V A it TABLE MOTION. 229 



Any particle G, beneath the surface of the water, is 

 pressed forward by the head AH = EG, and pressed back- 

 ward by the head BG ; hence the head which produces mo- 

 tion is EG BG = EB, or h, as before ; and therefore (2) 

 is true for any particle. Solving (2) for h, and adding the 

 resistance of friction, as given in (1), we have 



* = +/, (3) 



in which p, a, and v denote the mean values of the wetted 

 perimeter, the transverse section, and the velocity, respect- 

 ively. 



transverse sections, respectively, and Q the quantity of 

 water which flows through any section in a unit of time, we 

 have 



, 



and Q = a v = a^. 



From (5), we have 



Now if the water flowed with the velocity v , we would 

 have the head due to the resistance of friction, from (1), 



~ J a 2g' V 



and if it flowed with the velocity v ly we would have the 

 head due to the resistance of friction 



But the former expression is less, and the latter is greater 

 than the true head due to the resistance of friction; hence, 



