James Clerk Maxwell.

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this motion in an eccentric circle is sufficiently near for illustration. The
motion of the waves relative to the ring is the same as before. The waves
of the first kind travel faster than the ring itself, and overtake the satellites,
those of the fourth kind travel slower, and are overtaken by them.

In fig. 11 we have an exaggerated representation of a ring of twelve satel-
lites afiected by a wave of the fourth kind where m = 2. The satellites here lie in
an eUipse at any given instant, and as each moves round in its circle about
its mean position, the ellipse also moves round in the same direction with half
their angular velocity. In the figure the dotted line represents the position of
the ellipse when each satellite has moved forward into the position represented
by a dot.

Fig. 13 represents a wave of the first kind where m = 2. The satellites at
any instant lie in an epitrochoid, which, as the satellites revolve about their
mean positions, revolves in the opposite direction with half their angular velocity,
so that when the satellites come into the positions represented by the dots,
the curve in which they lie turns round in the opposite direction and forms the
dotted curve.

In fig. 9 we have the same case as in fig. 13, only that the absolute orbits
of the satellites in space are given, instead of their orbits about their mean
positions in the ring. Here each moves about the central body in an eccentric

366 ON THE STABILITY OF THE MOTION OF SATURN's RINGS.

circle, which in strictness ought to be an ellipse not differing much from the
circle.

As the satellites move in their orbits in the direction of the arrows, the
curve which they form revolves in the same direction with a velocity 1^ times
that of the ring.

By considering these figures, and still more by watching the actual motion
of the ivory balls in the model, we may form a distinct notion of the motions
of the particles of a discontinuous ring, although the motions of the model are
circular and not elliptic. The model, represented on a scale of one-third in figs.
7 and 8, was made in brass by Messrs. Smith and Ramage of Aberdeen.

We are now able to understand the mechanical principle, on account of
which a massive central body is enabled to govern a numerous assemblage of
satellites, and to space them out into a regular ring; while a smaller central
body would allow disturbances to arise among the individual satelHtes, and
collisions to take place.

When we calculated the attractions among the satellites composing the ring,
we found that if any satellite be displaced tangentially, the resultant attraction
will draw it away from its mean position, for the attraction of the satellites it
approaches will increase, while that of those it recedes from will diminish, so that
its equilibrium when in the mean position is unstable with respect to tangential
displacements ; and therefore, since every satellite of the ring is statically unstable
between its neighbours, the slightest disturbance would tend to produce coUisions
among the satellites, and to break up the ring into groups of conglomerated
sateUites-

But if we consider the dynamics of the problem, we shall find that this
effect need not necessarily take place, and that this very force which tends
towards destruction may become the condition of the preservation of the ring.
Suppose the whole ring to be revolving round a central body, and that one
satellite gets in advance of its mean position. It will then be attracted forwards,
its path will become less concave towards the attracting body, so that its distance
from that body will increase. At this increased distance its angular velocity
will be less, so that instead of overtaking those in front, it may by this means
be made to fall back to its original position. Whether it does so or not must
depend on the actual values of the attractive forces and on the angular velocity
of the ring. When the angular velocity is great and the attractive forces small,

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 367

the compensating process will go on vigorously, and the ring wiU be preserved.
When the angular velocity is small and the attractive forces of the ring great,
the dynamical effect wiU not compensate for the disturbing action of the forces
and the ring ^vill be destroyed.

If the satellite, instead of being displaced forwards, had been originally
behind its mean position in the ring, the forces would have pulled it backwards,
its path would have become more concave towards the centre, its distance from
the centre would diminish, its angular velocity would increase, and it would
gain upon the rest of the ring till it got in front of its mean position. This
effect is of course dependent on the very same conditions as in the former case,
and the actual effect on a disturbed satellite would be to make it describe an
orbit about its mean position in the ring, so that if in advance of its mean
position, it first recedes from the centre, then falls behind its mean position in
the ring, then approaches the centre within the mean distance, then advances
beyond its mean position, and, lastly, recedes from the centre till it reaches its
starting-point, after which the process is repeated indefinitely, the orbit being
always described in the direction opposite to that of the revolution of the
ring.

