in reply to M. Poisson. 167 



the centre of gravity. Now these surfaces are independent of 

 all the matter exterior to them ; and therefore they are not 

 level surfaces, and the mass of fluid is not in equilibrio. But 

 when the equations (2) and (3) exist at the same time, we 

 thence derive the equation (1), which is the true equation of 

 the level surfaces, and establishes the equilibrium. 



It appears that V(r) — V'(7) is the pressure upon the as- 

 sumed point in the interior of the fluid. The pressure there- 

 fore is wholly determined by the figure of equilibrium, and has 

 no immediate dependence upon the centrifugal force ; and it 

 is easy to see that this must be the case. For there is no cen- 

 trifugal force at the points where the axis of I'otation traverses 

 the several level surfaces : at these points, therefore, the centri- 

 fugal force has no influence on the pressure ; and, as the pres- 

 sure is the same over the whole of every level surface, it fol- 

 lows that it is independent of the centrifugal foi'ce in all the 

 interior of the fluid. 



The similarity of the interior surfaces and the property con- 

 tained in the fourth proposition, determine the figures with 

 which the equilibrium is possible; and it is found that they 

 can only be ellipsoids : the equation of the outer surface as- 

 certains the rotatory velocity that must take place in every 

 particular figure. We are therefore entitled, upon undoubted 

 evidence, to establish the following proposition : 



Prop. 5. — A homogeneous mass of fluid consisting of par- 

 ticles that attract one another inversely as the square of the 

 distance, and revolving upon an axis passing through the cen- 

 tre of gravity, cannot be in equilibrio nor maintain a perma- 

 nent figure, unless it have the figure of an ellipsoid. 



The whole of the preceding reasoning may be brought 

 within a narrow compass. If a homogeneous planet in a fluid 

 state be in equilibrio, all the level surfaces must be similar to 

 one another : but the ellipsoid is the only figure in which the 

 level surfaces are all similar; wherefore the planet must have 

 the figure of an ellipsoid. All this, it is presumed, is here 

 proved by exact reasoning. On the other hand, the usual 

 theory is grounded on an assumption without proof. If the 

 outer surface of a homogeneous mass of fluid be a level sur- 

 face, it is assumed, for no other reason than that there is no 

 distinction of densitj', that all the level surfaces in the interior 

 do necessarily exist and produce an equilibrium. But, if there 

 be no distinction of density, there is a general equation be- 

 longing to the interior level surfaces; and, in order to prove 

 the existence of these surliiCGS, on which the equilibrium de- 

 pends, it should be shown that their e<|uation is a necessary 

 consequence of that of the outer smface. But this has not 



been 



