When this is calculated for the surface, we shall be able to find the ellipticity of any stratum we like by the ratio of ellipticities found above. 106. To reduce this to numbers put, as before, qa=24605, = 140° 58′ 35′′; then tan qa=-0.8105, z=4.0359, q'a2 = 6·0541, q'a2÷z=1·5. Hence the above formula for e gives This is remarkably near the result obtained by geodesy. 107. COR. 1. We will find the numerical value of the ellipticity of some of the strata, according to this law of density, with a view of showing at what rate the ellipticity diminishes in descending towards the centre. We will suppose LAW OF ELLIPTICITY OF STRATA. 105 the mass divided into four shells and a central nucleus, the radius of the nucleus and the thickness of each shell being equal to one-fifth of the earth's radius. Taking the first value of qa found above, then for the outer surface and the four other surfaces we have qa=140° 58′ 35′′, 112°46′52′′, 84°35′ 9′′, 56°23′26′′, 28°11′43′′ =2·4605, 1·9684, 1.4763, 0.9842, 0·4921 in arc; qa tan qa 3 =-3.035906, -0.826756, 0·139920, 0.654135, 0.917944, z = 4.035906, 1·826756, 0·860080, 0.345865, 0·082056, 1 -= q*a* 1 3 2 q3a2 0.2478, 0.5474, 1.1627, 2.8913, 12.1868, = 0.4955, 0.7743, 1.3765, 3.0971, 12.3884. =-0.2477, -0.2269, -0.2138, -0.2058, -0.2016. 108. COR. 2. In a future Article we shall require to know the masses of the portions of the earth which lie within these surfaces to a first approximation, that is, neglecting their ellipticity. We will therefore calculate them here. Let E be the whole mass of the earth, and M the mass of a portion of it of which a is the mean radius. Then, if A sin qa÷a is the density, α qa α ピ 1 M = ["Awa3. A in ga da = Aπ4 (− = cos ga + sin ga); Απα. Α α Hence the ratios of the masses of the four shells and of the nucleus to the whole mass are 0.3372, 0.3257, 0.2237, 0.0981, "0.0153. And the ratios of the mean densities of these to the mean density of the whole earth equal these multiplied by the fractions Fourth Test. THE PRECESSION OF THE EQUINOXES. 109. The last test we shall apply is the amount of Precession to which the fluid theory leads. PROP. To find the amount of Precession of the Equinoxes in the Earth, supposing its mass arranged according to the fluid law. = = = I obliquity of the ecliptic 23° 28′ 18", i inclination of Moon's orbit to ecliptic=5°8′ 50′′, n and n' are the mean motions of the Earth round its axis and round the Sun, and their ratio = 365.26, n" the mean motion of the Moon round the Earth = 27.32 days, vratio of masses of Earth and Moon = 75. (See Mechanical Philosophy, Second Edition, Art. 470 also, changing the notation, Airy's Tracts, Fourth Edition, p. 213, Arts. 36, 38.) Substituting the above quantities, ELLIPTICITY FOUND FROM PRECESSION. 107 where A and C are the principal moments of inertia of the mass, the latter about the axis of revolution. To find these let xyz be the co-ordinates to any element of the mass, row be the polar co-ordinates to the same. Then the mass of this element pr'dudwdr, μ= cos 0. Also The terms are here arranged as Laplace's Functions. (See Art. 39, Ex. 4.) Now r-radius of any stratum = a 1 + € And putting p= 2 sin qa, and integrating by parts, α σ (2) = ["pa'da = Q ["aʼsin qada 3a2 6a 6 = 9 (− = cos ga + sin ga + cos ga— sin ga) = Qa2 qa Hence substituting z, as before, tan qa .. Annual Precession = 32162" (€ But the Precession = 50" 1 by observation; This is a satisfactory result, although it does not accord with the geodetic measure so closely as the direct calculation of the ellipticity in Art. 106 does. They both, however, bear testimony to the truth of the fluid theory and also in some measure of the law of density we have assumed. |