The proper implementation of the tau -p method for surface data excited by a point source requires a cylindrical slant stack. Usually the common (Cartesian) slant stack is computed instead as an approximation to the geometrically correct procedure. Here we describe a formulation of the cylindrical slant stack as a weighted sum of Cartesian slant stacks; our cylindrical slant stack is computationally efficient to perform. We show how, although the usefulness of the slant stack is most easily seen with Cartesian coordinates, it can also be used with Fourier-Bessel transforms.An example of the method shows results computed from data recorded on the West Florida Shelf. Severe edge-effect noise which overwhelms the Cartesian slant stack is attenuated by the cylindrical slant-stacking. Applications of the cylindrical slant stack to other seismological calculations, such as Lamb's problem, are also discussed. In particular, we prove that the plane-wave reflection coefficients apply exactly in the tau -p domain; hence an amplitude-versus-offset analysis is unambiguous in that domain.