In this paper we investigate the flow of hypersurfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact hypersurfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short-time existence theorem for hypersurfaces that are sufficiently close to a sphere and, using centre manifold analysis, the stability of the sphere as a stationary solution to the flow is determined. We will find that for initial hypersurfaces sufficiently close to a sphere, the flow will exist for all time and the hypersurfaces will converge exponentially fast to a sphere. This result was shown for the case where the symmetric function is the mean curvature and the constraint is on the (n+1)-dimensional enclosed volume by Escher and Simonett (1998).