  We study the restrictions of rank 2 semistable vector bundles E on P^2 to
conics. A Grauert-Mulich type theorem on the generic splitting is proven. The
jumping conics are shown to have the scheme structure of a hypersurface J_{2}
in P^5 of degree c_{2}(E) when c_{1}(E)=0 and of degree c_{2}(E)-1 when
c_{1}(E)=-1. Some examples of jumping conics and jumping lines are studied in
detail.
