Minimum Energy in Vortex Motion. 535 



of an additional portion of irrotational liquid. Continue this 

 process until the sponge occupies the whole enclosure. 



After that continue the process further, and the result will 

 be that each time the containing canister is allowed to go 

 round freely in space, the fluid will tend to a condition in 

 which a certain portion of the original vortex core gets filtered 

 into a position next to the boundary, (within a distance from 

 the axis which we shall denote by c), and the fluid in this space 

 tends to a more and more nearly uniform mixture of vortex 

 with irrotational fluid. This central vortex sponge, on repe- 

 tition of the process of preventing the canister from going 

 round, and again leaving it free to go round, becomes more 

 and more nearly irrotational fluid, and the outer belt of pure 

 vortex becomes thicker and thicker. The resultant motion is 

 now 



T y b 2 + c 2 — a 2 e 



1 = ? g r, for r < c, 



o 



a 2 —!) 2 

 T = ^-?^^, for r>c; 



and the moment of momentum is 



The final condition towards which the whole tends is a belt 

 constituted of the original vortex core now next the boundary ; 

 and the fluid which originally revolved irrotationally round it 

 now placed at rest within it, being the condition (16 above) 

 of absolute minimum energy. Begin once more with the con- 

 dition (15 above) of absolute maximum energy, and leave the 

 fluid to itself, whether with the canister free to go round some- 

 times, or always held fixed, provided only it is ultimately held 

 from going round in space ; the ultimate condition is always 

 the same, viz. the condition (16) of absolute minimum energy. 

 The enclosing rotational belt, being the actual substance of the 

 original vortex, is equal in its sectional area to irb 2 ; and 

 therefore c 2 = a 2 — b 2 . The moment of momentum is now 

 ^7rfZ> 4 , being equal to the moment of momentum of the 

 portion of the original configuration consisting of the then 

 central vortex. 



19. It is difficult to follow, even in imagination, the very 

 fine — infinitely fine — corrugation and drawing-out of the 

 rotational fluid \ and its intermingling with the irrotational 

 fluid ; and its ultimate re-separation from the irrotational 

 fluid, which the dynamics of § § 17 , 18 have forced on our 

 consideration. This difficulty is obviated, and we substitute 



