474 BEPOET — 1880. 



II. The condition for steady motion of an incompressible inviscid fluid filling a 

 finite fixed portion of space (that is to say, motion in which the velocity and direc- 

 tion of motion continue imchanged at every point of the space within which the 

 fluid is placed) is that, with given vorticity, the energy is a thorough maximum, 

 or a thorough minimum, or a minimax. The farther condition of stability is secured 

 by the consideration of energy alone for any case of steady motion, for which the 

 energy is a thorough maximum or a thorough minimum ; because when the boun- 

 dary is held fixed the energy is of necessity constant. But the mere consideration 

 of energy does not decide the question of stability for any case of steady motion in 

 which the energy is a minimax. 



III. It is clear that, commencing with any given motion, the energy may be 

 increased indefinitely by properly-designed operation on the boundary (understood 

 that the primitive boundary is returned to). Hence, with given vorticity, there is 

 no thorough maximum of energy m any case. There may also be co7nplete annid- 

 ment of the energy by operation on the boundary (with return to the primitive 

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



1. The case of two equal, parallel, and oppositely rotating vortex columns 

 terminated perpendicularly by two fixed parallel planes, which, by proper operation 

 on the boundary, may be so mixed (like two eggs ' whipped ' together) that, in- 

 finitely near to any portion of either, there shall be some of the other. 



2. The case of a single Helmholtz ring, reduced by diminution of its aperture to 

 an infinitely long tube coiled within the enclosure. 



3. The case of a single vortex column, with two ends on the boundary, bent 

 till its middle meets the boundary ; and farther bent and extended, till it is broken 

 into two equal and opposite vortex columns ; and then farther dealt with till these 

 two are whipped together to mutual annihilation. 



IV. 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 surface perpendicular to them, subjected to changes of figure (but always 

 truly cylindric and perpendicular to the planes). 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. 

 Wo shall, however, see farther (XI. below) that possibly in every case, except cases 

 of a narrow, well-defined character, and certainly in many cases, there is an infinite 

 number of stable steady motions. 



V. In the present case, clearly, though there are an infinite number of unstable 

 steady motions, there are only two stable steady motions — those of absolute maxi- 

 mum and of absolute minimum energy. 



VI. In every steady motion, when the boundary is circular, the stream lines 

 are concentric circles, and the fluid is distributed in co-axial cylindric layers 

 of equal vorticity. In the stable motion of maximum energy, the vorticity is 

 greatest at the axis of the cylinder, and is less and less outwards to the circumfer- 

 ence. In the stable motion of minimum energy the vorticity is smallest at the 

 axis, and greater and greater outwards to the circumference. To express the con- 

 ditions symboUcally, let T be the velocity of the fluid at distance r from the axis 

 (understood that the direction of the motion is perpendicular to the direction of r) f 

 the vorticity at distance r is — 



♦(f-ff> 



If the value of this expression diminishes from >• = to r = a, the motion is stable, 

 and of maximum energy. If it increases from ?• = to r = « the motion is stable 

 and of minimum energy. If it increases and diminishes, or diminishes and in- 

 creases, as r increases continuously, the motion is unstable. 



Vll. As a simplest subcase, let the vorticity be uniform through a given por- 

 tion of the whole fluid, and zero through the remainder. In the stable motion of 

 greatest energy, the portion of fluid having vorticity will be in the shape of a cii'cular 



