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33. To find the Loss of Energy due to internal friction in a hroad Fluid
Ring, the parts of which revolve about the Planet, each with the velocity of a
satellite at the same distance.

Conceive a fluid, the particles of which move parallel to the axis of x
with a velocity u, u being a function of z, then there will be a tangential pres-
sure on a plane parallel to xy

dU .. r.

= /x-y- on umt 01 area
'^ dz

due to the relative sliding of the parts of the fluid over each other.
In the case of the ring we have

The absolute velocity of any particle is tor. That of a particle at distance
{r-\-Zr) is

(ar + -j- {(ar) hr.



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 355

If the angular velocity had been uniform, there would have been no sliding,
and the velocity would have been

cji" + (ohr.
The sliding is therefore

d(o ^

r -J- or,
ar

and the friction on unit of area perpendicular to r is fir -p •

The loss of Energy, per unit of area, is the product of the sliding by the
friction,

or, /x?-*-^ Sr in unit of time.

The loss of Energy in a part of the Ring whose radius is r, breadth
Sr, and thickness c, is

27rr*c/x -j- Sr.
In the case before us it is f Tr/x/Scr"* Sr.

If the thickness of the ring is uniform between r = a and r = h, the whole
loss of Energy is

in unit of time.

Now half the vis viva of an elementary ring is
npcrhr r^oy = nfxSSr,
and this between the limits r = a and r = h gives

npcS (a — h).

The potential due to the attraction of 5 is twice this quantity with the
sign changed, so that

E=-TrpcS(a-b),

E dt~ ^ p ah'

45—2



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

Now Professor Stokes finds a/^ = 0-0564 for water,

^ P

and =0'116 for air,

taking the unit of space one English inch, and the unit of time one second.
We may take a = 88,209 miles, and ?> = 77,636 for the ring A) and a = 75,845,
and 6 = 58,660 for the ring B. We may also take one year as the unit of
time. The quantity representing the ratio of the loss of energy in a year to
the whole energy is

I dE 1 p .-L • ^

E W= 60,880,000,000,000 ^^' ^^' "^^ ^'

^^ 39,540,000,000,000 ^'' ^^^ ^^"^ ^'

showing that the efiect of internal friction in a ring of water moving with
steady motion is inappreciably small. It cannot be from this cause therefore
that any decay can take place in the motion of the ring, provided that no
waves arise to disturb the motion.



Recapitulation of the Tlieory of the Motion of a Rigid Ring.

The position of the ring relative to Saturn at any given instant is defined
by three variable quantities.

1st. The distance between the centre of gravity of Saturn and the centre
of gravity of the ring. This distance we denote by r.

2nd. The angle which the line r makes with a fixed line in the plane of
the motion of the ring. This angle is called 0.

3rd. The angle between the line r and a Hne fixed with respect to the
ring so that it coincides with r when the ring is in its mean position. This is
the angle <^.

The values of these three quantities determine the position of the ring so
far as its motion in its own plane is concerned. They may be referred to as
the radius vector, longitude, and angle of lihration of the ring.

The forces which act between the ring and the planet depend entirely upon
their relative positions. The method adopted above consists in determining the



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 357

potential ( V) of the ring at the centre of the planet In terms of r and <^. Then

the work done by any displacement of the system is measured by the change

of VS during that displacement. The attraction between the centre of gravity

(IV
of the Ring and that of the planet Is ~S , , and the moment of the couple

clV
tending to turn the ring about Its centre of gravity Is S-j-j,

It Is proved In Problem V, that if a be the radius of a circular ring, r^^uf
the distance of its centre of gravity from the centre of the circle, and R the

mass of the ring, then, at the centre of the ring, ,- = 5/, -yj = 0.

(PV Ji
It also appears that T-^ = -k~3 {^ +9)> "which is positive when g > —I,

d'V R
and that -n\=^—f'(^—g), which is positive when ^<3.

d'V . . .

