gravity and fluid pressure on the displaced body is the couple of forces up and down in verticals through G and E, and the correction due to the wedges. This correction consists of a force vertically upwards through the centre of gravity of A'IA, and downwards through that of BIB'. These forces are equal [§ 704 (1)], and therefore constitute a couple which [704 (2)] has the axis of the displacement for its axis, and which [§ 704 (3)] has its moment equal to owk A if A be the area of the plane of flotation, and k its radius of gyration (§ 235) round the principal axis in question. But since GE, which was vertical (G'E') in the position of equilibrium, is inclined at the infinitely small angle to the vertical in the displaced body, the couple of forces W in the verticals through G and E has for moment Who, if h denote GE; and is in a plane perpendicular to the axis, and in the direction tending to increase the displacement, when G is above E. Hence the resultant action of gravity and fluid pressure on the displaced body is a couple whose moment is

(wAk3 – Wh)0, or w (Ak3 – Vh)0,

if V be the volume immersed. It follows that when Ak2> Vh the equilibrium is stable, so far as this displacement alone is concerned. Also, since the couple worked against in producing the displacement increases from zero in simple proportion to the angle of displacement, its mean value is half the above; and therefore the whole amount of work done is equal to

¿w (Ak3 - Vh)0".

708. If now we consider a displacement compounded of a vertical (downwards) displacement z, and rotations through infinitely small angles 0, 'round the two horizontal principal axes of the plane of flotation, we see (§§ 706, 707) that the work required to produce it is equal to

{w[Az3 + (Ak3 – Vh) 0a + (Ak'2 – Vh) 013],

and we conclude that, for complete stability with reference to all possible displacements of this kind, it is necessary and sufficient that

709. When the displacement is about any axis through the centre of inertia of the plane of flotation, the resultant of fluid pressures is equal to the weight of the body; but it is only when the axis is a principal axis of the plane of flotation that this resultant is in the plane of displacement. In such a case. the point of intersection of the resultant with the line originally vertical, and through the centre of gravity of the body, is called the Metacentre. And it is obvious, from the above investigation, that for either of these planes of displacement the condition of stable equilibrium is that the metacentre shall be above the centre of gravity.

710. We shall conclude with the consideration of one case of the

equilibrium of a revolving mass of fluid subject only to the gravitation of its parts, which admits of a very simple synthetical solution, without any restriction to approximate sphericity; and for which the following remarkable theorem was discovered by Newton and Maclaurin :—

711. An oblate ellipsoid of revolution, of any given eccentricity, is a figure of equilibrium of a mass of homogeneous incompressible fluid, rotating about an axis with determinate angular velocity, and subject to no forces but those of gravitation among its parts.

The angular velocity for a given eccentricity is independent of the bulk of the fluid, and proportional to the square root of its density.

712. The proof of this proposition is easily obtained from the results already deduced with respect to the attraction of an ellipsoid and the properties of the free surface of a fluid.

M

P

We know, $538, that if APB be a meridian section of a homogeneous oblate spheroid, AC the polar axis, CB an equatorial radius, and Pany point on the surface, the attraction of the spheroid may be resolved into two parts; one, Pp, perpendicular to the polar axis, and varying as the ordinate PM; the other, Ps, parallel to the polar axis, and varying as PN. These components are not equal when MP and PN are equal, else the resultant attraction at all points in the surface would pass through C; whereas we know that it is in some

18

B

n N

such direction as Pf, cutting the radius BC between B and C, but at a point nearer to C than n the foot of the normal at P. Let then

where a and B are known constants, depending merely on the density (p), and eccentricity (e), of the spheroid.

Also, we know by geometry that Nn = (1-e) CN.

