530 



Sir "W. Thomson on Maximum ant 



referred to in § 3 III. above, be complete annulment of the 

 energy by operation on the boundary (with return to the pri- 

 mitive boundary), as we see by the following illustrations : — 



(a) Two equal, parallel, and oppositely rotating, vortex 

 columns terminated perpendicularly by two fixed parallel 

 planes. By proper operation on the cylindric boundary, they 

 may, in purely two-dimensional motion, be thoroughly and 

 equably mixed in two infinitely thin sheets. In this condition 

 the energy is infinitely small. 



(6) A single Helmholtz ring, reduced by diminution of its 

 aperture to an infinitely long tube coiled within the enclosure. 

 In this condition the energy is infinitely small. 



(c) A single vortex column, with two ends on the boundary, 

 bent till its middle infinitely nearly meets the boundary; and 

 further bent and extended till it is broken into two equal and 

 opposite vortex columns, connected, one end of one to one end 

 of the other, by a vanishing vortex ligament infinitely near 

 the boundary ; and then further dealt with till these two 

 columns are mixed together to virtual annihilation. 



12. To avoid, for the present, the extremely difficult general 

 question illustrated (or suggested) by the consideration of such 

 cases, confine ourselves now to two-dimensional motions in a 

 space bounded by two fixed parallel planes and a closed 

 cylindric, not generally circular cylindric, surface perpen- 

 dicular to them, subjected to changes of figure (but always 

 truly cylindric and perpendicular to the planes) . Also, for 

 simplicity, confine ourselves for the present to vorticity either 

 positive or zero, in every part of the fluid. It is obvious that, 

 with the limitation to two-dimensional motion, the energy 

 cannot be either infinitely small or infinitely great with any 

 given vorticity and given cylindric figure. Hence, under 

 the given conditions, there certainly are at least two stable 

 steady motions — those of absolute maximum and absolute 

 minimum energy. The configuration of absolute maximum 

 energy clearly consists of least vorticity (or zero vorticity, if 

 there be fluid of zero vorticity) next the boundary and greater 

 and greater vorticity inwards. The configuration of absolute 

 minimum energy clearly consists of greatest vorticity next 

 the boundary, and less and less vorticity inwards. If there 

 be any fluid of zero vorticity, all such fluid will be at rest 

 either in one continuous mass, or in isolated portions sur- 

 rounded by rotationally moving fluid. For illustration, see 

 figs. 4 and 5, where it is seen how, even in so simple a case as 

 that of the containing vessel represented in the diagram, there 

 can be an infinite number of stable steady motions, each with 

 maximum (though not greatest maximum) energy ; and also 



