  Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated
$R$-module of dimension $d$. Let $\underline{x} = x_1, ..., x_d$ be a system of
parameters of $M$ and $\underline{n} = (n_1, ..., n_d)$ a $d$-tuple of positive
integers. In this paper we study the length of generalized fractions $M
(1/(x_1, ..., x_d, 1))$ which was introduced by Sharp and Hamieh in
\cite{ShH85}. First, we study the growth of the function $J_{\underline{x},
M}(\underline{n}) = \ell(M (1/(x_1^{n_1}, ..., x_d^{n_d}, 1))) - n_1...n_d
e(\underline{x};M)$. Then we give an explicit calculation for the function
$J_{\underline{x}, M}(\underline{n})$ in the case where $M$ admits a
Macaulayfication. Most previous results on this topic are now easy to
understand and to improve.
