1920-21.] Stable Flow of a Fluid in a Uniform Channel. 141 
§ 7. In practice the ideal case where the vortices are formed exactly in 
symmetrical positions is, of course, never realised. The question immedi- 
ately arises whether, if the exact arrangement is one of equilibrium or 
steady motion, it is also one of stability ; or more precisely, under what 
conditions is the symmetrical arrangement of two equal and opposite 
vortices in a uniform channel along which fluid is moving steadily with 
the parabolic distribution in velocity of a viscous fluid across the channel, 
one of stability ? 
§ 8. Examination of the Stability of the Vortex Pair. — The general 
motion in the channel is given by u = U(l — y 2 /a 2 ). Let the centres of 
the two vortices P and Q of strengths —k and -f k respectively be situated 
initially at the points (0, a — a) and (0, — a Pa). We may dispense with 
the walls and deal with the infinite fluid provided an infinite row of 
vortices of equal but alternating strengths +k and — k be placed along 
the y- axis at the points whose co-ordinates are given by 
+ k; (0, (in ± l)a + a} ; - k ; {0, (in±l)a — a}. 
Let P receive small displacements (£, f) and Q (£', f) parallel to the 
x and y axes respectively ; then, in virtue of the fact that the walls are rigid 
and that the fluid does not leave them, the displacements of the images 
are immediately determined. 
Regarding the field as a complex plane, the co-ordinates of the system 
of vortices now become 
+ k ; $+i(in+la, + a- rj), £' + i(in - la + a + rj')} 
“->?') J 
( 8 ) 
— k ; £+i(in + la - a + 77), £' + i(4rc— la 
the vortices P and Q corresponding to 
£ + i(a — a + f) and £' + i( — a + a + rf) respectively. 
If (u, v) and (u' v') are the component velocities of the vortices P and 
Q, they will be composed of two types of terms : 
(a) the contribution in velocity due to direct effect of the infinite row 
of vortices ; 
(, b ) the general translational motion in the channel. 
Consider first the effect of (a). The contribution to —u + iv is 
1 
' 27 r T^{£ + i( a - a + y)} ~ {£ + i(in+la + a-r))} 
-{- etc. 
(9) 
four terms in all, in each of which the first member in double brackets in 
the denominator is the same, while the second member is successively each 
of the four terms in (8), with the appropriate sign for K in front of the 2. 
Expanding each typical term and neglecting quantities of higher order 
