Laplace's hypothetical law of density within the earth. Conse quences Multiply by r, and again differentiate; the result is d'v dr2+( K so that u, is known from (6). - —; cos (+C′')] ΚΑ K (7) (8, Now we have already proved that u increases from the centre outwards, so that we must have C'= 0, for otherwise u, would be infinite at the centre. Thus we have finally The constants are, of course, to be determined by the known values of the ellipticity of the surface and of the angular velocity of the mass. Now (5) becomes, at the surface, We dp drdr e may next eliminate p, and q by means of (3), (4), (6), and (8), and substitute everywhere re, for u. Also, if m be the ratio () of centrifugal force to gravity at the equator, is to be eliminated by means of the equation from which p is to be removed by (3). By the help of these substitutions (10) becomes transformed as follows: Let r K K 13 3 r sin Cos 3F K K K (12). be the mean density, and q, as before, the surface-density, If we put ƒ for this ratio of the mean density to the surfacedensity, a quantity which may be determined by experiment, Ratio of (13) gives From this equation may be found by approximation, and then (12) gives in terms of known quantities. In fact, it becomes. From (13) and (14) the numbers in columns iv. and v. of the following table are easily calculated. Column vii. shows the ratio of the moment of inertia about a mean diameter, on the assumed law of density, to what it would be if the earth were homogeneous: mean, to surface, density. Dynamical 825. The phenomena of Precession and Nutation result 3(C-A) sin &cos & Da (14), where S denotes the mass of the disturbing body, D its dis- origin of and Nuta will be proved in Vol. II., that Precession and Nutation are the Dynamical same as they would be if the earth were symmetrical about an Precession axis, and had for moment of inertia round equatorial diameters, tion. the arithmetical mean between the real greatest and least values. From (12) of § 539 we see that in general the differences of the moments of inertia round principal axes, or, in the case of symmetry round an axis, the value of C-A, may be determined solely from a knowledge of surface or external gravity, or [SS 794, 795] from the figure of the sea level, without any data regarding the internal distribution of density. Equating § 539 (12) to § 794 (17), in which, when the sea level is supposed symmetrical, F(0, 4) becomes simply (-cos 0), we find Similarly we may prove the same formula to hold for the real case, in which the sea level is an ellipsoid of three unequal axes, one of which coincides with the axis of rotation; provided e denotes the mean of the ellipticities of the two principal sections of this ellipsoid through the axis of rotation, and A the mean of the moments of inertia round the two principal axes in the plane of the equator. gives infor to the dis the earth's surface not. 826. The angular accelerations produced by the disturbing Precession couples are (§ 281) directly as the moments of the couples, mation as and inversely as the earth's moment of inertia round an equa- tribution of torial diameter. But (Vol. II.) the integral results, observed in mass, while Precession and Nutation, would, if the earth's condition varied, gravity does vary directly as CA, and inversely as C. We have seen (§ 794) that if the interior distribution of density were varied. in any way subject to the condition of leaving the surface [and consequently (§ 793) the exterior] gravity unchanged, CA remains unchanged. But it is not so with C, which will be the less or the greater, according as the mass is more condensed in the central parts, or more nearly homogeneous. to within a small distance of the surface: and thus it is that a comparison between dynamical theory and observation of Precession and Nutation gives us information as to the interior distribution of the earth's density (just as from the rate of gives infor mation as to the dis the earth's mass. Precession acceleration of balls or cylinders rolling down an inclined plane we can distinguish between solid brass gilt, and hollow tribution of gold, shells of equal weight and equal surface dimensions); while no such information can be had from the figure of the sea level, the surface distribution of gravity, or the disturbance of the moon's motion, without hypothesis as to primitive fluidity or present agreement of surfaces of equal density with the surfaces which would be of equal pressure were the whole deprived of rigidity. The constant of Preces sion deduced from Laplace's law. 827. But we shall first find what the magnitude of the C-A terrestrial constant of Precession and Nutation would be, C if Laplace's were the true law of density in the interior of the earth; and if the layers of equal density were level for the present angular velocity of rotation. Every moment of inertia involving the latter part of this assumption will be denoted by a Saxon capital. The moment of inertia about the polar axis is, by § 281, the first factor under the integral sign being an element of the mass, the second the square of its distance from the axis. For the moment of inertia about another principal axis (which may be any equatorial radius, but is here taken as that lying in the plane from which is measured), we have where r denotes the mean radius of the surface of equal density passing through r, 0, ; whence |