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When the velocity of the forced wave lies between the velocities of the
two middle free waves, or is greater than that of the swiftest, or less than
that of the slowest, then the radial displacement due to a radial disturbing
force is in the same direction as the force, but the tangential displacement
due to a tangential disturbing force is in the opposite direction to the force.

The radial force therefore in this case produces a positive forced wave, and
the tangential force a negative forced ivave.

When the velocity of the forced wave is either between the velocities of
the first and second free waves, or between those of the third and fourth, then
the radial disturbance produces a forced wave in the contrary direction to that
in which it acts, or a negative wave, and the tangential force produces a positive

The coefficient of the forced wave changes sign whenever its velocity passes
through the value of any of the velocities of the free waves, but it does so
by becoming infinite, and not by vanishing, so that when the angular velocity
very nearly coincides with that of a free wave, the forced wave becomes very
great, and if the velocity of the disturbing force were made exactly equal t-o
that of a free wave, the coefficient of the forced wave would become infinite.
In such a case we should have to readjust our approximations, and to find
whether such a coincidence might involve a physical impossibility.


The forced wave which we have just investigated is that which would main-
tain itself in the ring, supposing that it had been set agoing at the commence-
ment of the motion. It is in fact the form of dynamical equiUbrium of the
ring under the influence of the given forces. In order to find the actual motion
of the ring we must combine this forced wave with all the free waves, which
go on independently of it, and in this way the solution of the problem becomes
perfectly complete, and we can determine the whole motion under any given
initial circumstances, as we did in the case where no disturbing force acted.

For instance, if the ring were perfectly uniform and circular at the instant
when the disturbing force began to act, we should have to combine with the
constant forced wave a system of four free waves so disposed, that at the given
epoch, the displacements due to them should exactly neutralize those due to the
forced wave. By the combined effect of these four free waves and the forced
one the whole motion of the ring would be accounted for, beginning from its
undisturbed state.

The disturbances which are of most importance in the theory of Saturn's
rings are those which are produced in one ring by the action of attractive
forces arising from waves belonging to another ring.

The effect of this kind of action is to produce in each ring, besides its
own four free waves, four forced waves corresponding to the free waves of the
other ring. There will thus be eight waves in each ring, and the corresponding
waves in the two rings will act and react on each other, so that, strictly speak-
ing, every one of the waves will be in some measure a forced wave, although
the system of eight waves will be the free motion of the two rings taken
together. The theory of the mutual disturbance and combined motion of two
concentric rings of satellites requires special consideration.

18. On the motion of a ring of satellites when the conditions of stability
are not fulfilled.

We have hitherto been occupied with the case of a ring of satellites, the
stability of which was ensured by the smaUness of mass of the satellites com-
pared with that of the central body. We have seen that the statically unstable
condition of each satellite between its two immediate neighbours may be com-
pensated by the dynamical effect of its revolution round the planet, and a planet
of sufiicient mass can not only direct the motion of such satellites round its


own body, but can likewise exercise an influence over their relations to each
other, so as to overrule their natural tendency to crowd together, and distribute
and preserve them in the form of a ring.

We have traced the motion of each satellite, the general shape of the
disturbed ring, and the motion of the various waves of disturbance round the
ring, and determined the laws both of the natural or free waves of the ring,
and of the forced waves, due to extraneous disturbing forces.

We have now to consider the cases in which such a permanent motion of
the ring is impossible, and to determine the mode in which a ring, originally
regular, will break up, in the different cases of instability.

The equation from which we deduce the conditions of stability is —
U = n'-i(o' + -E(2K+L-N)\n'-4:(o-EMn

+ hco' - R{2K + L)\-RN -\r'M' = 0.

The quantity, which, in the critical cases, determines the nature of the
roots of this equation, is N. The quantity M in the third term is always
small compared with L and N when m is large, that is, in the case of the
dangerous short waves. We may therefore begin our study of the critical cases
by leaving out the third term. The equation then becomes a quadratic in n\
and in order that all the values of n may be real, both values of n' must be
real and positive.

