324 Mr. Ivory on the Figure of the Planets, 



there be no centrifugal force, the equation will belong to a 

 sphere; and the question is to determine what the figure will 

 be, when « is a very small fraction. Let a denote the radius 

 of a sphere equal in volume to the whole fluid mass, or the 

 radius of the sphere which would be the figure of the fluid in 

 the absence of all centrifugal force ; then we may assume with 

 Legendre, that the radius of the oblate spheroid is expressed 

 by this formula, 



r=:a(l +«j/) (2) 



y being a quantity depending upon the position of r, nega- 

 tive at the poles where the spheroid falls within the sphere, 

 and positive at the equator where the spheroid rises above the 

 sphere. It will be sufiicient for our present purpose if we ad- 

 mit, in order to fix our ideas, that the spheroid is a figure of 

 revolution ; and then y will be a function of the arc 6 alone. 

 What we have now to do is to determine the expression of y 

 by means of the equation (1); and we shall suppose that this 

 problem is solved, and that we have found y in a series of 

 terms proceeding by the powers of a. Conceive now that the 

 whole sphei-e having its radius equal to a, is divided into in- 

 numerable concentric spheres; put a' for the radius of one of 

 the interior spheres, and form this expression, 



?■" = a' (1 + ay). 

 It is manifest that the equation (1) will be equally true at 

 the surfaces of the two spheroids of which r and r' are the 

 radii ; for y is the same in both these quantities, and a and c', 

 in which consists their only difference, are entirely arbitrary. 

 It follows, therefore, that the whole fluid sphere, and all the 

 concentric spheres, are changed by the centrifugal force into 

 spheroids similar to one another and similarly posited about 

 the common centre. If any one of these spheroids taken se- 

 parately from the rest be in equilibria, they will all be in 

 equilibrio independently of one another ; for the equation (1), 

 which is made the only condition of equilibrium, applies alike 

 to all. It thus appears that the analytical assumptions in the 

 process of Legendre bring us necessarily to the same conclu- 

 sions already obtained in the first and second propositions. 

 But there is no proof that any one of the spheroids is in equi- 

 libria. On the contrary, we learn from the third proposition, 

 that there can be no equilibrium unless a new condition be 

 superadded to the equation of the fluid's surface. It is neces- 

 sary tliat all the interior similar surfaces be surfaces of equal 

 pressure, which cannot happen unless the fluid mass be an 

 elliptical spheroid exclusively of all other figures. 



It has now been shown, that the analytical process of Le- 

 gendre 



