Minimum Energy in Vortex Motion. 



531 



an infinite number of stable steady motions of minimum 

 (though not least minimum) energy. 



13. That there can be an infinite number of configurations of 

 stable motions, each of them having the energy of a thorough 

 minimum (as said in § 12), we see, by considering the case 

 in which the cylindric boundary of the containing canister 

 consists of two wide portions communicating by a narrow 

 passage, as shown in the drawings. If such a canister be 

 completely filled with irrotationally moving fluid of uniform 

 vorticity, the stream-lines must be something like those indi- 

 cated in fig. 4. 



Fit?. 4. 



Hence, if a not too great portion of the whole fluid is irro- 

 tational, it is clear that there may be a minimum energy, and 

 therefore a stable configuration of motion, with the whole of 

 this in one of the wide parts of the canister ; or the whole in 

 the other ; or any proportion in one and the rest in the other. 



Fiff. 5. 



Single intersection of stream-lines in rotational motion 

 may be at any angle, as shown in fig. 4. It is essentially 

 at right angles in irrotational motion, as shown in fig. 5, 

 representing ihe stream-lines of the configuration of maxi- 

 mum energy, for which the rotational part of the liquid is 

 in two equal parts, in the middles of two wide parts of the 

 enclosure. There is an infinite number of configurations of 



