36 
[167 
t 
167. 
APROPOS OF PARTITIONS. 
[From the Quarterly Mathematical Journal, vo! I. (1857), pp. 183—184.] 
Let II (1 — x a ) = (1 — aJ) (1 — x b )...(/c factors) and assume that 
1 
the part of which involves negative powers of 1 — x, then 
n (1 - x a )_ 
is 
Coefficient x q 
m [n(i- 
cc a )_ 
= coefficient z K ~ x in 
n - ) 
i-* v(l-A 3+1 1 — (1 - z) a J ’ 
which suggests the question of the expansion in powers of z, of the function 
II 
(1 - z) q+1 1 - (1 - *)« ‘ 
Now by the definition of Bernoulli’s numbers 
t 3 
1 =T-3 + £> A--B. 
e‘-l t 2’ r "'1.2 " s 1.2.3.4 + 7i *1.2.3.4.5.6 
from which it is easy to deduce 
^ _ ii_ - Bl iT2» + ^ a 1.2.8.42“- B3 1.2.3.4.5.№‘ +&C ’ ; 
l-e-« _e 
and, writing in this formula t = — a log (1 — z), we have 
a log (1-Q _ -^o g (l- г )«-A 1 ^^ ) « 3 +J B 2 1 1 ^g|^a 4 + & c. 
t 5 
— &c. 
i-(i-O a 
= e