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In this case r and </> are constant, and therefore V and its differential
coefficients are given. Equation (7) becomes,

which shews that the angular velocity is constant, and that

dey R+S dV , ,^.

r- = <o\ say (9).

dtj Rr dr

Hence, -71 = 0, and therefore by equation (8),

%-^ • •■•••(-)•

Equations (9) and (10) are the conditions under which the uniform motion
is possible, and if they were exactly fulfilled, the uniform motion would go on
for ever if not disturbed. But it does not follow that if these conditions were
nearly fulfilled, or that if when accurately adjusted, the motion were slightly
disturbed, the motion would go on for ever nearly uniform. The effect of the
disturbance might be either to produce a periodic variation in the elements
of the motion, the ampUtude of the variation being small, or to produce a
displacement which would increase indefinitely, and derange the system altogether.
In the one case the motion would be dynamically stable, and in the other it
would be dynamically unstable. The investigation of these displacements while
still very small wiU form the next subject of inquiry.

Prob. II. To find the equations of the motion when slightly disturbed.
Let r = r„ = o}t and (f) = (f>^ in. the case of uniform motion, and let

r=ro +r„


when the motion is slightly disturbed, where r^, 6^, and ^1 are to be treated

as small quantities of the first order, and their powers and products are to be

dV dV

neglected. We may expand -j-^ and -j-r by Taylor's Theorem,

dV_dV drV d'V

dr ~dr "^ di^ '''"*■ cZrc/t^"^^'

d<f>~'d<f'^drd<t>''''^ d<i>''^''


where the values of the differential coeflBcients on the right-hand side of the
equations are those in which i\ stands for r, and ^^ for ^.

CaJlmg ^=A ^^^ = M^ ^^=N,

and taking account of equations (9) and (10), we may write these equations,

a^= -sirs'" +^''+^^"

Substituting these values in equations (6), (7), (8), and retaining all small
quantities of the first order while omitting their powers and products, we have
the following system of linear equations in r^, O^, and ^i,

E (2r,co^ + r,^^^y{E + S)(Mr, + N<f.,) =0 (11),


d% , „ de\


(o%-2r,(o-^]-{R + S){L7\ + M<f>,) = (12),

RlH'^^ + ^-SiMr^ + N^:) =0 (13).

df ' df

Prob. III. To reduce the three simultaneous equations of motion to the
form of a single linear equati


Let us write n instead of the symbol -j- , then arranging the equations in
terms of i\, 6^, and j>^, they may be written:

{2R,o>n + (R + S)M}r, + (Rr:n')e, + {R + S)N<i>, =0 (14),

{Rn'-R<^'^-(R + S) L}r,-(2Rr,con)d,^{R + S)M<f>, = (15),

- (SM) r, + (Rk'n') 0, + {RUrv -SN)<j>, =0 (16).

Here we have three equations to determine three quantities r,, 6„ ^i ; but
it is evident that only a relation can be determined between them, and that
in the process for finding their absolute values, the three quantities will vanish
together, and leave the following relation among the coefiicients,


-{2Rr,oin+ {R + S)^r} [2R)\(on] [Rlcrc'-SN}
+ {Rn' - Rco' -(R + S) L] {Rh'rf} {(R + >S') N]

+ {SM) {Rrjn') {R + S)M- (SM) {2Rr,<on) (R + S)Xi=0 (17).

+ {2Rr,<on + (R + S)M} {RLni'} {(R + S) if}
- {Rn' - Rxo' -{R + S)} {Rr.'if} {RJc'n' - SN}
By multiplying up, and arranging by powers of n and dividing by Rn\
this equation becomes

Aii* + B)v+C=0 (18),


B = SRr-r:i''<o-'-R{R + S)Lr:Jc'-R{{R + S)]if + Si''}N- i (19).

