146 Proceedings of the Royal Society of Edinburgh. [Sess. 
viscosity will tend to force up the value of U Ji/k further into the region of 
stability, and to maintain the vortices in their equilibrium positions. No 
such corresponding assertion can, however, be made when U h/ic is initially 
below the critical, and the vortices move off along paths determined by 
their initial displacements. 
§ 11. For the evaluation of the velocities in the fluid when the vortices 
are in their undisturbed position, we may most conveniently specify the 
positions of the latter thus : — 
+ k; (2n+ la + a)i. - k ; (2n+ la — a)i. 
Hence for the effect of the vortices alone 
10 
= {l°g [} — (2w + la + a)z] - log [z - (2 n + 1 a - a)?!]} 
Ik , 
= 2^° S 
v K i 
=? — lot 
+,oo 
z - (2n + la + a)i 
-z— (2/z + la — a)i 
(z — ai) 2 
1 + 
o 1 + 
lK -I 
= o' lo S 
2tt 
\2n + l) 2 a 2 
(z + ai) 2 
(2n + l) 2 a 2 
cosh ~(z — ai) 
2 a> ’ 
cosh Z-(z + ai) 
2 a 
. dw ii< 
— U + IV = - — 
dz 4 a 
tanh — (z — ai) - tanh ^-(z + ai) 
2a 2 a 
. ira 
sm — 
k a 
2 a i 7 tZ , 7ra 
cosh — + cos — 
a a 
To this must be added at every point the motion due to the steady stream- 
ing in the channel ; thus 
— u + iv = 
. tv a 
sm — 
k a 
2 a i 7rZ 7r a 
cosh — + cos — 
a a 
§ 12. The motion in the channel at great distances from the vortices 
approaches the steady undisturbed motion of the fluid. Along the walls 
y = 
-u a + iv a = 
K 
. rra 
sin — 
a 
cosh ( — 
\ a 
+ l7r) + COS 
