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these forces tend to change the figure of the fluid, that theory is in- 

 adequate to give an exact determination of the equilibrium in those 

 cases. 



In the second part of the paper, the author applies his theory of 

 the equilibrium of fluids to the determination of the figure of the pla- 

 nets, under the supposition that they are composed wholly of fluid 

 materials. For this purpose he first considers the problem of deter- 

 mining the equilibrium of a homogeneous mass of fluid entirely at 

 liberty, when the accelerating forces are known functions of the co- 

 ordinates at their point of action. In the investigation of this pro- 

 blem, he supposes that the centre of gravity is at rest, and undis- 

 turbed by the action of any accelerating force. He then supposes the 

 fluid to be in equilibrium, and that three planes are laid down, inter- 

 secting one another at right angles in the centre of gravity of the 

 mass, to which planes the particles of the fluid are referred by rectan- 

 gular co-ordinates. The algebraical consequences of this supposition 

 are then pursued, the conditions necessary to equilibrium pointed 

 out, and the conclusion deduced, that the resultant of the accelera- 

 ting forces is perpendicular to the outer surface, and also to the in- 

 terior level surfaces of the fluid, at every point of which there is the 

 same intensity of pressure. The figure of the fluid being determined, 

 it remains to inquire, whether the equilibrium is secure ; and the 

 resultof the inquiry furnishes an equation which proves that thepar- 

 ticles have no tendency to move, from any inequality of pressure. 



A further discussion is entered into in order to prove that the pres- 

 sures propagated from the surfaces into the interior parts balance and 

 destroy one another, which completely establishes the permanence of 

 the figure of the fluid. It is also shown that the mass of fluid, under 

 these circumstances, has no tendency to turn upon an axis. 



To illustrate the foregoing problem, the author applies it to the de- 

 termination of the figure of equilibrium of a homogeneous mass of 

 fluid entirely at liberty, of which the particles attract one another with 

 a force directly proportional to the distance, at the same time that 

 they are urged by a centrifugal force caused by rotation about an 

 axis. 



He then enters upon the investigation of the second problem, in 

 which the law of attraction of the particles is that of the inverse du- 

 plicate ratio of the distance; and finally arrives at the conclusion, 

 that the form of the fluid in equilibrium is, exclusively of all other 

 figures, an oblate elliptical spheroid of revolution, and that its axis of 

 rotation is the lesser axis of the spheroid. He also shows that within 

 the spheroid there are no more than two sets of surfaces equally 

 pressed by the action of the exterior fluid; and no more than two diffe- 

 rent spheroids of equilibrium answering to the same rotatory motion. 

 If the whole spheroid be one of small oblateness, the greatest of the 

 interior surfaces of equable pressure, which is not a level surface, 

 stands upon the equator ; and the rest are within this, and are simi- 

 lar to it, and similarly posited. When it is very oblate, the greatest 

 of these surfaces is described about the lesser axis; and the rest are 

 within it, and are similar to it, and similarly posited. The existence 



