189] STEADY MOTION. 505 



Combining this with equation 36), 

 38) #i-i> a = -S0 



which determines q in terms of the difference of pressures. The 

 flux in unit time is then 



The theorem expressed by equation 33) is known as Daniel Bernoulli's 

 theorem. 



For gases expanding isothermally 



P = a log p = a logp -f const. 

 Consequently equation 33) becomes 



39) a logp -}- - q 2 = const. 



This formula may be used to calculate the velocity of efflux through 

 an orifice from a vessel containing gas under pressure. If the pressure 

 in the vessel at a point so remote from the orifice that the air may 

 be considered at rest is p and if the pressure of the atmosphere at 

 the orifice where the velocity is q is p , we have 



alogp = alogp Q + y 2 , 



\^ 



40) (f = 2 a log 



If the efflux is adiabatic, as in practice it nearly is, by 178, 18) 

 Accordingly 



41) q 2 = \ 



which is the usual formula for the efflux of gases. 



If the external force is gravity V=g0, so that equation 33) 

 becomes for an incompressible fluid, 



42) -f qz + -^ q 2 = const. 



Q 2 



If we consider efflux from a reservoir whose upper free surface is 

 so large that q is negligible, the pressure being that of the atmo- 



