Approximate solutions to the one-velocity neutron transport equation for an infinite cylinder with isotropic scattering and spatially piecewise constant cross sections are obtained by Fourier expansion of the neutron distribution function in one of the angular variables. An infinite coupled set of equations for the expansion coefficients is derived and general properties of the solutions to the truncated set of equations are discussed. A scheme for solving these equations by Gauss quadratures is given, and, as an example, the solution to the bare infinite cylinder critical problem is given in three orders of approximation. Excellent accuracy is obtained with a fairly small investment of analytical effort. The extension of the method to include the effects of anisotropic scattering is sketched.