10 
LORD RAYLEIGH ON THE CIRCULATION 
The solution to a first approximation is readily obtained. As in (10), (11), we have 
= x Pi~^~'p 2 = Qini cos kx (Ae - ^+Ce -/ ^).(31), 
in which we may take as before 
\/ ^■(l+f)=^(l + i).(32). 
By the condition at y— 0, 
A=-|C, C=^ 
so that 
. v D e iat cos fee f fe , , „ 1 . 
,+e ' y \ .( 33 )> 
. (34) . 
In passing to real quantities it will be convenient to write 
. m- 
Thus throwing away the imaginary parts of (33), (34), we get 
\fj= cos kx j — cos (nf + e +i^) + e ~^cos (?^+e—/3y)j . . . (3G), 
u=y/2.(3 H cos Tex je -/ ^ C os (7i^+e+3^) — e“^cos (iit-\-e-\-\TT—fiy) j . (37), 
v=H sin &rj — ySv/2 e~^cos (^+e+^7r)+^e - ^cos (nt-\-e— /3y)j . . (38). 
From (32), (35), the approximate value of H is —and that of e is — ^7r. 
More exact values will however be required later. We find 
H= _ v / !(/3-^V+/3 i J = ( 1+ 2^). (39 ^’ 
cos e= VK^W+W]" = Vi{ l ~w) .^ 40) ' 
The values of u and v above expressed give u=0, v—v 0 sin Tex cos nt, when y—0. 
This is sufficient for a first approximation, but in proceeding further we must remember 
