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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.
We know, $ 538, that if APB be a meridian section of a homogeneous oblate spheroid, AC the polar axis, CB an equatorial radius, and P any point on the surface, the attraction of the spheroid may be resolved into two parts;
A one, Pp, perpendicular to the polar axis, and vary
9 ing as the ordinate PM; the other, Ps, parallel to the polar axis, and varying as PN. These components are not equal
N when MP and PN are equal, else the resultant attraction at all points in
f the surface would pass through C; whereas we know that it is in some 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
Pp=a.PM, and Ps=ß.PN,
where a and ß are known constants, depending merely on the density (), and eccentricity (e), of the spheroid.
Also, we know by geometry that Nn=(1-e) CN.
Hence; to find the magnitude of a force Pa perpendicular to the axis of the spheroid, which, when compounded with the attraction, will bring the resultant force into the normal Pn: make pr=P9, and we must have Pr Nn
=(a-(1-3) PM. Now if the spheroid were to rotate with angular velocity w about AC, the centrifugal force, f$ 39, 42, 225, would be in the direction Pa; and would amount to
w2PM. Hence, if we make
w=a-(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 Auid in equilibrium. Now, ($ 522 of our larger work)
(1) This determines the angular velocity, and proves it to be proportional to np.
713. If, after Laplace, we introduce instead of e a quantity € defined by the equation
(2) =tan (sin-le),
62 When e, and therefore also e, is small, this formula is most easily calculated from
* + etc.
(4) of which the first term is sufficient when we deal with spheroids so little oblate as the earth.
The following table has been calculated by means of these simplified formulae. The last figure in each of the four last columns is given to the nearest unit. The two last columns will be explained a few sections later :
19,780 15,730 13,022 11,096 9,697 8,804
3287 '2917 2506 *2030
94 '95 '96 97
0'0027 'OIIO '0258 '0490 '0836 1356 2172 3588 6665 '7198 *7813 .8533 '9393 I'045 I'179 I'359 1:627 2'113
8,729 8,718 8,732 8,783 8,891 9,098 9,504 10,490
w? From this we see that the value of increases gradually from
2πρ zero to a maximum as the eccentricity e rises from zero to about 0.93, and then (more quickly) falls to zero as the eccentricity rises from 0.93 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) 2πρ 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.
715. It may be useful, for special applications, to indicate briefly how p 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 o its mean density, expressed in terms of the unit just defined. Taking 20,900,000 feet as the value of R, we have o=0.000000368=3.68 x 10-7.
(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
10-7= 6.7 x 10-8.
5.5 716. 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 V55; 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 4s we find er =0.093, which corresponds to an ellipticity of about to
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=pa , and the moment of momentum
A=1 Tpwa 11 -6%. These equations, along with (2), determine the three quantities, a, e,
Eliminating a between the two just written, and expressing e as before in terms of e, we have
A2 * w*(1 +€2)$
k This gives
2TP (1+e?) where k is a given multiple of pš. Substituting in 771 (2) we have
€3 Now the last column of the table in § 713 shows that the value of this function of € (which vanishes with c) continually increases with e, and becomes infinite when e is infinite. Hence there is always one, and only one, value of e, and therefore of e, which satisfies the conditions of the problem.
718. All the above results might without much difficulty have been obtained analytically, by the discussion of the equations; but we have preferred, for once, to show by an actual case that numerical calculation may sometimes be of very great use.
719. No one seems yet to have attempted to solve the general problem of finding all the forms of equilibrium which a mass of homogeneous incompressible fluid rotating with uniform angular velocity may assume. Unless the velocity be so small that the figure differs but little from a sphere, the problem presents difficulties of an exceedingly formidable nature. It is therefore of some importance to know that we can by a synthetical process show that another form, besides that of the ellipsoid of revolution, may be compatible with equilibrium ; viz. an ellipsoid with three unequal axes, of which the least is the axis of rotation. This curious theorem was discovered by Jacobi in 1834, and seems, simple as it is, to have been enunciated by him as a challenge to the French mathematicians?. For the proof we must refer to our larger work.
See a Paper by Liouville, Journal de l'Ecole Polytechnique, cahier xxiii., footnote to p. 290.