The singular eigenfunction expansion technique is used to solve rigorously the equation of radiative transfer describing the interlocking-doublet model in the theory of line scattering. The Planck function, represented as a linear function of the optical variable, is assumed to be independent of frequency over the range of interest, and the ratios ƞj of the line-scattering coefficients to the continuous absorption coefficient are taken to be constants. The normal-mode expansion technique, in conjunction with the appropriate particular solution, is used to obtain a rigorous analytical solution for the radiation intensity valid anywhere in a semi-infinite medium subjected to zero incident radiation. Half-range completeness and orthogonality theorems applicable for the basis set used are employed to effect the desired solution with a minimum of effort, and, as a procedure alternative to older techniques, the Case method is used to construct the half-space S-matrix, useful when surface quantities are of principal interest.