In an earlier paper on this subject the author (1921) proposed a theory of the rings which showed satisfactory agreement with the observed measurements of the rings. The mathematical method was, however, subjected to criticism. In the present paper the subject is again attacked by an entirely different method which is free from the objections raised against the first method. A family of periodic orbits of the particles forming the ring, when perturbed by a satellite, is constructed, and the stability of these orbits is examined by the method of small displacements. Stability determined in this way is shown to have a real meaning when applied to the problem in hand. The positions of instability of the particles lead to the divisions in the ring and the inner and outer boundaries, close agreement with observation being obtained. The analysis, though quite different from that of the earlier paper, reproduces its main features, and introduces further points of interest.