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Font size within the fluid F, = - 2nkz' -, ^ = - inJcz

at the bounding planes F= — iirkc^ ; ->- = T 47r^c

beyond them V, = 27r^c ( + 2z ± c) ; -y- = =F ^nkc

.(86);

ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 341

the upper sign being understood to refer to the boundary at distance +c, and
the lower to the boundary at distance — c from the origin.

Having ascertained the potential of the undisturbed stratum, we find that
of the disturbance by calculating the effect of a stratum of density k and
thickness t„ spread over each surface according to the law of thickness already
found. By supposing the coeJB&cient A small enough, (as we may do in calcu-
lating the displacements on which stabiUty depends), we may diminish the
absolute thickness indefinitely, and reduce the case to that of a mere " super-
ficial density," such as is treated of in the theory of electricity. We have here,
too, to regard some parts as of negative density ; but we must recollect that we
are dealing with the difference between a disturbed and an undisturbed system,
which may be positive or negative, though no real mass can be negative.

Let us for an instant conceive only one of these surfaces to exist, and let
us transfer the origin to it. Then the law of thickness is

l, = mcABm.'mx (83),

and we know that the normal component of attraction at the surface is the
same as if the thickness had been uniform throughout, so that

on the positive side of the surface.
Also, the solution of the equation

d'V dyv_

dx" "^ dz' ~ '

consists of a series of terms of the form Ce'" sin ix.

Of these the only one with which we have to do is that in which i= —m.
Applying the condition as to the normal force at the surface, we get

V=2'irkce''^Asmmx (87),

for the potential on the positive side of the surface, and

V=27rkce'^ABm7nx (88),

on the negative side.

[ (89)

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

Calculating the potentials of a pair of such surfaces at distances +c and —c
from the plane of xy, and calling V the sum of their potentials, we have for
the space between these planes

F/ = 2TrkcA sin mxe""" (e"^ + e-*^)
beyond them F/ = 27rZ;c^ sinma!;e^"^(e'^ + e~'^)

the upper or lower sign of the index being taken according as z is positive or
negative.

These potentials must be added to those formerly obtained, to get the
potential at any point after displacement.

We have next to calculate the pressure of the fluid at any point, on the
supposition that the imaginary planes protect each shce of the fluid from the
pressure of the adjacent sHces, so that it is in equilibrium under the action of
the forces of attraction, and the pressure of these planes on each side. Now
in a fluid of density h, in equilibrium under forces whose potential is V, we
have always —

so that if we know that the value of p is 2\ where that of F is F^, then at

any other point

jD=^„ + ^(F-F„).

Now, at the free surface of the fluid, ]p = 0, and the distance from the
free surface of the disturbed fluid to the plane of the original surface is ^, a
small quantity. The attraction which acts on this stratum of fluid is, in the
first place, that of the undisturbed stratum, and this is equal to A^irkc, towards
that stratum. The pressure due to this cause at the level of the original
surface will be AnJifcC, and the pressure arising from the attractive forces due
to the displacements upon this thin layer of fluid, will be small quantities of
the second order, which we neglect. We thus find the pressure when z = c to be,

Pa = AvJc^c^mA sin mx.

The potential of the undisturbed mass when z = c is

V,= -2TTkc\
and the potential of the disturbance itself for the same value of z, is

F; = 2TrkcA sin mx (1 + e""^).

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 343

So that we find the general value of jp at any other point to be

^ = 27r^^ (c' - z') + 27r/:'c^ sin ?7ia; {2c»i - 1 - €- ^"^ + e"^ (e"- + e""^)} . . . (90).

This expression gives the pressure of the fluid at any point, as depending
on the state of constraint produced by the displacement of the imaginary planes.
The accelerating effect of these pressures on any particle, if it were allowed to
move parallel to x, instead of being confined by the planes, would be

_1 dp
k dx'

The accelerating effect of the attractions in the same direction is

dV
dx'

so that the whole acceleration parallel to cc is

X= -lirkmcA cos 7nx {2mc - e''^ - I) (91).

It is to be observed, that this quantity is independent of z, so that every
particle in the slice, by the combined effect of pressure and attraction, is urged
with the same force, and, if the imaginary planes were removed, each slice
would move parallel to itself without distortion, as long as the absolute dis-
placements remained small. We have now to consider the direction of the
resultant force X, and its changes of magnitude.

