Minimum Energy in Vortex Motion. 537 



from all such considerations at present by our limitation (§12) 

 to two-dimensional motion. 



22. Annul now the Laplacian attraction and return to our 

 purely adynamic system of incompressible fluid acted on only 

 by pressure at its bounding surface, and by mutual pressure 

 between its parts, but by no "applied force" through its 

 interior. For any given momentum between the extreme 

 possible values ir^b 2 (a 2 — \b 2 ) and \tt% b*, there is clearly, 

 besides the §§ 17, 18 solution (minimum energy), another 

 determinate circular solution, viz. the configuration of circular 

 motion, of which the energy is greater than that of any other 

 circular motion of same vorticity and same moment of 

 momentum. This solution clearly is found by dividing the 

 vortex into two parts — one a circular central column, and the 

 other a circular cylindric shell lining the containing vessel ; 

 the ratio of one part to the other being determined by the con- 

 dition that the total moment of momentum have the prescribed 

 value. But this solution (as said above, § 14 and footnote) 

 may be proved to be unstable. 



I hope to return to this case, among other illustrations of 

 instability of fluid motion — a subject demanding serious con- 

 sideration and investigation, not only by purely scientific 

 coercion, but because of its large practical importance. 



23. For the present I conclude with the complete solution, 

 or practical realization of the solution (only found within the 

 last few days, and after §§ 10-18 of the present article were 

 already in type) of a problem on which I first commenced trials 

 in 1868 : to make the energy an absolute maximum in two- 

 dimensional motion with given moment of momentum and given 

 vorticity in a cylindric canister of given shape. The solution is, 

 in its terms, essentially unique ; " absolute maximum ■' mean- 

 ing the greatest of maximums. But the same investigation 

 includes the more extensive problem : To find, of the sets of 

 solutions indicated in § 12, different configurations of the 

 motion having the same moment of momentum. For each 

 of these the energy is a maximum, but not the greatest 

 maximum, for the given moment of momentum, The most 

 interesting feature of the practical realization to which I have 

 now attained is the continuous transition from any one steady 

 or periodic solution, through a series of steady or periodic 

 solutions, to any other steady or periodic solution, produced by 

 a simple mode of operation easily understood, and always under 

 perfect control. The operating instrument is merely a stirrer, 

 a thin round column, or rod, fitted perpendicularly between 

 the two end plates, and movable at pleasure to any position 



Phil. Mag. S. 5. Vol. 23. No. 145. June 1887. 2 