The condition of the values of n^ being real is

oj* + co'-R{AK + 2L-UN) + \b'{2K+L-\-NY>0 (61),

which shews that ay must either be about 14 times at least smaller, or about 14
times at least greater, than quantities like - RN.

That both values of if may be positive, we must have
co' + -R{2K + L-N)>0

i3co'' - R(2K-^L)\-RN>0



We must therefore take the larger value 6£ oi\ and also add the condition
that N be positive.


We may therefore state roughly, that, to ensure stability, , the coefficient

of tangential attraction, must lie between zero and -^oi\ If the quantity be
negative, the two small values of n will become _pwre impossible quantities. If
it exceed ^oi\ all the values of n will take the form of mixed impossible

If we write x for - RN, and omit the other disturbing forces, the equation

becomes U=n*-{(o'-x)n' + Sco'x = (63),

whence n' = ^{co'-x)±^^/<o*-U(o'x + x' (64).

If X be small, two of the values of n are nearly ±<o, and the others are
small quantities, real when x is positive and impossible when x is negative.


If x be greater than {7-^IS)ar, or ^ nearly, the term under the radical
becomes negative, and the value of ?i becomes

n= ±^^fjT2^ + o}'-x±^/^-^'Jl2co'x-ajr + x (65),

where one of the terms is a real quantity, and the other impossible. Every
solution may be put under the form

n=p±J^^q (66),

where ry = for the case of stability, p = for the pure impossible roots, and p
and q finite for the mixed roots.

Let us now adopt this general solution of the equation for n, and determine
its mechanical significance by substituting for the impossible circular functions
their equivalent real exponential functions.

Substituting the general value of n in equations (34), (35),

p = A[cos {ms +(p + 'J^^q)t + a} + cos {ms + ip- J -lq)t + a}] ... (67),

^^_^MP+±zlAsm{,ns + (p + ^^q)t + a} ]
(p + J-lqf + x

_^MEpdIi^sm{ms+(p-sr^lq)t + a} \
{p-'J -IqY + x J


Introducing the exponential notation, these values become

p = A(^^ + €-''')co3(ms-{-pt + a) (69),

W r 2) (^' + r/ + x) (€«' + €-«') sin (771,5 +j9« + a) 1

We have now obtained a solution free from impossible quantities, and applicable
to every case.

When ^ = 0, the case becomes that of real roots, which we have already
discussed. When p = 0, we have the case of pure impossible roots arising from
the negative values of if. The solutions corresponding to these roots are

/3 = ^ (e«' + €-«') cos (m5 + a) (71).

o-=-^r^^^(€''-e-^0cos(m5 + a) (72).

The part of the coefficient depending on e"'' diminishes indefinitely as the
time increases, and produces no marked effect. The other part, depending on
€^', increases in a geometrical proportion as the time increases arithmetically, and
so breaks up the ring. In the case of x being a small negative quantity, q' is
nearly 3x, so that the coefficient of cr becomes

It appears therefore that the motion of each particle is either outwards and
backwards or inwards and forwards, but that the tangential part of the motion
greatly exceeds the normal part.

It may seem paradoxical that a tangential force, acting towards a position
of equilibrium, should produce instability, while a small tangential force from that
position ensures stability, but it is easy to trace the destructive tendency of
this apparently conservative force.

Suppose a particle slightly in front of a crowded part of the ring, then
if X is negative there will be a tangential force pushing it fonvards, and this
force will cause its distance from the planet to increase, its angular velocity U>
diminish, and the particle itself to fall back on the crowded part, thereby
increasing the irregularity of the ring, till the whole ring is broken up. In
the same way it may be shewn that a particle hehiiid a crowded part will be
pushed into it. The only force which could preserve the ring from the effect


of tills action, is one which would prevent the particle from receding from the
planet under the influence of the tangential force, or at least prevent the dimi-
nution of angular velocity. The transversal force of attraction of the ring is of
this kind, and acts in the right direction, but it can never be of sufficient magni-
tude to have the required effect. In fact the thing to be done is to render the
last term of the equation in w positive when N is negative, which requires


and this condition is quite inconsistent with any constitution of the ring which
fiilfils the other condition of stability which we shall arrive at presently.

