186 Canon Moseley on the steady Flow of a Liquid. 



U= work done per unit of time on the liquid which enters 

 the pipe by the pressure of that in the reservoir, 



Uj = work carried away per unit of time by the liquid which 

 flows from the extremity of the pipe, 



U 2 — work expended on the various resistances which are op- 

 posed to the descent of the liquid in the reservoir and to 

 its passage from the reservoir through its aperture into 

 the pipe, 



U 3 = work expended on the resistance of the internal surface 

 of the pipe to the flow of the liquid along it, 



U 4 = internal work of the resistance of the films to the flowing 

 of each film over the surface of the next in succession ; 

 then, by the principle of virtual velocities, 



U = U 1 + U 2 + U 3 + U 4 (2) 



Let v = velocity of any film, 



v = velocity of the filament which coincides with the axis 

 of the pipe, 



Y= velocity of the film which is in contact with the sur- 

 face of the pipe, 



R= internal radius of pipe, 



r= radius of the film whose velocity is v, 



p == resistance per unit of surface to the sliding of the film 

 whose radius is r over that whose radius is r-\-dr, 



h= height of the liquid in the reservoir above the centre 

 of the aperture, 



Z= length of the pipe, 



w— weight of cubic unit of liquid. 



Let the unit of length in all the above measurements be the 

 French metre, and the unit of weight the French kilogramme. 



The weight of the liquid which flows out of the pipe per unit 

 of time is represented by 



\ wvi^irrdr). 



This weight of liquid is therefore that which descends through 

 the height h in the reservoir per unit of time ; 



.*. U = A I wv{2irrdr) = ^7rwh\ vrdr. ... (3) 



Jo Jo 



Also the work U 1 which the liquid flowing out of the pipe 

 carries away with it per unit of time, is represented by half its 

 vis viva. 



But the weight of liquid which flows per unit of time between 

 two films whose radii are r and r + dr is w{27rrdr)v. Half the 



