1914-15.] Eesistance to Motion of a Body in a Fluid. 105 
and therefore for 
^^ = cosh~^^3 (46) 
the equilibrium is neutral for this disturbance. 
Since the stable arrangement must be so for every displacement that 
can be given to it, condition (46) must necessarily be satisfied. We may 
therefore now proceed to impress other disturbances upon the vortices on 
this assumption. Suppose none of the vortices are displaced except the 
two corresponding to ^ = 0 and q = 0. P and Q are then zero except for 
these values of p and q, when they are both unity. In this case we find 
■ • (47) 

• • (48) 
C--C'-- * 
. • (49) 
and equations (12) to (15) now become 
27t _ ' ji n ' 
■ • (BO) 
^ ' 4 - Br? ' 
~ ^ dt~ • 
■ • (51) 
27t d^^ _ Cr) 
. (52) 
2^ dvo p 
( dt~ • 
. ■ (53) 
Hence 
■■ 47t2^ ^ " 47tV^’^_^,2V 
• • (54) 
and similar equations for the other co-ordinates. Equation (54) shows at 
once that this arrangement of vortices with the relation ~ = 
= cosh ^ ^3 is 
stable for this special disturbance. Consider a slightly more 
general type 
of disturbance by assuming 
P cos (2/. + !)(/) 
cos 
• • (55) 
^ cos (2g'+ l)<^' 
cos 
. (56) 