If -y— is positive, then the attraction between the centres decreases as the

distance increases, so that, if the two centres were kept at rest at a given

d'V . . .
distance by a constant force, the equilibrium would be unstable. If -t-t; is positive,

then the forces tend to increase the angle of libration, in whichever direction
the libration takes place, so that if the ring were fixed by an axis through its
centre of gravity, its equilibrium round that axis would be unstable.

In the case of the uniform ring with a heavy particle on its circumference
whose weight ="82 of the whole, the direction of the whole attractive force of
the ring near the centre will pass through a point lying in the same radius as
the centre of gravity, but at a distance from the centre = fa. (Fig. 6.)

If we call this point 0, the line SO will indicate the direction and position
of the force acting on the ring, which we may call F.

It Is evident that the force F, acting on the ring in the line OS, will tend
to turn it round its centre of gravity R and to increase the angle of libration
KRO. The direct action of this force can never reduce the angle of libration
to zero again. To understand the indirect action of the force, we must recollect
that the centre of gravity (i?) of the ring is revolving about Saturn in the
direction of the arrows, and that the ring is revolving about its centre of gravity



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

with nearly the same velocity. If the angular velocity of the centre of gravity
about Saturn were always equal to the rotatory velocity of the ring, there
would be no libration.

Now suppose that the angle of rotation of the ring is in advance of the
longitude of its centre of gravity, so that the line RO has got in advance of
SRK by the angle of libration KRO. The attraction between the planet and
the ring is a force F acting in SO. We resolve this force into a couple, whose
moment is FRN, and a force F acting through R the centre of gravity of the
ring.

The couple affects the rotation of the ring, but not the position of its centre
of gravity, and the force RF acts on the centre of gravity without affecting the
rotation.

Now the couple, in the case represented in the figure, acts in the positive
direction, so as to increase the angular velocity of the ring, which was already
greater than the velocity of revolution of R about S, so that the angle of
libration would increase, and never be reduced to zero.

The force RF does not act in the direction of >S', but behind it, so that it
becomes a retarding force acting upon the centre of gravity of the ring. Now
the effect of a retarding force is to cause the distance of the revolving body to
decrease and the angular velocity to increase, so that a retarding force increases
the angular velocity of R about S.

The effect of the attraction along SO in the case of the figure is, first, to
increase the rate of rotation of the ring round R, and secondly, to iacrease the
angular velocity of R about S. If the second effect is greater than the first,
then, although the line RO increases its angular velocity, SR will increase its
angular velocity more, and will overtake RO, and restore the ring to its original
position, so that SRO will be made a straight line as at first. If this accelerat-
ing effect is not greater than the acceleration of rotation about R due to the
couple, then no compensation will take place, and the motion will be essentially
unstable.

If in the figure we had drawn ^ negative instead of positive, then the
couple would have been negative, the tangential force on R accelerative, r would
have increased, and in the cases of stability the retardation of 6 would be greater
than that of (^ + <^), and the normal position would be restored, as before.



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 359

The object of the investigation is to find the conditions under wliich this
compensation is possible.

It is evident that when SRO becomes straight, there is still a difference
of angular velocities between the rotation of the ring and the revolution of
the centre of gravity, so that there will be an oscillation on the other side,
and the motion will proceed by alternate oscillations without limit.

If we begin with r at its mean value, and <^ negative, then the rotation
of the ring will be retarded, 7* will be increased, the revolution of r will be
more retarded, and thus <f> will be reduced to zero. The next part of the
motion will reduce r to its mean value, and bring (f) to its greatest positive
value. Then r will diminish to its least value, and (f> will vanish. Lastly r
will return to the mean value, and <f) to the greatest negative value.

It appears from the calculations, that there are, in general, two different
ways in which this kind of motion may take place, and that these may have
different periods, phases, and amplitudes. The mental exertion required in follow-
ing out the results of a combined motion of this kind, with all the variations of
force and velocity during a complete cycle, w^ould be very great in proportion to
the additional knowledge we should derive from the exercise.

