TBANSACTIONS OF SECTION A. 475 



cylinder rotating like a solid round its own axis, coinciding with the axi^ of the 

 enclosure ; and the remainder of the fluid will revolve irrotationally around it, so 

 as to fulfil the condition of no finite slip at the cylindrical interface hetween the 

 rotational and irrotational portions of the fluid. The expression for this motion in 

 symbols is 



T = Cr from »• = to r = 6 ; 



and T = ^ — from r = 6 to r = «. 

 r 



VIII. In the stable motion of minimum energy the rotational portion of the 

 fluid is in the shape of a cylindric shell, inclosing the irrotational remainder, which 

 in this case is at rest. The symbolical expression for this motion is 



T = 0, when r <^ («- - 6'-) and T = C (^r - '^^), 

 when r > ^ {cc^ — b"), 



IX. Let now the liquid be given in the configuration VII. of greatest energy, and 

 let the cylindrical 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 resistance 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 re- 

 main truly cylindrical. Let now the cylindric boundary be slighly deformed and 

 left to itself, and lield 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 cylindric bounding 

 sheet, by which energy is consumed, and moment of momentum taken out of the 

 fluid. Let this process go on until a certain quantity of moment of momentmn 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 is initiallv 



It is now 



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 cen- 

 tral vortex column. The process goes on until what may be called a vortex sponge 

 is formed ; a mixture homogeneous on a large scale, but consisting of portions of 

 rotational and irrotational fluid, more and more finely mixed together as time ad- 

 vances. The mixture is, as indicated above, altogether analogous to the mixture of 

 the substances of two eggs whipped together in the weU-known culinary operation. 

 Let b' be the radius of the cylindric vortex sponge, b being as before the radius of 

 the original vortex column 



X. Once more, hold the cylindric case from going round in space, and continue 

 holding it until some more moment of momentum is stopped from the fluid. Then 

 leave it to itself agam. The vortex sponge will swell by the mingling with it of an 

 additional portion of irrotational liquid. Continue this process vmtil the sponge 

 occupies the whole enclosure. 



* See Proceedings of the Royal Society of EdinUirgh for 1880, or Pkiloso2>Jdcal Maga- 

 zine for 1880 : ' Vibrations of a Columnar Vortex : ' Wm. Thomson. 



