  Let $|L|$ be a linear system on a smooth complex Enriques surface $S$ whose
general member is a smooth and irreducible curve of genus $p$, with $L^ 2>0$,
and let $V_{|L|, \delta} (S)$ be the Severi variety of irreducible
$\delta$-nodal curves in $|L|$. We denote by $\pi:X\to S$ the universal
covering of $S$. In this note we compute the dimensions of the irreducible
components $V$ of $V_{|L|, \delta} (S)$. In particular we prove that, if $C$ is
the curve corresponding to a general element $[C]$ of $V$, then the codimension
of $V$ in $|L|$ is $\delta$ if $\pi^{-1}(C)$ is irreducible in $X$ and it is
$\delta-1$ if $\pi^ {-1}(C)$ consists of two irreducible components.
