308 HYDRAULICS AND ITS APPLICATIONS 



Then applying Bernoulli's equation of energy, we have 





Let the difference in the level of the water surface over the points A 

 and B be r. 



Then Z A + y A = Z R + y K + r 



.'. Z A Z R = r + y r> y A 



If the stream is sensibly parallel over the length A B, as will be the 

 case if 6 is not large, we have 



^ = ^andf=^ 



r 2 r 2 



If now we imagine the area A divided into n elementary sections, each 

 equal to we get, for each stream tube of area : 



^ ^ v 2 v 2 



n' n' 2 g n ' A B 



and, summing these over the whole section, 



'A r,; 2 



If we assume that the velocity is constant over any cross section, the 

 above equation reduces to 



where A H P is the total frictional loss between the cross sections at A 

 and B. 



This is still true of the whole mass of water in the stream if the 

 velocity at a cross section is not uniform, provided that the distribution 

 of velocity is such that the total kinetic energy at that section is equal to 

 the mass of water multiplied by the square of the mean velocity at the 

 section. In this case V K and V A become the mean velocities. Experi- 

 ments by Messrs. Fteley and Stearns on the flow of water in the Sudbury 

 conduit, 9 feet wide and 3 feet deep, in which the velocity was measured 

 at 97 different points in a cross section, gave results showing that the 

 error in assuming this to be true was less than 1 per cent. The error 



