Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/623

have alluded to above may be obtained. The principles of hydrostatics show us that if X, Y, Z be the components parallel to three rectangular axes of the forces acting on a particle of a fluid mass at the point x, y, z, then, p being the pressure there, and p the density, dp =p(Xdx + Y<fy + Zdz) ; and for equilibrium the necessary conditions are, that p(X.dx + Ydy + Zdx) be a complete differential, and at the free surface Kdx + Ydy + Zdz = 0. This equation implies that the resultant of the forces is normal to the surface at every point, and in a homogeneous fluid it is obviously the differential equation of all surfaces of equal pressure. If the fluid be heterogeneous then it is to be remarked that for forces of attraction according to the ordinary law of gravitation, if X, Y, Z be the components of the attraction of a mass whose potential is V, then dV, dV, dV, Xdx + Ycty + Zdz - -r-dx + ~dy + -r-dz , ax ay dz which is a complete differential. And in the case of a fluid rotating with uniform velocity, in which the so-called centrifugal force enters as a force acting on each particle proportional to its distance from the axis of rotation, the corresponding part of Xdx + ~Ydy + Zdz ia obviously a complete differential. Therefore for the forces with which we are now concerned ~Kdx + Ydy + Zdz = cHJ, where U is some function of x, y, z, and it is necessary for equilibrium that dp = pdV be a complete differential ; that is, p must be a function of U or a function of p, and so also p a function of U. So that cZU = is the differential equation of surfaces of equal pressure and density. We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium. It may be proved that the attraction of the ellipsoid o; 2 + # 2 + 2 2 (l + e 2 ) = c 2 (l + e 2 ) upon a particle P of its mass at x, y, z has for components X- Ax, Y = Ay, Z = Cz, where A = tan - tan Besides the attraction of the mass of the ellipsoid, the centrifugal force at P has for components -#uj 2 , -yw 2 , 0; then the condition of fluid equilibrium is (A - u"-)xdx + (A- u^ydy + Czdz = , which by integrating gives (A-ar)(* 2 + 2/ 2 ) + Cs 2 = constant. This is the equation of an ellipsoid of rotation, and there fore the equilibrium is possible. The equation coincides with that of the surface of the fluid mass if we make which <nves 3 + 2-rrp tan" If we would determine the maximum value of w from this equation, we find that it corresponds to the value of e determined by the condition tan 1 , = (l hence it may be shown that if the angular velocity exceed 2 that calculated from ^r = > 2247, equilibrium is impos sible for the form of an ellipsoid of revolution. If w fall short of this limit, there are two ellipsoids which satisfy the condition of equilibrium; in one of these the eccentricity is 601 greater and in the other less than 93. In the case of the earth, which is nearly spherical, we get by expanding the expression for w 2 in powers of e 2 , rejecting the higher powers, and remarking that the ellipticity e = Je 2 , 2 _ 4 2 _ 8 2^~15 ~15 e> Now, if m be the ratio of the centrifugal force at the equator to gravity there, ~ ^TrpC-Cu* 2irp 3 I + TO In the case of the earth it is a matter of observation that m = -%!?, hence the ellipticity 5 1 so that the ratio of the axes on the supposition of a homogeneous fluid earth is 230: 231, as announced by Newton. Now, to come to the case of a heterogeneous fluid, we shall assume that its surfaces of equal density are spheroids, concentric and having a common axis of rotation, and that the ellipticity of these surfaces varies from the centre to the outer surface, the denmty also varying. In other words, the body is composed oT homogeneous spheroidal shells of variable density and ellipticity. On this supposition we shall express the attraction of the mass upon a particle in its interior, and then, taking into account the centrifugal force, form the equation expressing the condition of fluid equilibrium. The attraction of the homogeneous spheroid x 2 + y z + 2 2 (1 + 2e) = c 2 (l + 2e), where e is the ellipticity, of which the square is neglected, on an internal particle, whose co-ordinates are #=/, y = Q, z = h, has for its x and z components the. Y component being of course zero. Hence we infer that the attraction of a shell whose inner surface has an ellipticity e, and its outer surface an ellipticity e + de, the density being p, is expressed by 4 2 44 dX = - - - irpfda , dZ = - vphde . To apply this to our heterogeneous spheroid ; if we put c t for the semiaxis of that surface of equal density on which is situated the attracted point P, and c for the semiaxis of the outer surface, the attraction of that portion of the body which is exterior to P, namely, of all the shells which inclose P, has for components r - lo dc , 16 15 de , < r dc . dc both e and p being functions of c. Again the attraction of a homogeneous spheroid of density p on an external point /, h has the components . where X = ^ Now e being considered a function of c, we can at once express the attraction of a shell (density ( o) contained between the surface defined by c + dc, e + de and that defined by c, e upon an external point ; the differentials with respect to c, viz. dX." dZ", must then be integrated with p under the integral sign as being a function of c. The integration will extend from c=-0 to c = c r Thus the components of the attraction of the heterogeneous spheroid

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