of rotating spheroid geneous The liquid. a linear differential equation, of the second order, for u, with of heterocoefficients and independent term, known functions of r. general solution, as is known, is of the form where a is a function of r satisfying the equation (18), Differential equation to be integrated. (19), C and C' are two arbitrary constants, and P and P' two distinct functions of r. Equation (15) requires that C=0 and C'=0; in other words, u, if satisfying it, is fully determinate. This is best shown by remarking that if, instead of (15), we have tion of con where K, K' are any two constants, these constants disappear in the differentiations, and we have still the same differential equation, (17); and that the two arbitrary constants C and C' of the Determinageneral solution (18) of this are determined by (20) when any stants to two values are given for K and K'. In fact, the expression (18), the required used for u, reduces (20) to Co(P)+Co(P')=Kr'+K'r-i-1 (21), which shows that σ¿(P) and σ¡(P') must be distinct linear functions of r and r-i-1, and determines C and C'. Thus we see that whatever be A; we have, in the integration of the differential equation (19), and the determination of the arbitrary constants (14), the complete solution of our problem. complete solution. tion of the law of force Unless it is desired, as a matter of analytical curiosity, or for Introducsome better reason, to admit the supposition that N is any arbi- Newtonian trary function of r, it is unnecessary to retain both and pas two distinct given functions. For the external force of the nucleus, or that part of it of which N is the potential, being by hypothesis symmetrical, relatively to the centre, it must in nature vary inversely as the square of the distance from this point; that Equilibrium of rotating spheroid of heterogeneous liquid. Simplifica Using this last in (17), and reducing by differentiation, we have dlogy, 2 dur 2d{r=iAi)} (20 Another form, convenient for cases in which the disturbing force is due to external attracting matter, or to centrifugal force of the fluid itself, if rotating, is got by putting, in (17), r―i+1ui=li and reducing by differentiation. Thus (27), +20 + 1 d + ei = r dr dr dr ate deviation from sphericity. Layers of greatest and least proportionate deviation from sphericity. dr2 dlogy, i+1 dei, 2(i-1) dlog dr With this notation the intermediate integral, obtained from (15) by the first step of the process of differentiating executed in the order specified, gives Important conclusions, readily drawn from these forms, are that if Q is a solid harmonic function (as it is when the disturbance is due either to disturbing bodies in the core, or in the space external to the fluid, or to centrifugal force of the fluid rotating as a solid about an axis); then (1.) e, regarded as positive, and as a function of r, can have no maximum value, although it might have a minimum; and (2.) if the disturbance is due to disturbing masses outside, or to any other cause (as centrifugal force) which gives for potential a solid harmonic of order i with only the r term, and no term ri-1, e; can have no minimum except at the centre, and must increase outwards throughout the fluid. To prove these, we must first remark that necessarily diminishes outwards. To prove this, let n denote the excess of the mass of the nucleus above that of an equal solid sphere of density s equal to that of the fluid next the nucleus. we may put (24) under the form For stability it is necessary that n and s-p' be each positive; and therefore the last term of the second member is positive, and diminishes as r increases, while the second term of the same is negative, and in absolute magnitude increases, and the first term and therefore the second member of (28) vanishes. (31), geneous Hence if, liquid. and is therefore positive, which proves (1.) Lastly, when the force is such as specified in (2.), we have A¡ = Kr simply, and therefore the second member of (29) vanishes. This equation then gives, for values of r exceeding a by infinitely little, Proportionate deviation for case of centrifugal force from without. which is positive. Hence e; commences from the nucleus in- force, or of creasing. But it cannot have a minimum (1.), and therefore it increases throughout, outwards. 823. When the disturbance is that due to rotation of the Case of liquid, the potential of the disturbing force is 2 (x2+y3), which is equal to a solid harmonic of the second degree with a constant added. From this it follows [SS 822, 779] that the surfaces of equal density are concentric oblate ellipsoids of revolution, with a common axis, and with ellipticities diminishing from the surface inwards. centrifugal force. neglecting terms of the second order because w, and therefore Equilibrium Thus the sphere, whose radius was r, has become an oblate ellipsoid of revolution whose ellipticity [§ 822 (27)] is Case of centrifugal force. Laplace's hypothetical and its equatorial diameter is increased by; the volume re maining unaltered. In order to find the value of us, we must have data or assumptions which will enable us to integrate equation (15). These may be given in many forms; but one alone, to which we proceed, has been worked out to practical conclusions. 824. To apply the results of the preceding investigation to law of den- the determination of the law of ellipticity of the layers of sity within the earth. equal density within the earth, on the hypothesis of its original fluidity, it is absolutely essential that we commence with some assumption (in default of information) as to the law which connects the density with the distance from the earth's centre. For we have seen (§ 821) how widely different are the results obtained when we take two extreme suppositions, viz., that the mass is homogeneous; and that the density is infinitely great at the centre. In few measurements hitherto made of the Compressibility of Liquids (see Vol. 1, Properties of Matter) has the pressure applied been great enough to produce condensation to the extent of one-half per cent. The small condensations thus experimented on have been found, as might be expected, to be very approximately in simple proportion to the pressures in each case; but experiment has not hitherto given any indication of the law of compressibility for any liquid under pressures sufficient to produce considerable condensations. In default of knowledge, Laplace assumed as a hypothesis the law of compressibility of the matter of which, before its solidification, the earth consisted, to be that the increase of the square of the density is proportional tween den to the increase of pressure. This leads, by the ordinary equation of hydrostatic equilibrium, to a very simple expression for the law of density, which is still further simplified if we assume that the density is everywhere finite. Assumed relation be sity and pressure. Neglecting the disturbing forces, we have (§§ 822, 752) dp=pd(V+N) Laplace's hypothetical (1). law of den sity within But, by the hypothesis of Laplace, as above stated, k being some the earth. μ (2). or, by § 822 (5), = 4 = [', r'p'dr' + =#= [ ' r'2 p'dr' + ". Multiplying by r, and differentiating, we get r a If we suppose the whole mass to be liquid, i.e., if there be no solid core, or, at all events, the same law of density to hold from surface to centre, G must vanish, else the density at the centre would be infinite. Hence, in what follows, we shall take We are now prepared to find the value of u, in § 823, which depends the ellipticity of the strata. For (15) of becomes, by (23) of that section and the late equation (4), (4), Determina ticities of upon tion of ellip822 surfaces (5) where u' is the mass of fluid, following the density law (3), which of equal density. |