I 



Andy US of the Mecanique Celeste ofM. La Place. 207 



figure satisfies the equilibrium of a homogeneous fluid mass, 

 eiulowcd with an uniform rotatory motion about a fixed 

 axis; but in order to resolve this problem completely, we 

 must determine a prion' all the possible figures of equili- 

 brium, or ascertain that the elliptic figure is the only one 

 which fulfils these conditions. Wc perceive besides, that 

 in the inquiry into the figure of the planets we ought not 

 lo confine ourselves to the case of homogeneity ; but then, 

 this inquiry, considered in a general point of view, becomes 

 extremely difficult. Fortunately it is simplified relatively 

 to the planets and satellites, on account of the small diffe- 

 rence which exists between the figure of these bodies and 

 that of the sphere, which admits of our neglecting the 

 square of this difitrence and the quantities which depend 

 on it. In order to treat this problem in the most general 

 manner, the author considers the equilibrium of a fluid 

 mass covering a body formed of layers of variable densities, 

 endowed wiih a rotatory motion about a fixed axis, and 

 solicited by the action of foreign bodies ; and he establishes 

 the general equation of this equilibrium, when the covered 

 spheroid dift'rrs very little from a sphere. This spheroid 

 may besides be entirely fluid ; and it may be formed of a 

 solid nucleus covered by a fluid : in these two cases, which 

 are reduced to one only, if the spheroid be homogeneous, 

 the preceding equation determines its figure, that of the 

 fluid strata which cover it, and also gives, by simple diffe- 

 rentiation, the variation of the gravity at its surface. When 

 there are no foreign bodies, in which case the spheroid, sup- 

 posed lo be homogeneous and of the same density as the 

 fluid, is only solicited I)y the attraction of its molecules and 

 the centrifugal force of its rotatory motion ; its figure be- 

 comes that of an ellipsoid of revolution, on which the in- 

 creases of gravity and the diminutions of radii are propor- 

 tional to the squares of the sines of the latitudes ; from all 

 which the author concludes that the elliptic figure is then 

 the only one which satisfies the equilibrium. "^Ihis demon- 

 stration rests enlircly on the sole hypothesis, that the figure 

 of the spheroid difiiirs very little from the sphere; but it 

 requires the development of the radius of tliis spheroid in a 

 series of iunctious of a particular kind, whicfi the author 

 has demonstrated above to be always possible : but in order 

 to avoid all the difliculties wliich this development might 

 give rise to, lie resumes the same problem by a direct me- 

 thod, and independent of series ; this metliul consists in the 

 first place in transforming the equation of equilibrium in 

 tutth a way as to leader it linear with rc-pect lo the radius 



Vector 



