Analysis ofihe Mccanujne Celeste of M. La Place. 205 



roids, supposed to be fluid.^, would assume in consequence 

 of the mutual attraction oF all their parts, and of the other ' 

 forces which act upon them. For this purpose he seeks 

 the figure which satisfie-s the equilibrium of a homogeneous 

 fluid masSj eudo-.vedwithan uniforrn rotatorv motion about 

 a fixed axis. Ue supposes this figure to be that of an el- 

 lipsoid of revolution, whose rotatory axis is the axis of re- 

 volution itself. He determines the attractive and centri- 

 fugal forces which flow from this hypothesis; and, substi- 

 tuting them in the equation of the equilibrium of fluids, he 

 thence obtains an equation independent of the co-ordinates 

 of the surfaci^, which establishes the relation that must 

 exist between the eccentricity of the spheroid and the polar 

 axis, in order that the equation of equilibrium may be satis- 

 fied. Hence it follows, that the elliptic figure satisfies the 

 conditions of equilibrium, at least when the ratio of the 

 eccentricity to the polar axis is properly determined in the 

 function of the centrifugal force, and of the density of the 

 body. On this supposition, the gravity at the pole, is to 

 the gravity at the equator, as the diaiTieter of the equa- 

 tor is to the polar axis, and we deduce from it the general 

 relation of the latitude to the gravitation. These results 

 also make known the ratio of the eccentricity to the polar 

 axis, and that of the centrifugal force to the density of the 

 body, by means of the lencrth of the seconds pendulum, 

 and the length of the degree of the meridian, both beinc 

 observed in the same point of latitude. The author applies 

 these formulce to the earth, supposed to be an ellipsoid of 

 revolution and homo2;eneous ; and fixes on this hvp.othesis 

 the ratio of the polar axis to the diameter of the equator. 



The author afterwards examines whether the equation 

 which gives the ratio of the eccentricity to the polar axis 

 be susceptible of several real roots. He shows that with 

 respect to the same iriotion of rotation, the number of these 

 real roots is reduced to two, whence it results, that to ilie 

 •amc angular motion of rotation two diflerent figures of 

 equilibrium answer; but the rapidity of l!:is njotion is li- 

 mited, for equilibrium could not take place with an elliptic 

 figure when the duration of rotation does not surpass the 

 product of one hour and 90 seconds, into ttie square root 

 of the ratio of the mean density of the earth to that of the 

 fluid mass. The time is here reckoned according to the 

 new division. 'l"hc observed rotations of Jupiter and of tlie 

 6un are within the limits of this duiation. 



One would suppose that lliis limit is the one al which the 



fluid 



