The problem of calculating the incompressible two-dimensional flow of a gas past porous surfaces is shown to involve the determination of an analytic function satisfying a complicated non-linear relation between its real and imaginary parts on these surfaces. In “linearized” flow this boundary condition reduces to a linear relation, and it is possible to determine the analytic function defining the flow. The general problem of determining an analytic function subject to a linear relation between its real and imaginary parts on a single boundary is solved by reducing it to two simultaneous Dirichlet problems. Solutions of this problem have been obtained by Carleman, Gakhov and others (see1) for references), but the solution given here is more direct, as it does not require the continuation of the function to a “sectional holomorphic” function1) beyond the original boundary. The flow past an infinite wall, the “porosity” of which varies along its length, is calculated as a simple example of the theory. The theory of an aerofoil in a slotted tunnel is also outlined.