MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
139 
denote the portion which is made up of the simple fractions having powers of .r— j:, 
for their denominators, we have by a known theorem 
= coefficient - in 
faU, ^ 3C-X -Z f{x^+z) 
Now by a theorem of Jacobi’s and Cauchy’s, 
coefficient^ in F2= coefficient y in ; 
whence, writing \ we have 
^ I = coefficient 7 in - 
t x^-xe^ f{x^e-*) 
x^ — xe^ a7j(l — — e®) a?](l — e®) 
1 . 
Now putting for a moment x—x^e^, we have 
1 1 1 
and whence 
x^ — xe^ X 
|(/C/ X XaW i4/ j t4/ 
the general term of which is 
n(s 
1 
— 
— 1)' Xj — X 
Hence representing the general term of 
x^^(xje-^) 
by xXit % so that 
f{x^e-‘) 
m • I • C ^X,(p(x,e~‘) 
xx, = coefficient y in ^ f{x,e-*) ’ 
we find, writing down only the general term 
■ ■ • + + ■ ■ 
where the value of x/Ti depends upon that of s, and where A’eKtends from .v— 1 to sz=k. 
Suppose now that the denominator is made up of factors (the same or different) of 
the form l—x"". And let a be any divisor of one or more of the indices m, and let 
k be the number of the indices of which a is a divisor. The denominator contains 
the divisor [l— a;®]*, and consequently if § be any root of the equation [l — a'“]=0, 
the denominator contains the factor {^—xy. Hence writing § for .r, and taking the 
sum with respect to ail the roots of the equation [l— a'“]=0, we find 
_ 
n(s 
where 
Xi= coefficient] in f 
