Oer — Stability or Instability of Motions of a Viscous Liquid. 71 



is known that the fundamental free disturbances have stability of the 

 common exponential type, but that it would not be true if the contrary were 

 the case ; and in Art. 8, pp. 86-88, a simple instance is taken of a system 

 having only one coordinate in which this argument is seen to be correct. 



In Art. 9, p. 88, it is pointed out besides, that, except at the boundaries, 

 it is not known that the " forced " disturbance, i), docs diminish indefinitely. 



It is accordingly held that Lord Kelvin has not proved stability, even for 

 infinitesimal disturbances. 



As the fundamental modes of disturbance do, as is shown in Chapter II., 

 possess stability of the simple exponential character, the " special " solution 

 is, I believe, as a matter of fact, the solution for a given initial disturbance ; 

 if this be a simple trigonometrical function of the coordinates, the form of v is 

 simple ; but that of the " forced " disturbance, 'o, in no case appears capable 

 of being readily calculated. It is urged, however, in Art. 10, pp. 88-90, that 

 this solution actually proves that for sufficiently small viscosity or sufficiently 

 great velocity the motion is unstable ; for under such circumstances v, 

 considered alone, will increase very much if the constants are properly 

 chosen, the possible ratio being limited only by friction ; and it is held 

 that the fact that v violates the boundary-conditions is of little importance 

 if the wave-lengths in all directions are sufficiently small. The bou.ndary- 

 conditions being that the velocity perpendicular to the depth of the stream 

 and its gradient in the same direction should yanish, it is seen moreover 

 that it is quite easy to add to v a term which gives a solution satisfying 

 either one of these conditions or the other, but not both. (If the former 

 be chosen, the solution thus obtained includes as a limiting case that given 

 in Part I. for the same problem in the absence of viscosity.) 



In Art. 11, pp. 90-92, numerical values corresponding to the circumstances 

 under which instability has been actually observed to set in under somewhat 

 similar circumstances are substituted in the two-dimensioned form of the first 

 of these two modifications of the " special" solution ; it appears that it would 

 not be possible for the kinetic energy of the relative motion of any disturbance 

 of the simple type in question to increase to more than about four times its 

 original value. 



And in Art. 12, p. 93, the same is done for the second modification; and 

 it is seen that an initial disturbance of the same type, but with different 

 constants, might increase about ten-thousand-fold. 



In Chapter II., pp. 95-121, the fundamental free disturbances of this same 

 steady motion are discussed. 



The preliminary analysis is, of course, substantially that given by Lord 

 Kelvin in the "special" solution: supposing the plane boundaries to be 



[10^^] 



