THE REACTION WHEEL; BARKER'S MILL. 289 



direction with the velocity F. The absolute velocity of the 

 water is therefore 



v V = VT* + fyh - V. (2) 



Now the useful work done per second by each pound of 

 water must equal the work due to the height h, diminished 

 by the work remaining in the water after leaving the 

 machine. Hence, 



* i T. ( v ~ F > 2 



useful work = h - = ' 



The whole work expended by the water fall is h foot- 

 pounds per second ; consequently, to find the efficiency of 

 the machine, we divide (3) by h (Anal. Mechs., Art. 216), 

 and get 



~~ - F) F 



> 



- 

 efficiency = v - -JL -- > (5) 



= 1 _ A + etc. (6) 



(by the Binomial Theorem), 



which increases towards the limit 1 as F increases towards 

 infinity. Neglecting friction, therefore, the maximum 

 efficiency is reached when the wheel has an infinitely great 

 velocity of rotation. But this condition is impracticable to 

 realize ; and even at practicable but high velocities of rota- 

 tion, the prejudicial resistances, arising from the friction of 

 the water and the friction upon the axis, would considera- 

 bly reduce the efficiency. Experiment seems to show that 

 the best efficiency of these machines is reached when the 

 velocity is that due to the head, so that F 8 = 2gh. 



