536 Sir W. Thomson on Maximum and 



for the "vortex sponge *' a much easier (and in some respects 

 more interesting) conception, vortex spindrift, if (quite arbi- 

 trarily, and merely to help us to understand the minimum- 

 energy-transformation of vortex column into vortex shell) we 

 attribute to the rotational portion of the fluid a Laplacian* 

 mutual attraction between its parts " insensible at sensible 

 distances " and between it and the plane ends of the con- 

 taining vessel of such relative amounts as to cause the inter- 

 face between rotational and irrotational fluid to meet the end 

 planes at right angles. Let the amount of this Laplacian 

 attraction be exceedingly small — so small, for example, that 

 the work required to stretch the surface of the primitive 

 vortex column to a million million times its area is small in 

 comparison with the energy of the given fluid motion. 

 Everything w r ill go on as described in §§ 17, 18 if, instead of 

 " run out into fine laminae of liquid " (§ 17, line 31) we sub- 

 stitute break off into millions of detached fine vortex columns ; 

 and instead of " sponge " {passim) we substitute " spin- 

 drift." 



20. The solution of minimum energy for given vorticity 

 and given moment of momentum (though clearly not unique, 

 but infinitely multiplex, because magnitudes and orders of 

 breaking- off" of the millions of constituent columns of the 

 spindrift may be infinitely varied) is fully determinate as to 

 the exact position of each column relatively to the others ; and 

 the cloud of spindrift revolves as if its constituent columns 

 were rigidly connected. The viscously elastic containing 

 vessel, each time it is left to itself, as described in §§ 17, 18, 

 flies round with the same angular velocity as the spindrift 

 cloud within ; and so the whole motion goes on stably, without 

 loss of energy, until the containing vessel is again stopped or 

 otherwise tampered with. 



21. It might be imagined that the Laplacian attraction 

 would cause our slender vortex columns to break into detached 

 drops (as it does in the well-known case of a fine circular jet 

 of water shooting vertically downwards from a circular tube, 

 and would do for a circular column of water given at rest in 

 a region undisturbed by gravity), but it could not, because the 

 energy of the irrotational circulation of the fluid round the 

 vortex column must be infinite before the column could 

 break in any place. The Laplacian attraction might, how- 

 ever, make the cylindric form unstable ; but we are excluded 



* So called to distinguish it from the " Newtonian " attraction, because, 

 I believe, it was Laplace who first thoroughly formulated " attraction in- 

 sensible at sensible distances," and founded on it a perfect mathematical 

 theory of capillary attraction. 



