Page:Mécanique céleste Vol 2.djvu/14

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VllI . CONTENTS OF THE SECOND VOLUME.

A very simple result of this equation is, the relation between the terms of the series, expressing

the attraction of a spheroid, upon an external point, and those of the radius of the spheroid,

developed in a series of functions of a particular kind [1463, 1467]. These functions are given

• by the' nature of the attraction [1465'] ; they are of very great use in the theory of the figures

and motions of spheroids, as well as in the oscillations of the fluids which cover them. . § 11

General theorem [1470] ; expressing the double integral of the differentials which depend upon the product of two of these functions. Simplification of the expressions of the radius of the spheroid, and of its attraction, when the origin of the radius is fixed at the centre of gravity of the spheroid [1480', 1483^*] §12

On the attraction of a spheroid upon a point placed within it [1496] ; also of a stratum upon a point placed within it [1501]. Conditions which are necessary, in order that the point may be equally attracted in every direction [1502] §13

On the attractions of a spheroid, differing but little from a sphere, and formed of strata, varying according to any law [1505, 1506] § 14

Extension of the preceding researches to any spheroid whatever ; reduction of its attraction to a series of a very simple form ; new solution of the problems of the attractions of elliptical spheroids which results from this method [1507 — 1563] § 15, 16, 17

CHAPTER III. ON THE FIGURE OF A HOMOGENEOUS FLUID MASS IN EaUILIBRIUM, AND ENDOV^^ED

WITH A ROTATORY BIOTION 199

General equation of the surface in a state of equilibrium [1564]. The ellipsoid satisfies this equation [1574"]. Determination of this ellipsoid. The gravities at the pole and at the equator are in the ratio of the diameter of the equator to the axis of the poles [1578"]. Two elliptical figures, and no more, can satisfy a given angular motion of rotation [1.597"] ; and relatively to the earlh, supposing it to be homogeneous, the equatorial diameter is to the polar axis as 680,49 to 1, in the most oblate ellipsoid [1603"] ; and as 231,7 to 230,7 [1592"], in the least oblate ellipsoid. A homogeneous fluid mass cannot be in equilibrium with an elliptical figure, unless the duration of its rotation exceed the product of 0day,1009 [=2h 25m], by the square root of the ratio of the mean density of the earth to that of the mass [1605'] § 18, 19, 20

If the primitive duration of rotation be less than this limit, it will increase by the flattening of the fluid mass ; and whatever be the primitive impressed force, the fluid, by means of the tenacity of its particles will finally attain a permanent elliptical figure, of a particular form, to be determined by the nature .of these forces. The axis of rotation is that passing through the centre of gravity, and which was, at the commencement of the motion, the axis of the greatest momentum of the forces [1607"— 1614""] § 21

CHAPTER IV. ON THE FIGURE OF A SPHEROID," DIFFERING BUT LITTLE FROM A SPHERE, AND

COVERED BY A FLUID STRATUM IN EaUILIBRIUM 241

General equation of equilibrium [1615] § 22

Development of this equation, when the forces acting upon the fluid depend upon the centrifugal force of the rotatory motion, upon the attractions of the fluid and of the spheroid, and upon the attraction of foreign bodies [1616^'"'] § 23

Equation of equilibrium, when the spheroid and fluid are homogeneous and of the same density [1636]. Expression of the radius of the spheroid [1644], and the gravity of its surface [1647]. 97.121.167.179 00:28, 12 February 2025 (UTC)]]]]