C=R{(R + S)l''- 3Sr:} oy + (R + S) {{R + S) t + Sr^} (Z.V- IP) J
Here we have a biquadratic equation in ?i which may be treated as a
quadratic in ?r, it being remembered that ?i stands for the operation -j- .

Prob. IV. To determine whether the motion of the ring is stable or
unstable, by means of the relations of the coefficients A, B, C.

The equations to determine the forms of r^, 6^, and <^i are all of the form
. d*u -r, dhi ^ ^ /^^\

^*+-^*+^"=» (-°''

and if n be one of the four roots of equation (18), then

will be one of the four terms of the solution, and the values of i\, 6^, and
<^i will differ only in the values of the coefficient D.

Let us inquire into the nature of the solution in different cases.

(1) If n be positive, this term would indicate a displacement which
must increase indefinitely, so as to destroy the arrangement of the system.

(2) If n be negative, the disturbance which it belongs to would gradually
die away.

(3) If n be a pure impossible quantity, of the form ±aj —\, then there
will be a term in the solution of the form D cos [at + a), and this would indi-


cate a periodic variation, whose amplitude is D, and period ^^ .


(4) If n be of the form b±J'^a, the first term being positive and
the second impossible, there will be a term in the solution of the form

De^' cos {at + a),
which indicates a periodic disturbance, whose amplitude continually increases
till it disarranges the system.

(5) If n be of the form -h±s/-la, a negative quantity and an im-
possible one, the corresponding term of the solution is

i>e"*'cos {(it + a),
which indicates a periodic disturbance whose amplitude is constantly diminishing.

It is manifest that the first and fourth cases are inconsistent with the
permanent motion of the system. Now since equation (18) contains only even
powers of n, it must have pairs of equal and opposite roots, so that every
root coming under the second or fifth cases, implies the existence of another
root belonging to the first or fourth. If such a root exists, some disturbance
may occur to produce the kind of derangement corresponding to it, so that
the system is not safe unless roots of the first and fourth kinds are altogether
excluded. This cannot be done without excluding those of the second and fifth
kinds, so that, to insure stability, aU the four roots must be of the third kind,
that is, pure impossible quantities.

That this may be the case, both values of n" must be real and negative,
and the conditions of this are —

1st. That A, B, and C should be of the same sign,

2ndly. That R>iAC.

When these conditions are fulfilled, the disturbances will be periodic and
consistent with stability. When they are not both fulfilled, a small disturbance
may produce total derangement of the system.

Prob. V. To find the centre of gravity, the radius of gyration, and the
variations of the potential near the centre of a circular ring of small but variable

Let a be the radius of the ring, and let 6 be the angle subtended at the
centre between the radius through the centre of gravity and the line through
a given point in the ring. Then if /i be the mass of unit of length of the


ring near the given point, ft will be a periodic function of 6, and may there-
fore be expanded by Fourier's theorem in the series,

li = — {1 + 2/cos^ + §^cos2^ + §/isin2^ + 2ico3(3^ + a) + &c.} (21),

where/, g, h, &c. are arbitrary coefficients, and R is the mass of the ring.

(1) The moment of the ring about the diameter perpendicular to the
prime radius is

R)\= r ficr cos ecW = Raf,
therefore the distance of the centre of gravity from the centre of the ring,

(2) The radius of gyration of the ring about its centre in its own plane
is evidently the radius of the ring =a, but if k be that about the centre of
gravity, we have

.'. Af = a=(l-f).

(3) The potential at any point is found by dividing the mass of each
element by its distance from the given point, and integrating over the whole

Let the given point be near the centre of the ring, and let its position be
defined by the co-ordinates r and xjj, of which r is small compared with a.

The distance (p) between this point and a point in the ring is
i = i {1 + %03 (^ - 0) + i (Q' + 1 (3' cos 2{i,-0)+&c.}.