We may observe that the waves belonging to the two real values of n,
±(D, must be conceived to be travelling round the ring during the whole time
of its breaking up, and conducting themselves like ordinary waves, till the
excessive irregularities of the ring become inconsistent with their uniform propa-

The irregularities which depend on the exponential solutions do not travel
round the ring by propagation among the sateUites, but remain among the same
satellites which first began to move irregularly.

We have seen the fate of the ring when x is negative. When x is small
we have two small and two large values of n, which indicate regular waves,
as we have already shewn. As x increases, the small values of n increase, and
the large values diminish, till they meet and form a pair of positive and a
pair of negative equal roots, having values nearly +"68w. When x becomes
greater than about -^(o", then all the values of n become impossible, of the
form ^j-F-n/ — Ig", q being small when x first begins to exceed its limits, and p
being nearly + '6S(o.

The values of p and cr indicate periodic inequalities having the period — ,

but increasing in amplitude at a rate depending on the exponential e''. At the
beginning of the motion the oscillations of the particles are in eUipses as in the
case of stability, having the ratio of the axes about 1 in the normal direction
to 3 in the tangential direction. As the motion continues, these ellipses increase
in magnitude, and another motion depending on the second term of cr is com-
bined with the former, so as to increase the ellipticity of the oscillations and to


turn the major axis into an inclined position, so that its fore end points a little
inwards, and its hinder end a little outwards. The oscillations of each particle
round its mean position are therefore in ellipses, of which both axes increase
continually while the eccentricity increases, and the major axis becomes sUghtly
inclined to the tangent, and this goes on till the ring is destroyed. In the
mean time the irregularities of the ring do not remain among the same set of
particles as in the former case, but travel round the ring^ with a relative angular

velocity - ^^ Of these waves there are four, two travelling forwards among the

satellites, and two travelling backwards. One of each of these pairs depends
on a negative value of q, and consists of a wave whose amplitude continually
decreases. The other depends on a positive value of q, and is the destructive
wave whose character we have just described.

19. We have taken the case of a ring composed of equal satellites, as
that with which we may compare other cases in which the ring is constructed
of loose materials diiferently arranged.

In the first place let us consider what will be the conditions of a ring
composed of satellites of unequal mass. We shall find that the motion is of
the same kind as when the satellites are equal.

For by arranging the satellites so that the smaller satellites are closer
together than the larger ones, we may form a ring which will revolve uni-
formly about Saturn, the resultant force on each satellite being just sufficient
to keep it in its orbit.

To determine the stability of this kind of motion, we must calculate the
disturbing forces due to any given displacement of the ring. This calculation
will be more complicated than in the former case, but will lead to results of
the same general character. Placing these forces in the equations of motion,
we shall find a solution of the same general character as in the former case,
only instead of regular waves of displacement travelling round the ring, each
wave will be split and reflected when it comes to irregularities in the chain of
satellites. But if the condition of stability for every kind of wave be fulfilled,
the motion of each satellite will consist of small oscillations about its position
of dynamical equilibrium, and thus, on the whole, the ring will of itself assume
the arrangement necessary for the continuance of its motion, if it be originally
in a state not very different from that of equilibrium.


