188 Canon Moseley on the steady Flow of a Liquid. 

 dimensions compared with the first, 



AU 4 =-2rfJ%r<J)*j (6) 



.*. by equation (2), 



r r 



2irwh\ vrdr 



= — ( R v 3 rdr + U 2 + 2ttZR (/*, + ^V 2 ) V- 2tt/ f * /*r (^ W ; 



or considering separately the case of a portion of the liquid 

 bounded by a film whose radius is r } 



2irwh I vrdr — — \ v 3 rdr + U 9 — 2irl \ ur( — )dr, 

 Jo ffl Jo V*"' 



in which equation the work of the bounding film of the portion 

 of the liquid which is considered separately from the rest, is in- 

 cluded in the last integral. Differentiating the above equation, 

 considering U 2 constant, and reducing, 



*-*-¥£) w 



If there were no resistance to the flow of the films over one an- 

 other, or to the flow of the liquid over the internal surface of the 

 pipe, the whole of the work done by the weight of the liquid in the 

 reservoir on that in the pipe per unit of time would be accumulated 

 in the liquid discharged per unit of time. Let v be the velocity 

 the liquid would under these circumstances acquire. Then 7rR 2 u 



represents the discharge per unit of time, and — - — - v 2 repre- 



9 

 sents the work accumulated in it. Also hw7rWv represents the 



work done upon it per unit of time by the pressure of the liquid 



in the reservoir, W7r ^ v 



.'. — - tr = rtl07T.K. 1/ : 



2g 



.-. v 2 = 2gh (8) 



The same result is arrived at, as it ought to be, by making 



/uL = in equation (7), and substituting v for v. By equation (7), 



w \dr J 

 (dv\ 1 w 



' * \dr').v(v i -v>) ~~ - 2gt^ 



But 1 = lrl I/_j_ 1 Y] 



z>(u 2 -v 2 ) u 2 [_v + 2\v-v v + vJ] '' 



" \dr)[y 2\v—v v + v)j ~ 2gl^ ~pl 



