Page:Mécanique céleste Vol 2.djvu/15
CONTENTS OF THE SECOND VOLUME. IX
If there be no foreign attraction, this surface will be elliptical, and the ellipticity will be f of the ratio of the centrifugal force to gravity [1648] ; the diminution of the radius of the spheroid, from the equator to the poles, will be proportional to the square of the sine of * "
th'e latitude [1G4S'"], and if we take the radius, and the gravity at the poles, for the unit of measure and of gravity respectively, we shall find, that the increment of gravity is equal to the decrement of the radius [1648, 1648'J §24,25
A direct demonstration, independent of series, that the elliptical figure is then the only one which corresponds to the state of equilibrium [1649" — 1676""] §26
In some cases, a homogeneous fluid mass, surrounding a sphere, can have an infinity of different figures of equilibrium. Determination of these figures [1676^ — 1701] §27,28
General equations of equilibrium of the fluid strata, of variable densities, which cover a spheroid [1702] §29
Examination of the case in which the spheroid is wholly fluid [1709'"]. If there be no foreign attractions, the spheroid will then be an ellipsoid of revolution [1731']. The densities will diminish, and the ellipticities increase from the centre to the surface [1731"]. The limits of the oblateness are between -f and g- of the ratio of the centrifugal force to gravity [1732"']. Equation of the curve, whose elements are in the direction of gravity, from the centre to the surface [1734] §30
Simplification of the expression of the radius of a spheroid covered by a fluid in equilibrium, supposing the origin of the radius to be fixed at the centre of gravity of the whole mass [1734'"], and that it turns about one of the principal axes [1755] § 31, 32
Very simple expressions of the force of gravity [1769], of the length of a pendulum [1770], and of the length of a degree upon the surface of the spheroid [1774], in terms of the radius [1765], An easy method, which results, for verifying by observation, any hypothesis that may be imagined, relative to the laws of the variations of the degrees and of gravity [1777^"]. The hypothesis of Bouguer, according to which the variation of the degrees from the equator to the poles is proportional to the fourth power of the sine of the latitude, is incompatible with the observations of the pendulum [1787"]. Reason why the aberrations from the elliptical figure are much more sensible in the degrees of the meridian, than in the lengths of the pendulum [1777"] §33
If the strata of the spheroid be supposed elliptical, the figure of the surrounding fluid will also be elliptical. The variations of the radii of the earth, of the degrees of the meridian and of gravity, will then be proportional to the square of the sine of the latitude [1795 — 1796]. The whole variation of gravity from the equator to the pole, divided by the whole expression of gravity, will be as much above or below f of the ratio of the centrifugal force to gravity at the equator, as the ellipticity is below or above the same quantity respectively [1806]. . § 34
Expressions of the attractions of elliptical spheroids upon an external point [1811", &c.]. § 35
Of the law of gravity at the surface of a homogeneous fluid spheroid, the attraction being as any power of the distance [1817] § 36
Method of noticing the terms depending on the square and higher powers of the centrifugal force, in the investigation of the figures of spheroids, covered by a fluid in equilibrium [1820" — 1839]. We can satisfy ourselves that the equilibrium of the fluid is rigorously possible ; although we cannot ascertain this figure, except by successive approximations
[1839"] §37
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