240 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OE THE E-GON AND E-ACE. 
and since 
H-agdi 
Re {r, by XX^TIL, 
11. agdi 
and 
X (r, by XXXIIL, 
when r is divisible by n, 
—ag 
X (n/i:)„=i{D(2+a„eO-R,^, (r, /5:) J 
n — 2h 
If now we put in (B.) 2h we can obtain 11 ^ (^? ^)ni a^nd ever)- number k)„ 
in which n, r, and Jc are multiples of 2 A, including the case of A=l. 
We shall now find 3lc“^(8, Jc)^ and Il 4 e^( 8 , 
For 7^=6 in equations (B.), we can only put "^ =2, A=l, and ^’=2 ; then 
r = 8=6 +2.1(l-f0)=6+2.1(0+l) 
A:=2 = 6-2-j-2.1(0-l)=6-f2.1(-l-f0) 
A,=2D(3, 0)=2 
E“^(8, 2)«=i2 = l. 
If ^^=4, ^* 2 ^= integer gives only {r>n) and being integer in the formula 
~ag 
for , gives k—% or else k—i. Also, by the same formula, 
Ef( 8 , 2).=i{D(4, 0)-R«“(4, 0))=i(l-l)=0 
E?( 8 , 4),=i{D(4, 1)-2(E‘'(4, 1)+E"f»(4, l))}=i(2-0)=l ; 
for the only entries in our register for r=4 are 
0)=1 and R^'"(4, I)=I. 
n — 2h 
When w =6 in equations (B'.), is not integer for an even value of li ; but when 
-agdi 
2 ? — 4, A =2 = 2 , and by the formula for (r, A:)„, 
2)4=R^“^''‘'(4, 0) = 1; 
for ^=2 is the only value that finds integer in our register. 
XXXIX. Problem h. To find Ro'‘®(r, k)„, the number of {l-\-'k)-partitioned v-gons 
ha ving h agonal axes only of reversion, all loaded axes. 
71 2 /?- 
In any one of these there are 2A equidistant sides, the w^th, the ^th, the 5 )jth, &c. of 
the w-gon loaded, half with a (l-|-eo)-partitioned (4-l-2ao)-gon, and the rest, alternating 
with these, loaded with a (1+^5) -partitioned ^44-2aj^j-gon, all having at least one agonal 
axis of reversion, which is also one in the w-gon. The numbers So, as well as a^. 
