494 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



which supposes that the centre of gravity of the whole body of fluid continues at rest 

 and free from the action of any forces. Thus it appears that G, the only point of a 

 fluid in equilibrium not acted upon by any force, is no other than the centre of gravity 

 of the mass. 



2. The equilibrium of a fluid entirely at liberty will not be disturbed by a pressure 

 of the same intensity applied to all the parts of the exterior surface. 



By the intensity of a pressure is meant the amount of it when applied to some 

 given surface, most conveniently to the unit of surfaces. A constant pressure, or one 

 acting uniformly with the same intensity, is proportional to the surface to which it is 

 applied. 



This being understood, what is affirmed above is an immediate consequence of the 

 fundamental property of an incompressible fluid to transmit a pressure exerted upon 

 its surface in all directions without any loss of intensity. The inward pressure upon 

 any part of the surface thus produces an equivalent outward pressure upon every other 

 part, which is balanced by the contrary pressure supposed to act over the whole sur- 

 face. Wherefore if a mass of fluid be in equilibrium, it will continue in equilibrium, 

 supposing a pressure of the same intensity to be applied to all parts of the surface. 



3. The action of the forces upon the particles in the interior parts of the body of 

 fluid is next to be considered. 



Take any point {x y z) of the mass, and draw through it in any direction a plane 

 surface w infinitely small and of any figure ; from the same point {x y z) draw the 

 infinitely short line S s perpendicular to w, and construct an upright prism upon the 

 base w with the height ^ s. The forces acting upon a particle at the point {x y z) 

 being represented as before by X, Y, Z, and the coordinates of the end of ^ * being 

 X + ^ -^^ 3/ + ^2/j ^ + ^ ^3 we shall have this identical equation. 



(X^f +Y-|f- + z4^)x^^X«^=(X$^ + Y^3/ + Z^^) X 



w 



or by introducing a new symbol, 



F X ^ * X w; = (X ^ x + Y ^y + Z ^ ;s) X 2^. 



% X ^ y 8 s 

 ^^^ T7' T7' T7^ ^^^ cosmes of the angles which the directions of the forces 



make with I s : wherefore X y^ , ^17' ^ fl* ^^^ *^^ partial forces urging the par- 

 ticle {xyz) in the direction of ^* ; and the whole accelerating force in the same direc- 

 tion is equal to F. The density being constant, and represented by unit, the mass of 

 the prism will be equal to^s X iv\ and as this may be as small as we plBase, we may 

 assume that every particle of it is urged by the same force F ; so that F X ^ * X «^ is 

 the effort of the prism to move from the point {xy z) in the direction of ^^. Let/?, a 



