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- B cos {ms + 13) ^ sin 2m6 cot 6]

I_i2_
/x 4 sin ^

(?)•

314

Let us put

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

sin* mO cos'' 6 cos' m6\

K=t

f^va. 2m6 cos
\ 4sin*^

/sin" md cos' 6

sin' mt
2 sin^

1

+ i

sin'?n^

2sin^

(8)^:

where the summation extends to all the sateUites on the same side of F,
that is, every value of 6 of the form - tt, where x is a whole number less

than

The radial force may now be written

P = ~R {K+ LA cos ims + a) - MB cos {'tm + ^)}

(9).

* Tlie following values of several quantities which enter into these investigations are calculated for a
ring of 36 satellites.

A' =24-5.

^ sin- md cos- \$ ^ cos^ md ^

if

.V

sinS d sin e

m=

43 - 43

ni= 1

32 32 -16

16

37

m= 2

107 28 26

25

115

m — 3

212 25 81

28

221

;u= 4

401 24 177

32

411

vi= 9

975 20 468

30

986

/ft- 18

1569 18 767

1582

r gi-eat,

- Z - -5259 when m -
= '4342 „ TO =
= -3287 „ m =

"3'

ON THE STABILITY OF THE MOTION OF SATURN 8 RINGS.

315

The tangential force may be calculated in the same way, it is

T=- R{MAam(iiis-\-a) + NBsm(7ns + IB)} (10).

The normal force is

Z= -^-RJC cos (ms + y) (11).

G. We have found the expressions for the forces which act upon each
member of a system of equal satellites which originally formed a uniform ring,
but are now aflfected with displacements depending on circular functions. If
these displacements can be propagated round the ring in the form of waves

with the velocity — , the quantities a, y8, and y will depend on t, and the

complete expressions will be

p = ^ cos (ms + nt-\- a) '

a = Bam(ms + nt+^) ■ (12).

^ = Ccos (ms + nt + y).
Let us find in what cases expressions such as these will be true, and
what will be the result when they are not true.

Let the position of a satellite at any time be determined by the values
of r, (j), and C, where r is the radius vector reduced to the plane of reference,
<t> the angle of position measured on that plane, and ^ the distance from it.
The equations of motion will be

[dtj df ^ r-^-^

dr d4 d^_^
^Tt dt ^"^ df~

d^

df'

1^

.(13).

If we substitute the value of ^ in the third equation and remember that r
is nearly = 1 , we find

(14).

As this expression is necessarily positive, the value of n' is always real,
and the disturbances normal to the plane of the ring can always be propa-

31G ON THE STABILITY OF THE MOTION OF SATURN's RINGS.

gated as waves, and therefore can never be the cause of instability. We
therefore confine our attention to the motion in the plane of the ring as
deduced from the two former equations.

Putting r = 1 4- /) and (f> = <ot + s + a; and omitting powers and products of
p, cr and their differential coeflScients,

''+'">+2-t-t='^-2«''+^

-l+§=^

(15).

Substituting the values of p and cr as given above, these equations become
oi'-S - RK+ U- -]-2S - EL + 7f)A cos (ttis + nt + a)

+ (2(071 + -RM)B COS (ins + nt + ^) = ...(16),

H'

(2(071 + - EM) A sin (ins + nt + a) + (if +-RN)Bam(7ns + nt-\-^) = 0.... (17).
p p

Putting for (ins + nt) any two diflferent values, we find from the second
equation (17)

a=)8 (18),

and (2(on + -E]\f)A + (n'+-EN)B = (19),

and from the first (16) ((o' + 2S - EL + iv) A + (2(on + - EM) B = (20),

and (o'-S - EK=0 (21).

p

Eliminating A and B from these equations, we get
n'-{S(o'-2S + -E(L-N)}n^

-'4(o-EMn + ((o' + 2S - EL)-EN - ,E'M' = (22),

a biquadratic equation to determine n.

For every real value of n there are terms in the expressions for p and o-

of the form

A cos (nis + nt + a).

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 317

For every pure impossible root of the form ±7 — In' there are terms of
the forms

^e^^'cos (ms + a).

Although the negative exponential coefficient indicates a continually diminlshmg
displacement which is consistent with stability, the positive value which neces-
sarily accompanies it indicates a continually increasing disturbance, which would
completely derange the system in course of time.

For every mixed root of the form ±n/ — In' + n, there are terms of the form

.46*"'' cos {ms + nt + a).

