12 
LORD RAYLEIGH ON THE CIRCULATION 
rik H 2 sin 2 kxe Py \ (2(3 
d{(3yf 4A/3 2 
— 1 ) sin (By— cos (By — 2/3// sin (By-\-2e Py 
} • («). 
whence 
And 
r- 
niiiL- sin 2 kx e Py \ [ B 
4v-/3 :i 
— 21 +%) sin fBy+i cos (3y+±/By sin (3y+±e ^ . (45). 
d-yjr %&H 2 sin2&r ? \ ( (3 
dy 
4w/3 2 
/B 
— 2 )sin/3y —— cos(By—\fBy sm(3y-\-\(By cos (By—\e ^ M46J 
2 k j ™ 2 k 
To obtain the value of u at the surface of the plate it will be sufficient to put y— 0 
in (4G). Thus 
?i/dP sin 2kxJ 
4P/3 2 | 
(47). 
By (32), (39) 
nkW_W( k\_yl( k\ 
4 P/3 2 2 n\^(3) 2V\ i ~£/’ 
if as before we put V for k/n. Thus in (47) 
u= 4y{~k~?) ®in 2kx .( 48 )- 
To obtain the complete value of u at the surface of the plate, corresponding to (37), 
(46), we have to add to (48) that given in (41). The term of lowest order disappears, 
and we are left simply with 
Q V ~ 
u— — sin 2 kx .(49). 
In like manner we find for the complete value of v at the surface of the plate 
corresponding to (38), (45), 
11 Vr?k cos 2 kx 
v=v Q sin kx cos nt- 
8/3 V 
(50). 
The values of u and v expressed in (49) and the second part of (50) must be can¬ 
celled by a suitable choice of the complementary function, satisfying vS J= 0, so that 
to the second order of approximation the fluid in contact with the plate may have no 
relative motion. 
The complementary function is 
whence 
i// = (A -j- By )e sin 2 kx, 
