  We study several subseries of the space of second order theta functions on
the Jacobian of a non-hyperelliptic curve. In particular, we are interested in
the subseries P\Gamma_{00} consisting of 2theta-divisors having multiplicity at
least 4 at the origin, or, equivalently, containing the surface C-C, and in its
analogues consisting of 2theta-divisors having higher multiplicities at the
origin, containing the four-fold Sym^2C-Sym^2C, or singular along the surface
C-C. We use rank 2 vector bundles with a given number of global sections to
prove canonical isomorphisms between quotients of the above introduced
subseries and vector spaces defined by the canonical divisor.