If we take the positive exponential, we have a series of m waves travelling

with velocity — and increasing in amplitude with the coefficient e"^"'. The

negative exponential gives us a series of m waves gradually dying away, but
the negative exponential cannot exist without the possibility of the positive one
having a finite coefficient, so that it is necessary for the stability of the motion
that the four values of n be all real, and none of them either impossible
quantities or the sums of possible and impossible quantities.

We have therefore to determine the relations among the quantities K, L,
M, N, R, S, that the equation

n'-lS+^RidK+L-N)]?^

'-4<o-RMn + {SS+ - R (K-L)} - RN- \ R'M'^ U=0

may have four real roots.

7. In the first place, U is positive, when tz is a large enough quantity,
whether positive or negative.

It is also positive when 7i=;0, provided S be large, as it must be, com-
pared with - RL, - RM and - RN.

If we can now find a positive and a negative value of n for which U
is negative, there must be four real values of n for which U=0, and the four
roots will be real.

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

Now if we put n= ±J^JS,

U= -^S' + l -R{7N±ij2M-L-dK) S+ \r{KN-LN^M%

which is negative if >S be large compared to R.

So that a ring of satellites can always be rendered stable by increasing
the mass of the central body and the angular velocity of the ring.

The values of L, M, and N depend on m, the number of undulations in

the ring. When m = ^, the values of L and N will be at their maximum

and M=0. If we determine the relation between S and R in this case so
that the system may be stable, the stability of the system for every other
displacement will be secured.

8. To find the mass which must be given to the central body in order
that a ring of satellites may permanently revolve round it.

We have seen that when the attraction of the central body is sufficiently
great compared with the forces arising from the mutual action of the satellites,
a permanent ring is possible. Now the forces between the satellites depend on
the manner in which the displacement of each satellite takes place. The con-
ception of a perfectly arbitrary displacement of all the satellites may be rendered
manageable by separating it into a number of partial displacements depending
on periodic functions. The motions arising from these small displacements will
take place independently, so that we have to consider only one at a time.

Of all these displacements, that which produces the greatest disturbing
forces is that in w^hich consecutive satellites are oppositely displaced, that is,

when m = -, for then the nearest satellites are displaced so as to increase as

z

much as possible the effects of the displacement of the satellite between them.
If we make /x a large quantity, we shall have

2™^<^ = e;(l + 3-' + 5- + &c.) = ^.(l-0518).
sm^ n' ^ TT

M=0, N=2L, J5r very small.

IT

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 319

Let - RL = X, then the equation of motion will be
/A

n*-{S-x)n' + 2x{'iS-x)=U=0 (23).

The conditions of tliis equation having real roots are

S>x (24),

(S-xY>^x{'iS-x) (25).

The last condition gives the equation

6:'-26*Sx + 9ar>0,

whence S>2Q-U2x, or>S<0-351a; (26).

The last solution is inadmissible because S must be greater than x, so that

the true condition is »S>25*649a:,

> 25-649 i 72^3 -5259,

/X IT

S>-ASd2im'R (27).

So that if there were 100 satellites in the ring, then

5>4352i2
is the condition which must be fulfilled in order that the motion arising from
every conceivable displacement may be periodic.

If this condition be not fulfilled, and if S be not sufiadent to render the
motion perfectly stable, then although the motion depending upon long undu-
lations may remain stable, the short undulations wiill increase in amplitude till
some of the neighbouring satellites are brought into collision.

9. To determine the nature of the motion when the system of satellites
is of small mass compared with the central body.

The equation for the determination of n is

^ /x ^ /x

+ {Zoy - R{2K+L)]~ RN -\R'M'=^0 (28).

F' r" r"

When R is very small we may approximate to the values of n by assuming
that two of them are nearly ± co, and that the other two are small.

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

If we put n= ±(0,

dU

dn

= ±2g>' + &c.

Therefore the corrected values of n are

n^±{<o + ^R(2K + L-.m)} + ^RM.

(29).

The small values of n are nearly ±/3-i2iV^: correcting them in the
way, we find the approximate values

n=±./3^EN^2~RM

same

(30).

The four values of n are therefore

1

^1= -<o-^-E{2K+L^iM-4N)

RN- — RM

fXCt)

^z=+J^-RN- — RM

(31),

^4= +o>+^ - R(2K+L + iM-4N)

and the complete expression for p, so far as it depends on terms containing ms,
is therefore P = A, cos {ms + n^t + a^)-\-A^ cos (ws + n^t + c^)

+ A^co&(ms + nJ, + a^-{-A^coB{ms-\-nJ^ + a^) (32),

and there will be other systems, of four terms each, for every value of m in
the expansion of the original disturbance.

