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hypotheses the appearances both of the bright Rings and the recently discovered dark Ring may
be most satisfactorily explained; and to indicate any causes to which a change of form, such as
is supposed from a comparison of modern with the earlier observations to have taken place, may
be attributed.

E. GUEST, rice-Chancellor.

J. CHALLIS.

S. PARKINSON.

W. THOMSON.
March 23, 1855.



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 289



CONTENTS.

Nature of the Problem 290

Laplace's investigaticm of the Equilibrium of a Ring, otuI its minimum density 29i

Hit proof that Oie platie of tlie Rings will follow that of Saturn's Equator — that a solid uniform Ring is

necessarily unstable 293

Further investigation required— Theory of an Irregular Solid Ring leads to the result that to ensure stability

the irregularity mitst be enormous 294

Theory of a fluid or discontinuous Ring resolves itself into the investigation of a series of waves . . 295

PART I.

ON THE MOTION OF A RIGID BODY OF ANY FORM ABOUT
A SPHERE.

Equations of Motion 296

Problem I. To find the conditions under which a uniform motion of the Ring is possible . . . 298

Problem II. To find the equations of the motion when slightly distxirbed 299

Problem III. To reduce the three siinultaneou^ equations of motion to the form of a single linear equation 300
Problem IV. To determijie whether the motion of the Ring is stable or unstable, by meayis of the relations

of the coefficients A, B, C 301

Problem V. To find the centre of gravity, the radius of gyration, and the variations of the potential Tieaf

the centre of a circular ring of small but variable section 302

Problem VI. To determine the condition of stability of the motion in terms of the coeffilcierits f, g, h, which

indicate the distribution of mass in the ring 306

RB6ULTS. I«^, a uniform ring is unstable. 2nd, a ring varying in section according to the law of sines is
unstable. 3rd, a uniform ring loaded with a heavy particle may be stable, provided the mass of the
particle be between 'SlSSeS and "8279 of the whole. Case in which the ring is to the particle as 18
«o 82 307

PART II.

ON THE MOTION OF A RING, THE PARTS OF WHICH ARE NOT
RIGIDLY CONNECTED.

1. General Statemeiii of the Problem, and limitation to a nearly uniform ring 310

2. Notation 311

3. Expansion of a function in terms of sines and cosines of multiples of the variable . . . . 311

4. Magnitude and direction of attraction between two elements of a disturbed Ring 312

5. Resultant attractions on any one of a ring of equal satellites disturbed in any way .... 313
Note. Calculated values of these attractions in particular cases . . . . . . . . 314

6. Equations of motion of a satellite of the Ring, and biquadratic equation to determine the wave-velocity 31.5

7. A ring of satellites may always be rendered stable by increasing the mass of the central body . . 317

8. Relation between the number and mass of satellites and the mass of the central body necessary to

ensure stability. S>-4352,x2R 318

9. Solution of the biquadratic equation when the mass of t/ie Ring is small ; and complete e.rprcssions

for the Tnotion of each satellite .............. 319

10. Each satellite moves {relatively to tJie ring) in an ellipse 321



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

11. Each satellite moves absolutely/ in S2>ace in a curve which is nearly an ellipse for the large values

of n, and a spiral of many nearly circular coih when n is small 321

12. The form of the ring at a given instant is a series of undulations 322

13. These uiidvlations travel round the ring with velocity relative to the ring, and a absolutely 323

14. General Solution of the Problem — Given the position and motion of every satellite at any one time,
to calculate the position and motion of any satellite at any other time, provided that the condition

of stability is fulfilled 323

15. Calculation of the effect of a periodic external disturbing force 326

16. Treatment of disturbing forces in general 328

17. Theory of free waves and forced waves 329

18. Motion of the ring when the conditions of stability are net fulfilled. Two different ways in which

the ring may be broken up 330

19. Motion of a riyig of unequal satellites 335

20. Motion of a ring composed of a clowi of scattered particles 336

21. Calculation of the forces arising from the displacements of such a system 337

22. Application to the case of a ring of this kind. The mean density must be excessively s^nall, which

is iTiconsistent with its moving as a whole ............ 338

23. On the forces arising from inequalities in a thin stratum of gravitating incompressible fluid of
indefinite extent 338

24. Application to the case of a flattened fluid ring, moving with uniform angular velocity. Such a

ring will be broken up into portions which may continue to revolve as a ring of satellites . . . 344

ON THE MUTUAL PERTUKBATIONS OF TWO RINGS.