We must remember that the original displacement is A cos 7nx, if therefore
(2mo-e~"^— 1) be positive, X will be opposed to the displacement, and the
equilibrium will be stable, whereas if that quantity be negative, X will act
along with the displacement and increase it, and so constitute an unstable
condition.

It may be seen that large values of nic give positive results and small
ones negative. The sign changes when

2mc = l'lA7 (92),

which corresponds to a wave-length

\ = 2c^^^ = 2c{5'i7l) (93).

The length of the complete wave in the critical case is 5*471 times the
thickness of the stratum. Waves shorter than this are stable, longer waves
are unstable.

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

The quantity 2mc{2mc-e-^-l),

has a minimum when 2mc = '607 (94),

and the wave-length is 10 '3 5 3 times the thickness of the stratum.

In this case 2mc (2mc-e-^"^- 1)= - '509 (95),

and X='5097rMcosmx (96).

24. Let us now conceive that the stratum of fluid, instead of being infinite
in extent, is limited in breadth to about 100 times the thickness. The pressures
and attractions will not be much altered by this removal of a distant part of
the stratum. Let us also suppose that this thin but broad strip is bent round
in its own plane into a circular ring whose radius is more than ten times the
breadth of the strip, and that the waves, instead of being exactly parallel to
each other, have their ridges in the direction of radii of the ring. We shall
then have transformed our stratum into one of Saturn's Kings, if we suppose
those rings to be liquid, and that a considerable breadth of the ring has the
same angular velocity.

Let us now investigate the conditions of stability by putting
x= - 27rkmc (2mc - e"^ - 1)

into the equation for n. We know that x must lie between and ^^ to

ensure stabihty. Now the greatest value of x in the fluid stratum is -50917^-.
Taking Laplace's ratio of the diameter of the ring to that of the planet, this
gives 42-5 as the minimum value of the density of the planet divided by that
of the fluid of the ring.

Now Laplace has shewn that any value of this ratio greater than 1-3 is
inconsistent with the rotation of any considerable breadth of the fluid at the
same angular velocity, so that our hypothesis of a broad ring with uniform
velocity is untenable.

But the stabihty of such a ring is impossible for another reason, namely,
that for waves in which 2mc> 1-147, x is negative, and the ring will be destroyed
by these short waves in the manner described at page (333).

When the fluid ring is treated, not as a broad strip, but as a filament of
circular or elliptic section, the mathematical difiSculties are very much increased.

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 345

but it may be shown that in this case also there will be a maximum value
of X, which will require the density of the planet to be several times that of
the ring, and that in all cases short waves will give rise to negative values
of X, inconsistent with the stability of the rmg.

It appears, therefore, that a ring composed of a continuous liquid mass
cannot revolve about a central body without being broken up, but that the
parts of such a broken ring may, under certain conditions, form a permanent
ring of satellites.

On the Mutual Perturbations of Two Rings.

25. We shall assume that the difference of the mean radii of the rings
is small compared with the radii themselves, but large compared with the
distance of consecutive satellites of the same ring. We shall also assume that
each ring separately satisfies the conditions of stability.

We have seen that the effect of a disturbing force on a ring is to produce
a series of waves whose number and period correspond with those of the dis-
turbing force which produces them, so that we have only to calculate the
coefficient belonging to the wave from that of the disturbing force.

Hence in investigating the simultaneous motions of two rings, we may
assume that the mutually disturbing waves travel with the same absolute
angular velocity, and that a maximum in one corresponds either to a maximum
or a minimum of the other, according as the coefficients have the same or
opposite signs.

Since the motions of the particles of each ring are affected by the disturbance
of the other ring, as well as of that to which they belong, the equations of
motion of the two rings will be involved in each other, and the final equation
for determining the wave-velocity will have eight roots instead of four. But as
each of the rings has four free waves, we may suppose these to originate forced
waves in the other ring, so that we may consider the eight waves of each ring
as consisting of four free waves and four forced ones.

In strictness, however, the wave- velocity of the "free" waves will be
affected by the existence of the forced waves which they produce in the other
ring, so that none of the waves are really " free " in either ring independently,
though the whole motion of the system of two rings as a whole is free.

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

We shall find, however, that it is best to consider the waves first as free,
and then to determine the reaction of the other ring upon them, which is such
as to alter the wave-velocity of both, as we shall see.

The forces due to the second ring may be separated into three parts.

