6 THE EQUATIONS OF MOTION. [CHAP. I. 



element of Art. 6, the parts of this surface-integral due to the two 

 ^-faces are 



\ + ^ dx / tydz* an d - (u J dx\ dydz, 



which give together -7- dxdydz. Calculating in the same way 



the parts due to the other faces, we find 



du dv dw 



Since we have also V= dxdydz, (6) becomes 



dv 



or, as it may be written, 



dp d.pu d.pv d. pw ( . 



-7: H ---- 7 -- 1 -- 7 -- 1 -- 7 = ................ (o). 



at dx dy dz 



This is called the equation of continuity. 



If the fluid be incompressible though not necessarily of uniform 

 density, the value of p does not alter as we follow any element, 



i. e. x~ = 0, so that (7) becomes 



du dv dw , . 



~r + -7- + -T- = ........................ ( 9 )- 



dx dy dz 



The expression 



du dv dw 



dx dy dz 



which, as we have seen, measures the rate of increase of volume of 

 the fluid at the point (x, y, z), is very conveniently termed the 

 expansion at that point. 



9. There are certain restrictions as to the values of the 

 dependent variables in the foregoing equations. 



Thus u, v, w, p, p are essentially single-valued functions. 



The quantities u } v, w must be finite, and in general continuous, 

 though we may have isolated surfaces at which the latter restriction 

 does not hold. If the fluid move so as always to form a continuous 

 mass, a certain condition, given in Art. 10, must be satisfied at such 

 a surface. 



The quantity p is necessarily continuous, and finite. It is also 



