HOMOGENEOUS FLUID AT LIBERTY. 515 



nearly spherical, d V d will be more and more flattened ; so that, the centrifugal force 

 being zero and A B C a perfect sphere, a' V d will be an infinitely thin circle of fluid 

 particles in the plane of the equator. 



The problem that has been solved leads to a consideration which it is important to 

 notice, because it relates to a principle of equilibrium that has been very generally 

 adopted. It has been shown that the equation of the surface (2.) is indeterminate, 

 and admits of innumerable solutions ; but in every figure which satisfies that equa- 

 tion, the other equation (1.), viz. 



p =/^ + I- (y2 + ^2) _ C, 



will hold at every interior point {x y z) of the mass of fluid. In this latter equation, 

 /? is the pressure of any canal issuing from the point {xy %) and extending to the sur- 

 face of the fluid ; and therefore, in every figure which satisfies the equation of the 

 surface, every such canal will exert the same pressure upon a molecule placed at the 

 point {xy %). Now of the innumerable figures that satisfy the equation of the surface 

 there is only one that is in equilibrium ; and thus it is proved, that a mass of fluid, 

 without being in equilibrium, may assume many figures in which every interior par- 

 ticle is pressed with equal intensity by all the canals issuing from it and terminating 

 in the surface. And as neither the equation of the surface, nor the equal pressure of 

 all the canals extending from a molecule to the surface, is sufficient to secure the 

 equilibrium except when the forces are explicit functions of the coordinates ; so 

 neither of the two properties can be employed in any other hypothesis respecting the 

 forces, to verify an equilibrium, that is, to prove that a proposed figure will be in 

 equilibrium. 



8. In the following problem the forces in action are known functions of the co- 

 ordinates, and the solution is deduced immediately from the theorem in No. 5. 



Problem II. 



To determine the figure of equilibrium of a fluid at liberty, the particles being sup- 

 posed to 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 revolving about 

 an axis. 



As the attractions of the particles balance one another at the centre of gravity, in 

 order to free that point from the action of any forces the axis of rotation must pass 

 through it. 



Let X, y, z denote the coordinates of an attracted particle, and .r', y', z' those of an 

 element dm oi the mass, the origin being at the centre of gravity, and x, x' being 

 parallel to the axis of rotation ; adopting for the unit of mass the whole given mass 

 of fluid, and for the unit of force the attraction of the whole mass collected in a point 

 upon a particle at the distance 1, the attraction of dm upon the assumed particle at 