The other terms contain powers of — higher than the second.
We have now to determine the value of the integral,

Jo P
and in multiplying the terms of (/i) by those of f-J , we need retain only
those which contain constant quantities, for all those which contain sines or


cosines of multiples of {^1^ — 0) will vanisti when integrated between the limits.
In this way we find

^=- {l+/%osr/; + i^'(l-4-5rcos2i/, + ^sin2tA)} (22).

The other terms containing higher powers of — .

In order to express V in terms of r, and (f)„ as we have assumed in the
former investigation, we must put

r' C09 xjj= — Tj + ^r^^/,

^=§{^-f'i^it^^+9) + i^fr.<f>. + ir<l>n^-9)} (23).

From which we find , ,




These results may be confirmed by the following considerations applicable to
any circular ring, and not involving any expansion or integration. Let af be
the distance of the centre of gravity from the centre of the ring, and let
the ring revolve about its centre with velocity o). Then the force necessary
to keep the ring in that orbit will be —Rafoi^.

But let >S be a mass fixed at the centre of the ring, then if

every portion of the ring will be separately retained in its orbit by the attrac-
tion of S, so that the whole ring will be retained in its orbit. The resultant
attraction must therefore pass through the centre of gravity, and be

-^ a}

therefore ^^^rL,

dr a:


cl'V cfV d'V
The equation 3^ + rf^- + dz' + *'P = «

is true for any system of matter attracting according to the law of gravitation.
If we bear in mind that the expression is identical in form with that which
measures the total efflux of fluid from a differential element of volume, where

-J- , -J- , -7- are the rates at which the fluid passes through its sides, we may

easily form the equation for any other case. Now let the position of a point
in space be determined by the co-ordinates r, ^ and z, where z is measured
perpendicularly to the plane of the angle <j>. Then by choosing the directions
of the axes x, y, z, so as to coincide with those of the radius vector r, the per-
pendicular to it in the plane of <^, and the normal, we shall have
dx = dr^ dy = rd^, dz = dz,

dV^dV dV^ldV dV^dV
dx~ dr ^ dy r d<l>' dz dz

The quantities of fluid passing through an element of area in each direction are
-T- rd(paz, -j-7 - ardz, -p rdcpdr,

so that the expression for the whole efflux is

1 dV d^V 1 d^V d^V

r dF^d^^7 df^d^ ^^^'

which is necessarily equivalent to the former expression.

Now at the centre of the ring -r^ may be found by considering the attrac-
tion on a point just above the centre at a distance z,

dV_ p z

dz {a'->tz'f'

d'V R .

-^= - 3,whenz = 0.

Ai 1 \ dV R , .

Also we know ^ = — ^ , and r = aj,

V (XV (Xi

so that m any curcular rmg "^^^^ d^^ a^ ^ **

an equation satisfied by the former values of L and N.


By referring to tlae original expression for the variable section of the ring,
it appears that the effect of the coefficient / is to make the ring thicker on
one side and thinner on the other in a uniformly graduated manner. The eflfect
of ^ is to thicken the ring at two opposite sides, and diminish its section in
the parts between. The coefficient h indicates an inequality of the same kind,
only not symmetrically disposed about the diameter through the centre of

Other terms indicating inequalities recurring three or more times in the
circumference of the ring, have no effect on the values of X, M and N. There is
one remarkable case, however, in which the irregularity consists of a single
heavy particle placed at a point on the circumference of the ring.

Let P be the mass of the particle, and Q that of the uniform ring on
which it is fixed, then R = P-{-Q,

■> K'


••• 3 = ^ = 3/- (27)-

Prob. VI. To determine the conditions of stability of the motion in terms
of the coefficients/, g, h, which indicate the distribution of mass in the ring.

The quantities which enter into the differential equation of motion (18)
are R, S, k", i\, (o", L, M, N. We must observe that S is very large compared
with R, and therefore we neglect R in those terms in which it is added to S,
and we put


Substituting these values in equation (18) and dividing by H'a*/-, we obtain
{l-P)n* + (l-y^ + y^g)nW + (^-&r-lg^-lh^ + 2fg)<.^ = (28).

