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THE FIGURE OF THE EARTH, ON THE SOLE HYPOTHESIS
OF THE SURFACE BEING A SURFACE OF EQUI
LIBRIUM AND NEARLY SPHERICAL.
88. UPON the hypothesis of the Earth being a fluid mass it was shown by Clairaut, in his celebrated work Figure de la Terre, published in 1743, that the increase of gravity in passing from the equator to the poles varies as the square of the sine of the latitude, and that a certain relation must necessarily subsist between the ellipticity and the amount of gravity, a relation which has been ever since known as Clairaut's Theorem. Laplace demonstrated the same, on the simpler hypothesis of the surface only being a surface of equilibrium, and the interior being solid or fluid, but consisting of strata nearly spherical. Professor Stokes, in an investigation published in the Cambridge Philosophical Transactions for 1849, has done the same, without making any assumption whatever regarding the constitution of the interior of the mass, but assuming only that the surface is a spheroid of equilibrium, of small ellipticity. The present Chapter is borrowed wholly from Mr Stokes's investigation. Clairaut's Theorem is valuable as it gives us the means of determining the ellipticity by means of pendulum oscillations, the times of which measure the force of gravity at the several stations where experiments are made.
PROP. To find the law of gravity at the surface of a spheroid of equilibrium and of small ellipticity.
89. Let V be the potential of the mass.
Then because the surface is a surface of equilibrium,
const. = V + fw* (1 – M°) . By a process precisely like that in Art. 23 we have, for an external point, dry d
dV 1 d V +
du T*1-2 dw?
Let V be expanded in a series of Laplace's Functions,
V+V, + ... + Vi+... Then since the above equation is linear with respect to V, and a series of Laplace's Functions cannot equal zero unless the Functions are separately zero (see Art. 35), we have, by substituting the above series for V and remembering the condition given by Laplace's Equation,
where Wi and Z are independent of r. The complete value of V becomes
Now V evidently vanishes, from its very definition, when r is infinite. Hence Z=0, Z=0, Z= 0 ...
2 + go2 703
If the surface were spherical, this expression would be reduced to its first term. Hence in our case W,, W must be all small quantities of the first and higher orders.
Equate the sums of Laplace's Functions of the same order
i. W = a const.
I wa', W.=0,
WE=0, W.= 0 ...
g be gravity. Then since the angle between the radius vector r and the normal varies as the ellipticity and therefore its cosine must be taken =1, the value of gravity
dV is the part of the centrifugal force resolved along r
Substituting for r and omitting small quantities of the second order, W
The first portion of this is evidently the mean value of g, because if g be multiplied by an element of the surface and integrated throughout, the latter part will disappear. Let the first part be G; therefore also w.a=m. G, since m is small;
Hence the increase of gravity in passing from the equator to the poles varies as the square of the sine of the latitude; and also polar gravity – equatorial gravity equatorial gravity
x ratio of centrifugal force at equator to gravity.
90. This is Clairaut's Theorem; which is thus demon. strated by Professor Stokes without making any assumption regarding the interior of the mass. Nothing can be inferred, therefore, from any numerical value we may obtain for gravity, and therefore for the ellipticity, by pendulum experiments, regarding the Earth's mass having been originally fluid or not.
For a valuable and interesting account of pendulum experiments made in places in all latitudes, and the result regarding the Figure of the Earth, we must refer the reader to Major General Sabine's work on the subject, Account of Experiments to determine the Figure of the Earth by means of the Pendulum vibrating Seconds in different Latitudes. London, 1825*. The ellipticity thus deduced is zo, rather greater than that obtained by the geodetic and other methods. In consequence of the irregularities of the surface of the Earth the experiments with pendulums need various corrections before they can be properly applied to determine the ellipticity. The principal ones depend upon the elevation of the station above the sea-level (for which Dr Young gave a formula of correction, see Phil. Trans. for 1819), and the excess or defect of matter in table-lands or the sea in the neighbourhood of the station. If the station, for instance, be on a rock in an island in the midst of a sea, such as St Helena, the correction for this second disturbing cause will be different from what it would be for a station at the same elevation from the sea-level in the midst of a continent. This effect depends, as may be gathered from Art. 52, not upon the height of the station above the sea-level, but upon the excess or defect of matter however arranged. Professor Stokes has fully considered the influence of these causes of derangement in his paper above referred to. He shows, that the effect of these corrections for the irregularities of the surface, and for the different elevations and other local circumstances of the stations where the experiments are made, is to reduce the value of the ellipticity, and make it nearer to do.
* See also his latest remarks on the subject in the Notes to his translation of the Cosmos, Vol. IV. Part I.
91. Cor. 1. The investigation in the last Proposition gives, after making substitutions,
E being the mass of the Earth. W is evidently equal to the mass, because as r becomes infinitely great the second term vanishes with reference to the first, and we know that in that case the value of the potential must be the mass divided by the distance.
92. Cor. 2. Laplace first pointed out that the ellipticity of the Earth would have an effect upon the Moon's motion. The expression in the last Article leads to the following formula for the change in the Moon's latitude produced by this cause,
sin 21 sin (nt +e),
2ha? 2 where a = distance of Moon, h= mean motion of the node, n = mean motion, e = the epoch, or longitude when t=0, I the obliquity of the ecliptic. (See Mechanical Philosophy, Second Edition, Art. 556 ; also, changing the notation, Airy's Tracts, Fourth Edition, p. 188, Art. 84.) This has been