504 MR. IVORY ON THE EQUILIBRIUM OF A MASS OP 



the equilibrium of the fluid will obviously be a consequence of the theorem. It is 

 therefore demonstrated, with respect to problems of the second class, that the equa- 

 tion of the upper surface of the fluid is not sufficient by itself to determine the equi- 

 librium of the mass. 



In the theorem, the term level surface is liable to no ambiguity ; but in the more 

 complex state of the forces that occurs in the second division of problems, two dif- 

 ferent systems of surfaces in which the pressure is constant require attention ; for the 



pressure caused by the forces j^, -^, j^, is constant in all interior surfaces de- 

 termined by the equation 



and the pressure caused by the forces ^, j—, -^, is constant in all surfaces of 



which the general equation is 



d . ?)' {x, y, z) = 0. 



It will therefore conduce to clearness if the meaning of a level surface be restricted, by 

 adding to the two properties of being perpendicular to the resultant of the forces 

 acting on the particles contained in it, and being pressed at all its points with the 

 same intensity, the further condition of being deduced by varying the constant in 

 the equation of the upper surface of the fluid. The effect of the equation 



d . (p' (x, y, z) — d . (p (x, y, z) = 



is to verify the two differential equations above mentioned at all the points of the 

 samesurfaci: it implies that the two systems of surfaces of constant pressure are 

 blended in one ; and as this is a necessary condition of equilibrium, it distinguishes 

 from all other figures those which are alone susceptible of an equilibrium. 



7. The general theory of the equilibrium of homogeneous fluids at liberty having 

 been explained at sufficient length, it is next to be applied to some of the principal 

 problems. 



Problem I. 



To determine the equilibrium of a homogeneous fluid at liberty, the particles attract- 

 ing one another with a force inversely proportional to the square of the distance, 

 at the same time that they are urged by a centrifugal force caused by revolving 

 about an axis. 



The mass of fluid being in equilibrium, the centre of gravity will be free from the 

 action of any forces ; and as the attractive forces balance one another at that point, 

 there must be no centrifugal force at the same point ; that is, the axis of rotation 

 must pass through it. 



The origin of the coordinates being placed in the centre of gravity, let x, y, z, de- 

 note the rectangular coordinates of a particle of the fluid, and x', y', z', those of a 

 molecule dm of the mass, the two coordinates x and x' being parallel to the axis of 



