EEV. T. P. KIEKMAN ON THE K-PiJETITIONS OF THE E-GON AND ErACE. 231 
Thus it is evident that we can register Rf from A=| to A=l, for all values 
that make r— w—2A, and multiples of Ah. 
For a simple and complete example, to be carried out through all our investigation, we 
may take the partitions of the octagon. Here we are to seek Rf“^'^*(8, Jc)^. Evidently 
n<% (if ^>0), and =6 or =4. 
0 
When w=6, — - ^- =1, the only value, gives A™!, and Rf^*’(8, whatever be /r, 
, 8—6 , 
because — is not mteger. 
A — lh 
When w=4, — — ^ has no integer value >0; wherefore, since r>w, Rf''^*'(8, 
for all values of k. Hence 
Rf“^"‘(8, ^}=0, 
whatever be the nucleus, except that 
Rr"‘( 8 , 0 ) 3=1 (art. III.). 
XXIX. Problem c. To find Rdc“®^(r, k)^, the number of {l-\-\i)’fafUtmi€d r-^£ws, bidlt 
on an n-gonal nucleus, having h loaded agonal, and h clear diagonal, axes of rexersion. 
The investigation to be made differs from that in the five preceding articles in that 
the 2h sides of the w-gon, bisected by the agonal axes, are all loaded with a certain 
(l4-So)-partitioned (4+2ao)-gon xAo, having at least one agonal axis of reversion 
(Theorem G.).' This must be so posited that its agonal axis shall be in a line with that 
of the w-gon (Theorem N.), and must be placed alike on the 2h equidistant sides, since 
all agonal axes of the r-gon must bisect the same configuration at each end (Theorem C.). 
The remaining sides of the w-gon may be loaded, the ??ith from Aq, with any (l + 6„,)-par- 
titioned (or divisioned) (2+«,„)-gon so as to form a figure reversible about the 27i axes, 
and so that the equations foUoiving be satisfied : — 
r — n-\-’2t]i{^-\-^a^-\- Ah^i-\- a ^-\~ . . 
where «„>0, ; and e,^= — l, if «„,= 0 ; 
(A'.) 
the numbers Ko and being any that we may select from our register of polygons having 
at least one agonal axis, which is supposed complete for values of (2-}-2ao) not greater 
T^“Tt 71 
than All that is said of the loading of the — sides between the axes, and about 
the import and necessity of equations (A.), applies to equations (A'.). 
The sides being thus loaded, we have an f-gon which has only clear diagonal axes, and 
is reversible about 2h equidistant and alternate agonal and diagonal axes. Leaving now 
the 2 A agonally reversible (l + 2Q53)-gons undisturbed, we can vaiy the positions of the 
diagonals, exactly as in (XXVI.), so as to produce about the agonal axis through the 
nth side of the r-gon 
D(2-{-cfi, Cl) € 2 ) . . .T)(2-\-an-^t A a- 2 /t 
' -ih. Ah ) Ah 
