54-56] ENERGY IN CYCLIC REGIONS. 63 



denotes the work done by the impulsive forces applied to that 

 membrane ; and so on. Hence (5) expresses the fact that the 

 energy of the motion is equal to the work done by the whole 

 system of impulsive forces by which we may suppose it generated. 



In applying (5) to the case where the fluid extends to 

 infinity and is at rest there, we may replace the first term of 

 the third member by 



-<ft (6), 



where the integration extends over the internal boundary only. 

 The proof is the same as in Art. 46. When the total flux across 

 this boundary is zero, this reduces to 



^d8 (7). 



an 



The minimum theorem of Lord Kelvin, given in Art. 45, may 

 now be extended as follows : 



The irrotational motion of a liquid in a multiply-connected 

 region has less kinetic energy than any other motion consistent 

 with the same normal motion of the boundary and the same value 

 of the total flux through each of the several independent channels 

 of the region. 



The proof is left to the reader. 



Sources and Sinks. 



56. The analogy with the theories of Electrostatics, the 

 Steady Flow of Heat, &c., may be carried further by means of the 

 conception of sources and sinks. 



A * simple source ' is a point from which fluid is imagined to 

 flow out uniformly in all directions. If the total flux outwards 

 across a small closed surface surrounding the point be 4-Trra*, then 

 m is called the ' strength ' of the source. A negative source is 

 called a 'sink.' The continued existence of a source or a sink 

 would postulate of course a continual creation or annihilation of 

 fluid at the point in question. 



* The factor 4?r is introduced to keep up the analogy referred to. 



