352 Sir W. Thomson on the Propagation of Laminar 



in the diagram. Annul now the rigidity of the sides of the 

 boxes. The motion continues unchangedly steady. But is 

 it stable, now that the rigid partitions are done away with ? 

 No proof has yet been given that it is. If it is, laminar 

 waves, such as waves of light, could be propagated through it ; 

 and the velocity of propagation would beRv/2/3 if the sides 

 of the ideal boxes parallel to the undisturbed planes of the rings 

 are square (which makes ave It 2 = ave w 2 ), and if the dis- 

 tance between the square sides of each box bears the proper 

 ratio to the side of the square to make ave v 2 = ave U 2 = ave w 2 . 



23. Consider now, for example, plane waves, or laminar 

 vibrations, in planes perpendicular to the undisturbed planes 

 of the rings. The change of configuration of the vortices in 

 the course of a quarter period of a harmonic standing vibra- 

 tion, f(y, t) = sin cot cos Kg (which is more easily illustrated 

 diagram matically than a wave or succession of waves), is illus- 

 trated in fig. 2, for a portion of the fluid on each side of y = 0. 

 The upper part of the diagram represents the state of affairs 

 when t = ; the lower when t = 7r/(2co). But it must not be 

 overlooked, that all this §§ 22, 23 depends on the unproved 

 assumption that the symmetrical arrangement is stable. 



24. It is exceedingly doubtful, so far as I can judge after 

 much anxious consideration from time to time during these 

 last twenty years, whether the configuration represented in 

 fig. 1, or any other symmetrical arrangement, is stable when 

 the rigidity of the ideal partitions enclosing each ring sepa- 

 rately is annulled throughout space. It is possible that the 

 rigidity of two, three, or more of the partitions may be an- 

 nulled without vitiating the stability of the steady symmetric 

 motion ; but that if it be annulled through the whole of space, 

 for all the partitions, the symmetric motion is unstable, and 

 the rings shuffle themselves into perpetually varying relative 

 positions, with average homogeneousness, like the ultimate 

 molecules of a homogeneous liquid. 1 cannot see how, under 

 these conditions, the "vitiating rearrangement" referred to 

 at the end of § 20 can be expected not to take place within 

 the period of a wave or vibration. To suppose the overall 

 diameter of each ring to be very small in proportion to its 

 average distances from neighbours, so that the crowd would be 

 analogous rather to the molecules of a gas than to those of a 

 liquid, would not help us to escape the vitiating rearrange- 

 ment which would be analogous to that investigated by Max- 

 well in his admirable kinetic theory of the viscosity of gases. 

 I am thus driven to admit, in conclusion, that the most 

 favourable verdict I can ask for the propagation of laminar 

 waves through a turbulently moving inviscid liquid is the 

 Scottish verdict of not proven. 



