Page:EB1911 - Volume 08.djvu/832

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both e\! and \rho\! being functions of c\!. Again the attraction of a homogeneous spheroid of density \rho\! on an external point f, h\! has the components

\mathrm{X}'' {{=}} -\frac{4}{3}\pi k^2\rho fr^{-3} {c^3(1 + 2e) - \lambda ec^{5}},\!

\mathrm{Z}'' {{=}} -\frac{4}{3}\pi k^2\rho hr^{-3} {c^3(1 + 2e) - \lambda'ec^{5}},\!

\text{where }\lambda {{=}} \frac{3}{5}(4h^2 - f^2) / r^{4}, \qquad \lambda' {{=}} \frac{3}{5}(2h^2 - 3f^2) / r^{4}, \qquad\text{ and }r^2 {{=}} f^2 + h^2.\!

Now e\! being considered a function of c\!, we can at once express the attraction of a shell (density \rho\!) contained between the surface defined by c + dc, e + de\! and that defined by c, e\! upon an external point; the differentials with respect to c\!, viz. d\mathrm{X}'' d\mathrm{Z}''\!, must then be integrated with \rho\! under the integral sign as being a function of c\!. The integration will extend from c {{=}} 0\! to c {{=}} c_{1}\!. Thus the components of the attraction of the heterogeneous spheroid upon a particle within its mass, whose co-ordinates are f, 0, h\!, are

\mathrm{X} {{=}} -\frac{4}{3}\pi k^2f \left[\frac{1}{r^{3}}\int^{c1}_{0} \rho d{c^3(1 + 2e)} - \frac{\lambda}{r^{3}}\int^{c1}_{0} \rho d(ec^{5}) + \frac{2}{5} \int^{c0}_{c1} \rho de\right],

\mathrm{Z} {{=}} -\frac{4}{3}\pi k^2h \left[\frac{1}{r^{3}}\int^{c1}_{0} \rho d{c^3(1 + 2e)} - \frac{\lambda'}{r^{3}}\int^{c1}_{0} \rho d(ec^{5}) + \frac{4}{5} \int^{c0}_{c1} \rho de\right].

We take into account the rotation of the earth by adding the centrifugal force f\omega^2 {{=}} \mathrm{F}\! to \mathrm{X}\!. Now, the surface of constant density upon which the point f, 0, h\! is situated gives (1 - 2e) fdf + hdh {{=}} 0\!; and the condition of equilibrium is that (\mathrm{X} + \mathrm{F})df + Zdh {{=}} 0\!. Therefore,

(\mathrm{X} + \mathrm{F}) h {{=}} Zf (1 - 2e),\!

which, neglecting small quantities of the order e^2\! and putting \omega^2t^2 {{=}} 4\pi^2k^2\!, gives

\frac{2e}{r^3}\int^{c1}_{0} \rho d{c^3(1 + 2e)} - \frac{6}{5r^{5}}\int^{c1}_{0} \rho d(ec^{5}) - \frac{6}{5}\int^{c1}_{0} \rho de {{=}} \frac{3\pi}{t^2}.

Here we must now put c\! for c_{1}, c\! for r\!; and 1 + 2e\! under the first integral sign may be replaced by unity, since small quantities of the second order are neglected. Two differentiations lead us to the following very important differential equation (Clairault):

\frac{d^2e}{dc^2} + \frac{2\rho c^2}{\int \rho c^2 dc}\cdot\frac{de}{dc} + \left(\frac{2\rho c}{\int \rho c^2 dc} - \frac{6}{c^2}\right) e {{=}} 0.

When \rho\! is expressed in terms of c\!, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow.

If we put \mathrm{M}\! for the mass of the spheroid, then

\mathrm{M} {{=}} \frac{4\pi}{3}\int^{c}_0 \rho d{c^3(1 + 2e)}; \qquad\text{ and }m {{=}} \frac{c^3}{\mathrm{M}}\cdot\frac{4\pi^2}{t^2},

and putting c {{=}} c_{0}\! in the equation expressing the condition of equilibrium, we find

\mathrm{M}(2e - m) {{=}} \frac{4}{3}\pi \cdot\frac{6}{5c^2}\int^{c}_0 \rho d(ec^{5}).

Making these substitutions in the expressions for the forces at the surface, and putting r/c {{=}} 1 + e - e(h/c)^2\!, we get

\mathrm{G} \cos \phi {{=}} \frac{Mk^2}{ac}\left\{1 - e - \frac{3}{2}m + \left(\frac{5}{2}m - 2e\right)\frac{h^2}{c^2}\right\}\frac{f}{c}

\mathrm{G} \sin \phi {{=}} \frac{Mk^2}{ac}\left\{1 + e - \frac{3}{2}m + \left(\frac{5}{2}m - 2e\right)\frac{h^2}{c^2}\right\}\frac{h}{c}.

Here \mathrm{G}\! is gravity in the latitude \phi\!, and a\! the radius of the equator. Since

\sec \phi {{=}} (c/f){1 + e + (eh^2/c^2)},\!

