  The paper determines the change of moduli spaces of rank $2$ sheaves on
surfaces with $p_g=0$ under change of polarization and the corresponding change
of the Donaldson invariants. In this revised version we have made some minor
stylistic changes in the previous text. In addition we have added a final
chapter of about 20 pages (announced in the previous version), in which the six
lowest order terms (three of them non-zero) of the change are computed
explicitely using computations in the cohomology of Hilbert schemes of points.
