Maxwell determined the conditions of stability of a single ring of small particles moving round a large primary. He also made some incomplete remarks on the effects of introducing a second ring. The present investigation considers in greater detail the stability of two rings of particles moving about a primary and subject to the gravitational attractions of the primary and of each other. It is shown that such a system, under conditions satisfied by the Saturnian system, is stable, the particles oscillating finitely about their mean positions. It is inferred that the Saturnian system, considered as a number of such rings, is therefore also stable.