Minimum Energy in Vortex Motion. 533 



16. In the stable motion of minimum energy the rotational 

 portion of the fluid is in the shape of a cylindric shell, en- 

 closing the irrotational remainder, which in this case is at rest. 

 The symbolical expression for this motion is 



T = 0,whenr< V(« 2 -^ 2 ), 

 and T=f(r- ^^\ when r> ^{ot-b*). 



17. Let now the liquid be given in the configuration (14) 

 of greatest energy, and let the cylindric boundary be a sheet 

 of a real elastic solid, such as sheet-metal with the kind of 

 dereliction from perfectness of elasticity which real elastic 

 solids present ; that is to say, let its shape when at rest be a 

 function of the stress applied to it, but let there be a resist- 

 ance to change of shape depending on the velocity of the 

 change. Let the unstressed shape be truly circular, and let 

 it be capable of slight deformations from the circular figure 

 in cross section, but let it always remain truly cylindrical. 

 Let now the cylindric boundary be slightly deformed and left 

 to itself, but held so as to prevent it from being carried round 

 by the fluid. The central vortex column is set into vibration 

 in such a manner that longer and shorter waves travel round 

 it with less and greater angular velocity*. These waves cause 

 corresponding waves of corrugation to travel round the cylin- 

 dric bounding sheet, by which energy is consumed, and 

 moment of momentum taken out of the fluid. Let this pro- 

 cess go on until a certain quantity M of moment of momentum 

 has been stopped from the fluid, and now let the canister run 

 round freely in space, and, for simplicity, suppose its material 

 to be devoid of inertia. The whole moment of momentum 

 was initially — 



it is now 



7r?& 2 (a 2 -|6 2 )-M, 



and continues constantly of this amount as long as the 

 boundary is left free in space. The consumption of energy 

 still goes on, and the way in which it goes on is this : the 

 waves of shorter length are indefinitely multiplied and exalted 

 till their crests run out into fine laminae of liquid, and those 

 of greater length are abated. Thus a certain portion of the 

 irrotationally revolving water becomes mingled with the 

 central vortex column. The process goes on until what may 



* See ' Proceedings of the Royal Society of Edinburgh' for 1880, 

 or ' Philosophical Magazine ' for 1880, vol. x. p. 155 : " Vibrations of a 

 Columnar Vortex : " Win. Thomson. 