We now understand what would happen to a disturbed satellite, if all the
others were preserved from disturbance. But, since all the satellites are equally
free, the motion of one will produce changes in the forces acting on the rest,
and this will set them in motion, and this motion will be propagated from one
satellite to another round the ring. Now propagated disturbances constitute
waves, and all waves, however complicated, may be reduced to combinations of
simple and regular waves; and therefore all the disturbances of the ring may
be considered as the resultant of many series of waves, of different lengths, and
travelling with different velocities. The investigation of the relation between
the length and velocity of these waves forms the essential part of the problem,
after which we have only to split up the original disturbance into its simple
elements, to calculate the effect of each of these separately, and then to combine
the results. The solution thus obtained will be perfectly general, and quite
independent of the particular form of the ring, whether regular or irregular at
starting. § 14.

We next investigated the effect upon the ring of an external disturbing
force. Having split up the disturbing force into components of the same type

368 ON THE STABILITY OF THE MOTION OF SATURN's RINGS.

with the waves of the ring (an operation which is always possible), we found
that each term of the disturbing force generates a " forced wave " travelling with
its own angular velocity. The magnitude of the forced wave depends not only
on that of the disturbing force, but on the angular velocity with which the dis-
turbance travels round the ring, being greater in proportion as this velocity
more nearly coincides with that of one of the "free waves" of the ring, "We
also found that the displacement of the satellites was sometimes in the direction
of the disturbing force, and sometimes in the opposite direction, according to
the relative position of the forced wave among the four natural ones, producing
in the one case positive, and in the other negative forced waves. In treating
the problem generally, we must determine the forced waves belonging to every
term of the disturbing force, and combine these with such a system of free
waves as shall reproduce the initial state of the ring. The subsequent motion
of the rmg is that which would result from the free waves and forced waves
together. The most important class of forced waves are those which are pro-
duced by waves in neighbouring rings. § 15.

We concluded the theory of a ring of satellites by tracing the process by
which the ring would be destroyed if the conditions of stability were not
fulfilled. We found two cases of instability, depending on the nature of the
tangential force due to tangential displacement. If this force be in the direction
opposite to the displacement, that is, if the parts of the ring are statically
stable, the ring will be destroyed, the irregularities becoming larger and larger
mthout being propagated round the ring. When the tangential force is in the
direction of the tangential displacement, if it is below a certain value, the
disturbances will be propagated round the ring without becoming larger, and
we have the case of stability treated of at large. If the force exceed this value,
the disturbances will still travel round the ring, but they will increase in ampli-
tude continually till the ring falls into confusion. § 18.

We then proceeded to extend our method to the case of rings of different
constitutions. The first case was that of a ring of satellites of unequal size.
If the central body be of suflScient mass, such a ring will be spaced out, so that
the larger satellites will be at wider intervals than the smaller ones, and the
waves of disturbance will be propagated as before, except that there may be
reflected waves when a wave reaches a part of the ring where there is a change
in the average size of the satellites. § 19.

ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 369

The next case was that of an annular cloud of meteoric stones, revolving
uniformly about the planet. The avercige density of the space through which
these small bodies are scattered will vary with every irregularity of the motion,
and this variation of density will produce variations in the forces acting upon
the other parts of the cloud, and so disturbances will be propagated in this
ring, as in a ring of a finite number of satellites. The condition that such a
ring should be free from destructive oscillations is, that the density of the
planet should be more than three hundred times that of the ring. This would
make the ring much rarer than common air, as regards its average density,
though the density of the particles of which it is composed may be great.
Comparing this result with Laplace's minimum density of a ring revolving as
a whole, we find that such a ring cannot revolve as a whole, but that the inner
parts must have a greater angular velocity than the outer parts. § 20.

We next took up the case of a flattened ring, composed of incompressible
fluid, and moving with uniform angular velocity. The internal forces here arise
partly from attraction and partly from fluid pressure. We began by taking the
case of an intinite stratum of fluid affected by regular waves, and found the accurate
values of the forces in this case. For long waves the resultant force Is in the
same direction as the displacement, reaching a maximum for waves whose
length is about ten times the thickness of the stratum. For waves about five
times as long as the stratum is thick there is no resultant force, and for shorter
waves the force is in the opposite direction to the displacement. § 23.

Applying these results to the case of the ring, we find that it will be
destroyed by the long waves unless the fluid is less than -^ of the density of
the planet, and that in all cases the short waves will break up the ring into
small satellites.

