Jenike's radial solution, widely used in the design of materials-handling equipment, is a similarity solution of steady-state continuum equations for the flow under gravity of granular material through an infinite, right-circular cone. In this paper we study how the geometry of the hopper influences this solution. Using perturbation theory, we compute a first-order correction to the (steady-state) velocity resulting from a small change in hopper geometry, either distortion of the cross section or tilting away from vertical. Unlike for the Jenike solution, all three components of the correction velocity are nonzero; i.e., there is secondary circulation in the perturbed flow.