MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
129 
Now u being any quantity whatever and x being aroot of [l —x”"'] =0,we have identically 
[l — = — x){u — x'^)..{u — 
and therefore putting m— 1, we have w=(l —.»)(! 
and therefore 
whence 
Sx=z—, 
— &C. +- rT— 
(1 — x^){\ — a;^)..(l — x'^) [1 — a?”*] 
Again, m being as before a prime number, assume 
_ D „ I 
(1 — a:')(l — x ^)..(\ — a?”*) " ' [1 — a?™]’ 
we have in this case for [l — .r”*]=0. 
6x— 
'1 — x)‘^{\ — a;^) .. (1 — a?™"*)’ 
1 ] 
which is immediately reduced to ^x=- — . Now 
•' m 1—x 
[1 — M*”] [1 — M*”] — [1 —a;’"] 
u—x u—x 
or putting m = 1, 
(1+^+..+^*” ^) + (l +M.. -f-M’" 
1 —X 
=m — 1 d-m — 2x..-\-x” 
and substituting this in the value of dx, we find 
1 
(1 — x) (1 — x ^) .. (1 — a?™) 
P 1 (m— 1) + (m — 2)a:.. +a;’”~^ 
= &c. H — 2 
[1 — a?”‘] 
(t>X 
The preceding decomposition of the fraction ^ gives very readily the expansion of 
JX 
the fraction in ascending powers of x. For, consider a fraction such as 
Sx 
[1-x‘^y 
where the degree of the numerator is in general less by unity than that of the deno- 
minator ; we have 
i—x°= [i — ^'']n[i — x“'], 
where a' denotes any divisor of a (including unity, but not including the number 
a itself). The fraction may therefore be written under the form 
SxU [1 — a?“'] 
1 — a;“ 
where the degree of the numerator is in general less by unity than that of the deno- 
minator, i. e. is equal to a—l. Suppose that b is any divisor of a (including unity, 
but not including the number a itself), then 1— a?Ms a divisor of n[l— jr“'], and 
