The following report describes and foretells the situation in medium in which neutrons are being (a)produced as fast neutrons, (b) slowed down to thermal speeds by impacts with nuclei, and (c) absorbed by nuclei in such a manner that sometimes fresh fast neutrons appear at the place where a thermal neutron has just appeared.

Abstract. Successive approximations ae obtained from transport theory for the neutron distribution around a black sphere surrounded by a spherical gap and a non-absorbing medium extending to infinity. Six cases are calculated numerically to show the difference between this treatment and simple diffusion theory. The extrapolation distance and a quantity that is essentially the thermal utilization are calculated in different approximations. It is found that the neutron density in the gap, instead of being a constant as predicted by the simple theory, may vary by a factor of one-third in cases of practical interest.