  We find a complete set of relations for the rational cohomology ring of the
moduli space of rank three stable bundles over a Riemann surface of genus g and
also show that the Pontryagin ring vanishes in degree 12g-8 and greater. The
results are obtained by introducing some 'dual' Mumford relations and
generalising Kirwan's proofs of the Mumford and Newstead conjectures in the
rank two case. (In this revised version of the paper the vanishing degree of
the Pontryagin ring of the moduli space has been improved from `in and above
degree 12g-4' to `in and above degree 12g-8'. This degree is now known to be
sharp.)