The condition of stability is that this equation shall give both values of n*
negative, and this renders it necessary that all the coefficients should have the
same sign, and that the square of the second should exceed four times the
product of the first and third.

(1) Now if we suppose the ring to be uniform, /, g and h disappear,
and the equation becomes

n' + nV + | = (29),

which gives impossible values to n' and indicates the instability of a uniform

(2) If we make g and A = 0, we have the case of a ring thicker at one
side than the other, and varying in section according to the simple law of sines.
We must remember, however, that / must be less than ^, in order that the
section of the ring at the thinnest part may be real. The equation becomes

(l_/=),,* + (l.|/^)^V + (|-6/>* = (30).

The condition that the third term should be positive gives


The condition that n' should be real gives

71/^-112/^ + 32 negative,

which requires/" to be between "37445 and 1'2.

The condition of stability is therefore that /^ should lie between

•37445 and '375,

but the construction of the ring on this principle requires that /- should be
less than "25, so that it is impossible to reconcile this fonn of the ring with
the conditions of stability.

(3) Let us next take the case of a uniform ring, loaded with a heavy
particle at a point of its circumference. We have then g = Sf, h = 0, and the
equation becomes

(l-/=)n^ + (l-|/^ + f/ViV+(|-y/'+6/>^ = (31).


Dividing each term by 1 -/, we get

(l+/)n^+(l+/-f/0^^V + f{3(l+/)-8/=}a,^ = O (32).

The first condition gives /less than '8279.

The second condition gives / greater than '8 15865.

Let us assume as a particular case between these limits /= •82, which
makes the ratio of the mass of the particle to that of the ring as 82 to 18,
then the equation becomes

l-82 7i^ + '8114?iV+-9696a>' = (33),

which gives >J^^n= ±'5916(o or ±-3076w.

These values of n indicate variations of r^, O^, and ^i, which are com-
pounded of two simple periodic inequalities, the period of the one being 1"69
revolutions, and that of the other 3 '2 51 revolutions of the ring. The relations
between the phases and ampUtudes of these inequalities must be deduced from
equations (14), (15), (16), in order that the character of the motion may be
completely determined.

Equations (14), (15), (16) may be written as follows:

{Anco + hoi') ^ +2f7i%+f(3-g) (o"'(l>, = (34),

{ii^-l<o'^{S+g)}^'-2fcone,^ifh<o'<f>, = (35),

-/ho>^ '^ + 2 (1 -f^)n% + {2 (1 -f) n'-r {S-g) co^}<l>, = (36).

By eliminating one of the variables between any two of these equations,
we may determine the relation between the two remaining variables. Assuming
one of these to be a periodic function of t of the form A cos pt, and remem-
bering that n stands for the operation -7- , we may find the form of the other.

Tlius, eliminating 6^ between the first and second equations,

{n' + i7i<o'{5-g) + hoj'f-^+foy'{{3-g)<o-ym}cf>, = (37).



Assuming — =A^\wvt^ and <f)i = Q cos (ut — ^),

{-v' + ^vo)' (5 - g)} A cos pt + h(o^ A sin vt +fo/ (3 -rj) Qcos{vt - /3) + Ifhui'vQ sin {yt - /3).
Equating vt to 0, and to - , we get the equations

[v'-^voy (5 -g)} A =f<o'Q {(3 -g) cj cos /8 - ^/ii/ sin /3},

- h<o' A =fo)'Q {(3 - </) o) sin /8 + -l/ii/ cos ^8},

from which to determine Q and ^.

In all cases in which the mass is disposed symmetrically about the diameter
through the centre of gravity, A = and the equations may be greatly simplified.

Let 6i = P cos (vt — a), then the second equation becomes

{v' + ^0)' (3 + g)} A sin vt = 2Pfa}v sin {vt - a),

whence a = 0, P = ^^JtMiijO .4 (38).

