  This paper is the second in a series of three devoted to the smooth
classification of simply connected elliptic surfaces. In this paper, we study
the case where one of the multiple fibers has even multiplicity, and describe
the moduli space of stable rank two vector bundles with the appropriate first
Chern class needed to calculate Donaldson polynomials. The analysis is in many
ways parallel to the analysis in the case of vector bundles of trivial
determinant, but the asymmetry between the multiplicities also appears in the
moduli space. In this way, the first coefficient of an appropriate Donaldson
polynomial determines one of the multiplicities.
