Canon Moseley on the steady Flow of a Liquid. 187 

 vis viva of this liquid is therefore — ^— — v 2, ; 



w j-E^lV^f^ . . (4) 

 Jo 2 9 9 J 



The work U 3 expended per unit of time on the resistance of 

 the internal surface of the pipe to the flow of the liquid is equal 

 to the entire resistance of the surface multiplied by the velocity V 

 of the liquid in contact with it, since V is the distance through 

 which that resistance is overcome per unit of time. But the 

 resistance of the pipe per unit of surface is 



/^i + V^ 2 ( see equation 1), 

 and the surface is 2ttJI! ; 



/. U 3 =27tR/(^ 1 +A, 1 V 2 )V (5) 



To determine U 4 , which represents the aggregate internal 

 work of the mutual resistances of the successive films of liquid, 

 let it be observed that, as v represents the velocity of the film 



whose radius is r, v — (j~ jdr represents that of the film whose 



radius is r-\-dr, the negative sign being taken because as r in- 

 creases v diminishes. The distance by which one film slips over 



the next in the unit of time is therefore represented by — ( -7- Jdr. 



But the resistance opposed to this slipping is 2irrlp } 



To determine the unit of resistance p which is opposed by the 

 film whose radius is r + dr to the motion over it of that whose 



radius is r, let the velocity V ~~\-T,)dr of the former film be sup- 

 posed to be communicated in an opposite direction to both. 

 The resistance of the one film opposed to the motion over it of 

 the other will not thus be changed, but the former will be 

 brought to rest, and the other will move over it with the velocity 



dv 



j- dr. The case will thus become the same with that of the 



dr 



film which moves in contact with that fixed to the internal sur- 

 face of the pipe, except that the constants //,] and \ x will have 

 different values. Let these values be ft and X, then 



in which the second term may be neglected as of infinitely small 



