HOMOGENEOUS FLUID AT LIBERTY. 495 



function of x, y, z, represent the intensity of pressure at the point (^y z), and p + lp 

 will be the intensity at the other end of Is: the external pressures acting upon the 

 opposite ends of the prism are therefore p X u; and {p -i-^p) X w \ and the difference 

 of these, ovlp X w, is the impulse causing the prism to move towards the point {xy z) 

 in the direction of I s. Now, the prism being at rest, the impulses ¥ X^s X w and 

 Ip X w, which tend to move it in opposite directions, must be equal ; wherefore, 

 taking the foregoing value of F X ^ * X «^, and suppressing the factor w, which is 

 common to the equal quantities, the non-effect of the opposite forces requires this 

 equation, 



--lp=^y:.liv~\-Yly-\-Zlz, 



which expresses that the effort of the accelerating forces to move the prism in any 

 direction is counterbalanced by the contrary action of the pressure. The equation 

 must hold at every point of the mass, without any relation being supposed between 

 the infinitely small quantities Ix, ly, Iz ; which condition requires that 



^Ix -^Yly -\- Zlz 



be the variation of a function in which the three variables x, y, z, are independent of 

 one another. If this function be represented by ^' {x,y, %,), so that 



fOLdx + Ydy -\-Zdz) = ^^ {x,y, z,), 



we shall have 



C — p = (p' {x,y,z,). 



The forces respectively parallel to x, y, z, are now thus expressed : 



_ d . (p' (^, y, z,) _ d'.<p^ {x,y,z,) „ _ d .(p^ {x,y,z,) 



^ — dx ' * — dy > ^ — dz 



The differentials of 9' {x, y, z,) vanishing at the centre of gravity, the function will 

 increase on every side in receding from that point ; and when it becomes equal to C, 



we shall have 



C = (p' {x, y, z,), 



which is the equation of the surface of the fluid, the pressure p being equal to zero 

 at all the points of that surface. 



If an ijifinitely narrow canal of any figure be extended from the point {x y z) to the 

 surface of the fluid, the intensity with which all the fluid in the canal presses at the 

 point {xy z) will be equal to the function p. Let the whole 

 length of the canal be divided into small parts, ^ *, § ^, ^ *", 

 &c. ; and at every point of division draw the sections w, 

 w\ «;", &c., perpendicular to the sides of the canal, which 

 will thus be divided into an infinite number of small prisms, 

 to every one of which the foregoing investigation will ap- 

 ply. Wherefore, the variation of the intensity of pressure, 



or Ip, in the length of any prism, will be just equal to the action of the accelerating 

 forces upon the particles of the prism ; and the intensity with which all the fluid in the 



3 s 2 



