NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 405 
oc • • 
Now the first portion of this, since jo= — j, is explicit in terms of x, but the other 
requires further development, for which we nmst have recourse to equation (16.), 
putting Q^=(1 + A) and P^=^=^(0.^^+ ] &c.), where we find 
(1 + A)’ = (l + A)«.^,+ (1+A)\5,_.+ (1+A)"..y_,+ &c. 
But we also have by equation (18.), 
Therefore, subtracting and dividing by A, and applying each term of operation to the 
term <p{a—pb — h.O) of quantity. 
1-(1+A) 
[ 
A (p{a—pb — b.0)=Y0.s^ 
i-(i+Ar 
1-(1 + A)' 
1-(1 + A) 
1 
f^_iH ^ i\_2+kc.j(p{a—pb-jrb.O) 
■O.s^ 
1-(1 + A) 
<p(^a—pb — b.0).s^_,-] (p{a-pb — b.0).s^_^-^kc. . (31.) 
(19.) The expression for in the last article is general and entirely independent 
of any particular values assigned to x, a, b, s, the only relation established being that 
OC 
expressed by the equation y— =. Suppose therefore that in a certain proposed case 
we should have 
and therefore 
a=^x-\-s — 1; Z>= — S’, 
pb=X', a — pb=s—\, 
and the expression for the sum of the series becomes 
X 
Sj,= ^ (p(,rd-5 — l+'J-O) 
1-(1 + A)'^ 
^ <p(.s-l-l-5.0)..y^_, 
l-(l + A)^ 
■(p{s—l-\-s.0).s^_2-\- &c. 
(32.) 
in which expression the first member or non-periodical part is an explicit function of 
X and s, and the periodical part has all its coefficients independent on x and functions 
of s alone. 
(20.) The periodical p^rt of is however susceptible of another form, better adapted 
for numerical calculation, into which it may be thrown by making in equa- 
tion (9.), when it becomes 
^^> ( 0 ) + G+o), 
