Abstract We have examined the slowing down of neutrons in a lattice consisting of alternate slabs of noncapturing moderator and nonmoderating fuel whose cross-sections vary arbitrarily with energy. The Boltzmann equation is transformed to an integral equation into which appropriate boundary conditions are incorporated. Expansion of the flux functions in terms of orthogonal polynomials in space and angle co-ordinates yields an infinite set of coupled integral equations in energy. In the simplest approximation and for a hydrogen moderator these equations are easily integrated to give an expression for the resonance escape probability identical with the one recently derived—from quite different considerations—by J. Chernick. We show that higher-order terms in our equations should contribute little to neutron absorption in those capture resonances which contribute most to the integrated capture probability. Thus, while our simplest approximation is not a good one for determining flux distributions, it will be quite good for determining resonance escape probabilities. Numerical results will be given.