On the radiative equilibrium of a stellar atmosphere. XVII.

In this paper a systematic study is made of the various functional equations which characterize the transfer problems in plane-parallel atmospheres. First, a functional equation relating the angular distribution of `the emergent radiation from a semi- infinite atmosphere and the law of diffuse reflection by the same atmosphere is derived. This equation arises in consequence of the invariance of the emergent radiation from a semi-infinite atmosphere to the addition (or removal) of layers of arbitrary optical thickness to (or from) the atmosphere. Next, four functional equations governing the problem of transmission and diffuse reflection by an atmosphere of finite optical thickness are formulated. These equations have been derived under very general conditions; and they have been further reduced to a basic system of functional equations for the case in which scattering takes place in accordance with a phase function expressible as a series in Legendre polynomials. And, finally, further functional equations are formulated which determine the radiation field in the in- terior in terms of the scattering and transmission functions of atmospheres of finite optical thicknesses