25. Application of the general theory of free and forced waves 345

26. To determine the attractions between the rings 346

27. To form the equations of motion 348

28. Method of determining the reaction of the forced wave on tlve free wave which produced it . . 349

29. Cases in which the perturbations increase indefinitely . . ........ 351

30. Application to the theory of an indefinite number of concentric rings 352

31. On the effect of long-continued disturbances on a system of rings 352

32. On the effect of collisions among the parts of a revolving system ....... 354

33. On the effect of internal friction in a fluid ring 354

Ilecapitulation of the Theory of the Motion of a Rigid Ring. Reasons for ryecting tlie hypothesis of

rigidity 356

Recapitulation of the Theory of a Ring of Equ/il Satellites 360

Description of a working model shewing the motions of such a system 363

Theory of Rings of various constitutions 367

Mutual action of Two Rings 370

Case of many concentric Rings, &c. . . 371

General Conclusions 372

Appendix. Extract of a letter from Professor W. Thomson, of Glasgow, giving a solution of t/ie Pro-
blem of a Rigid Ring 374



There are some questions in Astronomy, to which we are attracted rather
on account of their pecuHarity, as the possible illustration of some unknown
principle, than from any direct advantage which their solution would afford to



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 291

mankind. The theory of the Moon's inequalities, though in its first stages it
presents theorems interesting to all students of mechanics, has been pursued into
such intricacies of calculation as can be followed up only by those who make
the improvement of the Lunar Tables the object of their lives. The value of
the labours of these men is recognised by all who are aware of the importance
of such tables in Practical Astronomy and Navigation. The methods by which
the results are obtained are admitted to be sound, and we leave to professional
astronomers the labour and the merit of developing them.

The questions which are suggested by the appearance of Saturn's Rings
cannot, in the present state of Astronomy, call forth so great an amount of
labour among mathematicians. I am not aware that any practical use has been
made of Saturn's Rings, either in Astronomy or in Navigation. They are too
distant, and too insignificant in mass, to produce any appreciable effect on the
motion of other parts of the Solar system; and for this very reason it is diflS-
cult to determine those elements of their motion which we obtain so accurately
in the case of bodies of greater mechanical importance.

But when we contemplate the Rings from a purely scientific point of view,
they become the most remarkable bodies in the heavens, except, perhaps, those
still less useful bodies — the spiral nebulae. When we have actually seen that
great arch swung over the equator of the planet without any visible connexion,
we cannot bring our minds to rest. We cannot simply admit that such is the
case, and describe it as one of the observed facts in nature, not admitting or
requiring explanation. We must either explain its motion on the principles of
mechanics, or admit that, in the Saturnian realms, there can be motion regu-
lated by laws which we are unable to explain.

The arrangement of the rings is represented in the figure (l) on a scale
of one inch to a hundred thousand miles. S is a section of Saturn through
his equator, A, B and C are the three rings. A and B have been known for
200 years. They were mistaken by Galileo for protuberances on the planet itself,
or perhaps satellites. Huyghens discovered that what he saw was a thin flat
ring not touching the planet, and Ball discovered the division between A and B.
Other divisions have been observed splitting these again into concentric rings,
but these have not continued visible, the only well-established division being one
in the middle of A. The third ring C was first detected by Mr Bond, at
Cambridge U.S. on November 15, 1850; Mr Dawes, not aware of Mr Bond's
discovery, observed it on November 29th, and Mr Lassel a few days later. It



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

gives little light compared with the other rings, and is seen where it crosses
the planet as an obscure belt, but it is so transparent that the limb of the
planet is visible through it, and this without distortion, shewing that the rays
of light have not passed through a transparent substance, but between the
scattered particles of a discontinuous stream.

