167] 
APROPOS OF PARTITIONS. 
37 
i.e. 
1 — (1 — z) a a Vlog (1 — z)J 
g-Jlog(l-»)a—&c. 
I e log (№^))- ilOB(1 ^ )a - &C -, 
a 
and putting p K for abc... and S 1} S 2 ... for the sums of the powers, we have, taking the 
product 
i-(i-*y- p . e 
whence also 
(1 - z)* +1 
n =—~— r = - e' l0s(i 
1 — (1 — z) a p K 
J) -(«■+1 '+ iSÙlog(1 -*)-8,+B,# &-&c. 
from which the development may be found. 
The index of e is 
(q + l—^K + ^S 1 )z 
+ (iq + \ — k + S 2 ) & 
+ (iç + ÏÏ — & K + — 2¥ &) 2:3 
+ &c. 
and developing the exponential, 
1°. The coefficient of z is 
q + ^ (fc — 2). 
2°. The coefficient of z 2 is 
iq 2 + q {ISi ~ i 0 - 3)} + ^ S 2 - \ (* ~ 3) Si + (k - 3) (¿/e - £), 
and so on. 
The peculiarity is the appearance of the factors k — 2, « — 3, &c. If we neglect 
these terms, and consider as well q as a, b, c... to be each of them of the dimension 
unity, the coefficients will be homogeneous.