2j(DV ^ '

The first equation becomes

^Aoiv cos vt - 2Pfv- cos vt + Qf (3 -g) w' cos (I'f - /S) = 0,

whence ^ = 0, <? = '^"t.f ' w^^-^ (S^)-

In the numerical example in which a heavy particle was fixed to the cir-
cumference of the ring, we have, when /= '82,


^ 1-3076

/•5916 P_r3-21 Q_f-l-229
t-3076' A~\b-72' A~\- 797'

so that if we put (ot = 0^ = the mean anomaly,

^ = .4sin(-5916(9o-a)+^sin(-3076 6'o-^) (40),

^1 = 3-21^ cos (-5916(90- a) + 5-72^ cos (-3070 ^0-/3) (41),

<^,= -l-229^cos(-5916l9o-a)-5-7975cos(-30766',-/3) ... (42).

These three equations serve to determine 1\, 6^ and <^i when the original
motion is given. They contain four arbitrary constants A, B, a, /3. Now since


the original values 1\, 0^, <^i, and also their first differential coefficients with
respect to t, are arbitrary, it would appear that six arbitrary constants ought
to enter into the equation. The reason why they do not is that we assume
r„ and 0^ as the Tiiean values of r and 6 in the actucd motion. These quantities
therefore depend on the original circumstances, and the two additional arbitrary
constants enter into the values of ^o and d^. In the analytical treatment of the
problem the differential equation in n was originally of the sixth degree with a
solution n- = 0, which implies the possibihty of terms in the solution of the
form Ct + D.

The existence of such terms depends on the previous equations, and we find
that a term of this form may enter into the value of 6, and that r^ may contain
a constant term, but that in both cases these additions will be absorbed into
the values of 0, and r,.



1. In the case of the Ring of invariable form, we took advantage of the
principle that the mutual actions of the parts of any system form at all times
a system of forces in equilibrium, and we took no account of the attraction
between one part of the ring and any other part, since no motion could result
from this kind of action. But when we regard the different parts of the ring
as capable of independent motion, we must take account of the attraction on
each portion of the ring as affected by the irregularities of the other parts, and
therefore we must begin by investigating the statical part of the problem in
order to determine the forces that act on any portion of the ring, as depending
on the instantaneous condition of the rest of the ring.

In order to bring the problem within the reach of our mathematical methods,
we limit it to the case in which the ring is nearly circular and uniform, and has
a transverse section very small compared with the radius of the ring. By
analysing the difficulties of the theory of a linear ring, we shall be better able
to appreciate those which occur in the theory of the actual rings.


The ring which we consider is therefore small in section, and very nearly
circular and uniform, and revolving with nearly uniform velocity. The variations
from circular form, uniform section, and uniform velocity must be expressed by a
proper notation.

2. To express the position of an element of a variable ring at a given time
in terms of the original position of the element in the ring.

Let S (fig. 3) be the central body, and SA a direction fixed in space.

Let SB be a radius, revolving with the mean angular velocity w of the
ring, so that ASB = (ot.

Let n be an element of the ring in its actual position, and let P be the
position it would have had if it had moved uniformly with the mean velocity w
and had not been displaced, then BSP is a constant angle =s, and the value
of 5 enables us to identify any element of the ring.

The element may be removed from its mean position P in three different

(1) By change of distance from S by a quantity l^TT = p.

(2) By change of angular position through a space Pp = a.

(3) By displacement perpendicular to the plane of the paper by a quantity C

p, a- and ^ are all functions of s and t. If we could calculate the attrac-
tions on any element as depending on the form of these functions, we miglit
determine the motion of the ring for any given original disturbance. We cannot,
however, make any calculations of this kind without knowing the form of the
functions, and therefore we must adopt the following method of separating the
original disturbance into others of simpler form, first given in Fourier's Tmitc
de Chaleur.

