Oi.'R — Stability or Instahility of Motions of a Viscous Liquid. 85 



that the motion is stable by the aid of a special solution not proportional to 

 a simple exponential function of the time. If we retain the supposition of 

 the present paper that the disturbance as a function of the time is pro- 

 portional to e"'*, we obtain an equation [(52) in Lord Kelvin's paper] which 

 has been discussed by Stokes. Erom his results it appears that it is not 

 possible to find a solution applicable to an unlimited fluid which shall be 

 periodic with respect to x, and remain finite when 7/ = ± go , and this whether 

 n be real or complex. The cause of the failure would appear to lie in the 

 fact indicated by Lord Kelvin's solution, that the stability is ultimately of a 

 higher order than can be expressed by any simple exponential function of the 

 time." 



Art. 6. No Proof of StaMlity in either Solution. A tacit Assumption in the 



special one. 



Lord Eayleigh's objection to the argument in Lord Kelvin's latter paper 

 appears unanswerable. The precise point of failure in the solution is that it 

 does not in reality satisfy the most general conditions which may be assigned, 

 just as, in the problem of the pendulum which Lord Eayleigh instances, the 

 most general conditions cannot be satisfied without the introduction of the 

 terms 



Ae^* + Be-^'K 



When the values of v and dvjdy are assigned at the bounding planes for all 

 values of x, z, t, Lord Kelvin's solution is evidently an absolutely determinate 

 one ; but the initial state of things in the interior may be arbitrarily pre- 

 scribed ; and to allow this to be done there must evidently be added solutions 

 which make v and clvjcly always zero at the bounding places : in other words, 

 free disturbances. 



Now, the special solution which Lord Eayleigh accepts in the second 

 passage quoted (Art. 5), contains no reference to the free disturbances any 

 more than does the solution which he rejects ; and, on examination, it must, 

 I think, be held that neither does it afford a proof of the stability of the 

 motion. The value of lo in it, like that of v in the other, is completely deter- 

 mined by the boundary conditions (8) without any reference to the initial 

 condition (7) ; and the statement in the penultimate sentence of Lord Kelvin's 

 first investigation that w rises gradually from zero at ^ = thus involves an 

 unjustified assumption that the solution which satisfies (2), (3), (8) will 

 satisfy (7) also. 



R.I. A. PROC, VOL. XXVII., SECT, A. [12] 



