16 
LORD RAYLEIGH ON THE CIRCULATION 
whence 
(p\/~ — in)xp= 0 
(65). 
If we eliminate cr directly from the fundamental equations (56), we get 
. (6fi). 
If we now assume that as functions of x the quantities cr, if/, &c., are proportional 
to e ,Jix , equations (63), (65) may be written 
$-n„=o, 
where F 3 =F-- 
2 
• ( 67 ), 
d f~ k T =0 ’ 
where k' 2 =k?- j-—. 
V 
( 68 ). 
If the origin for y be in the middle between the two parallel boundaries, cr must be 
an even function of y, and xp must be an odd function. Thus we may write 
cr=A cosh h"y. e v,t e' lx -, 
i//=Bsinh k'y.e mt e dx . . . . (69), 
u—[ —~A cosh k" y-\-k'. B sinh k'y )e M & 
v-C—'h. sinh k"y —?X’B sinh k'y )e i,lf e lh 
> ■ 
• (70). 
If the fixed walls are situated at y= u and v must vanish for these values of y. 
Eliminating from (70) the ratio of A to B, we get as the equation for determining k, 
k 2 tanh k'y^k'k" tanh k"y x .(71), 
in which kf, k" are given as functions of k by (67), (68). We now introduce further 
approximations dependent upon the assumption that the direct influence of friction 
extends through a layer whose thickness is a small fraction only of y v On this suppo¬ 
sition k' is large, and k" is small, so that we may put tanh k'y y = il, tanh k”y x — - hk"y x - 
Equation (71) then becomes 
k*=k'k' 
y i 
(72), 
