1887.] Sir W. Thomson on Stability of Fluid Motion. 363 
investigation. The case of b = oo is specially important and in- 
teresting. 
31. The present communication is confined to the much simpler 
case in which the two hounding 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 
prescribed by the Examiners for the Adams Prize of 1888. I have 
ascertained, and I now proceed to give 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-dimen- 
sional motion of the admissible 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 = (3y . 
Thus equation (19) becomes reduced to 
where 
dt +Py Tx = ^ * { > 
C= v 2 « j 
and (18), (15), (16), (17) become 
2/3 dx = - V p 
du n du n o dp 
m + p y - + ji^ liV H- Tx 
dv n dv 
dt +/Sy di 
dw r. dw 
dt + ^dx 
9 dp 
= ^ V -dy 
9 wyv 
= y. y A W - -A 
dp 
dz 
(20). 
( 21 ); 
• • ( 22 ), 
• • (23), 
• • (24), 
. . (25). 
It may be remarked that equations (22) . . . (25) imply (1), and 
that any four of the five determines the four quantities u, v, w, p. 
