Bulletin des Sciences Mathematiques./or August 1830. 187 



of the stratum between the two surfaces upon the particles 

 within the lower surface. Such an attraction therefore can 

 have no place in the theory of Clairaut, which notices no forces 

 except those in action at the outer surface. It is implied in 

 the theory that the only forces urging a particle, whether si- 

 tuated in the surface or in the interior parts of the fluid, are 

 the centrifugal force and the attraction of the whole mass ; 

 these forces produce the pressures of the canals a A and b B ; 

 but they have no connection with the pressure 5, which has 

 quite a different origin. The theory of Clairaut is therefore 

 insufficient for determining the equilibrium, because it leaves 

 out some of the causes tending to change the figure of the 

 fluid. 



Further, I shall prove that the equation found by M. Pois- 

 son leads to two independent conditions for the figure of 

 equilibrium. These conditions are, first, the equation of the 

 outer surface, which is all that Clairaut's theory requires; se- 

 condly, the equality of pressure at all the points of every in- 

 terior surface, as a b c, similar and similarly posited to the 

 outer surface. For we may suppose that one end a of the 

 canal a b remains fixed, while the other end b is successively 

 applied to every point of the surface a b c ; in every position of 

 the canal we shall still have the equation, 



P = ?-&; 



from which it follows that q 8 is equal to the constant quan- 

 tity of p at every point of the surface. Now, q being the weight 

 of the canal b B, and 3 the effort of the canal a b towards 6, 

 caused by the attraction of the stratum, the intensity of pressure 

 at every point of the surface a be, will be the same. By means 

 of this property, the equation of the surface a b c will be de- 

 rived from the equilibrium of the whole mass ABC; the 

 same equation is deducible from the separate equilibrium of 

 the interior mass a be; and as the two equations must be iden- 

 tical, we thence obtain a condition which is independent of the 

 outer surface of the fluid. 



When the conditions for the equilibrium are more atten- 

 tively investigated, it will appear that the attraction of the 

 stratum upon the particles in the inside must produce no in- 

 ternal pressure. From this it follows that 8 = in M. Pois- 

 son's equation. In a paper in the Phil. Trans, for 1824, in 

 which I first considered this problem, I have fulfilled what is 

 physically required for the equilibrium by supposing that the 

 stratum attracts every particle in the inside with equal force 

 in all opposite directions. I have since found that this is not 

 exact in all laws of attraction. But the fluid within the stratum 



2B2 will 



