OF THE EQUILIBRIUM OF FLUIDS. IQo 



the infinitely small differences ax, Ay, az, be the sides of 

 the parallelepiped: let p be the mean pressure on the dif- 

 ferent points of the surface AyAz, which is perpendicular 

 to X, and p' the same quantity belonging to its opposite 

 surface : the parallelepiped will be urged in the direction 

 of X by a force equal fo {p—p') AyAz. Now {p' — p) is the 

 difference of p, taken on the supposition that x alone is 

 variable; for though ^ is supposed to act in the direction 

 contrary to that oi p, yet the pressure, th^t a point of a 

 fluid undergoes, being the same in all directions, we may 

 consider j9' — p as the difference of the two forces, acting 

 in the same direction, at an infinitely small distance from 

 each other : so that we hawep'—pzzA^cp, and {p—p')AyAZ 



=: — Aj:pAyAz:=z-^ AXAyAz, Let P, Q, and R be the three 



accelerating forces which act on the fluid particles, inde- 

 pendently of their connexions, in directions parallel to x, 

 y, and z: if we call the density of the parallelepiped ^ , its 

 mass will be ^AxAyAz, and the product of the force P by 

 this mass will represent the whole motive force derived 

 from it; consequently the whole force, acting in the direc- 

 tion of jr, will be (f P ^\ AX AyAz, For similar rea- 

 sons, the elementary system will be solicited, in directions 

 parallel to y and 2, by the forces {g Q ^ j ax AyAz, and 



Ay 



(§ R P\ AxAyAz. We shall therefore have, for the 



conditions of equilibrium (6) (251) 



5^z=f(P3x + Qay + i25z)[:since^=^,and^»z:^^x + 



ox AX wr 



O 2 



