where q^ = vr ^- v -^ w 



Here fdp/p can be evaluated when the law of variation of p with p is known. The 

 integration "constant" F{t) must be regarded as an arbitrary function of t, for the mathemati- 

 cal reason that the integration involved only x, y, and s, and also for the physical reason that 

 itself contains an arbitrary "constant" which may be supposed to vary with the time. The 

 presence of this arbitrary term in cf) limits the usefulness of Equation [9d] in the general case. 



If the density is uniform and constant, 



and 



/■ 



dp p 



— = — + constant 

 P P 



-^ =— --9^ -fi + F(0, [9e] 



P dt 2^ 



where the arbitrary constant in the integral has been absorbed in F(t). In this case the 

 pressure itself contains an arbitrary additive constant; for it is well known that a uniform 

 pressure p^ applied to the boundary of an enclosed mass of incompressible liquid merely 

 raises the pressure throughout by the amount p^ without affecting the motion. To fix the 

 pressure completely, therefore, its value on the boundary must be known. This value then 

 fixes F(t) after the arbitrary additive function of t that occurs in has been chosen. The 

 equation thus obtained is very useful. 



If the only external forces acting are those due to gravity, it is sometimes convenient to 

 simplify the last equation by considering the pressure to be made up of two parts, p = p^ + p^ 

 where p is the hydrostatic pressure that would exist if there were no motion and p^ is the 

 dynamic pressure due to changes in velocity. When there is no motion, Equation [9e] gives, 

 with p substituted for p and F (t) assumed to be constant, 



p ^ + pQ = constant. [9f] 



If, then, p is replaced by p^ + Pj in Equation [9e] and p^ then substituted from Equation [9f], 

 the result is 



P dt 2^ 



where the constant in Equation [9f] has been absorbed in F{t). In Equation [9g], p is some- 

 times written for p^. 



When the only external forces are gravitational, if the 2-axis is drawn vertically upward, 

 X = Y = 0, Z = -g, where g is the acceleration due to gravity. Hence fl is a function of z only, 

 and integration of [9c] gives 



i). = gz + constant. 



Dynamical units are to be understood in Equations [9e] and [9g]. The pressure p or p^ 

 may be in pounds per square foot, p in slugs per cubic foot, q in feet per second, and t in 

 seconds. The potential cb would then be in feet squared per second, since the dimensions of 

 (ji are those of length times velocity or L^ T~ ^ . 



16 



