2 TP WW1 – e* sine –e (1 –c)} 2,

{V1 – é sinte-e (1 – eo)}y, Ayon “

{e - V1 – é sine} z. Let these be represented by Ax, By, Cz. Let w be the angular velocity of the rotation, then w* Va* + y is the centrifugal force of the particle (wyz), and the resolved parts of it parallel to the axes of x, y, z are wʻx, wʻy, 0. Hence X, Y, Z, the forces acting on (xyz) parallel to the axes, are

X=-(A w*) x, Y=-(B w*) y, Z=- Cz. These make Xdx + Ydy + Zdz a perfect differential, and therefore so far the equilibrium is possible. The equation of fluid equilibrium gives


dp = Xdx + Ydy + Zdz р

=-(A - w*) (adx + ydy) Czdz; 22

=constant – (A – w*) (*c* + y) – Cz'.

At the surface p=0, and therefore

A - w

(xc* + y) +z=const. is the equation to the surface; and this is a spheroid, and therefore the equilibrium is possible, the form of the spheroid being properly assumed. The eccentricity is given by the condition

CA - 2012 1-e =


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1 Now observation shows that the ratio of the centri

289 fugal force at the equator to gravity at the equator. Hence




.. πρα φα



289 πρα


By expanding in powers of e and neglecting powers higher than the second, because we know that the earth is nearly spherical, we have

1 es


2 3 2.4 5


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This result is so much greater than that obtained by other methods, as we shall see, that it decides against our considering the earth’s mass to be homogeneous. Indeed it is a priori highly improbable that the mass should be homogeneous, since the pressure must increase in passing towards the centre and the matter be in consequence compressed.


P. A.

69. Another value of e, nearly =1, satisfies the equation. But this does not give the figure of any of the heavenly bodies, since none of them are very elliptical.

Since there are two values of e which satisfy the equation, it might be supposed that the equilibrium of the mass under one of these forms would be unstable, and, upon any derangement taking place, the fluid would pass to the other as a stable form. But Laplace has shown (Méc. Céles. Liv. III. § 21) that for a given primitive impulse there is but one form. In fact it is easily seen that for a given value of w, the angular velocity, the vis viva of two equal masses, so different in their form as to have e small and nearly equal unity, must be very different, and that therefore the mass cannot pass from one form to the other without a new impulse from without being given to its parts. 70. The relation between w and e in Art. 68, shows that

dw as w alters e alters, and vice versâ. By putting -0, we

de find the greatest value of w which is consistent with equilibrium. This after some long numerical calculations gives

17197 e=


and time of rotation = 0.1009 day.

71. Before proceeding to calculate the ellipticity on the hypothesis of the earth’s mass being heterogeneous we will take the following extreme case. The density increases as we pass down towards the centre. Suppose that at the centre it is infinitely greater than elsewhere: that is, suppose the whole force resides in the centre. The case of nature must lie between this hypothesis and that of the earth's being homogeneous.

Prop. To calculate the ellipticity of a mass of fluid revolving about a fixed axis and attracted by a force residing wholly in the centre of the fluid and varying inversely as the square of the distance.

72. Let M be the mass of the fluid; the other quantities as before;


,, X=-
+w*x, Y

-+w'y, Z=-

78 Then the equation Xdə + Ydy + Zdz = 0 becomes

M go3

(adx + ydy + zdz) wo (adx + ydy) = 0;

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C 1 x2 + yo Na + y + z M580 By reversing this, squaring, expanding, and neglecting the

this is seen to be the equation to a spheroid.


square of


When x=) and y=0, then =c; when z=0, x* + y = a;

1 c 1 c 1 1 580
M' M

580 a 581


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1 581

This value of € is too small (as we might have expected), 1

is too large, to agree with the form deduced in other 232 ways.


PROP. To find the equation of equilibrium of a heterogeneous mass of fluid consisting of strata each nearly spherical, and revolving about a fixed axis passing through the

centre of gravity with a uniform angular velocity.

73. Let XYZ be the sums of the resolved parts of all the forces which act upon any particle (xyz) of the fluid, parallel to the axes of co-ordinates, p the density at that point, p the pressure. Then the equation of fluid equilibrium is


= Xdx + Ydy + Zdz.

р At the surface, and also throughout any internal stratum of equal pressure and therefore of equal density, in passing from point to point dp=0. Hence

Xdx + Ydy + Zdz = 0 is the differential equation to the exterior surface and to the surfaces of all the internal strata ; the particular value assigned to the constant after integration determining to which surface the integral belongs.

The following property belongs to all these surfaces. If ds be the element of any curve drawn on the surface through (xyz), and R be the resultant of XYZ; then the equation may be written

X da Y dy, 2 dz

+ +
Rds R ds Rds


which shows that the resultant force is at right angles to any line in the surface, and therefore to the surface itself at the point (wyz).

The equilibrium will be the same if we suppose the rotatory motion not to exist, but apply to each particle a force equal to the centrifugal force caused by the rotation. The forces then acting on the fluid will be the centrifugal force and the mutual attraction of the parts of the fluid. Let V be the potential Art. 18) for this mass, then

dV dV AV
dx' dy' dz

are the attractions parallel to the three axes tending towards the origin of co-ordinates. Let w be the angular velocity of

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