Abstract
A rapidly convergent solution is obtained for the equations of radiative transfer. An absorbing atmosphere having an arbitrary variation of mixing ratio, temperature and pressure with height is considered. The Lorentz shape of the spectral lines together with the dependence of their line strengths and half‐widths on pressure and temperature are taken into account in the calculations. It is assumed that the lines do not overlap and the range of validity of this approximation is stated. It is shown that the expression for the absorption of radiation over a path between any two heights can be written in a tractable form even after three‐fold integration over the variable conditions along the path, over frequency and over all angles in the hemisphere. When the mass per unit area, along the path, is small or the line intensity is small or both, thus causing the absorption of radiation to be small even at the centre of the line, the integrated absorption is proportional to the dimensionless constant gamma that occurs in atmospheric radiation problems; when these quantities are large, thus causing the absorption to be complete for several half‐widths from the centre of the line, the integrated absorption is proportional to the square‐root of gamma. This is the generalization to variable path conditions of the linear and square‐root relations of infra‐red spectroscopy. Some examples of the method developed here are given when the variable conditions along the atmospheric path can be represented with sufficient accuracy as polynomials. An approximation to the H~2~O and CO~2~ distributions shows that the absorption of radiation from the ground at a given height depends greatly on the variation of the mixing ratio with height.