532 Sir W. Thomson on Maximum and 



maximum energy in which the rotational part of the fluid is 

 unequally distributed between the two wide parts of the 

 enclosure. *"*M#j 



14. In every steady motion, when the boundary is cir- 

 cular, 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 

 circumference. In the stable motion of minimum energy the 

 vorticity is smallest at the axis, and greater and greater out- 

 wards to the circumference. To express the conditions sym- 

 bolically, let T be the velocity of the fluid at distance r from 

 the axis (understood that the direction of the motion is per- 

 pendicular to the direction of r), and let a be the radius of the 

 boundary. The vorticity at distance r is 



T dT\ 



2 \r 



+ TrY 



If the value of this expression diminishes from r = to r=a, 

 the motion is stable, and of maximum energy. If it increases 

 from r = to r = a, the motion is stable and of minimum 

 energy. If it increases and diminishes, or diminishes and 

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

 15. As a simplest subcase, let the vorticity be uniform 

 through a given portion 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 cir- 

 cular cylinder rotating like a solid round its own axis, coin- 

 ciding with the axis 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 cylindric interface between 

 the rotational and irrotational portions of the fluid. The 

 expression for this motion in symbols is 



T = £V from r = to r = 5, 



y 7 2 



and T = — from r = b to r = a. 



* This conclusion I had nearly reached in the year 1875 by rigid mathe- 

 matical investigation of the vibrations of approximately circular cylindric 

 vortices ; but 1 was anticipated in the publication of it by Lord Rayleigh, 

 who concludes his paper " On the Stability, or Instability, of certain Fluid 

 Motions" ('Proceedings of the London Mathematical Society,' Feb. 12, 

 1880) with the following statement : — " It may be proved that, if the fluid 

 move between two rigid concentric walls, the motion is stable, provided 

 that in the steady motion the rotation either continually increases or 

 continually decreases in passing outwards from the axis, 7 ' — which was 

 unknown to me at the time (August 28, 1880) when I made the com- 

 munication to Section A of the British Association at Swansea. 



