140] RESEARCHES ON THE PARTITION OF NUMBERS, 
and, the second term being 
/y»2 
— coefficient x m in 
247 
(1 — 0p) 2 (l — X 4 ) ’ 
we have P (0, 1, 2, 3) w |m = coefficient x m in 
and similarly it may be shown, that 
1 + x 4 
(1 — ¿c 2 ) 2 (1 — x 4 ) ’ 
P (0, 1, 2, 3) m £(3ra- 1) = coefficient x m in ^ ^ ■ 
As another example, take 
which is equal to 
coefficient ¿ 5m in 
— coefficient x? m in 
+ coefficient x m in 
P'(0, 1, 2, 3, 4, 5)fm, 
1 
(1 - X 4 ) (1 - X 6 ) (1 - X 8 ) (1 - Æ 10 ) 
a? 
(1 — x 2 ) (1 — x 4 ) (1 — x 6 ) (1 — X s ) 
(1 — x?) (1 — x 4 ) (1 — x 4 ) (1 — x 6 ) ' 
The multiplier for the first fraction is 
(1 - x 20 ) (1 - x 30 ) (1 - x 40 ) 
(1 — X 4 ) (1 — x 6 ) (1 — cP) ’ 
which is a function of x 2 of the order 36, the coefficients of which are 
1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 4, 4, 6, 4, 6, 5, 7, 5, 7, 5, 7, 5, 6, 4, 6, 4, 4, 3, 4, 2, 3, 1, 2, 1, 1, 0, 1, 
and the first part becomes therefore 
. 1 + X? + 4# 1 + 5x 6 + 7îc® + 4# 10 4- 3& 12 
coefficient m —- -r-r- —r-~ rr—. 
(1 - x 2 ) (1 — æ 4 ) (1 — x 6 ) (1 - x 8 ) 
The multiplier for the second fraction is 
(1 - æ 6 )(1 - # 12 )(1 - x 24 ) 
(1 -^)(1 -«*)(1 - a«)’ 
which is a function of x 2 of the order 14, the coefficients of which are 
1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1 ; 
and the second term becomes 
^ M 2x 2 + 2X 4 + Sxf + a? + x lf> 
— coefficient x m m 
(1 -0P) 2 (1-®*)(1 -oP) ’