It is difficult to estimate the thickness of the system ; according to the
best estimates it is not more than 100 miles, the diameter of A being 176,418
miles; so that on the scale of our figure the thickness would be one thousandth
of an inch.

Such is the scale on which this magnificent system of concentric rings is
constructed; we have next to account for their continued existence, and to
reconcile it with the known laws of motion and gravitation, so that by rejecting
every hypothesis which leads to conclusions at variance with the facts, we may
learn more of the nature of these distant bodies than the telescope can yet
ascertain. We must account for the rings remaining suspended above the planet,
concentric with Saturn and in his equatoreal plane ; for the flattened figure of the
section of each ring, for the transparency of the inner ring, and for the gradual
approach of the inner edge of the ring to the body of Saturn as deduced
from all the recorded observations by M. Otto Struvd {Sur les dimensions des
Anneaux de Saturne — Recueil de Memoires Astronomiques, Poulkowa, 15 Nov.
1851). For an account of the general appearance of the rings as seen from the
planet, see Lardner on the Uranography of Saturn, Mem. of the Astronomical
Society, 1853. See also the article "Saturn" in Nichol's Cyclopcedia of the
Physical Sciences.

Our curiosity with respect to these questions is rather stimulated than
appeased by the investigations of Laplace. That great mathematician, though
occupied with many questions which more imperiously demanded his attention,
has devoted several chapters in various parts of his great work, to points con-
nected with the Saturnian System.

He has investigated the law of attraction of a ring of small section on a
point very near it {Mec. Cel. Liv. iii. Chap, vi.), and from this he deduces the
equation from which the ratio of the breadth to the thickness of each ring is

to be found,

E' p X(X-l)

^~3a'p (\+l) (3X^+1)'

where R is the radius of Saturn, and p his density; a the radius of the ring,



ON THE STABIUTY OF THE MOTION OF SATURN's RINGS. 293

and p its density; and X the ratio of the breadth of the ring to its thick-
ness. The equation for determining X when e is given has one negative root
which must be rejected, and two roots which are positive while e<0"0543, and
impossible when e has a greater value. At the critical value of e, X = 2-594
nearly.

The fact that X is impossible when e is above this value, shews that the
ring cannot hold together if the ratio of the density of the planet to that of
the ring exceeds a certain value. This value is estimated by Laplace at I'S,
assuming a = 2R.

We may easily follow the physical interpretation of this result, if we observe
that the forces which act on the ring may be reduced to —

(1) The attraction of Saturn, varying inversely as the square of the dis-
tance from his centre.

(2) The centrifugal force of the particles of the ring, acting outwards, and
varying directly as the distance from Saturn's polar axis.

(3) The attraction of the ring itself, depending on its form and density,
and directed, roughly speaking, towards the centre of its section.

The first of these forces must balance the second somewhere near the mea,n
distance of the ring. Beyond this distance their resultant will be outwards,
within this distance it will act inwards.

If the attraction of the ring itself is not sufl&cient to balance these residual
forces, the outer and inner portions of the ring will tend to separate, and the
ring will be split up ; and it appears from Laplace's result that this will be
the case if the density of the ring is less than ^ of that of the planet.

This condition applies to all rings whether broad or narrow, of which the
parts are separable, and of which the outer and inner parts revolve with the
same angular velocity.

Laplace has also shewn (Li v. v. Chap, iii.), that on account of the oblate-
ness of the figure of Saturn, the planes of the rings will follow that of Saturn's
equator through every change of its position due to the disturbing action of
other heavenly bodies.

Besides this, he proves most distinctly (Liv. iii. Chap, vi.), that a solid uni-
form ring cannot possibly revolve about a central body in a permanent manner,
for the slightest displacement of the centre of the ring from the centre of the
planet would originate a motion which would never be checked, and would



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

inevitably precipitate the ring upon the planet, not necessarily by breaking the
ring, but by the inside of the ring falling on the equator of the planet.

He therefore infers that the rings are irregular solids, whose centres of
gravity do not coincide with their centres of figure. We may draw the con-
clusion more formally as follows, "If the rings were solid and uniform, their
motion would be unstable, and they would be destroyed. But they are not
destroyed, and their motion is stable; therefore they are either not uniform or
not solid."

