Sir William Thomson on Vortex Statics. 105 



nate particular shape for each of them which will give steady 

 motion ; and I think we may confidently judge that the motion 

 is stable in each, provided only the core is sufficiently thin. 

 It is more easy to judge of the cases in which there are mul- 

 tiple sinuosities by a synthetic view of them (§3) than by 

 consideration of the maximum-minimum problem of § 8. 



17. It seems probable that the two- or three- or multiple- 

 threaded toroidal helix motions cannot be stable, or even steady, 

 unless I, fi, and IN" are such as to make the shortest distances 

 between different positions of the core or cores considerable 

 in comparison with the core's diameter. Consider, for example, 

 the simplest case (§12, fig. 5) of two simple rings linked 

 together. 



18. Go back now to the simple circular Helmholtz ring. It 

 is clear that there must be a shape of absolute maximum energy 

 for given vorticity and given impulse, if we introduce the re- 

 striction that the figure is to be a figure of revolution — that is 

 to say, symmetrical round a straight axis. If the given vor- 

 ticity be given in this determinate shape, the motion will be 

 steady ; and there is no other figure of revolution for which it 

 would be steady (it being understood that the impulse has a 

 single force resultant without couple). If the given impulse, 

 divided by the cyclic constant, be very great in comparison 

 with the two-thirds power of the volume of liquid in which the 

 vorticity is given, the figure of steadiness is an exceedingly 

 thin circular ring of large aperture and of approximately cir- 

 cular cross section. This is the case to which chiefly attention 

 is directed by Helmholtz. If, on the other hand, the impulse 

 divided by the cyclic constant be very small compared with 

 the two-thirds power of the volume, the figure becomes like a 

 long oval bored through along its axis of revolution and with 

 the ends of the bore rounded off (or trumpeted) symmetrically, 

 so as to give a figure something like the handle of a child's 

 skipping-rope, but symmetrical on the two sides of the plane 

 through its middle perpendicular to its length. It is certain 

 that, however small the impulse, with given vorticity the figure 

 of steadiness thus indicated is possible, however long in the 

 direction of the axis and small in diameter perpendicular to 

 the axis and in aperture it may be. I cannot, however, say at 

 present that it is certain that this possible steady motion is 

 stable ; for there are figures not of revolution, deviating infi- 

 nitely little from it, in which, with the same vorticity, there is 

 the same impulse and the same energy, and consideration of 

 the general character of the motion is not reassuring on the 

 point of stability when rigorous demonstration is wanting. 



19. Hitherto I have not indeed succeeded in rigorously 

 Phil. Mag. S. 5. Vol. 10. No. 60. Aug. 1880, I 



