Equations of Motion of a Viscors Liquid. 41 



dx 



uz 

 dp <5m 



or as usually written: 



pdz 'H 



So far we have obtained three relations between the five 

 quantities u, r, tv, p, and f>. When these equations are used 

 in application to an incompressible fluid, /> is constant; but 

 when gases are under consideration, a fourth equation is 

 supplied by Boyle's Law, 



p=kp. 



Continuity. — The condition of continuity furnishes a fifth 

 equation. If the fluid be continuous, then the increase within 

 a given space during any given time must equal the excess 

 of inflow over outflow across the boundaries in the given 

 time. 



The mass of fluid contained by the element at any mo- 



°^^^^' ^« p dx dy dz 



and its increase in the time dt is 



— dt dx dy dz. 

 dt ^ 



The flow over the surface dy dz in time dt is 



n p dt dy dz 

 and the flow over the opposite face is 



u p dt dy dz-\ — ^^ — - dt dy dz dx, 

 dx 



giving an increase in the element of 



■ dt dx dy dz. 



dx 



