The solution of the neutron slowing down equation in an infinite, homogeneous moderator (with nuclei of mass M) has been obtained by means of direct summation of transition probabilities between the initial and final energy states. It has been possible to obtain an exact formula for the distribution of neutrons in lethargy space after N collisions with moderator nuclei. The asymptotic expansion of the results for large and for very large N is in agreement with the Dancoff's formulas. The accuracy of the asymptotic expansion has been estimated.