MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
131 
a is a divisor. The particular results previously obtained show, that m being a 
prime number, 
and 
coefiicient in 
coefficient x^ in 
(l-a?2)(l-a^). .(!-«« 
1 
3 ;;^=&C. + -(1,-1,0, 0,..) per 
m. 
1, — 1, — 1,..) perm,. 
Suppose, as before, that the degree of (px is less than that of fx, and let the analy- 
tical expression above obtained for the coefficient of x"^ in the expansion in ascending 
(I)X 
powers of x of the fraction ^ be represented by ¥q, it is very remarkable that if we 
expand^ in descending powers of x, then the coefficient of x"^ in this new expansion 
{q is here of course negative, since the expansion contains only negative powers of x) 
is precisely equal to — ; this is in fact at once seen to be the case with respect to 
each of the partial fractions into which ^ has been decomposed, and it is conse- 
quently the case with respect to the fraction itself*. This gives rise to a result of 
some importance. Suppose that px and fx are respectively of the degrees N and D ; 
it is clear from the form of fx that we have f(^ = {—)°x~^fx ; and I suppose that px 
is also such that = ; then writing D—N=A, and supposing that^ 
is expanded in descending powers of x, so that the coefficient of x'^ in the expansion 
is — F^, it is in the first place clear that the expansion will commence with the 
term a;"*, and we must therefore have 
F ^=0 
for all values of q from q= — 1 to q= — {h — 1), 
Consider next the coefficient of a term where ^ is 0 or positive ; the coefficient 
in question, the value of which is —Y{ — h—q), is obviously equal to the coefficient 
of in the expansion in ascending powers of x of 
1 
e. to 
(i)^(~)° coefficient in 
x’‘ipx 
fx 
(px , 
or what is the same thing, to 
)° coefficient in ^ ; 
and we have therefore, q being zero or positive, 
F(-A-^)=-(±n-)-F^. 
In particular, when px=l, Fq—0 
* The property is a fundamental one in the general theory of developments. 