Passing to the case of narroiv rings, we should find a somewhat larger
maximum density, but we should still find that very short waves produce forces
in the direction opposite to the displacement, and that therefore, as already
explained (page 333), these short undulations would increase in magnitude without
being propagated along the ring, till they had broken up the fluid filament into
drops. These drops may or may not fulfil the condition formerly given for the
stability of a ring of equal satellites. If they fulfil it, they will move as a
permanent ring. If they do not, short waves will arise and be propagated among
the satellites, with ever increasing magnitude, till a sufficient number of drops

370 ON THE STABILITY OF THE MOTION OF SATURN's RINGS.

have been brought into collision, so as to unite and form a smaller number of
larger drops, which may be capable of revolving as a permanent ring.

We have already investigated the disturbances produced by an external
force independent of the ring ; but the special case of the mutual perturbations
of two concentric rings is considerably more complex, because the existence of a
double system of waves changes the character of both, and the waves produced
react on those that produced them.

We determined the attraction of a ring upon a particle of a concentric
ring, first, when both rings are in their undisturbed state ; secondly, when the
particle is disturbed ; and, thirdly, when the attracting ring is disturbed by a
series of waves. § 26.

We then formed the equations of motion of one of the rings, taking in the
disturbing forces arising from the existence of a wave in the other ring, and
found the small variation of the velocity of a wave in the first ring as dependent
on the magnitude of the wave in the second ring, which travels with it. § 27.

The forced wave in the second ring must have the same absolute angular
velocity as the free wave of the first which produces it, but this velocity of
the free wave is slightly altered by the reaction of the forced wave upon it.
We find that if a free wave of the first ring has an absolute angular velocity
not very different from that of a free wave of the second ring, then if both
fi:ee waves be of even orders (that is, of the second or fourth varieties of waves),
or both of odd orders (that is, of the first or third), then the swifter of the
two free waves has its velocity increased by the forced wave which it produces,
and the slower free wave is rendered still slower by its forced wave ; and even
when the two free waves have the same angular velocity, their mutual action
will make them both split into two, one wave in each ring travelling faster,
and the other wave in each ring travelling slower, than the rate with which
they would move if they had not acted on each other.

But if one of the free waves be of an even order and the other of an odd
order, the swifter free wave will travel slower, and the slower free wave will
travel swifter, on account of the reaction of their respective forced waves. If
the two free waves have naturally a certain small difference of velocities, they
will be made to travel together, but if the difference is less than this, they
will again split into two pairs of waves, one pair continually increasing in

ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 371

magnitude without limit, and the other continually diminishing, 30 that one
of the waves in each ring will increase in violence till it has thrown the ring
into a state of confusion.

There are four cases in which this may happen. The first wave of the
outer ring may conspire with the second or the fourth of the inner ring, the
second of the outer with the third of the inner, or the third of the outer with
the fourth of the inner. That two rings may revolve permanently, their distances
must be arranged so that none of these conspiracies may arise between odd
and even waves, whatever be the value of m. The number of conditions to
be fulfilled is therefore very great, especially when the rings are near together
and have nearly the same angular velocity, because then there are a greater
number of dangerous values of m to be provided for.

In the case of a large number of concentric rings, the stability of each pair
must be investigated separately, and if in the case of any two, whether con-
secutive rings or not, there are a pair of conspiring waves, those two rings will
be agitated more and more, till waves of that kind are rendered impossible by
the breaking up of those rings into some different arrangement. The presence
of the other rings cannot prevent the mutual destruction of any pair which
bear such relations to each other.

It appears, therefore, that in a system of many concentric rings there will
be continually new cases of mutual interference between different pairs of rings.
The forces which excite these disturbances being very small, they will be slow
of growth, and it is possible that by the irregularities of each of the rings the
waves may be so broken and confused (see § 19), as to be incapable of mounting
up to the height at which they would begin to destroy the arrangement of the
ring. In this way it may be conceived to be possible that the gradual dis-
arrangement of the system may be retarded or indefinitely postponed.

But supposing that these waves mount up so as to produce collisions among
the particles, then we may deduce the result upon the system from general
dynamical principles. There will be a tendency among the exterior rings to
remove further from the planet, and among the interior rings to approach the
planet, and this either by the extreme interior and exterior rings diverging
from each other, or by intermediate parts of the system moving away from the
mean ring. If the interior rings are observed to approach the planet, while it

47—2

372 ON THE STABILITY OF THE MOTION OF SATURN S RINGS.

is known that none of the other rings have expanded, then the cause of the
chancre cannot be the mutual action of the parts of the system, but the resistance
of some medium in which the rings revolve. § Si-
There is another cause which would gradually act upon a broad fluid ring
of which the parts revolve each with the angular velocity due to its distance
from the planet, namely, the internal friction produced by the slipping of the
concentric rings with different angular velocities. It appears, however (§ 33),
that the effect of fluid friction would be insensible if the motion were regular.

