514 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



from an algebraic equation ; it is attended with no difficulty, and is very fully dis- 

 cussed by all the authors who have written on the figure of the earth ; it would, there- 

 fore, be superfluous to treat of it here ; but it may not be improper to add a few 

 words for the purpose of explaining in what manner the number of solutions of the 

 problem is limited by the nature of the equilibrium. 



Let ABC represent an oblate elliptical spheroid of homogeneous fluid in equi- 

 librium by revolving about the axis P Q ; and a b c, an 

 interior level surface, which is therefore similar to the 

 upper surface ABC, and similarly posited about the cen- 

 tre : the stratum between the two surfaces will act upon 

 the fluid within the level surface in two ways, namely, by 

 pressure and by attraction. From the nature of the sphe- 

 roid, the attraction of the stratum upon every particle 

 within the level surface is zero ; and the pressure of the 

 exterior fluid acts upon every point of the same surface 

 with equal intensity : wherefore, the whole mass ABC being in equilibrium, if the 

 stratum be taken off", the remaining body of fluid ab c will be in equilibrium sepa- 

 rately. But another spheroid, a' b' c', of a different form, may be traced within A B C, 

 the less axes and the equators of the two figures coinciding, such that it will remain 

 in equilibrium separately, upon abstracting the exterior fluid. Every small portion 

 St of the surface a' b' c' is pressed inward by the exterior fluid ; it also sustains a 

 pressure from within outward, caused by the attraction of the fluid on the outside of 

 the surface a' b' c' upon the particles within that surface. Now, although each of the 

 two contrary pressures varies from one point of the surface to another, yet the sphe- 

 roid may be so determined, that their joint action, or their difference, shall be the 

 same at every point of the surface. When the spheroid a' b' c' has this figure, it will 

 be in equilibrium with respect to the action of the exterior fluid ; and, if that be 

 abstracted, it will be in equilibrium separately, because the whole mass ABC is in 

 equilibrium. What has been said may easily be proved by calculation ; for the sphe- 

 roid ABC being given, we know the pressure of the exterior fluid upon s t; wt know 

 also the attraction of the exterior fluid upon a particle of the spheroid a' V c', for it is 

 equal to the difference of the attractions of the spheroids ABC and a' V c' upon the 

 particle : and hence it is easy to deduce, that the relation between the oblateness and 

 the centrifugal force is expressed by the same equation in the spheroid d V d and in 

 the level surfaces. 



It thus appears, that in general there are two spheroids of the same matter, but not 

 more than two, which will be in equilibrium with the same rotatory velocity. If the 

 oblateness of A B C increase, that of a' V c' will decrease ; and the tv/o spheroids con- 

 tinually approaching the same figure, they will ultimately coincide in a limit at which 

 there is only one form of equilibrium. On the other hand, as A B C becomes more 