We are now able to determine the value of o- from equations (12), (20), by
putting /8 = a, and

2<an + - RM

5= —

(33).

n' + -RN

ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 321

So that for every term of p of the form

p = Acos (ms -{-111 + a) (34),

there is a corresponding term in a,

2w7i + - RM

7t' + -RN

A sin {ms-¥7it + a) (35).

10. Let us now fix our attention on the motion of a single satellite,
and determine its motion by tracing the changes of p and a- while t varies
and 5 is constant, and equal to the value of s corresponding to the satellite
in question.

We must recollect that p and a- are measured outwards and forwards from
an imaginary point revolving at distance 1 and velocity o, so that the motions
we consider are not the absolute motions of the satellite, but its motions
relative to a point fixed in a revolving plane. This being understood, we may
describe the motion as elliptic, the major axis being in the tangential direc-
tion, and the ratio of the axes being nearly 2 ^ , which is nearly 2 for n, and n,
and is very large for n^ and n^.

The time of revolution is — , or if we take a revolution of the ring as

the unit of time, the time of a revolution of the satellite about its mean

... . it)
position IS - .

The direction of revolution of the satellite about its mean position is in
every case opposite to that of the motion of the ring.

11. The absolute motion of a satellite may be found from its motion
relative to the ring by writing

r=l+p = l+^cos {ms + nt + a),
d = (ot + s-{-<T = (ot + s-2 -Asm{ms-\-nt-\-a).

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

When n is nearly equal to ±(0, the motion of each satellite in space is
nearly elliptic. The eccentricity is A, the longitude at epoch s, and the longi-
tude when at the greatest distance from Saturn is for the negative value n^

- — R{2K+L-iM-4N)t + {m+l)s + a,
and for the positive value n^

- — R{2K+L + 4M^4.N)t-{m+l)s-a.

We must recollect that in all cases the quantity within brackets is negative,
so that the major axis of the ellipse travels forwards in both cases. The chief
difference between the two cases lies in the arrangement of the major axes of
the ellipses of the different satellites. In the first case as we pass from one
satellite to the next in front the axes of the two ellipses lie in the same
order. In the second case the particle in front has its major axis behind that
of the other. In the cases in which n is small the radius vector of each
satellite increases and diminishes during a periodic time of several revolutions.
This gives rise to an inequality, in which the tangential displacement far exceeds
the radial, as in the case of the annual equation of the Moon.

12. Let us next examine the condition of the ring of satellites at a given
instant. We must therefore fix on a particular value of t and trace the changes
of p and <r for different values of s.

From the expression for p we learn that the satellites form a wavy line,
which is furthest from the centre when (ms + nt + a) is a multiple of 27r, and
nearest to the centre for intermediate values.

From the expression for cr we learn that the satellites are sometimes in
advance and sometimes in the rear of their mean position, so that there are
places where the satellites are crowded together, and others where they are
drawn asunder. When n is positive, ^ is of the opposite sign to A, and the
crowding of the satellites takes place when they are furthest from the centre.
When n is negative, the satellites are separated most when furthest from the
centre, and crowded together when they approach it.

The form of the ring at any instant is therefore that of a string of beads
forming a re-entering curve, nearly circular, but with a small variation of distance

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 323

from the centre recurring m times, and forming m regular waves of trans-
vei-se displacement at equal intervals round the circle. Besides these, there are
waves of condensation and rarefaction, the effect of longitudinal displacement.
When n is positive the points of greatest distance from the centre are points
of greatest condensation, and when n is negative they are points of greatest
rarefaction.

13. We have next to determine the velocity with which these waves of
disturbance are propagated round the ring. We fixed our attention on a par-
ticular satellite by making s constant, and on a particular instant by making t
constant, and thus we determined the motion of a satellite and the form of the
ring. We must now fix our attention on a phase of the motion, and this we
do by making p or a- constant. This implies

ms + nt + a = constant,

ds _ n
dt~ m*

So that the particular phase of the disturbance travels round the ring with an

angular velocity = relative to the ring itself. Now the ring is revolving

in space with the velocity w, so that the angular velocity of the wave in space is

tj- = w (36).

m

Thus each satellite moves in an ellipse, while the general aspect of the
ring is that of a curve of m waves revolving with velocity ct. This, however,
is only the part of the whole motion, which depends on a single term of the
solution. In order to understand the general solution we must shew how to
determine the whole motion from the state of the ring at a given instant.

