  Let $C$ be a nonsingular irreducible projective curve of genus $g\ge2$
defined over the complex numbers. Suppose that $1\le n'\le n-1$ and
$n'd-nd'=n'(n-n')(g-1)$. It is known that, for the general vector bundle $E$ of
rank $n$ and degree $d$, the maximal degree of a subbundle of $E$ of rank $n'$
is $d'$ and that there are finitely many such subbundles. We obtain a formula
for the number of these maximal subbundles when $(n',d')=1$. For $g=2$, $n'=2$,
we evaluate this formula explicitly. The numbers computed here are
Gromov-Witten invariants in the sense of a recent paper of Ch. Okonek and A.
Teleman (to appear in Commun. Math. Phys.) and our results answer a question
raised in that paper. In this revised version some references are added.