3. Let C/" be a function of s, it is required to express U in a series of
sines and cosines of multiples of s between the values 5 = and .s = 2t.

Assume U=A,coss + A., cos 2*- + &c. -f A ^ cos nis + A „ cos ns

+ B, sin ,s + B, cos 2.s + &c. + B,„ sin ms + B„ sin ns.


Multiply by coa Tusds and integrate, then all terms of the form
J cos ms cos nsds and / cos ms sin nsds
will vanish, if we integrate from s = to s = 27r, and there remains

I U COS msds= IT A^, Ua\-D.msds = 'TrB^.

If we can determine the values of these integrals in the given case, we
can find the proper coefficients A^, B^, &c., and the series will then represent
the values of U from s = to 5 = 27r, whether those values be continuous or
discontinuous, and when none of those values are infinite the series will be

In this way we may separate the most complex disturbances of a ring into
parts whose form is that of a circular function of s or its multiples. Each of
these partial disturbances may be investigated separately, and its efiect on the
attractions of the ring ascertained either accurately or approximately.

4. To find the magnitude and direction of the attraction between two
elements of a disturbed ring.

Let P and Q (fig. 4) be the two elements, and let their original positions
be denoted by s^ and 5j, the values of the arcs BP, BQ before displacement.
The displacement consists in the angle BSP being increased by ctj and BSQ
by 0*2 , while the distance of P from the centre is increased by p, and that of
Q by Pj. We have to determine the effect of these displacements on the distance
PQ and the angle SPQ.

Let the radius of the ring be unity, and 5j — .9i = 2^, then the original
value of PQ will be 2 sin 0, and the increase due to displacement

= (/>2 + Pi) sin ^ + (o-j - (Ti) cos 6.

We may write the complete value of PQ thus,

PQ = 2Bme{l+i{p, + p,)+^{(T,-(T,)cot0\ (1).

The original value of the angle SPQ was -^-6, and the increase due to

displacement is i{Pi — Pi) cot ^ - ^ (o-j - Ci),



30 that we may write the values of sin SPQ and cos SPQ,

Gin SFQ = cos e {I +i{p,-p,)-i {a-,- a,) ta,n0} (2),

cos SPQ = am e {I -i(p,-p,)coVd + i (a-,- a-,) cot 6} (3).

If we assume the masses of P and Q each equal to - R, where P is the

mass of the ring, and p, the number of satellites of which it is composed, the
accelerating effect of the radial force on P is

li}22^ = l - «_^{l_(p. + p,)_i(p._p,)eof^-iK-.T.)cot3}...(4),

and the tangential force

I j^sinSPQ li^COS^-. , \ / + ^ , l x mi /r:\

]1^ PQ ^^H^i^I^-^/^^-f/^^-l^'-^Olcot^ + itan^)} (5).

1 L — l
The normal force is -R ^ . , \.
p. 8 sm^ 6

5. Let us substitute for p, or and { their values expressed in a series of
sines and cosines of multiples of 5, the terms involving ms being

Pi = A cos {ms + a), pi = A cos (ms + a + 20),

o-, = -Bsin(m5 + ^), cr. = B sin {7}is-\-fi + 20),

C, = C cos (ms + y), C2 = Ccos {ms + y + 26).

The radial force now becomes

1 — ^ cos {ms + a) ( 1 + cos 2m0) + A sin {ms + a) sin 2md i

+ ^A cos {ms + a) (1 - cos 2m6) cot' ^ - ^^ sin (t/i^ + a) sin 2ni6 cot" 6 \ (6).
+^B sin {ms + ft) {1 -cos 2m^) cot ^-^5cos(??i5 + /8) siii2w^cot^.

The radial component of the attraction of a corresponding particle on the
other side of P may be found by changing the sign of 6. Adding the two
together, we have for the effect of the pair

- ^-^ — ^ {1 — ^ COS {ms + a) (2 cos" md — sin' md cot' 6)

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