154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 509 
blit this equation gives 
and we have 
[1 — x a ] = (x — p) (x — p a *) ... (x — p a a), 
if 1, a 2 , ••• a a are the integers less than a and prime to it (a is of course the degree 
of [1 — ¿e a ]). Hence 
XP = 6p p a ~' n (1 - p«*- 1 ) ’ 
and therefore 
0p = - n (1 - p^- 1 ) xp ; 
or putting for XP its value 
dp = — p a n (1 — p ai_1 ) A_ s , 
where a is the degree of [1 — x a ~\ and 
of unity) less than a and prime to it. 
the equation [1 — p a ] = 0, may be reduced to 
a—1, and then by simply changing p into x 
Occ 
then by multiplication 
cii denotes in succession the integers (exclusive 
The function on the right hand, by means of 
an integral function of p of the degree 
we have the required function 6x. The 
fraction 
can 
of the terms by the proper factor be 
[1 — # a ] 
reduced to a fraction with the denominator 1 — x a , and the coefficients of the numerator 
of this fraction are the coefficients of the corresponding prime circulator ( ) per a q . 
Thus, let it be required to find the terms depending on the denominator [1 — af\ in 
1 
these are 
where 
(1 - x) (1 - x 2 ) (1 - x 3 ) (1 -x i )(l- x 5 ) (1 - x e ) ’ 
S-X&-, xd x S 
p — x p — X 
YjP = coeff. \ in v, P -, 
t f(per t ) > 
X?p = coeff. ^ in t P 
t f(p e ~ l ) 
1 1 
f(pe- f ) " (1 - per*) (1 - p 2 e- 2i ) (1 - p 4 e~ 4i ) (1 - p 5 e~ 5t ) (1 - e~ st ) (1 - e~ 6t ) 
and