I have not discovered"" either in the works of Laplace or in those of more
recent mathematicians, any investigation of the motion of a ring either not uni-
form or not solid. So that in the present state of mechanical science, we do
not know whether an irregular solid ring, or a fluid or disconnected ring, can
revolve permanently about a central body; and the Saturnian system still re-
mains an unregarded witness in heaven to some necessary, but as yet unknown,
development of the laws of the universe.

We know, since it has been demonstrated by Laplace, that a uniform solid
ring cannot revolve permanently about a planet. We propose in this Essay to
determine the amount and nature of the irregularity which would be required
to make a permanent rotation possible. We shall find that the stability of the
motion of the ring would be ensured by loading the ring at one point with a

* Since this -was written, Prof. Challis has pointed out to me three important papers in Gould's
Astronomical Journal: — Mr G. P. Bond on the Rings of Saturn (May 1851) and Prof. B. Pierce of
Harvard University on the Constitution of Saturn's Rings (June 1851), and on the Adams' Prize
Problem for 1856 (Sept. 1855). These American mathematicians have both considered the conditions
of statical equilibrium of a transverse section of a ring, and have come to the conclusion that the
rings, if they move each as a whole, must be very narrow compared with the observed rings, so
that in reality there must be a great number of them, each revolving with its own velocity. They
have also entered on the question of the fluidity of the rings, and Prof. Pierce has made an
investigation as to the permanence of the motion of an irregular solid ring and of a fluid ring.
The paper in which these questions are treated at large has not (so far as I am aware) been
pxiblished, and the references to it in Gould's Journal are intended to give rather a popular account
of the results, than an accurate outline of the methods employed. In treating of the attractions of
an irregular ring, he makes admirable use of the theory of potentials, but his published investi-
gation of the motion of such a body contains some oversights which are due perhaps rather to the
imperfections of popular language than to any thing in the mathematical theory. The only part of
the theory of a fluid ring which he has yet given an account of, is that in which he considers
the form of the ring at any instant as an ellipse; corresponding to the case where n = u), and
m=l. As I had only a limited time for reading these papers, and as I could not ascertain the
methods used in the original investigations, I am unable at present to state how far the results of
this essay agree with or differ from those obtained by Prof. Pierce.



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 295

heavy satellite about 4-^ times the weight of the ring, but this load, besides
being inconsistent with the observed appearance of the rings, must be far too
artificially adjusted to agree with the natural arrangements observed elsewhere,
for a very small error in excess or defect would render the ring again unstable.

We are therefore constrained to abandon the theory of a solid ring, and
to consider the case of a ring, the parts of which are not rigidly connected,
as in the case of a ring of independent satellites, or a fluid ring.

There is now no danger of the whole ring or any part of it being pre-
cipitated on the body of the planet. Every particle of the ring is now to be
regarded as a satellite of Saturn, disturbed by the attraction of a ring of
satellites at the same mean distance from the planet, each of which however is
subject to slight displacements. The mutual action of the parts of the ring will
be so small compared with the attraction of the planet, that no part of the
ring can ever cease to move round Saturn as a satellite.

But the question now before us is altogether different from that relating to
the solid ring. We have now to take account of variations in the form and
arrangement of the parts of the ring, as well as its motion as a whole, and
we have as yet no security that these variations may not accumulate till the
ring entirely loses its original form, and collapses into one or more satellites,
circulating round Saturn. In fact such a result is one of the leading doctrines
of the " nebular theory " of the formation of planetary systems : and we are
familiar with the actual breaking up of fluid rings under the action of "capil-
lary " force, in the beautiful experiments of M. Plateau.

In this essay I have shewn that such a destructive tendency actually exists,
but that by the revolution of the ring it is converted into the condition of
dynamical stability. As the scientific interest of Saturn's Rings depends at
present mainly on this question of their stability, I have considered their motion
rather as an illustration of general principles, than as a subject for elaborate
calculation, and therefore I have confined myself to those parts of the subject
which bear upon the question of the permanence of a given form of motion.

