In this paper the Rayleigh hypothesis in the theory of reflection by a cylindrical perturbation in a plane surface is investigated analytically. The hypothesis asserts that above the surface the scattered field may be expanded in terms of outward-going wave functions. As such, it is analogous to the assumption made by Lord Rayleigh in his treatment of diffraction by a reflection grating. We show that the validity of the Rayleigh hypothesis is governed by the distribution of singularities in the analytic continuation of the exterior scattered field. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. A procedure is presented that enables the validity of the Rayleigh hypothesis to be checked for a surface whose profile can be described by an analytic function. Numerical results are presented.