HOMOGENEOUS FLUID MASS. 69 than the second, because we know that the earth is nearly spherical, we have 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 à 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. 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 as w alters e alters, and vice versa. By putting dw de = 0, we find the greatest value of w which is consistent with equilibrium. This after some long numerical calculations gives 2 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 homo geneous. 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; Then the equation Xdx + Ydy + Zdz = 0 becomes By reversing this, squaring, expanding, and neglecting the 1 square of this is seen to be the equation to a spheroid. 2 580 When x=0 and y = 0, then z=b; when z=0, x2+y2=a2; This value of e is too small (as we might have expected), 1 232 as is too large, to agree with the form deduced by actual measurement by geodesy. § 2. The Earth considered to be a fluid heterogeneous mass. 73. From what has gone before it is clear that the earth's mass is not of uniform density throughout. This result indeed we might have anticipated. We shall now enter upon the more general theory of considering the mass to be heterogeneous in its density. PROP. To prove that if the mass of the earth is heterogeneous it must lie in strata nearly spherical about the earth's centre. 74. The truth of this Proposition rests upon these two facts, which are obtained from observation: (1) That the external surface is nearly spherical; (2) That the force of gravity tends nearly towards the earth's centre. Let r, 0, w be the co-ordi nates from the centre of any point of the surface, (cos = μ), and let r = a + a. u, where a is the mean radius, u a function of μ and w, and a a small constant, the square of which may be neglected because the surface is nearly spherical, r'0' w′ co-ordinates to any point in the interior of the mass, (cos e' =μ'), p' the density at this point. Then (Art. 19) the potential of the whole mass at the point on the surface is p'r'2dp'do'dr' √x2 +r'2 — 2rr'p /2 ·μ2 √1 — μ22 cos (w — w'). By expansion this becomes 1 2π 12 where P... P... are Laplace's Coefficients. Put p = R+B. U', where R' is a function of only, independent of μ and w', and U' is a function of all three r''w', B a constant. We have to prove that ẞ is a small quantity of the order of a. Also suppose ['"'r'top'dr′ = $('r') + B . & (r', μ', w') = $ (a + au') these being series of Laplace's Functions. Then remembering their property proved in Art. 27, by the property proved in Art. 32: Cis a constant. EARTH'S STRATA NEARLY SPHERICAL. 73 Now since the force of gravity acts very nearly towards the dV dv centre of the earth, the quantities (see Art. 20), and " αμ dw which depend upon the parts of gravity at right angles to r, must both be very small. Hence dui dvi dui dvi aB +B must be small: and this must be the case for all values of μ and w, that is for every spot on the earth's surface. This cannot be the case unless ẞ be small as well as α. ω, Hence the terms in p' which depend upon μ' and 'are very small. From this it follows that p' may be regarded as a function of r'+a.v' where v' is some function of r', ', w'. .. r' + a.v' = constant will be the general equation to layers of equal density. This is evidently the equation to a surface nearly spherical around the origin of r'. Hence the mass lies in strata nearly spherical about the earth's centre. 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. 75. 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 = Xd + Ydy+Zdz. 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. is the differential equation to the exterior surface and to the surfaces of all the internal strata; the particular value assigned |