1st. The constant attraction when both rings are at rest.

2nd. The variation of the attraction on the first ring, due to its own
disturbances.

3rd. The variation of the attraction due to the disturbances of the second
ring.

The first of these affects only the angular velocity. The second affects the
waves of each ring independently, and the mutual action of the waves depends
entirely on the third class of forces.

26. To deteivnine the attractions between two rings.

Let R and a be the mass and radius of the exterior ring, R and a' those
of the interior, and let all quantities belonging to the interior ring be marked
with accented letters. (Fig. 5.)

1st. Attraction between the rings when at rest.

Since the rings are at a distance small compared with their radii, we may
calculate the attraction on a particle of the first ring as if the second were an
infinite straight line at distance a' — a from the first.

7?'

The mass of unit of length of the second ring is - — > , and the accelerating

effect of the attraction of such a filament on an element of the first ring is

TV

— —, 7\ inwards (97).

na [a — a) ^

The attraction of the first ring on the second may be found by transposing
accented and unaccented letters.

In consequence of these forces, the outer ring will revolve faster, and the

inner ring slower than would otherwise be the case. These forces enter into

the constant terms of the equations of motion, and may be included in the
value of K.

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 347

2nd. Variation due to disturbance of first ring.

If we put a(l+p) for a in the last expression, we get the attraction
when the first ring is displaced. The part depending on p is

r-, TT, P inwards (98).

Tra [a-ay '^

This is the only variation of force arising from the displacement of the
first ring. It affects the value of X in the equations of motion.

3rd. Variation due to waves in the second ling.

On account of the waves, the second ring varies in distance from the
first, and also in mass of unit of length, and each of these alterations produces
variations both in the radial and tangential force, so that there are four things
to be calculated :

2nd, Radial force due to tangential displacement.

3rd. Tangential force due to radial displacement.

4th. Tangential force due to tangential displacement.

1st. Put a'(l+p') for a\ and we get the term in p

— -, \ ? ~ ,; p' inwards = XV> say (99).

ira (a -af ^ t^ > J v ^

2nd. By the tangential displacement of the second ring the section is
iced in the proportion
of the radial force equal to

reduced in the proportion of 1 to l - j , , and therefore there is an alteration

-yr inwards = — /x' -j-, say (100).

ird'(a — a') ds' '^ ds'

3rd. By the radial displacement of the second ring the direction of the
filament near the part in question is altered, so that the attraction is no longer
radial but forwards, and the tangential part of the force is

.5 '^ ^'=+/^' forwards (lOl).

ira (a-a) ds '^ ds

44—2

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

4th. By the tangential displacement of the second ring a tangential force

arises, depending on the relation between the length of the waves and the

distance between the rings.

"-ot' J f+«xsinp^ ,

If we make m — - = p, and m -i ax = H,

a ^ J-o.(l+x-y

/?'

the tangential force is a (a-a'Y ^^' ^ *''^' (102).

We may now write down the values of X, /x, and v by transposing accented
and unaccented letters.

g^(2a-a) R ^^ _?_ n (103).

ira (a-aj '^ TTa{a-a)' ira {a-af

Comparing these values with those of X', /x', and v, it will be seen that
the following relations are approximately true when a is nearly equal to a:

^'=-'i = ^ = |> (104).

X H' ^ R^

27. To form the equations of motion.

*The original equations were

^■' + o,'p + -2o,^-'^, = P = S+K-(2S-L)Ap-MBp + yp - y:'^,

Putting p = ^ cos {ills + nt), ar = B8m (ms + nt),

p' = A' cos {im + nt), cr'^R sin {ins + nt),
then u>' = S-vK

{(o'■V2S+n'-L)A + {2(on+M)B-XA' + |J:mB = 0^ ,^^^.

{2con + M)A + {n'^-N)B-ij:mA' + vR = o] ^ ''

The corresponding equations for the second ring may be found by trans-
posing accented and unaccented letters. We should then have four equations
to determine the ratios of A, B, A', B', and a resultant equation of the eighth
degree to determine n. But we may make use of a more convenient method,
since X', ix, and v are small. Eliminating B we find

An'-A(ai'^-lK+L-N)n'-iAo>Mn + AN{Zoy)\_ , .

(-X'A' + fx'mR)n' + {ix'mA' -v'B') 2<onj ^ ''

those which were applicable in the case of a ring of radius unity. Ed.]