\mathrm{G} {{=}} \frac{Mk^2}{ac}\left\{1 - \frac{3}{2}m +\left(\frac{5}{2}m - e\right) \sin^2 \phi \right\},

an expression which contains the theorems we have referred to as discovered by Clairault.

The theory of the figure of the earth as a rotating ellipsoid has been especially investigated by Laplace in his Mécanique celeste. The principal English works are:—Sir George Airy, Mathematical Tracts, a lucid treatment without the use of Laplace’s coefficients; Archdeacon Pratt’s Attractions and Figure of the Earth; and O’Brien’s Mathematical Tracts; in the last two Laplace’s coefficients are used.

In 1845 Sir G. G. Stokes (Camb. Trans. viii.; see also Camb. Dub. Math. Journ., 1849, iv.) proved that if the external form of the sea—imagined to percolate the land by canals—be a spheroid with small ellipticity, then the law of gravity is that which we have shown above; his proof required no assumption as to the ellipticity of the internal strata, or as to the past or present fluidity of the earth. This investigation admits of being regarded conversely, viz. as determining the elliptical form of the earth from measurements of gravity; if \mathrm{G}\!, the observed value of gravity in latitude \phi\!, be expressed in the form \mathrm{G} {{=}} g(1 + \beta \sin^2 \phi)\!, where g\! is the value at the equator and \beta\! a coefficient. In this investigation, the square and higher powers of the ellipticity are neglected; the solution was completed by F. R. Helmert with regard to the square of the ellipticity, who showed that a term with \sin^2 2\phi\! appeared (see Helmert, Geodäsie, ii. 83). For the coefficient of this term, the gravity measurements give a small but not sufficiently certain value; we therefore assume a value which agrees best with the hypothesis of the fluid state of the entire earth; this assumption is well supported, since even at a depth of only 50 km. the pressure of the superincumbent crust is so great that rocks become plastic, and behave approximately as fluids, and consequently the crust of the earth floats, to some extent, on the interior (even though this may not be fluid in the usual sense of the word). This is the geological theory of “Isostasis” (cf. Geology); it agrees with the results of measurements of gravity (vide infra), and was brought forward in the middle of the 19th century by J. H. Pratt, who deduced it from observations made in India.

The \sin^2 2\phi\! term in the expression for \mathrm{G}\!, and the corresponding deviation of the meridian from an ellipse, have been analytically established by Sir G. H. Darwin and E. Wiechert; earlier and less complete investigations were made by Sir G. B. Airy and O. Callandreau. In consequence of the \sin^2 2\phi\! term, two parameters of the level surfaces in the interior of the earth are to be determined; for this purpose, Darwin develops two differential equations in the place of the one by Clairault. By assuming Roche’s law for the variation of the density in the interior of the Earth, viz. \rho {{=}} \rho_{1} - k(c/c_{1})^2, k\! being a coefficient, it is shown that in latitude 45°, the meridian is depressed about 3¼ metres from the ellipse, and the coefficient of the term \sin^2\phi \cos^2\phi (= \tfrac{1}{4} \sin^22\phi )\! is -0.0000295. According to Wiechert the earth is composed of a kernel and a shell, the kernel being composed of material, chiefly metallic iron, of density near 8.2, and the shell, about 900 miles thick, of silicates, &c., of density about 3.2. On this assumption the depression in latitude 45° is 2¾ metres, and the coefficient of \sin^2\phi \cos^2\phi\! is, in round numbers, -0.0000280.[1] To this additional term in the formula for \mathrm{G}\!, there corresponds an extension of Clairault’s formula for the calculation of the flattening from \beta\! with terms of the higher orders; this was first accomplished by Helmert.

For a long time the assumption of an ellipsoid with three unequal axes has been held possible for the figure of the earth, in consequence of an important theorem due to K. G. Jacobi, who proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium.[2] It has been objected to the figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi’s theorem (c : a < 1/\sqrt{}2). Admitting this, it has to be noted, on the other hand, that Jacobi’s theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe; indeed the irregular distribution of continents and oceans suggests the possibility of a sensible divergence from a perfect surface of revolution. We may, however, assume the ellipsoid with three unequal axes to be an interpolation form. More plausible forms are little adapted for computation.[3] Consequently we now generally take the ellipsoid of rotation as a basis, especially so because measurements of gravity have shown that the deviation from it is but trifling.

Local Attraction.

In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. That

  1. O. Callandreau, “Mémoire sur la théorie de la figure des planètes,” Ann. obs. de Paris (1889); G. H. Darwin, “The Theory of the Figure of the Earth carried to the Second Order of Small Quantities,” Mon. Not. R.A.S., 1899; E. Wiechert, “Über die Massenverteilung im Innern der Erde,” Nach. d. kön. G. d. W. zu Gött., 1897.
  2. See I. Todhunter, Proc. Roy. Soc., 1870.
  3. J. H. Jeans, “On the Vibrations and Stability of a Gravitating Planet,” Proc. Roy. Soc. vol. 71; G. H. Darwin, “On the Figure and Stability of a liquid Satellite,” Phil. Trans. 206, p. 161; A. E. H. Love, “The Gravitational Stability of the Earth,” Phil. Trans. 207, p. 237; Proc. Roy. Soc. vol. 80.