1887 .] Sir W. Thomson on Stability of Fluid Motion. 361 
This is reduced to 
if we put 
dF d?F . _ 
dv d 2 v 
' ' ' 
U = v + \g sin I//x. ( b 2 - y 2 ) 
(»)• 
( 4 ) , 
( 5 ) . 
For terminal conditions (the hounding planes supposed to be ?/ = 0 
and y = b, we may have 
v = F(t) when y = 0 ) 
» V=*I 
where F and g denote arbitrary functions. These equations (4) 
and (6) show (what was found forty-two years ago by Stokes) that 
the diffusion of velocity in parallel layers, provided it is exactly in 
parallel layers , through a viscous fluid, follows Fourier’s law of the 
“linear” diffusion of heat through a homogeneous solid. Now, 
towards answering the highly important and interesting question 
which Stokes raised, — Is this laminar motion unstable in some 
cases'? — go hack to (1) and (2), and in them suppose u, v, w to he 
each infinitely small: (1) is unchanged ; (2) with U eliminated by 
(5), become 
where 
dw 
dt 
dv 
dt 
dw 
dt 
+ [„ + i c(b2-P)]^ + c(f?-f 
• (7), 
+ [v+lc(V- P)}% 
9 dp 
= U 2 v--f 
dy 
• (8). 
+ [v + i c(b*-f)] d £ 
1 
§ 
[> 
3. 
II 
• (9), 
c = g sinl/fjL . 
• (io), 
and, for brevity, p now denotes, instead of as before the pressure, 
the pressure + g cos I y. 
We will suppose v to he a function of y and t determined by (4) 
and (6). Thus (1) and (7), (8), (9) are four equations which, with 
proper initial and boundary conditions, determine the four unknown 
quantities u, v , w, p ; in terms of x, y, z, t. 
29. It is convenient to eliminate u and w ; by taking ^ 
of (7), (8), (9), and adding. Thus we find, in virtue of (1), 
