a Horizontal Layer of Fluid. 531 



instability is 4*508 times the diameter of the jet, exceeding 

 the wave-length at which instability first enters in the ratio 

 of about 3 : 2. Accordingly this is the sort of disintegration 

 to be expected when the jet is shielded as far as possible from 

 external disturbance. 



It will be observed that there is nothing in this theory 

 which could fix the phase of the predominant disturbance, 

 or the particular particles of the fluid which will ultimately 

 form the centres of the detached drops. There remains a 

 certain indeterminateness, and this is connected with the 

 circumstance that absolute regularity is not to be expected. 

 In addition to the wave-length of maximum instability we 

 must include all those which lie sufficiently near to it, and 

 the superposition of the corresponding modes will allow of 

 a slow variation of phase as we pass along the column. The 

 phase in any particular region depends upon the initial cir- 

 cumstances in and near that region, and these are supposed 

 to be matters of chance *. The superposition of infinite 

 trains of waves whose wave-lengths cluster round a given 

 value raises the same questions as we are concerned with 

 in considering the character of approximately homogeneous 

 light. 



In the present problem the case is much more compli- 

 cated, unless we arbitrarily limit it to two dimensions. The 

 cells of Benard are then reduced to infinitely long strips, 

 and when there is instability we may ask for what wave- 

 length (width of strip) the instability is greatest. The 

 answer can be given under certain restrictions, and the 

 manner in which equilibrium breaks down is then approxi- 

 mately determined. So long as the two-dimensional cha- 

 racter is retained, there seems to be no reason to expect the 

 wave-length to alter afterwards. But even if we assume a 

 natural disposition to a two-dimensional motion, the direc- 

 tion of the length of the cells as well as the phase could 

 only be determined by initial circumstances, and could not 

 be expected to be uniform over the whole of the infinite 

 plane. 



According to the observations of Benard, something of! this 

 sort actually occurs when the layer of liquid has a general 

 motion in its own plane at the moment when instability 

 commences, the length of the cellular strips being parallel 

 to the general velocity. But a little later, when the general 

 motion has decayed, division-lines running in the perpen- 

 dicular direction present themselves. 



* When a jet of liquid is acted on by an external vibrator, the reso- 

 lution into drops may be regularized in a much higher degree. 



2 2 



