developed in harmonics.

Using this in (6) according to §§ 544, 542, 536, we have

Harmonic development of disturbing potential

where R denotes the value of R, from the point (r', 0, ø) instead of (r, 0, $).

To complete the expansion of the hydrostatic equation (4) we may suppose the harmonic expression for Q to be either directly given, or be found immediately by Appendix B. (52), or by (8) of § 539, according to the form in which the data are pre-· sented. Thus let us have

according to the notation of App. B. (37) and (38), A“, B denoting known functions of r. Using now this and (8) in (4), we have

which merely shows the value of F(r), introduced temporarily in

(4) and not wanted again: and, by terms of order i,

Equation of equilibrium for general harmonic term:

Lastly, expanding R, (as above for the i term of Q) by App. B. (37), let us have

where u, v are functions of r, to the determination of which the problem is reduced. Hence equating separately the coefficients of cos sp, etc., on the two sides, and using u, to denote any

one of the required functions u, v, and A, any of the given leading to functions A, B, and u, u, the values of u, for r=r′ and r ==

equation for general coefficient

u., regarded as a function of r.

or, as it will be convenient sometimes to write it, for brevity, o.(u) = A..

where σ, denotes a determinate operation, performed on u any
function of r, continuous or discontinuous. To reduce (15) to a
differential equation, divide by ', differentiate, multiply by p2+2,
and differentiate again. If, for brevity, we put

.(15),

Equili brium of rotating spheroid of heterogeneous

liquid.

a linear differential equation, of the second order, for u,, with Differential

coefficients and independent terms known functions of r.
general solution, as is known, is of the form

* =CP+CP +a

where a is a function of r satisfying the integral equation

equation to

The be inte

....... (18),

(19) [(15) repeated];

C and C' are two arbitrary constants, and P and P' are two
distinct functions of r.

1=

Equation (15) requires that C = 0 and C' = 0; in other words, u, if satisfying it, is fully determinate. This is best shown by remarking that if, instead of (15), we take

grated.

where K, K' are any two constants, these constants disappear in
the differentiations, and we have still the same differential
equation, (17): and that the two arbitrary constants C and C'' Determina-
of the general solution (18) of this are determined by (20) when stants to
any two values are given for K and K'. In fact, the expression the required
(18), used for u,, reduces (20) to

Cơ ̧ (P) + C'v ̧ (P') = Kr' + K'r ̄`~1 ...... .(21),
which shows that σ(P) and σ(P') cannot either of them be

tion of con

complete

solution.

Determination of constants to complete the required solution.

Introduc

tion of the
Newtonian
law of force.

zero, and that they must be distinct linear functions of r1 and ri-, and determines C and C'.

Thus we see that whatever be A, we have, in the integration of the differential equation (19), and the determination of the arbitrary constants to satisfy (15), the complete solution of our problem.

Unless it is desired, as a matter of analytical curiosity, or for some better reason, to admit the supposition that N is any arbitrary function of r, it is unnecessary to retain both and p as two distinct given functions. For the external force of the nucleus, or that part of it of which N is the potential, being by hypothesis symmetrical relatively to the centre, it must in nature vary inversely as the square of the distance from this point; that is to say, dN μ

μ being a constant,

mass of the nucleus.

=

dr

.(22),

d (4r3)

dp

which gives 4πρ =

.(23).

(24),

r2dr

du

and 4π =r +4 ......(25). dr dr2 dr

Using this last in (17), and reducing by differentiation, we have

+2

Another form, convenient for cases in which the disturbing force is due to external attracting matter, or to centrifugal force of the fluid itself, if rotating, is got by putting, in (17),

With this notation the intermediate integral, obtained from (15)

by the first step of the process of differentiating executed in the Differential order specified, gives

equation for proportionate deviation from

sphericity.

brium of

geneous

greatest and

portionate

Important conclusions, readily drawn from these forms, are Equili that if is a solid harmonic function (as it is when the rotating spheroid disturbance is due either to disturbing bodies in the core, or in of heterothe space external to the fluid, or to centrifugal force of the liquid. fluid rotating as a solid about an axis); then (1) e,, regarded as Layers of positive, and as a function of r, can have no maximum value, least proalthough it might have a minimum; and (2) if the disturbance deviation from spheriis due to disturbing masses outside, or to any other cause (as city. centrifugal force) which gives for potential a solid harmonic of order i with only the term, and no term r1, e, can have no minimum except at the centre, and must increase outwards throughout the fluid.

To prove these conclusions, we must first remark that necessarily diminishes outwards. To prove this, let n denote the excess of the mass of the nucleus above that of an equal solid sphere of density s equal to that of the fluid next the nucleus. Then we may put (24) under the form

For stability it is necessary that n and s-p' be each positive; and therefore the last term of the second member is positive, and diminishes as r increases, while the second term of the same is negative, and in absolute magnitude increases, and the first term is constant. Hence diminishes as r increases. Again, when the force is of the kind specified, we must [App. B. (58)]

and therefore the second member of (28) vanishes. Hence if, for

any value of r,

de/dr = 0,

force is such as specified in (2), we have 4, Kr simply, and

=

VOL. II.

26

Proportionate devia

tion for case of centrifugal force, or of force

from without.

therefore the second member of (29) vanishes. This equation then gives, for values of r exceeding a by infinitely little,

which is positive.

Hence e, commences increasing from the nucleus. But it cannot have a minimum (1), and therefore it increases throughout, outwards.

823. When the disturbance is that due to rotation of the liquid, the potential of the disturbing force is w2 (x2 + y2), which is equal to a solid harmonic of the second degree with a constant added. From this it follows [SS 822, 779] that the surfaces of equal density are concentric oblate ellipsoids of revolution, with a common axis, and with ellipticities diminishing from the surface inwards.

neglecting terms of the second order because w, and therefore

also ur, are very small.

Thus the sphere, whose radius was r, has become an oblate ellipsoid of revolution whose ellipticity [§ 822 (27)] is

Its polar diameter is diminished by the fraction ur ore, and its equatorial diameter is increased by e,; the volume remaining unaltered.

In order to find the value of u, we must have data or assumptions which will enable us to integrate equation (15). These may be given in many forms; but one alone, to which we proceed, has been worked out to practical conclusions.