7-9] PHYSICAL EQUATIONS. 7 



across the remaining faces, we have for the total gain of mass, 

 per unit time, in the space SxSySz, the formula 



id. pu d.pv d. pw\ 

 - f H 7 - H f 

 \ dx dy a* J 



dy 



Since the quantity of matter in any region can vary only in 

 consequence of the flow across the boundary, this must be 

 equal to 





whence we get the equation of continuity in the form 

 dp ^d.pu [ d.pv [ 



dt dx dy dz 



9. It remains to put in evidence the physical properties of 

 the fluid, so far as these affect the quantities which occur in our 

 equations. 



In an * incompressible ' fluid, or liquid, we have Dp/Dt 0, in 

 which case the equation of continuity takes the simple form 



+ * + *2 = ........................ (1). 



dx dy dz 



It is not assumed here that the fluid is of uniform density, 

 though this is of course by far the most important case. 



If we wished to take account of the slight compressibility of 

 actual liquids, we should have a relation of the form 



P = K(P-P O )/P O ........................ (2), 



or p/p = l+p/H; ........................... (3), 



where K denotes what is called the f elasticity of volume.' 



In the case of a gas whose temperature is uniform and constant 

 we have the ' isothermal ' relation 



PlP=P/Po ........................... (4), 



where p , p are any pair of corresponding values for the tempera- 

 ture in question. 



In most cases of motion of gases, however, the temperature is 

 not constant, but rises and falls, for each element, as the gas is 

 compressed or rarefied. When the changes are so rapid that we can 

 ignore the gain or loss of heat by an element due to conduction 

 and radiation, we have the 'adiabatic' relation 



