In Bulletin No. 32 of the Observatory (Nov. 24, 1907) Percival had written: “Laplace first showed that the rings could not be, as they appear, wide solid rings inasmuch as the strains due to the differing attraction of Saturn for the several parts must disrupt them. Peirce then proved that even a series of very narrow solid rings could not subsist and that the rings must be fluid. Finally Clerk-Maxwell showed that even this was not enough and that the rings to be stable must be made up of discrete particles, a swarm of meteorites in fact. But, if my memory serves me right, Clerk-Maxwell himself pointed out that even such a system could not eternally endure but was bound eventually to be forced both out and in, a part falling upon the surface of the planet, a part going to form a satellite farther away.

“Even before this Edward Roche in 1848 had shown that the rings must be composed of discrete particles, mere dust and ashes. He drew this conclusion from his investigations on the minimum distance at which a fluid satellite could revolve around its primary without being disrupted by tidal strains.

“The dissolution which Clerk-Maxwell foresaw can easily be proved to be inevitable if the particles composing the swarm are not at considerable distances from one another, which is certainly not the case with the rings as witnessed by the light they send us even allowing for their comminuted form. For a swarm of particles thus revolving round a primary are in stable equilibrium only in the absence of collisions. Now in a crowded company collisions due either to the mutual pulls of the particles or to the perturbations of the satellites must occur. At each collision although the moment of momentum remains the same, energy is lost unless the bodies be perfectly elastic, a condition not found in nature, the lost energy being converted into heat. In consequence some particles will be forced in toward the planet while others are driven out and eventually the ring system disappears.

“Now the interest of the observations at Flagstaff consists in their showing us this disintegration in process of taking place and furthermore in a way that brings before us an interesting case of celestial mechanics.”

He examines the rings mathematically, as the result of perturbations caused by the two nearest of the planet’s satellites, Mimas and Enceladus.

The effect is the same that occurs in the case of Jupiter and the asteroids, Saturn taking the place of the Sun, his satellites that of Jupiter, and the rings that of the asteroids. In spite of repetition it may be well to state in his own words the principle of commensurate periods and its application to the rings:[36]

“The same thing can be seen geometrically by considering that the two bodies have their greatest perturbing effect on one another when in conjunction and that if the periods of the two be commensurate they will come to conjunction over and over in these same points of the orbit and thus the disturbance produced by one on the other be cumulative. If the periods are not commensurate the conjunctions will take place in ever shifting positions and a certain compensation be effected in the outstanding results. In proportion as the ratio of periods is simple will the perturbation be potent. Thus with the ratio 1:2 the two bodies will approach closest only at one spot and always there until the perturbations induced themselves destroy the commensurability of period. With 1:3 they will approach at two different spots recurrently; with 1:4 at three, and so on....

“We see, then, that perturbations, which in this case will result in collisions, must be greatest on those particles which have periods commensurate with those of the satellites. But inasmuch as there are many particles in any cross-section of the ring there must be a component of motion in any collision tending to throw the colliding particles out of the plane of the ring, either above or below it.

“Considering, now, those points where commensurability exists between the periods of particle and satellite we find these in the order of their potency:

With Mimas, 1:2
1:3
1:4
With Enceladus, 1:3