WORK OF THE POXCELET WATER WHEEL. 287 



the wheel. The water will therefore drop off the floats de- 

 prived of nearly all its kinetic energy. Nearly the whole of 

 the work of the stream must therefore have been expended 

 in driving the float ; and the water will have been received 

 without shock, and discharged without velocity. 



Let v and F be the velocities of the stream and float re- 

 spectively ; then the initial velocity of the stream relative 

 to the float is v V, and the water will continue to run up 

 the curved float until it comes to relative rest ; it will then 

 descend along the float, acquiring in its descent, under the 

 influence of gravity, the same relative velocity which it had 

 at the beginning of its ascent, but in a contrary direction. 

 Therefore the absolute velocity of the water leaving the 

 float is F (v V) = 2 F v. 



Now the useful work U done on the wheel must equal 

 the work stored in the water at first, diminished by the 

 work stored in the water on leaving the wheel ; hence 



W W 



-F)F. (1) 



Comparing this expression with (1) of Art 156, we see 

 that the work performed by the Poncelet wheel is double 

 that of the common undershot wheel. 



Sen. This wheel works to the best advantage when the 

 speed of the periphery is one-half that of the stream (Art. 

 154, Cor.). This conclusion also follows from the form of 

 the floats, as above described ; since if all the work is taken 

 out of the water when it leaves the floats, its velocity must 

 then be zero, and therefore 2 F v = 0, or F= y.* 



The efficiency of a Poncelet wheel has been found in ex- 



* The inventor, Poncelet, states that, in practice, the velocity of the water, in 

 order to produce its maximum effect, ought to be about 21 times that of the wheel, 

 and that the efficiency of the wheel is about 0,7 (Tate's Mech. Phil., p. 818). 



