NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 403 
oc cc 
= by a second integer division t, we have, putting y for 
=^— ^|o.-Sj+] -^'^-1+ — ('^ — l).s^,_^+i| 
— - jo .^^+1 (^— ; ( 25 .) 
and so on as far as we please. 
(14.) The periodic function O.^^+l &c. depends implicitly on x, because y is 
dependent on x. Its value however (as well as that of any other implicit periodical 
function) is very easily obtained by following out the process explained in (art. 11). 
Suppose, for example, we had the more general periodic function of y, 
q^=Za.ty■\-h.ty_■^-\- h.ty_t+^. 
Then we may write down the successive values of x, y, Qy in order thus : 
X 
0, 1, 2, .. 
. (.9—1) ; .9, 
.9-1-1, .. 
.. (2.9-1); 2s, ... 
y 
p 
p 
o 
V.* 
0; 1, 
1, .. 
1 ; 2,... 
t , 
qy 
a, a, a, .. 
a; h, 
h,.. 
Z> ; C , ... 
...; a, 
Thus we see that qy is a periodical function of x, having for its period st instead of 
either .s or ^ separately, the first s coefficients being all alike and each —a, the next 
s all alike and each =&, and so on ; or 
® ^ ('^0^-2s+ 1 } + + • ' • • 1 } • 
y 
(15.) Hence we are enabled to express the value of = explicitly as a periodic func- 
tion of X ; for by equation (25.), if we put st=n, 
\—n — .5^-1 +....5— 1 •-S;r-s+ij’ 
But by equation (21.) we have 
^^x~\~^'x—s~\~^x—2s'\~’"^'x—ns+s 
n^-ns+s-ii ; 
and by the equation (26.) of the foregoing article, 
ty ^X~\~^X—l I ' '^X—S+l 
^y—l s— I ~1“ • • 2s+l 
iy-2 — '^x-2s~\~^x~2s-\~\~ " •''^x-3s+\') &C. ; 
and consequently by substitution, 
.w^_i + 2.7?^_2+ w— (2/.) 
3 F 2 
(26.) 
