Laplace's hypotheti cal law of density within the earth.

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:

Ellipticity of strata of equal density.

824*. The table given in § 824 is principally of interest for application to the case of the earth, because it embraces those values of which correspond with values of f nearly equal to 2; and experiment has shown that the mean density of the earth is about twice that of superficial rocks. But the march of the functions and f, as we pass from the hypothesis of the homogeneity of the planet to that of infinitely small surface density, will afford an interesting illustration of the Laplacian theory, and will besides afford the means of application with some degree of probability, to some of the other planets.

* This section (§ 824′) is derived from a paper by Mr Darwin in the Monthly Notices of the R. Ast. Soc., Dec. 1876.

increases from unity to infinity, and 5m/2e from 2 to }π2.

Intermediate values of these functions, computed from the

formulæ of § 824, are given in the following table :

The numbers here given are applicable in two ways, viz. for determining the ellipticity of any internal stratum of the earth, and for application to the cases of the external figures of the other planets as above stated.

Ellipticity

of strata

of equal density.

Ellipticity of strata of equal density.

To determine the ellipticity of an internal stratum we write (12) § 824 in the following form:

The distri

bution of density in Jupiter and Saturn.

Dynamical

origin of

and Nuta

tion.

We must in (3) take as the same fraction of 0, as r, the radius of the stratum in question, is of r the earth's mean radius. Thus if for example, r = 5r, and if (as is probable in the case of the earth) f=2·1, 0=144o, we must take 60°. The table then shows that = 60° gives f=10817. By substitution in (3) we get e 120; which with =, gives e = 337.

=

In the cases of those planets which have satellites, m and - m are determinable from observation and from the theory of the satellites; so that 5m/2e is determinable. This function being known, the corresponding value of f is determinable from the table, or by direct computation. For example, MrG. H. Darwin has shown that in the case of Jupiter, where 5m/2¢ is 3·2646, we must have f = 68, 0 = 179° 11′ 20′′, and e = 1/16-022*. Different data, perhaps equally probable, give somewhat different results, but in all cases the physical conclusion is that the superficial density of the visible disk of Jupiter is very small compared with the mean density-a conclusion which appears to agree well with the telescopic appearance of that planet. A similar application to the planet Saturn points to a similar result, but the conclusion is less certain on account of the great uncertainty in the data.

825. The phenomena of Precession and Nutation result Precession from the earth's being not centrobaric (§ 534), and therefore attracting the sun and moon, and experiencing reactions from them, in lines which do not pass precisely through the earth's centre of inertia, except when they are in the plane of its equator. The attraction of either body transferred (§ 559, c) from its actual line to a parallel line through the earth's centre of inertia, gives therefore a couple which, if we first assume, for simplicity, gravity to be symmetrical round the polar axis,

*In the Méc. Cél. (v111. vii. § 23) Laplace uses values of m and t which make 5m/2c greater than 2. His determination of the Precessional Constant of the planet is thus vitiated.

tends to turn the earth round a diameter of its equator, in the Dynamical direction bringing the plane of the equator towards the dis- Precession turbing body. The moment of this couple is [§ 539 (14)] tion. equal to

origin of and Nuta

where S denotes the mass of the disturbing body, D its distance, and 8 its declination; and C and A the earth's moments of inertia round polar and equatorial diameters respectively. In all probability (§§ 796, 797) there is a sensible difference between the moments of inertia round the two principal axes in the plane (§ 795) of the equator: but it is obvious, and will be proved in Vol. II., that Precession and Nutation are the same as they would be if the earth were symmetrical about an axis, and had for moment of inertia round equatorial diameters, 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 [§§ 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 (3-cos2 ), 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 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.

Precession

gives infor

to the dis

the earth's

surface

not.

826. The angular accelerations produced by the disturbing mation as couples are (§ 281) directly as the moments of the couples, tribution of and inversely as the earth's moment of inertia round an equamass, while torial diameter. But the integral results, observed in Precession gravity does and Nutation, would, if the earth's condition varied, vary directly as C-A, and inversely as C. We have seen (§ 791) that if the interior distribution of density were varied in any way subject to the condition of leaving the superficial, and consequently (§ 793) the exterior, gravity unchanged, C - A 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 Precession acceleration of balls or cylinders rolling down an inclined mation as plane we can distinguish between solid brass gilt, and hollow tribution of gold, shells of equal weight and equal surface dimensions);

gives infor

to the dis

the earth's

mass.

The constant of

deduced from La

place's law.

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.

827. But we shall first find what the magnitude of the Precession terrestrial constant (C– A)/C of Precession and Nutation would be, 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 black-letter 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.