154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS, 
hence finally 
e i p=± i (i2 + 23p), = 315(42+ 23*); 
and the partial fraction is 
which is 
1 3 42 + 23# 
324 Xdx 1 + # + #2 ’ 
1 . 42 —19# — 23# 2 
324 ^ 13^ 
and gives rise to the prime circulator # (42, —19, —23) per 3 
The part depending on the denominator 1 — # is 
+ v #a ^ * ^~ 3 
1 — # 1 * 1 — # 1.2 
A — 9 1 / 3 \2 -^-—3 , 
+ {xd x f ; ...+ 
where 
1 — # 
1 
iTSTsWpi. 
(1 - e~ l ) (1 - e~ 2t ) (1 - e~ st ) (1 - e~ 4f j (1 - e~ 5t ) (1 - e~^) 
A 1 „ 1 , 1 p 
= - + A § - ... + A j - + &c. 
We have here 
lo Srh* = - log * + 5 * - Ft + Mso p '- &c ” 
and thence the fraction is 
21. 91 „ 455 
1 
7201 6 ' 
2 24 570 
which is equal to 
720 i 6 
11 24 + 1152 + •” 
i 455 ^ 
x(1 + 576* + 
1 1 7 1, 77 1 245 1 43981 1 199577 1 
= 720 t 6 + 480 t 5 + 1080 P + 1152 t 3 + 103680 P + 345600 t + 
and consequently the partial fractions are 
l 1 7 1 77 1 245 1 
86400 (xdx): ‘ l^x + IT520 ^ Xdx)i l^x + 6480 ^^l-x + 2304 {xdx) ~ l” 
+ 
43981 
103680 
from which the non-circulating part is at once obtained. 
(xd x ) 
+ 
199577 1 
1 - #‘ r 345600 1 -#’