The result of this theory of a rigid ring shows not only that a perfectly
uniform ring cannot revolve permanently about the planet, but that the irregu-
larity of a permanently revolving ring must be a very observable quantity, the
distance between the centre of the ring and the centre of gravity being between
•8158 and '8279 of the radius. As there is no appearance about the rings
justifying a belief in so great an irregularity, the theory of the solidity of the
rings becomes very improbable.

When we come to consider the additional difficulty of the tendency of the
fluid or loose parts of the ring to accumulate at the thicker parts, and thus
to destroy that nice adjustment of the load on which stability depends, we
have another powerful argument against solidity.

And when we consider the immense size of the rings, and their comparative
thinness, the absurdity of treating them as rigid bodies becomes self-evident.
An iron ring of such a size would be not only plastic but semifluid under the
forces which it would experience, and we have no reason to believe these rings
to be artificially strengthened with any material unknown on this earth.



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



Recapitulation of the Theory of a Ring of equal Satellites.

In attempting to conceive of the disturbed motion of a ring of unconnected
satellites, we have, in the first place, to devise a method of identifying each
satellite at any given time, and in the second place, to express the motion of
every satellite under the same general formula, in order that the mathematical
methods may embrace the whole system of bodies at once.

By conceiving the ring of satellites arranged regularly in a circle, we may
easily identify any satellite, by stating the angular distance between it and a
known satellite when so arranged. If the motion of the ring were undisturbed,
this angle would remain unchanged during the motion, but, in reality, the
satellite has its position altered in three ways : 1st, it may be further from
or nearer to Saturn; 2ndly, it may be in advance or in the rear of the position
it would have had if undisturbed ; 3rdly, it may be on one side or other of
the mean plane of the ring. Each of these displacements may vary in any way
whatever as we pass from one satellite to another, so that it is impossible
to assign beforehand the place of any satellite by knowing the places of the
rest. § 2.

The formula, therefore, by which we are enabled to predict the place of
every satellite at any given time, must be such as to allow the initial position
of every satellite to be independent of the rest, and must express all future
positions of that satellite by inserting the corresponding value of the quantity
denoting time, and those of every other sateUite by inserting the value of the
angular distance of the given satelUte from the point of reference. The three
displacements of the satellite will therefore be functions of two variables — the
angular position of the satellite, and the time. When the time alone is made
to vary, we trace the complete motion of a single satellite ; and when the time
is made constant, and the angle is made to vary, we trace the form of the
ring at a given time.

It is evident that the fonn of this function, in so far as it indicates the
state of the whole ring at a given instant, must be wholly arbitrary, for the
form of the ring and its motion at starting are limited only by the condition
that the irregularities must be small. We have, however, the means of breaking
up any function, however complicated, into a series of simple functions, so that
the value of the function between certain limits may be accurately expressed



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 361

as the sum of a series of sines and cosines of multiples of the variable. This
method, due to Fourier, is peculiarly applicable to the case of a ring returning
into itself, for the value of Fourier's series is necessarily periodic. We now
regard the form of the disturbed ring at any instant as the result of the
superposition of a number of separate disturbances, each of -which is of the nature
of a series of equal waves regularly arranged round the. ring. Each of these
elementary disturbances is characterised by the number of undulations in it, by
their amplitude, and by the position of the first maximum in the ring. § 3.

When we know the form of each elementary disturbance, we may calculate
the attraction of the disturbed ring on any given particle in terms of the con-
stants belonging to that disturbance, so that as the actual displacement is the
resultant of the elementary displacements, the actual attraction will be the
resultant of the corresponding elementary attractions, and therefore the actual
motion will be the resultant of all the motions arising from the elementary
disturbances. We have therefore only to investigate the elementary disturbances
one by one, and having established the theory of these, we calculate the actual
motion by combining the series of motions so obtained.