Hence; to find the magnitude of a force Pq perpendicular to the axis of the spheroid, which, when compounded with the attraction, will bring the resultant force into the normal Pn: make pr = Pq, and we must have

Now if the spheroid were to rotate with angular velocity w about AC, the centrifugal force, $$ 39, 42, 225, would be in the direction Pq, and would amount to

the whole force on P, that is, the resultant of the attraction and centrifugal force, will be in the direction of the normal to the surface, which is the condition for the free surface of a mass of fluid in equilibrium.

This determines the angular velocity, and proves it to be proportional to √p.

713. If, after Laplace, we introduce instead of e a quantity € defined by the equation

the expression (1) for w3 is much simplified, and

w3 3+ €2

=

2πρ

€3

When e, and therefore also e, is small, this formula is most easily calculated from

of which the first term is sufficient when we deal with spheroids so little oblate as the earth.

The following table has fied formulae. The last given to the nearest unit. few sections later:

been calculated by means of these simplifigure in each of the four last columns is The two last columns will be explained a

2πρ

increases gradually from

zero to a maximum as the eccentricity e rises from zero to about o'93, and then (more quickly) falls to zero as the eccentricity rises from o'93 to unity. The values of the other quantities corresponding to this maximum are given in the table.

714. If the angular velocity exceed the value calculated from

(5)

when for p is substituted the density of the liquid, equilibrium is impossible in the form of an ellipsoid of revolution. If the angular velocity fall short of this limit there are always two ellipsoids of revolution which satisfy the conditions of equilibrium. In one of these the eccentricity is greater than o'93, in the other less.

715. It may be useful, for special applications, to indicate briefly how Ρ is measured in these formulae. In the definitions of §§ 476, 477, on which the attraction formulae are based, unit mass is defined as exerting unit force on unit mass at unit distance; and unit volumedensity is that of a body which has unit mass in unit volume. Hence, with the foot as our linear unit, we have for the earth's attraction on a particle of unit mass at its surface

where R is the radius of the earth (supposed spherical) in feet, and ☛ its mean density, expressed in terms of the unit just defined. Taking 20,900,000 feet as the value of R, we have

σ = 0'000000368 = 3.68 × 1077.

(6)

As the mean density of the earth is somewhere about 5.5 times that of water, the density of water in terms of our present unit is

716. The fourth column of the table above gives the time of rotation in seconds, corresponding to each value of the eccentricity, p being assumed equal to the mean density of the earth. For a mass of water these numbers must be multiplied by √5'5; as the time of rotation to give the same figure is inversely as the square root of the density.

For a homogeneous liquid mass, of the earth's mean density, rotating in 23h 46m 45 we find e=0'093, which corresponds to an ellipticity of about 3ਰ•

717. An interesting form of this problem, also discussed by Laplace, is that in which the moment of momentum and the mass of the fluid are given, not the angular velocity; and it is required to find what is the eccentricity of the corresponding ellipsoid of revolution, the result proving that there can be but one.

It is evident that a mass of any ordinary liquid (not a perfect fluid, § 684), if left to itself in any state of motion, must preserve unchanged its moment of momentum, § 202. But the viscosity, or internal friction, § 684, will, if the mass remain continuous, ultimately destroy all relative motion among its parts; so that it will ultimately rotate as a rigid solid. If the final form be an ellipsoid of revolution, we can easily show that there is a single definite value of its eccentricity. But, as it has not yet been discovered whether there is any other form consistent with stable equilibrium, we do not know that the mass will necessarily assume the form of this particular ellipsoid. Nor in fact do we know whether even the ellipsoid of rotation may not become an unstable form if the moment of momentum exceed some limit depending on the mass of the fluid. We shall return to this subject in Vol. II., as it affords an excellent example of that difficult and delicate question Kinetic Stability, § 300.

If we call a the equatorial semi-axis of the ellipsoid, e its eccentricity, and w its angular velocity of rotation, the given quantities are the mass M = πра®√√1-e3,

These equations, along with (3), determine the three quantities, a, e, and w.

Eliminating a between the two just written, and expressing e as before in terms of e, we have