154] 
SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 
507 
in which formula [1 — x a ] denotes the irreducible factor of 1—oc?, that is, the factor 
which equated to zero gives the prime roots, and p is a root of the equation 
[1 — x a ] = 0; the summation of course extends to all the roots of the equation. The 
index s extends from s = 1 to s = k; and we have then the portion of the fraction 
depending on the denominator [1 — a?\ In the partition of numbers, we have <£# = 1, 
and the formula becomes therefore 
l/4[i—*“] 
where 
We may write 
= 1 S-3g-- 
II (s — 1/ p — x 
— _l 1 (xd ys-1 _ 
“■•• + IIO—1)° x) £l-a?]’ 
XP = coeff. j in t?* 1 
Ape-r 
fx = II (1 — x m ), 
where on has a given series of values the same or different. The indices not divisible 
by a may be represented by on, the other indices by cip, we have then 
fx — II (1 — x n ) II (1 — a? p ), 
where the number of indices ap is equal to k. Hence 
/(per*) = n (1 - p n e~ nt ) n (1 - pWe-^y, 
or since p is a root of [1 — x a ] = 0, and therefore p a = 1, we have 
f (pe~ l ) = n (1 — p n e~ nt ) n (1 — e~ apt ); 
and it may be remarked that if n = v (mod. a), where v < a, then instead of p n we 
may write p v , a change which may be made at once, or at the end of the process of 
development. 
We have consequently to find 
XP = coeff. ^ in t s ~ l ■ P 
n (1 - p n e~ 1lt ) H (1 - e~ apt )' 
The development of a factor is at once deduced from that of ^ 'A ce ~t > an( ^ 
l p 6 
a series of positive powers of t. The development of a factor ^ _ e - apt i s deduced from 
that of 
^ - and contains a term involving -. Hence we have 
1 — e f t 
n (1 - p n e~ nt ) n (1 - e~ apt ) 
=A —* ¿1 + ^-№-i) +k~i • • • + A_i - + A 0 + &c., 
and thence 
XP = P A ~s 
t k 
64—2