HOMOGENEOUS FLUID AT LIBERTY. 499 



and by these forces we may suppose that every particle in the small mass k X w is 

 urged. Now let 



^-\/m+m+i^r-' 



and by a well-known property, the cosines of the angles which the coordinates of the 

 lower surface of the stratum make with the normal of the same surface, will be re- 

 spectively equal to 



1 J ^ 1 d <^ \ d <p 



Y ^ ITx' 'Y ^ Ty' f ^ Jz ' 

 and hence the sum of the partial forces acting in the direction of k^ will be equal to 



¥'\dx) ~^F-\dj/) '^F'\dz) — *• 



wherefore, F X k X w will be the impulse or pressure exerted by the small mass 

 k X iv upon the small surface w, on which it insists. Again, the coordinates of the 

 end of k in the upper surface of the stratum, are 



k d <p Jc d (^ Jc d <^ 



and as the equations of the two curve surfaces are 



^ / , k d ^ , k d (p , k d <p\ 



we deduce. 



Wherefore, the pressure F X A^ X «^ of the mass k X w upon the small surface w, will 

 be equal to ^px ^^; which proves that the incumbent stratum exerts a constant 

 pressure upon the surface passing through A', the intensity at every point being equal 

 to I p. The like demonstration may be employed to show, that any other stratum 

 exerts a constant pressure upon the fluid below it ; and hence it follows, that all the 

 fluid above any of the interior surfaces, whatever be the number of strata it consists 

 of, presses with the same intensity at every point of the surface. Now the forces 

 urging the particles of the fluid decrease continually in approaching the centre of 

 gravity, at which point they are evanescent : wherefore the infinitely small mass, or 

 drop, contained within the surface nearest the centre, may be considered as free from 

 the action of any accelerating forces ; and, its surface being subjected to the constant 

 pressure of all the incumbent strata, these pressures, the directions of which ultimately 

 pass through the centre of gravity, will balance one another without any tendency to 

 produce either progressive or rotatory motion. 



If n be the number of strata above any of the interior surfaces, the intensity of 



