The general equations of motion are developed for a compressible, inviscid flow in which a non-uniform distribution of heat transfer is applied to the fluid or a non-uniform generation of heat per unit volume occurs. In general, vorticity can be created if the cross-products of the temperature and entropy gradients are finite. If the temperature gradients in the flow are small (first-order), then a non-uniform heat addition across the stream will produce a second-order change in vorticity. For this type of flow, solutions are obtained for the variations of velocity and density that occur in a two-dimensional plane flow and an axially symmetric three-dimensional flow. A simple expression is also obtained for the streamline displacement caused by the non-uniform addition of heat.