508 
SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 
[154 
The actual development, when k is small (for instance k = 1 or k = 2), is most readily 
obtained by developing each factor separately and taking the product. To do this we 
have 
c j. , c + ° 2 i ¿o c + 4c 2 + c 3 , ^ , 0 
t + 77 ^ ii 2 77 — it 3 + &C., 
1-ce-t 1 — c (l-c) 2 " ' (1 — c) 3 2 
(l-c) 4 
where by a general theorem for the expansion of any function of e*, the coefficient 
of is 
(-y 1 
11/ l-c(l + A) 
0/ 
(-)// 1 
+ 
+ 
of 
11/ U - c ' (1 - Gf ‘ (1 - c'y +1 
(where as usual AO^ = 1/ — Of, A 2 0^ = 2f — 2 . V + Of, &c.) and 
Of 
1 - er* t + 2 + 12 t 720 f + 30240 &C ” 
where, except the constant term, the series contains odd powers only and the coef 
ficient of ¿ 2/_1 is — > Bi> B-2, B 3 ... denoting the series ~ 
YY ¿¿J o oU 
numbers. 
of Bernoulli’s 
But when k is larger, it is convenient to obtain the development of the fraction 
from that of the logarithm, the logarithm of the fraction being equal to the sum of 
the logarithms of the simple factors, and these being found by means of the formulae 
c c 
t + 
c + c 2 t 3 c + 4c 2 + c 3 t 2 0 
X + —77 XT XT + &C. 
1 — c (1 — c) 2 2 (l-c) 3 6^ (1-c) 4 24 
l0g 1 - l0g t + 2 1 24 t2 + 2880 * 181440 V ' + &C ‘ 
The fraction is thus expressed in the form 
1 
gky-ykytz-y... • 
11 (1 — p n ) II (ap) t k 
and by developing the exponential we obtain, as before, the series commencing with 
. 1 
A -*w 
Kesuming now the formula 
XP = P A -*> 
which gives %p as a function of p, we have 
0x = u XP . 
[1 — x a ] p — x ’