There is a very general and very important problem in Dynamics, the solu-
tion of which would contain all the results of this Essay and a great deal
more. It is this —

"Having found a particular solution of the equations of motion of any
material system, to determine whether a slight disturbance of the motion indi-



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

cated by the solution would cause a small periodic variation, or a total
derangement of the motion."

The question may be made to depend upon the conditions of a maximum
or a minimum of a function of many variables, but the theory of the tests
for distinguishing maxima from minima by the Calculus of Variations becomes
so intricate when applied to functions of several variables, that I think it doubt-
ful whether the physical or the abstract problem will be first solved.



PART I.

ON THE MOTION OF A RIGID BODY OF ANY FORM ABOUT A SPHERE.

We confine our attention for the present to the motion in the plane of
reference, as the interest of our problem belongs to the character of this motion,
and not to the librations, if any, from this plane.

Let S (Fig. 2) be the centre of gravity of the sphere, which we may call
Satiu-n, and E that of the rigid body, which we may call the Ring. Join RS,
and divide it in G so that

SG : GR '.: R : S,

R and S being the masses of the Ring and Saturn respectively.

Then G will be the centre of gravity of the system, and its position will
be unaffected by any mutual action between the parts of the system. Assume G
as the point to which the motions of the system are to be referred. Draw GA
in a direction fixed in space.

Let AGR = e, and SR = r,

then ^^^'S+R^' ^^^ ^^^STR^'

so that the positions of S and R are now determined.

Let BRR be a straight line through R, fixed with respect to the substance
of the ring, and let BRK=^.



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 297

This determines the angular position of the ring, so that from the values
of r, 6, and ^ the configuration of the system may be deduced, as far as relates
to the plane of reference.

We have next to determine the forces which act between the ring and
the sphere, and this we shall do by means of the potential function due to
the ring, which we shall call V.

The value of V for any point of space S, depends on its position relatively
to the ring, and it is found from the equation



where dm is an element of the mass of the ring, and r is the distance of that
element from the given point, and the summation is extended over every element
of mass belonging to the ring. V will then depend entirely upon the position
of the point S relatively to the ring, and may be expressed as a function
of r, the distance of S from R, the centre of gravity of the ring, and ^, the
angle which the line SR makes with the line RB, fixed in the ring.

A particle P, placed at S, will, by the theory of potentials, experience a

dV . ... . \ dV

moving force P —p in the direction which tends to increase r, and P - -jj

in a tangential direction, tending to increase ^.

Now we know that the attraction of a sphere is the same as that of

a particle of equal mass placed at its centre. The forces acting between the

dV . .

sphere and the ring are therefore S -j~ tending to increase r, and a tangential

\ dV .

force S - -j-r , applied at S tending to increase <;^. In estimating the efiect of

this latter force on the ring, we must resolve it into a tangential force S - -jj-

dV
acting at R, and a couple S -j-r tending to increase (f).

We are now able to form the equations of motion for the planet and the
ring.



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

For the planet



^d jf Rr Ydd\ _R_ ^Jy , .

^ dt ]S^VRl dtj '- " S + R '^ d<f> ^'^'

«l(^)-^(f)'=^^' (^)-

For the centre of gravity of the ring,

j.d (f Sr Y ^^1 S dV , .

^dt\\S+-R) Ttr~STR^df ^ ^'

j.d^ f Sr \ Sr (d0Y_ dV , .

For the rotation of the ring about its centre of gravity,

^S(''+«=^f (5)'

where h is the radius of gyration of the ring about its centre of gravity.

Equation (3) and (4) are necessarily identical with (l) and (2), and shew
that the orbit of the centre of gravity of the ring must be similar to that
of the Planet. Equations (1) and (3) are equations of areas, (2) and (4) are
those of the radius vector.

Equations (3), (4) and (5) may be thus written,

M-'^T!-'-'^}-(^-^i'- («)'

-{§-©}-( - -)f - (^)-

-(f-^f) - ^ - («)•

These are the necessary and sufficient data for determining the motion of
the ring, the initial circumstances being given.

Prob. I. To find the conditions under which a uniform motion of the
ring is possible.

By a uniform motion is here meant a motion of uniform rotation, during
which the position of the centre of the Planet with respect to the ring does
not change.



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 299



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