38 Proceedings of the Royal Irish Academy. 



pressure also is to the first order of small quantities periodic, it may appear 

 paradoxical that the second cause should have any effect either. In com- 

 puting, however, to the second order of small quantities the rate at which 

 the pressures do work on the fluid, terms of the second order must be 

 retained in the pressure. And to this order the pressure is not periodic in x, 

 as is shown by the equation 



1 dp du ,^ . du dAi ^. 



for the product vdAildy involves sin^/a? and cos^fe. 



Akt, 12. The Motion is Stable, if Initial Disturhance he sufficiently small. 



It is evident, then, from what precedes, that Lord Eayleigh's analysis 

 is sufficient to include the most general disturbance. And as the former of 

 equations (30) is now equivalent to 



and leads to an expression for i// in terms of integrals which are obviously 

 finite, unless the end-conditions are extraordinary, it appears that, as long as 

 equations (29) represent the motion, a disturbance cannot increase indefinitely, 

 and accordingly that the motion is stable for the most general disturbance, 

 if siifficiently small initially. 



Equation (32) shows also that, in the case of a disturbance in three 

 dimensions, the same is true at least as far as v is concerned ; and it seems 

 reasonable to infer stability for a sufficiently small disturbance of this type 

 also. 



Indeed, if the disturbance is of definite wave-lengths in the o: and z 

 directions, but is of an arbitrary character in so far as it depends on y, it 

 may be seen that, if sufficiently small initially, the y velocity-component 

 eventually diminishes indefinitely as ^"^ and, in the two-dimensioned case at 

 least, the x component of relative velocity as t"^. 



If the disturbance has initially 



% == f{y) cos Ix cos nz, (48) 



tJien at time t we have, (see equation (23)) : 



2-^/003 nz = sinh A (b - y) 



sinh A», {Xy(»,) -/'(»,)} cos l(:c - 3./0 dn 



sinh Xy sinh X (5 - jj) {Vfin) -/"('?)} cos l(x - /3»jO dr^ 



jy 



+ terms derivable by changing x - (3rit into o: + ^rit, 

 where A- = ^* + n^. (49) 