20. We now pass to the case of a ring of an entirely different construc-
tion. It is possible to conceive of a quantity of matter, either solid or liquid,
not collected into a continuous mass, but scattered thinly over a great extent
of space, and having its motion regulated by the gravitation of its parts to
each other, or towards some dominant body. A shower of rain, hail, or cinders
is a familiar illustration of a number of unconnected particles in motion; the
visible stars, the milky way, and the resolved nebula?, give us instances of a
similar scattering of bodies on a larger scale. In the terrestrial instances we
see the motion plainly, but it is governed by the attraction of the earth, and
retarded by the resistance of the air, so that the mutual attraction of the
parts is completely masked. In the celestial cases the distances are so enor-
mous, and the time during which they have been observed so short, that we
can perceive no motion at all. StiU we are perfectly able to conceive of a
collection of particles of small size compared with the distances between them,
acting upon one another only by the attraction of gravitation, and revolving
round a central body. The average density of such a system may be smaller
than that of the rarest gas, while the particles themselves may be of great
density ; and the appearance from a distance will be that of a cloud of vapour,
with this difference, that as the space between the particles is empty, the rays
of light will pass through the system without being refracted, as they would
have been if the system had been gaseous.

Such a system will have an average density which may be greater in some
places than others. The resultant attraction wiU be towards places of greater
average density, and thus the density of those places wiU be increased so as
to increase the irregularities of density. The system will therefore be statically
unstable, and nothing but motion of some kind can prevent the particles from
forming agglomerations, and these uniting, till all are reduced to one solid

We have already seen how dynamical stability can exist where there is
statical instability in the case of a row of particles revolving round a central
body. Let us now conceive a cloud of particles forming a ring of nearly uni-
form density revolving about a central body. There will be a primary effect of
inequalities in density tending to draw particles towards the denser parts of the
ring, and this will ehcit a secondary effect, due to the motion of revolution,
tending in the contrary direction, so as to restore the rings to uniformity. The


relative magnitude of these two opposing forces determines the destruction or
preservation of the ring.

To calculate these effects we must begin with the statical problem : — To
determine the forces arising from the given displacements of the ring.

The longitudinal force arising from longitudinal displacements is that which
has most effect in determining the stability of the ring. In order to estimate ita
limiting value we shall solve a problem of a simpler form.

21. An infinite mass, originally of uniform density Tc, has its particles
displaced by a quantity f parallel to the axis of x, so that ^ = AcQ^mx, to
determine the attraction on each particle due to this displacement.

The density at any point will differ from the original density by a quantity
k' , so that

{k + k') (dx + d^) = kdx (73),

k'= —k-r- = Akm sin mx (74).

The potential at any point will be V+V, where V is the original potential,
and F' depends on the displacement only, so that

dT d'V d'V ^ ,, ^ ,^,,

^+-5^ + ^- + ^'^^=^ (^^)-

Now V is a function of x only, and therefore,

V = AirAk —sinmx (76),

and the longitudinal force is found by differentiating V with respect to x.

X= -,— = ink A cos mx = 'ink^ (77).

Now let us suppose this mass not of infinite extent, but of finite section
parallel to the plane of yz. This change amounts to cutting off all portions
of the mass beyond a certain boundary. Now the effect of the portion so cut
off upon the longitudinal force depends on the value of m. When m is large,
so that the wave-length is small, the effect of the external portion is insensible,
so that the longitudinal force due to short waves is not diminished by cutting
off a great portion of the mass.


22. Applying this result to the case of a ring, and putting s for x, and

a- for $ we have

cr = ^ cos ms, and T= AttJcA cos ms,

so that -RN=4:Trk,

when on is very large, and this is the greatest value of N.

The value of L has little effect on the condition of stability. If L and
M are both neglected, that condition is

(o'>27-S5e (2nk) (78),

and if L be as much as ^N, then

o>^>25-649 (27rk) (79),

so that it is not important whether we calculate the value of L or not.

The condition of stability is, that the average density must not exceed a
certain value. Let us ascertain the relation between the maximum density of
the ring and that of the planet.