ON THE STABILITY OP THE MOTION OF SATURN's RINGS. 349

Putting B = ^A, A' = xA, B' = ^A' = ^xA,

we have ii* - {o.' ( + 2 A") + X - iV} n' - 4(oMn + Sco'N] ^jj^^ / ^qj^x

~ = 47i'-2a;';i + &c (108),

an

-r = - ^''^' + H''ml3'}r + 2/»iw?i - 2u^'a)n (109),

28. If we were to solve the equation for n, leaving out the terms involving
X, we should find the wave-velocities of the four free waves of the first ring,
supposing the second ring to be prevented from being disturbed. But in reality
the waves in the first ring produce a disturbance in the second, and these in
turn react upon the first ring, so that the wave-velocity is somewhat difierent
from that which it would be in the supposed case. Now if x be the ratio
of the radial amplitude of displacement in the second ring to that in the first,
and if n be a value of n supposing cc = 0, then by Maclaurin's theorem,

n= Jfn + -j-x (Ill)-

The wave-velocity relative to the ring is , and the absolute angular

velocity of the wave in space is

n n I dn . ^-.

'ar = oi =0) j-x (112),

m m m ax '

= +p-qx (113),

, n , \ dn

where » = w , and o = — -j- .

^ m ^ m ax

Similarly in the second ring we should have

-=/-<z'^ (114);

and since the corresponding waves in the two rings must have the same abso-
lute angular velocity,

^ = ■25-', or 'p — qx^'p—ci - (115)-

350 ON THE STABILITY OF THE MOTION OF SATURN S RINGS,

This is a quadratic equation in x, the roots of which are real when

is positive. When this condition is not fulfilled, the roots are impossible, and
the general solution of the equations of motion will contain exponential factors,
indicating destructive oscillations in the rings.

Since q and q' are small quantities, the solution is always real whenever
p and p' are considerably different. The absolute angular velocities of the two
pairs of reacting waves, are then nearly

V -\ — ^^/ , and r) — ^^, ,

instead of p and p\ as they would have been if there had been no reaction
of the forced wave upon the free wave which produces it.

When 2^ and p' are equal or nearly equal, the character of the solution
will depend on the sign of qq. We must therefore determine the signs of q
and q' in such cases.

Putting P = —7-, we may write the values of q and q'

x/ ^ / /6> fO\ ,,(0 0)

X + 211 m — - - - 4i/ - -
n ^ \n 71/ 71 71

7n ' 4?i^ — 2<xr

Oi Ct/\ , Oi 0)

, _ n ^ \n 71 1 71 n

^~m" in"-2o)"

Referring to the values of the disturbing forces, we find that

X' IX V _ Ka
X iL V Ra"

(116).

TT g n 471* — 2&> Ra l^^*7\

Hence X = _^ — , —-, (117).

q n 4n'-2w* Ra

Since qq' is of the same sign as -^ , we have only to determine whether

2 '2

2n - , and 2n' -— , are of the same or of different signs. If these quantities
n 71 '

are of the same sign, qq is positive, if of different signs, qq' is negative.

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 351

Now there are four values of n, which give four corresponding values of

2n

72 1= -W + &C., 2?ii- is negative,
??j = — a small quantity, 2n^ is positive,

jjj = -f. a small quantity, 211^ is negative,

^3

n^ = oi — kc., 271^ is positive.

The quantity with which we have to do is therefore positive for the even
orders of waves and negative for the odd ones, and the corresponding quantity
in the other ring obeys the same law. Hence when the waves which act upon
each other are either both of even or both of odd names, qq will be positive,
but when one belongs to an even series, and the other to an odd series, qq
is negative.

29. The values of j) and p' are, roughly,

X>^ = oi + — — &c., ^o = w + &c., ^3 = (u — &c., ^4 = (o — — - + &c.

^j' = Co' H &C., p.' = 0) + &c., Pa' = co' — &c., Pi=Oi 1- &C.

(118).

<ji is greater than <u, so that j>^ is the greatest, and Pi the least of these
values, and of those of the same order, the accented is greater than the unac-
cented. The following cases of equahty are therefore possible under suitable
circumstances ;

P, =P,\ Pi =p/»

P4=P,' (when m=l), p,=2^3,

p.=p:,

In the cases in the first column qq' will be positive, in those in the second
column qq' will be negative.

352 ox THE STABILITY OF THE MOTION OF SATURN's RINGS.

