188, 189] EQUATION OF ACTIVITY. 503 



30) 



since the volume of integration is fixed. The first term of the integral 

 represents the kinetic energy and the second term, the potential 

 energy due to the applied forces. The term on the left in 29) is 

 accordingly the rate of increase of the energy, kinetic and potential. 

 Of the terms on the right, the amount of matter Qqcos(qn)dS 

 flowing through dS in unit time brings with it the energy 



so that the first part of the surface integral represents the total in- 

 flow of energy. The remaining surface integral and volume integral 

 containing p represent the work done by the pressure, for at the 

 surface the velocity q and the force pdS give the activity 



pqcos (qri)dS, 



so that the surface integral represents the activity of the pressure at 

 the surface. 



If we consider a small element of volume F, the work done in 

 compressing it by an amount dV is as above pdV, and the activity 



Q1 \ dV du . dv 



31) -r- = 



Putting V=dr and integrating, we find that 



rrr isu . a. 



is the activity of the pressure in producing changes of density in 

 the whole mass. Transposing this term we find that equation 29) 

 expresses the following: The rate of increase of energy of the fluid, 

 both kinetic and potential, due to the external forces plus the activity 

 of compression (production of intrinsic energy) is equal to the rate 

 of inflow of energy plus the activity of pressure at the surface. 

 Equation 29) is therefore the equation of activity or conservation of 

 energy. 



189. Steady Motion. Steady motion is defined as a motion 

 which is the same at all times. Assuming that not only X, T 9 Z, V 

 but^f, v,w,p, Q are independent of t, equations 27) for steady motion 

 become 



32) 



