Viscous Fluid between two Parallel Planes. 191 



These equations (15) . . . (19) are of course satisfied by 

 u = 0, v = 0, io = 0, p = 0. The question of stability is, Does 

 every possible solution of them come to this in time ? It 

 seems to me probable that it does ; but I cannot, at present 

 at all events, enter on the investigation. The case of b = co 

 is specially important and interesting. 



31. The present communication is confined to the much 

 simpler case in which the two bounding planes are kept moving 

 relatively with constant velocity ; including as sub-case, the 

 two planes held at rest, and the fluid caused by gravity to 

 move between them. But we shall first take the much simpler 

 sub-case, in which there is relative motion of the two planes, 

 and no gravity. This is the very simplest of all cases of the 

 general question of the Stability or Instability of the Motion 

 of a Viscous Fluid. It is the second of the two cases pre- 

 scribed by the Examiners for the Adams Prize of 1888. I 

 have ascertained, and I now give (§§ 32 . . . 39 below) the 

 proof, that in this sub-case the steady motion is wholly stable, 

 however small or however great be the viscosity ; and this 

 without limitation to two-dimensional motion of the admis- 

 sible disturbances. 



32. In our present sub-case, let /3b be the relative velocity of 

 the two planes; so that in (6) we may take F = 0, ^;-=/3b; and 

 the corresponding steady solution of (4) is 



v = fy (20). 



Thus equation (19) becomes reduced to 



da , a da „ j 



dt J dx n ' I f<m. 



where f * * ' ' [Zi)f 



a=\/ 2 v J 



and (18), (15), (16), (17) become 



f^g^^V^J. . . . (23), 



£+*TB =^~% <">, 



dw dw - dp ir%r . x 



It may be remarked that equations (22) . . . (25) imply (1), 

 and that any four of the five determines the four quantities 

 m, t?, w y p. It will still be convenient occasionally to use (1). 



