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Application of the moment method of L. V. Spencer and U. Fano is made to the penetration and slowing down of neutrons in infinite media. General considerations are restricted mainly to those modifications of the method necessary for the neutron problem; in particular, a special technique is given for reducing the integral term of the neutron transport equation. Calculations are carried out for point isotropic monoenergic neutron sources of 1, 100, 500 kev, 1, 2, 5, and 14 Mev in an infinite atmosphere of air. Best available numerical values of the cross sections are used, and the effects of absorption, anisotropic angular distributions, and inelastic scattering are included. Distances out to 10 mean free paths are considered.