14. Given the position and motion of every satellite at any one time, to
calculate the position and motion of every satellite at any other time, provided
that the condition of stability is fulfilled.

The position of any satellite may be denoted by the values of p and cr for
that satellite, and its velocity and direction of motion are then indicated by the

values of -r and -y- at the g:iven instant.
dt at

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

These four quantities may have for each satellite any four arbitrary values,
as the position and motion of each satellite are independent of the rest, at the
beginning of the motion.

Each of these quantities is therefore a perfectly arbitrary ftmction of s, the
mean angular position of the satellite in the ring.

But any function of s from s = to s = 27r, however arbitrary or discontinuous,
can be expanded in a series of terms of the form A cos (5 + a) + A' cos (2s + a') + &c.
See § 3.

Let each of the four quantities p, -^ , a, -j- he expressed in terms of such

a series, and let the terms in each involving ms be

p = Ecoa{'ms + e) (37),

^^=Fcos(ins+f) (38).

<T =G cos (ms+g) (39),

^ = Hco3{ms + h) (40).

These are the parts of the values of each of the four quantities which are
capable of being expressed in the form of periodic fimctions of ms. It is
evident that the eight quantities E, F, G, H, e, f, g, h, are all independent and
arbitrary.

The next operation is to tind the values of X, M, N, belonging to disturb-
ances in the ring whose index is m [see equation (8)], to introduce these
values into equation (28), and to determine the four values of n, (ti,, tIj, 1I3, n^).

This being done, the expression for p is that given in equation (32), which
contains eight arbitrary quantities (A,, A^, A3, At, «„ a^, a^, aj.

Giving t its original value in this expression, and equating it to Eco3{7m-\-e),
we get an equation which is equivalent to two. For, putting 7ns = 0, we have
^1 cos Oi + .^2 cos a, + -^3 cos a, + ^^ cos a^ = -E' cos e (41).

And putting ms= , we have another equation

-4i sin Oi + ^j sin aj + ^3 sin 03 + ^< sin a^ = ^ sin e (42).

ON THE STABILITY OF THE MOTION OF SATUBN's RINGS. 325

Differentiating (32) with respect to t, we get two other equations

- A^n^ Bina-kc.-F cos/ (43),

Aji^ cos a + &c.=F sin/ (44 ).

Bearing in mind that B„ B^, &c. are connected with A„ A^, &c. by equa-
tion (33), and that B is therefore proportional to A, we may write B = A^,
where

2o)n + - RM

P ^
P= 7

H'
^ being thus a fiinction of n and a known quantity.

The value of <r then becomes at the epoch

<r = ^i)8i sin (m5 4- Oi) -I- &c. = Gcoa('ms-\-g),
from which we obtain the two equations

^^1 sin Oi -I- &c. = 6^ cos g (45),

^^iC0Sai + &c. = —Geing (46).

Differentiating with respect to t, we get the remaining equations

A^jij^ cos Oj + &c. = ^ cos A (47),

^^iniSinai-l-&c. = iZ'sinA (48).

We have thus found eight equations to determine the eight quantities
^1, &c. and Oi, &c. To solve them, we may take the four in which -^iCosoi,
&c. occur, and treat them as simple equations, so as to find ^iCosoj, &c. Then
taking those in which ^isinoi, &c. occur, and determining the values of those
quantities, we can easily deduce the value of A^ and a,, &c. from these.

We now know the amplitude and phase of each of the four waves whose
index is m. All other systems of waves belonging to any other index must
be treated in the same way, and since the original disturbance, however arbitrary,
can be broken up into periodic functions of the form of equations (37 — 40),
our solution is perfectly general, and applicable to every possible disturbance of
a ring fulfilling the condition of stability (27).

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

15. We come next to consider the effect of an external disturbing force,
due either to the irregularities of the planet, the attraction of satellites, or
the motion of waves in other rings.

All disturbing forces of this kind may be expressed in series of which the
general term is

A cos {vt + ms + a),

where v is an angular velocity and m a whole number.

Let P cos {ins + vt +p) be the central part of the force, acting inwards, and
Q sin (ms + vt + q) the tangential part, acting forwards. Let p = A cos {tus + vt + a)
and a- = Bsm (ms + vt-]- fi), be the terms of p and a which depend on the
external disturbing force. These will simply be added to the terms depending
on the original disturbance which we have already investigated, so that the
complete expressions for p and <t will be as general as before. In consequence
of the additional forces and displacements, we must add to equations (16) and
(17), respectively, the following terms:

{Zar - R (2K+ L) + v"] A cos (m^-{-vt-\- a)

+ (2q)V -\- - RM) B COS (ms + vt + f3)-P cos (ms + vt-hp) = (49).

