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The integral equation for the isotropic scattering of radiation in a medium bounded laterally by planes ±y0 is solved as a series in powers of exp (−y0). This series is certainly convergent for y0 large enough, and rapidly convergent for most practical purposes.

Radiation from arbitrary directions incident on a thick, homogeneous, conservative atmosphere with isotropic scattering is mostly diffusely reflected. A smaller part diffuses through the atmosphere and emerges at the other side. Sobolev's equations, which give the asymptotic forms of the reflection and transmission functions for large thickness,b, in terms of the functionH(μ) known from the theory of semi-infinite atmospheres, are newly derived. It is shown by two examples that knowledge of these asymptotic forms makes it possible accurately to interpolate any quantity dependent onbbetweenb = 1 andb = ∞.

A temperature gradient is introduced into the density matrix equation by analogy with the Boltzmann transport equation. The electric and heat currents are defined in the case of isotropic scattering, in which particular case, it is shown that the Wiedemann-Franz law holds even in the details of the oscillations of both the electric and thermal conductivities.

THE problem of the radiation field in a plane-parallel multilayer system, whose outer boundary is irradiated by parallel rays, is considered. The case of non-isotropic radiation is studied. A formal solution of the problem is given, and using this, the spatial-angular distribution function of the intensity of the diffused radiation is constructed by applying the method of discrete ordinates. Consider a multilayer plane-parallel system, consisting of N layers in contact with one another which are irradiated by a parallel pencil of rays incident at some angle to the outward normal of the system. We assume that the boundaries of the layers are known and that the source functions are defined differently in each layer. It is required to determine the radiative field in any region of the system and find the law of the angular distribution of the outgoing radiation.

1. In (1) M. G. Smith solved the equationwherefor R > 0, ø( − R) = − ø(R) and 0 < ω ≤ 1. In order to determine a constant, A(R1), appearing in the solution, two expressions for q(R1) were obtained, namelyand

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

The kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.