203 Analysis of the Me:canique Celeste of M. La Place, 



vector of" the spheroid. Supposing afterwards the action of 

 foreign powers to be nothing, we deduce from this equa- 

 tion, and by differcntiaiions only, that if the spheroid 

 sought be one of revolution, it can onl) be an ellipsoid 

 which is reduced to a sphere, when there is no rotatory 

 motion ; so ihit the sphere is the only surface of revolution 

 which satislits the equilibrium of an immoveable homo- 

 geneous fluid mass. fJence it is afterwards concluded ge- 

 nerally, that if the fluid mass be solicited by any very small 

 forces whatever, ilu re is only one possible figure of equili- 

 brium ; for, by supposing that there are several, there would 

 of course be several different radii, which beine substituted 

 in the equation of equilibrium, would verify it; and as this 

 equation is hnear widi respect to these radii, the sum of any 

 two of theiTi would still satisfy equally as well as their dif- 

 ference. Hence the author ingeniouslv deduces, that this 

 diiference must be nothing; from which he concludes, 

 that the spheroid can be in equilibrium in one way only. 



Afterwards comes the consideration of the equilibrium 

 of a homogeneous t^iuid mass which covers a spheroid of a 

 different density. On this head he observes, that we may 

 regard this sphere as being of the same density with the 

 fluid, and place in its centre a force reciprocally as the 

 square of the distances. By means of this consideration, 

 we easily oiilain the equation of equilibrium for this sphe- 

 roid ; and it results that there are generally in this case, and 

 when the spheroid is one of revolution, two figures of equili- 

 brium. When there is no rotatory motion, and when we 

 consider the foreign forces as nothing that nuitually attract 

 the molecules of the body, one of these two figures is sphe- 

 rical, and both are so if the spheroid be homogeneous ; 

 which confirms the preceding results. 



After havin<j[ thus obtained the figures of revolution which 

 satisfy the equilibrium of a homogeneous fluid mass which 

 covers a sphere, the au'.hor gives the method of deducing 

 from it those which arc not of revolution. With this view 

 he transfers to any given point the origin of the angles 

 which determine the position of the radius vector in space, 

 aiK'les which were previously taken into account, reckoning 

 from the extremity of the axis of revolution. By this me- 

 thod all these angles enter into the expression of the radius 

 vector; and as, in consequence of what jjrecedes, this ra- 

 dius satisfies the equation of equilibrium, whatever may be 

 the positi(m of this new origin, it will still satisfy it when 

 we vary this ori'^in in any way whatever. This variation 

 only influences the excess of the radius vector of the sphe- 

 roid 