30. Now each of the four values of p is a function of w, the number
of undulations in the ring, and of a the radius of the ring, varying nearly
as cfl Hence m being given, we may alter the radius of the ring till any
one of the four values of p becomes equal to a given quantity, say a given
value of /, so that if an indefinite number of rings coexisted, so as to form
a sheet of rings, it would be always possible to discover instances of the
equality of x> ^^^ V among them. K such a case of equahty belongs to the
first column given above, two constant waves will arise in both rings, one
travelling a little faster, and the other a little slower than the free waves.
If the case belongs to the second column, two waves will also arise in each
ring, but the one pair will graduaUy die away, and the other pair wHl increase
in ampUtude indefinitely, the one wave strengthening the other till at last both
rino-s are thrown into confusion.

The only way in which such an occurrence can be avoided is by placing
the rings at such a distance that no value of m shall give coincident values
of _p and J), For instance, if w > 2a), but w < So), no such coincidence is possible.
For j)^ is always less than p./, it is greater than p, when m = 1 or 2, and less
than _p4 when m is 3 or a greater number. There are of course an infinite
number of ways in which this noncoincidence might be secured, but it is plain
that if a number of concentric rings were placed at small intervals from each
other, such coincidences must occur accurately or approximately between some
pairs' of rings, and if the value of [p-fj is brought lower than -^qq, there
will be destructive interference.

This investigation is applicable to any number of concentric rings, for, by
the principle of superposition of small displacements, the reciprocal actions of
any pair of rings are independent of all the rest.

31. On the effect of long-continued disturbances on a system of rings.

The result of our previous investigations has been to point out several
ways in which disturbances may accumulate till collisions of the different par-
ticles of the rings take place. After such a collision the particles wUl still
continue to revolve about the planet, but there will be a loss of energy in
the system during the colUsion which can never be restored. Such coUisions
however will not affect what is called the Angular Momentum of the system
about the planet, which will therefore remain constant.

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 353

Let M be the mass of tlie system of rings, and hm that of one ring
whose radius is r, and angular velocity (o = S^r~^. The angular momentum of
the ring is

half its vis viva is ^tuV'Sm = ^Sr~^ hm.

The potential energy due to Saturn's attraction on the ring is

-Sr-'hm.
The angular momentum of the whole system is invariable, and is

S'^%{r^hm) = A (119).

The whole energy of the system is the sum of half the vis viva and the
potential energy, and is

-^St{r-'hm) = E (120).

A is invariable, while E necessarily diminishes. We shall find that as E
diminishes, the distribution of the rings must be altered, some of the outer
rings moving outwards, while the inner rings move inwards, so as either to
spread out the whole system more, both on the outer and on the inner edge
of the system, or, without affecting the extreme rings, to diminish the density
or number of the rings at the mean distance, and increase it at or near the
inner and outer edges.

Let us put x = r^-,

then A-

= S-t{xdm) is constant.

Now let

tixdm)
^^~ t{dm) '

and

X = Xi + x\

then we may write

-^ = t(r-^Bm)=^t{x-'dm),
lb

= Sc^m(a: - 2|3 + 3|i-&c.),

= \t{dm)-^,X{xdm)-]-^,t(x'Bm)-kc (121).

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

Now t(dm) = M a constant, t(xdm) = 0, and t(x"-Bm) is a quantity which
increases when the rings are spread out from the mean distance either way,
X being subject only to the restriction t (xdm) = 0. But % (x'dm) may
increase without the extreme values of x being increased, provided some other
values be increased.

32. In fact, if we consider the very innermost particle as moving in an
ellipse, and at the further apse of its orbit encountering another particle
belonging to a larger orbit, we know that the second particle, when at the
same distance from the planet, moves the faster. The result is, that the
interior satellite will receive a forward impulse at its further apse, and will
move in a larger and less eccentric orbit than before. In the same way one
of the outermost particles may receive a backward impulse at its nearer apse,
and so be made to move in a smaller and less eccentric orbit than before.
When we come to deal with collisions among bodies of unknown number, size,
and shape, we can no longer trace the mathematical laws of their motion with
any distinctness. All we can now do is to collect the results of our investi-
gations and to make the best use we can of them in forming an opinion as
to the constitution of the actual rings of Saturn which are still in existence
and apparently in steady motion, whatever catastrophes may be indicated by
the various theories we have attempted.

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