paper entitled "On the Stability of Saturn's Rings," he summarizes his
deductions as follows:

"Let us now gather together the conclusions we have been able to draw
from the mathematical theory of various kinds of conceivable rings.

"We found that the stability of the motion of a solid ring depended
on so delicate an adjustment, and at the same time so unsymmetrical a
distribution of mass, that even if the exact conditions were fulfilled,
it could scarcely last long, and, if it did, the immense preponderance
of one side of the ring would be easily observed, contrary to
experience. These considerations, with others derived from the
mechanical structure of so vast a body, compel us to abandon any theory
of solid rings.

"We next examined the motion of a ring of equal satellites, and found
that if the mass of the planet is sufficient, any disturbances produced
in the arrangement of the ring will be propagated around it in the form
of waves, and will not introduce dangerous confusion. If the satellites
are unequal, the propagations of the waves will no longer be regular,
but disturbances of the ring will in this, as in the former case,
produce only waves, and not growing confusion. Supposing the ring to
consist, not of a single row of large satellites, but a cloud of evenly
distributed unconnected particles, we found that such a cloud must
have a very small density in order to be permanent, and that this is
inconsistent with its outer and inner parts moving with the same angular
velocity. Supposing the ring to be fluid and continuous, we found that
it will be necessarily broken up into small portions.

"We conclude, therefore, that the rings must consist of disconnected
particles; these must be either solid or liquid, but they must be
independent. The entire system of rings must, therefore, consist either
of a series of many concentric rings each moving with its own velocity
and having its own system of waves, or else of a confused multitude of
revolving particles not arranged in rings and continually coming into
collision with one another.

"Taking the first case, we found that in an indefinite number of
possible cases the mutual perturbations of two rings, stable in
themselves, might mount up in time to a destructive magnitude, and that
such cases must continually occur in an extensive system like that of
Saturn, the only retarding cause being the irregularity of the rings.

"The result of long-continued disturban