136 
MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
To transform the preceding expressions, I write when h is odd instead of x, and 
I put for shortness 6 instead of ^ or 2{\k—s), and y instead of or 
a-l-j(5+l) ; we have to consider an expression of the form 
coefficient x^”" in 
where Fx is the product of factors of the form l—x''. Suppose that a' is the least 
common multiple of a and 0, then (1 — J7“') -i-(l — is an integral function of x, equal 
XX suppose, and 1 -i-(l — x“)=%x-7- (1 — x“'). Making this change in all the factors of 
Fx which require it {i. e. in all the factors except those in which a is a multiple of 0), 
the general term becomes 
coefficient x^”' in 
Gx 
where Gj: is a product of factors of the form 1 — x“', in which a' is a multiple of 0, 
{. e. Gx is a rational and integral function of x^. But in the numerator x^Wx we may 
reject, as not contributing to the formation of the coefficient of x®”*, all the terms in 
which the indices are not multiples of^; the numerator is thus reduced to a rational 
and integral function of x^, and the general term is therefore of the form 
coefficient a;®’" in 
or what is the same thing, of the form 
coefficient x^ in — • 
XX 
Where )cx is the product of factors of the form 1 — and Xx is a rational and integral 
function of x, the particular value of the fraction depends on the value of s; and 
uniting the different terms, we have an expression 
which is equivalent to 
coefficient x^in S. (— )* — j 
® 'XX 
coefficient a?™ in 
where yj? is a product of factors of the form 1— j?", and <px is a rational and integral 
function of x. And it is clear that the fraction will be a proper one when each 
of the fractions in the original expression is a proper fraction, t. e. in the case of 
P(0, l,2..A')'"^(/rm— a), when for^even cc<^k{k-{-2), and for^odd a<:^(A-+ 1)(^+3); 
and in the case of P'(0, \,2 ..ky'^{km—a,), when for k even a+1 <\k{k-\-2), and for 
k odd a+1 <^(Ar+l)(A:+3). 
We see, therefore, that 
P(0, 1,2 a), 
and 
P'(0, 1 , 2 .. k)”'\{km — ci), 
are each of them of the form 
coefficient x” in 
tx 