Let h be the radius of the planet, that of the ring being unity, then the
mass of Saturn is ^Trh'k' = o)"' if k' be the density of the planet. If we assume
that the radius of the ring is twice that of the planet, as Laplace has done,
then h = ^ and

1 = 334-2 to 307-7 (80),

so that the density of the ring cannot exceed 3^ of that of the planet. Now
Laplace has shewn that if the outer and inner parts of the ring have the same
angular velocity, the ring will not hold together if the ratio of the density of
the planet to that of the ring exceeds 1-3, so that in the first place, our ring
cannot have uniform angular velocity, and in the second place, Laplace's ring
cannot preserve its form, if it is composed of loose materials acting on each
other only by the attraction of gravitation, and moving with the same angular
velocity throughout.

23. On the forces arising from inequalities of thickness in a thin stratum
of fluid of indefinite extent.

The forces which act on any portion of a continuous fluid are of two kinds,
the pressures of contiguous portions of fluid, and the attractions of all portions of
the fluid whether near or distant. In the case of a thin stratum of fluid, not


acted on by any external forces, the pressures are due mainly to the component
of the attraction which is perpendicular to the plane of the stratum. It is
easy to shew that a fluid acted on by such a force will tend to assume a
position of equilibrium, in which its free surface is plane ; and that any irregu-
larities will tend to equalise themselves, so that the plane surface will be one
of stable equilibrium.

It is also evident, that if we consider only that part of the attraction
which is parallel to the plane of the stratum, we shall find it always directed
towards the thicker parts, so that the effect of this force is to draw the fluid
from thinner to thicker parts, and so to increase irregularities and destroy

The normal attraction therefore tends to preserve the stability of equilibrium,
while the tangential attraction tends to render equilibrium unstable.

According to the nature of the irregularities one or other of these forces
will prevail, so that if the extent of the irregularities is small, the normal
forces will ensure stability, while, if the inequaUties cover much space, the
tangential forces will render equilibrium unstable, and break up the stratum into

To fix our ideas, let us conceive the irregularities of the stratum split up
into the form of a number of systems of waves superposed on one another,
then, by what we have just said, it appears, that very short waves will disap-
pear of themselves, and be consistent with stability, while very long waves will
tend to increase in height, and will destroy the form of the stratum.

In order to determine the law according to which these opposite effects
take place, we must subject the case to mathematical investigation.

Let us suppose the fluid incompressible, and of the density k, and let it
be originally contained between two parallel planes, at distances +c and — c
from that of (xy), and extending to infinity. Let us next conceive a series of
imaginary planes, parallel to the plane of {ijz), to be plunged into the fluid
stratum at infinitesimal distances from one another, so as to divide the fluid
into imaginary slices perpendicular to the plane of the stratum.

Next let these planes be displaced parallel to the axis of x according to this
law — that if x be the original distance of the plane from the origin, and ^ its
displacement in the direction of x,

i=A cosmx (81).


According to this law of displacement, certain alterations will take place in
the distances between consecutive planes ; but since the fluid is incompressible,
and of indefinite extent in the direction of y, the change of dimension must
occur in the direction of z. The original thickness of the stratum was 2c. Let
its thickness at any point after displacement be 2c + 2^, then we must have

.+i)=2^ («2)'

1= — c -r-=cmA sinwa; (83).

(2c + 20 (l

Let us assume that the increase of thickness 2^ is due to an increase of C,
at each surface ; this is necessary for the equilibrium of the fluid between the
imaginary planes.

We have now produced artificially, by means of these planes, a system of

waves of longitudinal displacement whose length is — and amplitude A ; and

we have found that this has produced a system of waves of normal displace-
ment on each surface, having the same length, with a height =cmA.

In order to determine the forces arising from these displacements, we must,
in the first place, determine the potential function at any point of space, and
this depends partly on the state of the fluid before displacement, and partly
on the displacement itself We have, in all cases —

d'V d'V d'V
^^+^ + ^=-^^^ («^)-

Within the fluid, p = k; beyond it, p = 0.
Before displacement, the equation is reduced to

d^' = -'-p («^)-

Instead of assuming F=0 at infinity, we shall assume F=0 at the origin,
and since in this case all is symmetrical, we have

Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 29 of 50)