  Let $P \cup P'$ be the two component Prym variety associated to an \'etale
double cover $\tilde{C} \to C$ of a non-hyperelliptic curve of genus $g \geq 6$
and let $|2\Xi_0|$ and $|2\Xi_0'|$ be the linear systems of second order theta
divisors on $P$ and $P'$ respectively. The component $P'$ contains canonically
the Prym curve $\tilde{C}$. We show that the base locus of the subseries of
divisors containing $\tilde{C} \subset P'$ is scheme-theoretically the curve
$\tilde{C}$. We also prove canonical isomorphisms between some subseries of
$|2\Xi_0|$ and $|2\Xi_0'|$ and some subseries of second order theta divisors on
the Jacobian of $C$.
