HOMOGENEOUS FLUID AT LIBERTY. 



527 



Let ABC represent a body of homogeneous fluid which 

 revolves about the axis H K, passing through the centre of 

 gravity G ; and describe an interior surface a h c, similar to 

 the upper surface ABC, and similarly posited about the point 

 G : if we suppose that the mass A B C is in equilibrium by the 

 action of the centrifugal force, and the attraction of its parti- 

 cles in the inverse proportion of the square of the distance, it 

 is a property derived from the particular law of attraction and 

 the nature of a centrifugal force, that every other body of the 

 same fluid, as aft c, similar to ABC, similarly posited about the common centre of 

 gravity G, and revolving about the same axis K H, will be separately in equilibrium 

 by the centrifugal force of its particles and the attraction of its own mass. It would 

 be superfluous to repeat the demonstration of this proposition here, as it is attended 

 with no difficulty, and has not been contested. And because the body of fluid a ft c is 

 separately in equilibrium with respect to the centrifugal force of its particles and the 

 attraction of its mass, it must likewise be in equilibrium with respect to the other 

 forces that act upon it : for if it were not so, the whole body of fluid ABC would 

 not be in equilibrium. 



Now the only force external to the mass ah c, and tending to change the figure of 

 that mass, is the attraction of the exterior stratum upon the interior particles. Let 

 M be any particle within the stratum : the several forces 

 which act upon it are, first, the centrifugal force ; secondly, 

 the attraction of the mass a ft c ; and, thirdly, the attraction 

 of the exterior stratum. On account of the separate equili- 

 brium of the mass a ft c, the combined action of the two first 

 forces has no tendency to move the molecule M ; and there- 

 fore the equilibrium of the whole mass ABC requires that 

 the attraction of the exterior stratum be ineffective to move 

 the same molecule. Thus every molecule M within the surface ahc must be urged 

 equally by the pressures which the attraction of the stratum causes in all canals 

 originating at the molecule, and terminating in the surface ah c. This is the same 

 condition to which every other mode of investigation has led ; and as the mathemati- 

 cal application of this property to determine the figure of equilibrium has already 

 been fully detailed, it need not be repeated here. 



In order to leave nothing unexplained, it will be proper to remark, that the interior 

 surface a ft c is a level surface, that is, it is perpendicular at every point to the re- 

 sultant of all the forces which act on a particle contained in it ; for the centrifugal 

 force of a particle at «, and the attraction upon it of the mass a ft c, have their re- 

 sultant perpendicular to the surface a ft c, because the body of ffuid a ft c is separately 

 in equilibrium : and the attraction of the stratum upon the particle at a is perpendi- 

 cular to the surface ahc, because the sum of all the molecules of the stratum, divided 



3 y2 



