to the constant after integration determining to which surface the integral belongs. The following property belongs to all these surfaces. If ds be the element of any curve drawn on the surface through (xyz), and R be the resultant of XYZ; then the equation may be written X dx Ydy Zdz + The first side of this is the cosine of the angle between the resultant and the line ds, and as it equals zero it shows that the resultant force is at right angles to any line in the surface, and therefore to the surface itself at the point (xyz). The equilibrium will be the same if we suppose the rotatory motion not to exist, but apply to each particle a force equal to the centrifugal force caused by the rotation. The forces then acting on the fluid will be the centrifugal force and the mutual attraction of the parts of the fluid. Let V be the potential (Art. 18) for this mass, then are the attractions parallel to the three axes tending towards the origin of co-ordinates. Let w be the angular velocity of rotation about the axis of z, taken as the fixed axis; w2.x and w2.y will be the centrifugal force at the point (xyz). Then dV dp - (dr+w2x) dx + (2 + wy) dy + d = dy is the equation to the surface and the strata. dz dz, Let be the distance of the point (xyz) from the origin, and the angle r makes with the axis of z, and cos 0=μ: then x+y=r2 sin2 0 = (1 — μ2) r2. Also let m be the ratio of the centrifugal force at the equator to gravity at the EQUATION OF EQUILIBRIUM. 75 equator (or 1); let a' be the mean radius of the stratum through (xyz); a the radius of the equator; then and M = 4π [^ p'a”da' = 1}π4(a) suppose, 3 the strata being considered spherical because of the smallness of the numerator in the value of m; this arrangement being made, because the second and third terms as they now stand, are Laplace's Functions of the order 0 and 2. (See Art. 39, Ex. 1.) a fo r=a (1+ Y2+ ... Y;+...) and p'a'da′ = }$ (a), 1 as before. Then substitute this value of V in the equation to the strata and equate terms of the order i. (See Art. 35.) except when i=2, in which case the second side is m a2 (a) 6 a3 ...... (2) By these equations Y is to be calculated, and then the form of the stratum of which the mean radius is a is known by the formula r = a(1+Y12+Y2+ ... + Y¿ + ...). PROP. To prove that Y=0, excepting the case of i= 2. 76. Since Y, and p are functions of a, they may be expanded into ascending series of the form Y;= WaR + ..., p = D+ D'a" + ..., p= where D is the density at the centre of the earth, and is as well as W and D' independent of a s, n ... must not be negative, otherwise Y, and p would be infinite at the centre. Now when these and the corresponding series obtained by putting a' for a, are substituted in the equation of the strata in the last Article, and the first side arranged in powers of a, the various coefficients ought to vanish; excepting when i=2, because then the second side is not zero. We shall therefore substitute these series, and search for values of W and s which satisfy the condition. SIMPLIFICATION OF THE RADIUS. 77 After two easy integrations the equation of the strata becomes No value of s will cause these terms to vanish. The only apparent case is when i=1, for then by putting s=i-2 the part in the brackets vanishes: but in this particular case s=1, and is negative and therefore inadmissible. Hence the only way of satisfying the condition is by putting W=0; this shows that Y, has no first term, that is, that it has no term at all and is therefore zero. PROP. To prove that the strata are all spheroidal, concentric, and have a common axis. 77. By the last two Articles it appears that the equation. to the surface r=a (1+ Y2), and the equation for calculating Y, is Suppose Y (and similarly Y) is expanded in a series of powers of, with indeterminate coefficients, to be ascertained by the condition that they shall satisfy the above equation. These coefficients will be functions of a only, as it is seen from the right-hand side of the equation that @ does not enter into the value of Y,. It is clear that Y, consists of only one term, that involving the simple power of-u. Let it be ε (3-μ3), e being a small quantity of the order of m. 2 Hence r = a {1+ε (} − μ2)}, μ = sin (latitude) = sin l -- = a (1 − } e) (1 + € cos37), since € is small. This is the equation to a spheroid from the centre, e being the ellipticity. The axis-minor coincides with the axis of revolution of the whole mass. Hence the strata are concen tric spheroids, the minor-axes of which coincide with the axis of revolution of the whole mass. 78. The fluid theory therefore teaches us (1) that gravity is everywhere perpendicular to the surface (Art. 75); (2) that the exterior surface and the surfaces of the strata are all concentric spheroids, with the axis of each coincident with the axis of the earth (Art. 77). These spheroids are of a definite form, depending upon the velocity of rotation and the law of density, as we shall see in Art. 80, and when we come to calculate their ellipticity. These spheroids so determined are called "spheroids of equilibrium," because they are the forms which the mass will assume when in equilibrium if it be fluid throughout. This term is used whether the mass has subsequently become solid or not, and refers solely to the form, not the condition. And in general when we say a body has a "surface of equilibrium,” we mean that the surface though solid would retain its form if it became fluid, all other things remaining the same. 79. Since the strata are all concentric spheroids with a common axis in the axis of rotation, it follows that the centre of the earth's mass coincides with the centre of the volume, and that the axis of rotation is one of the principal axes of the mass. For this is true of each of the spheroidal strata separately, and is therefore true also of the aggregate or the whole mass. PROP. To obtain an equation for calculating the ellipticity of the strata. in 80. Substitute e (2) for Y2 and e' (3 — μ3) for Y' equation (3) of Art. 75, and we have, after dividing by Divide both sides by a, and differentiate with respect to a; then multiply by ao, and differentiate again, and divide by the d'e coefficient of i da2; |