1920-21.] Stable Flow of a Fluid in a Uniform Channel. 143 
Expressions (10) and (12) now reduce to 
, . ik r 
- U + IV= — 
JL . cot — - (£-Q + % + F) # + (£~£') + i(v-y) sec 2 ™ 
2ai a 
16 
a" 
16 
ci u 2a 
, lr]T7 2 9 7Ta 
+ — - — x cosec 2 — 
8a' 2 2a 
_ p 
ITT 
[_2a 
cot " + (f - ?)* tan 2 ” + tnl( tan 2 ” + 2 cosec 2 ™ 
a ' lba 2 2a i6a 2 \ 2a 2a 
*S(-E + >)] ■ ™ 
u + iv — — 
2tt 
- ^ cot. a7r + ^ / ~^ + ^ + ^ 7r2 _ (^-^) + ^(F-^) ^ 2 SPP 2 7ra 
16 
16 
. — sec" 
8a 2 
cosec z 
i« f In , air , u 7 r 2 , 9 7ra , ^W 2 /, , 9 7ra\ 
= 7T ^r-cot— +(£-D__t a n ; + U r -( 1 +sec : 2 — 
27r|_2a a 16a 2 2a lba 2 \ 2 a/ 
— TTT^ f tan 2 — + 2 cosec 2 — Y 
16a 2 \ 2a 2a) _ 
(18) 
For the contribution to —u + iv due to (b), it is to be noted that since 
the vortices move with the fluid, and the total velocity in the undisturbed 
channel at a point x + iy is U^l — there must be added to (17) and (18) 
the terms 
/ 1 _a-a + 7 ]\ 
and 
- u( 
V a 2 / 
\ 
a -f a + r\ 
respectively ; or, retaining terms up to the first order only, 
C{! 
a - a 
a 2 
>-) 
a 2 ) 
and 
Tj/l- a ~ a 
+ 7?( a - a >) ' < 19 > 
If u 0 v 0 , u Q ' v 0 ' are the component velocities of P and Q when in the 
undisturbed position, these may be derived from (17), (18), and (19) by 
making £=y = £ / = r]' = 0, and equating real and imaginaries. Hence 
“ o= s cot ¥ +u ( 1_ ^) = “ o ' ; v °=°=< ■ ■ ( 20) 
indicating that the two vortices in the undisturbed position would move 
steadily along the channel with the velocity given by (20). For a given 
value of U Qj/k (positive) the vortex will move up channel only if a/a is 
greater than some definite number itself greater than J, for only if a/a>^ 
can the first term in u 0 become negative, while the second term is always 
positive. 
§ 9. Let the axes of co-ordinates be in steady motion with the speed- 
given by (20), so that in the undisturbed position the vortices would 