Assuming the motion of the satellites in one of the elementary disturbances
to be that of oscillation about a mean position, and the whole motion to be
that of a uniformly revolving series of undulations, we find our supposition to
be correct, provided a certain biquadratic equation is satisfied by the quantity
denoting the rate of oscillation. § 6.

When the four roots of this equation are all real, the motion of each
satellite is compounded of four difierent oscillations of difi'erent amplitudes and
periods, and the motion of the whole ring consists of four series of undulations,
travelling round the ring with different velocities. When any of these roots
are impossible, the motion is no longer oscillatory, but tends to the rapid
destruction of the ring.

To determine whether the motion of the ring is permanent, we must assure
ourselves that the four roots of this equation are real, whatever be the number
of undulations in the ring; for if any one of the possible elementary distui'b-
ances should lead to destructive oscillations, that disturbance might sooner or
later commence, and the ring would be destroyed.

Now the number of undulations in the ring may be any whole number
from one up to half the number of satellites. The forces from which danger



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

is to be apprehended are greatest when the number of undulations is greatest,
and by taking that number equal to half the number of satellites, we find the
condition of stability to be

S>.A352tiR,

where S is the mass of the central body, R that of the ring, and /x the number
of sateUites of which it is composed. § 8. If the number of satelHtes be too
great, destructive oscillations will commence, and finally some of the satellites
will come into coUision with each other and unite, so that the number of
independent satellites will be reduced to that which the central body can retain
and keep in discipline. When this has taken place, the satellites will not only
be kept at the proper distance from the primary, but will be prevented by its
preponderating mass from interfering with each other.

We next considered more carefully the case in which the mass of the ring
is very small, so that the forces arising from the attraction of the ring are
small compared with that due to the central body. In this case the values
of the roots of the biquadratic are all real, and easUy estimated. § 9.

If we consider the motion of any satellite about its mean position, as
referred to axes fixed in the plane of the ring, we shall find that it describes
an ellipse in the direction opposite to that of the revolution of the ring, the
periodic time being to that of the ring as o> to n, and the tangential ampli-
tude of oscillation being to the radial as 2(0 to n. § 10.

The absolute motion of each satellite in space is nearly elliptic for the large
values of n, the axis of the ellipse always advancing slowly in the direction of
rotation. The path of a satellite corresponding to one of the small values of
n is nearly circular, but the radius slowly increases and diminishes during a
period of many revolutions. § 11.

The form of the ring at any instant is that of a re-entering curve, having
m alternations of distance from the centre, symmetrically arranged, and m points
of condensation, or crowding of the satellites, which coincide with the points of
greatest distance when n is positive, and with the points nearest the centre
when n m negative. § 12.

This system of undulations travels with an angular velocity relative to

the ring, and co in space, so that during each oscillation of a satellite a

complete wave passes over it. § 14.



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 363

To exhibit the movements of the satellites, I have made an arrangement
by which 36 little ivory balls are made to go through the motions belonging
to the first or fourth series of waves. (Figs. 7, 8.)

The instrument stands on a pillar A, in the upper part of which turns
the cranked axle CC. On the parallel parts of this axle are placed two wheels,
RR and TT, each of which has 36 holes at equal distances in a circle neai-
its circumference. The two circles are connected by 36 small cranks of the
fonn KK, the extremities of which turn in the corresponding holes of the two
wheels. That axle of the crank K which passes through the hole in the wheel
S is bored, so as to hold the end of the bent wire which carries the satellite >S'.
This wire may be turned in the hole so as to place the bent part carrying
the satellite at any angle with the crank. A pin F, which passes through the
top of the pillar, serves to prevent the cranked axle from turning ; and a pin Q,
passing through the pillar horizontally, may be made to fix the wheel R, by
inserting it in a hole in one of the spokes of that wheel. There is also a
handle H, which is in one piece with the wheel T, and serves to turn the axle.

