form pairs making angles bisected by the intervening axis of reversion. The solu- 
tion before us of (A.) being undisturbed in both values and order of the numbers ^> 1 , 
&c., we can combine every (l-f^^j-division of the (2+«„)-gon in the interval with 
every (lfi-6|p)-division of the (2-l-«p)-gon, &c.; that is, we can make C take 
6 ]) . D( 2 ^ 2 } • B( 2 ^ 3 ) • • • B( 2 — 2 A , 6 n-2h) — ^ n-2h 
4h 4h 4h 
different forms, by mere variation of arrangement of diagonals, the same variation being 
made about every agonal axis, and consequently the same variation occuiTing about ever\- 
diagonal axis. Of these An-sh configurations no two can be alike ; for the intenals of 
4^ 
2h • • • • • 
sides between two adjoining axes are irreversible configurations, if we look at the 
axes which limit them, because one is un agonal and the other a diagonal axis ; so that, 
in fact, no disturbance of one or more diagonals of N can produce N' a repetition of N. 
If the limiting axes were alike, it might possibly occur that N' should be a reflexion of 
N, the configuration of one read from right to left in the interval being that of the other 
read from left to right. 
It is to be observed here, and in all products that we may handle of the form of 
A»- 2 a, that when a^=0 and consequently D( 2 -|-«ot 5 ^m) is either to be omitted. 
4h 
or counted imity, so that AO is always =1. 
XXVII. Next, let us make a variation in om solution of equation (A.), either by 
altering two or more values of or of oi* of both; or by exchanging the 
places of certain values without altering them. It is evident that C', the residt of such 