(2a)i; 4- - EM) A sin (ms + vt + a)

+ (v" + - EN) B Bm(ms + vt + fi)-¥Q sin (ms + vt + q) = (50).

Making 7ns + vt = in the first equation and - in the second,

{S(o' - E (2K+L) + if} A cos a + (2(ov + -E3f) B cos fi-P coap = (51).

(2a>v + - EM) A cosa + (v' + - EN) B COB fi + Qcosq = (52).

Then if we put
U' = v'-{oj' + -E(2K+L-N)}v'-A-EMv

+ {Sa>' - E(2K+L)}-EN-\E'M' (53),

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 327

we shall find the value of A cos a and B coa fi ;

v' + -RN 2cov-i-~RM
A cosa = ft; P coa p + t4 Qcoaq (54).

2(ov 4- - RM y' + 3<o' - R {K+ L)
Bcoafi= j^ Pcoap jp Qcoaq (55).

Substituting sines for cosines in equations (51), (52), we may find the
values of A sin a and B sin ^.

Now U* is precisely the same function of v that Z7 is of ?i, so that if u
coincides with one of the four values of n, U' will vanish, the coefiicients A
and B will become infinite, and the ring will be destroyed. The disturbing
force is supposed to arise from a revolving body, or an undulation of any kind

which has an angular velocity relatively to the ring, and therefore an

absolute angular velocity = w .

If then the absolute angular velocity of the disturbing body is exactly or
nearly equal to the absolute angular velocity of any of the free waves of the
ring, that wave will increase till the ring be destroyed.

The velocities of the free waves are nearly

l+i\ a> + i /s-i^.V, o> - /s-i^iV^, and 0) fl-i) (56).

When the angular velocity of the disturbing body is greater than that of
the first wave, between those of the second and third, or less than that of
the fourth, U' is positive. When it is between the first and second, or between
the third and fourth, U' is negative.

Let us now simplify our conception of the disturbance by attending to the
central force only, and let us put ^ = 0, so that P is a maximum when ms + vt
is a multiple of 27r. We find in this case a = 0, and /8 = 0. Also

if+^- RN
^=—^P (57),

2cjv + -RM
B= ^. P (58).

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

When U' is positive, A will be of the same sign as P, that is, the parts
of the ring wlU be furthest from the centre where the disturbing force towards
the centre is greatest. When U' is negative, the contrary will be the case.

When V is positive, B will be of the opposite sign to A, and the parts
of the ring furthest from the centre will be most crowded. When v is negative,
the contrary will be the case.

Let us now attend only to the tangential force, and let us put ^' = 0. We
find in this case also a = 0, )3 = 0,

2(ov+-RM

^= — tr — ^ (^^)'

B= ^. Q (60).

The tangential displacement is here in the same or in the opposite direc-
tion to the tangential force, according as £/"' is negative or positive. The
crowding of sateUites is at the points farthest from or nearest to Saturn
according as -y is positive or negative.

16. The effect of any disturbing force is to be determined in the following
manner. The disturbing force, whether radial or tangential, acting on the ring
may be conceived to vary from one satellite to another, and to be different at
different times. It is therefore a perfectly arbitrary function of s and t.

Let Fourier's method be applied to the general disturbing force so as to
divide it up into terms depending on periodic functions of s, so that each term
is of the form F (t) cos {ms + a), where the function of i is still perfectly arbitrary.

But it appears from the general theory of the permanent motions of the
heavenly bodies that they may all be expressed by periodic functions of t
arranged in series. Let vt be the argument of one of these terms, then the
corresponding term of the disturbance will be of the form

P cos (ttis + vt + a).

This term of the disturbing force indicates an alternately positive and
negative action, disposed in m waves round the ring, completing its period

ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 329

relatively to eaxih particle in the time — , and travelling as a wave among

the particles with an angular velocity , the angular velocity relative to fixed

space being of course oj — - . The whole disturbing force may be split up into
terms of this kind.

17. Each of these elementary disturbances will produce its own wave in
the ring, independent of those which belong to the ring itself. This new wave,
due to external disturbance, and following different laws from the natural waves
of the rincy, is called the farced wave. The angular velocity of the forced wave
is the same as that of the disturbing force, and its maxima and minima coin-
cide with those of the force, but the extent of the disturbance and its direction
depend on the comparative velocities of the forded wave and the four natural

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