Now suppose the pin P taken out, so as to allow the cranked axle to
turn, and the pin Q inserted in its hole, so as to prevent the wheel R from
revolving; then if the crank C be turned by means of the handle H, the
wheel T will have its centre carried round in a vertical circle, but will remain
parallel to itself during the whole motion, so that every point in its plane will
describe an equal circle, and all the cranks K will be made to revolve exactly
as the large crank C does. Each satellite will therefore revolve in a small
circular orbit, in the same time with the handle H, but the position of each
satellite in that orbit may be arranged as we please, according as we turn the
wire which supports it in the end of the crank.

In fig. 8, which gives a front view of the instrument, the satelHtes are so
placed that each is turned 60^ further round in its socket than the one behind
it. As there are 36 satellites, this process will bring us back to our starting-
point after six revolutions of the direction of the arm of the satellite; and
therefore as we have gone round the ring once in the same direction, the ami
of the sateUite will have overtaken the radius of the ring five times.

Hence there will be five places where the satellites are beyond their mean
distance from the centre of the ring, and five where they are within it, so
that we have here a series of five undulations round the circumference of the

46—2



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

ring. In this case the satellites are crowded together when nearest to the centre,
so that the case is that of the first series of waves, when m = 5.

Now suppose the cranked axle C to be turned, and all the small cranks
K to turn with it, as before explained, every satellite will then be carried
round on its own arm in the same direction ; but, since the direction of the
arms of different satellites is different, their phases of revolution will preserve
the same difference, and the system of satellites will still be arranged in five
undulations, only the undulations will be propagated round the ring in the
direction opposite to that of the revolution of the satellites.

To understand the motion better, let us conceive the centres of the orbits
of the satellites to be arranged in a straight line instead of a circle, as in
fig. 10. Each satellite is here represented in a different phase of its orbit, so
that as we pass from one to another from left to right, we find the position
of the satellite in its orbit altering in the direction opposite to that of the
hands of a watch. The satellites all lie in a trochoidal curve, indicated by
the line through them in the figure. Now conceive every satellite to move in
its orbit through a certain angle in the direction of the arrows. The satellites
will then lie in the dotted line, the form of which is the same as that of
the former curve, only shifted in the direction of the large arrow. It appears,
therefore, that as the satellites revolve, the undulation travels, so that any
part of it reaches successively each satellite as it comes into the same phase
of rotation. It therefore travels from those satellites which are most advanced
in phase to those which are less so, and passes over a complete wave-length
in the time of one revolution of a satellite.

Now if the satellites be arranged as in fig. 8, where each is more advanced
in phase as we go round the ring in the direction of rotation, the wave will
travel in the direction opposite to that of rotation, but if they are arranged
as in fig. 12, where each satellite is less advanced in phase as we go round
the ring, the wave will travel in the direction of rotation. Fig. 8 represents
the first series of waves where m = 5, and fig. 12 represents the fourth series
where m = 7. By arranging the satellites in their sockets before starting, we
might make w equal to any whole number, from 1 to 18. If we chose any
number above 18 the result would be the same as if we had taken a number
as much below 18 and changed the arrangement from the first wave to the
fourth.



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 365

In this way we can exhibit the motions of the satellites in the first and
fourth waves. In reality they ought to move in ellipses, the major axes being
twice the minor, whereas in the machine they move in circles : but the character
of the motion is the same, though the form of the orbit is diflferent.

We may now show these motions of the satellites among each other, com-
bined with the motion of rotation of the whole ring. For this purpose we
put in the pin P, so as to prevent the crank axle from turning, and take
out the pin ^ so as to allow the wheel R to turn. If we then turn the
wheel T, all the small cranks will remain parallel to the fixed crank, and the
wheel R will revolve at the same rate as T. The arm of each satellite will
continue parallel to itself during the motion, so that the satellite will describe
a circle whose centre is at a distance from the centre of R, equal to the arm
of the satellite, and measured in the same direction. In our theory of real
satellites, each moves in an ellipse, having the central body in its focus, but



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