ment by

about an

plane of

pressure on it is W upwards through E corrected by the amount Displace(upwards) due to the additional immersion of the wedge AIA', rotation and the amount (downwards) due to the extruded wedge B'IB. axis in the Hence the whole action of gravity and fluid pressure on the flotation. 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 [§ 763 (1)], and therefore constitute a couple which [§ 763 (2)] has the axis of the displacement for its axis, and which [§ 763 (3)] has its moment equal to OwkA, if A be the area of the plane of flotation, and kits radius of gyration (§ 281) round the principal axis in question. But since GE, which was vertical (as shown by 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 Whe, 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

(wAkWh) 0, or w(Ak2 – Vh) 0,

if V be the volume immersed. It follows that when Ak> Vh the equilibrium is stable, so far as this displacement alone is concerned.

in this dis

Also, since the couple worked against in producing the dis- Work done placement increases from zero in simple proportion to the placement. angle of displacement, its mean value is half the above; and therefore the whole amount of work done is equal to

placement.

767. If now we consider a displacement compounded of a General disvertical (downwards) displacement z, and rotations through infinitely small angles 0, ' round the two horizontal principal axes of the plane of flotation, we see (§§ 765, 766) that the Work rework required to produce it is equal to

źw [Az2 + (Ak2 − Vh) 02 + (Ak22 — Vh) 0’3],

quired.

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

of stability.

The meta

centre.

its exist

ence.

768. When the displacement is about any axis through the

Condition of centre of inertia of the plane of flotation, the resultant of fluid pressure 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.

A homogeneous

a figure of

769. The spheroidal analysis with which we propose to conclude this volume is proper, or practically successful, for hydrodynamic problems only when the deviations from spherical symmetry are infinitely small; or, practically, small enough to allow us to neglect the squares of ellipticities (§ 801); or, which is the same thing, to admit thoroughly the principle of the superposition of disturbing forces, and the deviations produced by them. But we shall first consider a case which admits of very simple synthetical solution, without any restriction to approximate sphericity; and for which the following remarkable theorem was discovered by Newton and Maclaurin: :

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

of a rotating

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

771. The proof of these propositions is easily obtained from

geneous

the results already deduced with respect to the attraction of an A homoellipsoid and the properties of the free surface of a fluid as ellipsoid is follows:

P

M

a figure of equilibrium of a rotating

We know, from § 522, that if APB be a meridional section liquid mass. of a homogeneous oblate spheroid, OC the polar axis, OA an equatorial radius, and P any point on the surface, the attraction of the spheroid may be resolved into two components; 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 result

ant attraction at all

points in the surface

A

n N

would pass through 0; whereas we know that it is in some
such direction as Pf, cutting the radius OA between O and A,
but at a point nearer to O than n the foot of the normal at P.
Let then
Pp=a. PM,

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

Also, we know by geometry that Nn = (1-e2) ON.

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 prPq, and we must have

A homogeneous ellipsoid is a figure of equilibrium

of a rotating liquid mass.

Now if the spheroid were to rotate with angular velocity w about OC, the centrifugal force (§§ 32, 35a, 259), would be in the direction Pq, and would amount to w2PM.

Hence, if we make w2 = a − (1 − e2) y....

.(1) ;

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.

The square This determines the angular velocity, and proves it to be pro

of a requi

site angular portional to √p.

velocity is

as the den- When e, and therefore also f, is small, this formula is most easily calculated from

sity of the liquid.

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

772. The following table has been calculated by means of these simplified formula. The last figure in each of the four last columns is given to the nearest unit. The two last columns will be explained in §§ 775, 776.

From this we see that the value of w2/2πp increases gradually from zero to a maximum as the eccentricity e rises from zero to

* Remark that the "e" of § 527 is not the eccentricity of the oblate spheroid which we now denote by e, and that with ƒ as there and e as here we have 1-e2=1/(1+ƒ2).

about 093, and then (more quickly) falls to zero as the eccen- Table of cor

responding values of ellipticities and angular velocities.

tricity rises from 0.93 to unity. The values of the other quantities corresponding to this maximum are given in the table.

773. If the angular velocity exceed the value calculated from

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 0.93, in the other less.

* Calculated from the mass and density, by the formula