The paper gives an exact solution to the problem of determining the intensity of radiation emerging from a semi-infinite plane-parallel atmosphere when the Planck function B(v, T) varies linearly with the optical depth, the ratio of line to continuous absorption is constant with depth, and when the line scattering is partly coherent and partly non-coherent. The non-coherent part of the scattering is taken to be proportional to the coefficient of line absorption and to the mean intensity of radiation at the centre of the line, a process which is a simplified form of completely non-coherent scattering. The combination of both scattering processes in a suitable ratio describes the formation of subordinate lines and, to a certain degree of approximation, resonance lines broadened by collisions. The solution in the general case involves auxiliary functions which satisfy two simultaneous linear integral equations, but when the non-coherent part of the scattering alone is present, the solution only depends on the H -functions, their moments, and the quadrature of a simple integral form.