[154 
510 
SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS, 
where it is easy to see that 
A_ s = 
18 (1 — p)(l ~ P 2 ) (1 — P 4 ) (1 — p 5 ) ’ 
^-i = 
il 1 
P + V + V + V 
and we have 
(1 — p) Cl — /D 2 ) (1 — P 4 ) (1 — P 5 ) (4 18 VI — P 1-p 2 1-p 4 1 — p s ; 
e 2 p = - p 2 (i - p) a_ 2 , 
0ip = - p 2 (1 -p)A._ 1 . 
But [1 — p s ] = 1 + p + p 2 = 0. Hence p 3 = 1, and therefore 
(1 - pH 1 - p ! ) (1 - p 4 ) (1 - p') = (1 - p)» (1 - p»)> = 9. 
Hence 
whence 
e -2p 162 ^ ^ ^ 162 ^ ~ 162 + 
= 162 ( 2 + ®)’ 
162 
and the partial fraction is 
2 + ¿r 
which is 
162 1 + ¿c + a* 2 ’ 
1 2 — x — x 3 
= 162 1 -# 3 ’ 
and gives rise to the prime circulator (2, —1, — 1) per 3 5 . 
loz 
The reduction 6 x p is somewhat less simple; we have 
4 18 1 - p 18 1 - p 2 
= ^ C 1 “ P 2 ) ^ - ¿P C 1 “ P 2 ) - ll PH 1 “ P)1 
54 
972 (1 - P 2 ) (51 - 10p - 14p 2 ) 
9^(61 + 4p-65^);