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 unsymmetrieal a distribution
of mass, that even if the exact condition 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 arrange-
ment of the ring will be propagated round it in the form of waves, and will not
introduce dangerous confusion. If the satellites are unequal, the propagation 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. Sup-
posing the ring to consist, not of a single row of large satellites, but of 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 may 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 systems of waves, or else
of a confused multitude of revolving particles, not arranged in rings, and
continually coming into collision with each other.

ON THE STABILITY OF THE MOTION OF rfATUKN S RINGS. 373

Taking the first case, we tbund 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 possible irregularity of the rings.

The result of long-continued disturbance was found to be the spreading
out of the rings in breadth, the outer rings pressing outwards, while the inner
rings press inwards.

The final result, therefore, of the mechanical theory is, that the only system
of rings which can exist is one composed of an indefinite number of unconnected
particles, revolving round the planet with different velocities according to their
respective distances. These particles may be arranged in series of narrow rings,
or they may move through each other irregularly. In the first case the destruc-
tion of the system will be very slow, in the second case it will be more rapid,
but there may be a tendency towards an arrangement in narrow rings, which
may retard the process.

We are not able to ascertain by observation the constitution of the two
outer divisions of the system of rings, but the inner ring is certainly transparent,
for the limb of Saturn has been observed through it. It is also certain, that
though the space occupied by the ring is transparent, it is not through the
material parts of it that Saturn was seen, for his limb was observed without
distortion ; which shows that there was no refraction, and therefore that the
rays did not pass through a medium at all, but between the solid or liquid
particles of which the ring is composed. Here then we have an optical argument
in favour of the theory of independent particles as the material of the rings.
The two outer rings may be of the same nature, but not so exceedingly rare
that a ray of light can pass through their whole thickness without encounterino^
one of the particles.

Finally, the two outer rings have been observed for 200 years, and it appears,
from the careful analysis of all the observations by M. Struve, that the second
ring is broader than when first observed, and that its inner edge is nearer the
planet than formerly. The inner ring also is suspected to be approaching the
planet ever since its discovery in 1850. These appearances seem to indicate
the same slow progress of the rings towards separation which we found to be
the result of theory, and the remark, that the inner edge of the inner ring is

374 ON THE STABILITY OF THE MOTION OF SATURN S RINGS.

most distinct, seems to indicate that the approach towards the planet is less
rapid near the edge, as we had reason to conjecture. As to the apparent
unchangeableness of the exterior diameter of the outer ring, we must remember
that the outer rings are certainly far more dense than the inner one, and that
a small change in the outer rings must balance a great change in the inner
one. It is possible, however, that some of the observed changes may be due
to the existence of a resisting medium. If the changes already suspected should
be confirmed by repeated observations with the same instruments, it will be
worth while to investigate more carefully whether Saturn's Rings are permanent
or transitionary elements of the Solar System, and whether in that part of
the heavens we see celestial immutability, or terrestrial corruption and generation,
and the old order giving place to new before our own eyes.

APPENDIX.

On the Stability of the Steady Motion of a Rigid Body about a Fixed Centre of Force.
By Peofessor W. Thomson {communicated in a letter).

The body will be supposed to be symmetrical on the two sides of a certain plane
containing the centre of force, and no motion except that of parts of the body parallel
to the plane will be considered. Taking it as the plane of construction, let G (fig. 14)
be the centre of gravity of the body, and a point at which the resultant attraction of
the body is in the line OG towards G. Then if the body be placed with coinciding
with the centre of force, and set in a state of rotation about that point as an axis, with

an angular velocity equal to A/Ajr. (where / denotes the attraction of the body on a

unit of matter at 0, S the amount of matter in the central body, M the mass of the
revolving body, and a the distance OG), it will continue, provided it be perfectly undis-
turbed, to revolve uniformly at this rate, and the attraction Sf on the moving body will
be constantly balanced by the centrifugal force oi'aM of its motion.

Let us now suppose the motion to be slightly disturbed, and let it be required to
investigate the consequences. Let X, S, Y, be rectangular axes of reference revolving

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