  A `coherent system' $(\Cal E,V)$, consists of a holomorphic bundle plus a
linear subspace of its space of holomorphic sections. Based on the usual notion
in Geometric Invariant Theory, a notion of slope stability has been defined for
such objects (by Le Potier, and also by Rhagavendra and Vishwanath). In this
paper we show that stability in this sense is equivalent to the existence of
solutions to a certain set of gauge theoretic equations. One of the equations
is essentially the vortex equation (i.e. the Hermitian-Einstein equation with
an additional zeroth order term), and the other is an orthonormality condition
on a frame for the subspace $V\subset H^0(\Cal E)$.
