162 Mr. Ivory on the Figure of the Planets, 



any misapprehension or error that may have impeded the 

 progress of this branch of mechanical philosophy, is at least 

 deserving of serious attention, and ought not to be fastidiously 

 thrown aside merely because it is opposed to what has re- 

 ceived the sanction of great names. 



In what follows, homogeneous fluids only are understood ; 

 and it is supposed that the accelerating forces which urge the 

 particles, are expressed by functions of the coordinates that 

 satisfy the criterion of integrability. It will also be proper to 

 notice a distinction depending on the nature of the accelera- 

 ting forces, inattention to which has been the cause of no little 

 perplexity in this theory. The forces mentioned may be either 

 explicitly given, so as to be entirely and absolutely known when 

 we know the coordinates of a molecule of the fluid upon which 

 they act; or the same forces may be relative to the unknown 

 figure of the mass of fluid. 



If the particles of a fluid are urged by attractions to fixed 

 centres and by centrifugal forces, the problem of equilibrium 

 falls under the first division. In such cases all the level sur- 

 faces have the same differential equation, and they are dif- 

 ferent only in the different pressures they sustain. When 

 any one of those surfaces is known, all the rest are derived 

 from it merely by varying the pressure, and the figure of the 

 whole mass in equilihrio, is necessarily determined. At the 

 outer surface of the fluid there is no pressure ; and, therefore, 

 when the outer surface satisfies the differential equation com- 

 mon to all the level surfaces, or, which is the same thing, 

 when the resultant of the accelerating forces urging the par- 

 ticles is perpendicular to that surface, all that is required for 

 solving the problem is fulfilled. 



When the particles of a fluid attract one another, the acce- 

 lerating forces urging them will depend upon the place they 

 occupy and upon the figure of the whole mass. As the at- 

 tractive forces upon points differently situated must vary, all 

 the level surfaces will not, in this case as before, have the same 

 differential equation. These surfaces will now vary from one 

 another on two accounts ; namely, the different pressures they 

 sustain, and the variation of the attractive forces according to 

 the situation of the surfaces in the mass. It appears, there- 

 fore, impossible in this case to establish any general relation 

 between the several level surfaces without the special con- 

 sideration of the figure of the fluid. We cannot infer that all 

 the level surfaces will exist in the interior of the fluid, and 

 will satisfy, each its peculiar equation, merely because gravity 

 acts in directions perpendicular to the outer surface. There 



appears 



