The theory of the perturbation of those compressible flows in which there are only two independent variables is developed, with special reference to steady plane transonic flow. Expressions are found for variations of both the physical flow and of its hodograph and relations obtained between the two sorts of variation. It is then proved that even if the Jacobian of the hodograph of a flow should vanish on the boundary the flow need not be critical. Using the expressions for variations in the physical plane of flow, boundary-value problems for perturbations can be discussed directly in the physical plane. The methods are applied to the case of circulatory flow outside circular or nearly circular cylinders. it is proved that instability can arise by a resonance process near certain critical speeds, measured on the boundary of the flow, and that at these speeds Dirichlet's problem for the perturbation stream function has no unique solution.
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