  We investigate the jumping conics of stable vector bundles $E$ of rank 2 on a
smooth quadric surface $Q$ with the first Chern class $c_1=\Oo_Q(-1,-1)$ with
respect to the ample line bundle $\Oo_Q(1,1)$. We show that the set of jumping
conics of $E$ is a hypersurface of degree $c_2(E)-1$ in $\PP_3^*$. Using these
hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on
$Q$ in the cases of lower $c_2(E)$.
