In some recent papers, the solutions of some problems of line formation involving non-coherent scattering and the interlocking of multiplet lines have been based on the principle of invariance governing the law of diffuse reflection. Hitherto this principle had been applied only in cases of coherent scattering and its applicability to problems involving non-coherent scattering processes has been questioned. In this paper the solutions for completely non-coherent scattering and for interlocked multiplet lines as given in (3) and (5) are obtained by a mathematical method, thus providing an independent verification of the principle of invariance in these cases. The problem of non-coherent scattering solved here is more general than that given in (3). It allows for the possibility of thermal emission associated with the line absorption; it applies when the frequencies involved are restricted to a finite interval (v1, v2) and it allows for incident radiation on the surface of the star.