104 Proceedings of the 'Royal Society of Edinburgh. [Sess. 
So far the only restriction that must be imposed upon P and Q is that for 
p = 0 q = 0, P = Q = 1 in order that ~ for q) = 0 and for ^ = 0. 
We may suppose, in order to solve equations (26) to (35), that rjq, 
t 
, rjQ increase with the time as where a = ~ • From (26) to (29) 
27T 
the equation for X is obtained 
as 
-X 
-A 
-B' 
-C' 
-A 
-X 
-C' 
-B' 
-B 
C 
-X 
A' 
C 
B 
A' 
-X 
which, when expanded, becomes 
- A2(A2 + A'2 + 2BB' - 2CC') + A^A'^ + B^B'2 + C^C'^ + B2C'2 + B'2C2 
- 2 AA'CC' - 2AA'BB' = 0 (37) 
The arrangement of vortices will be stable, provided that for arbitrarily 
assigned functions P and Q, and for some definite relation between h and I, 
all the values of \ obtained from equation (22) are either purely imaginary 
or else complex and such that does not increase indefinitely with 
the time. 
As a particular case, suppose that merely the vortex corresponding to 
y) = 0 is displaced, all the others being undisturbed. Q then is everywhere 
zero, and P also, except for p = 0. In this case 
hi 
2 * 
B 
B' = 0 
C = - 
+ /i^ 
- - w 
W' 
C' = 0 
The equation for X then becomes 
X4-2A2X2 + A4 = 0 
X= + A twice . 
(38) 
(39) 
(40) 
(41) 
(42) 
(43) 
(44) 
A being real, it follows that this arrangement of vortices is also unstable 
for this disturbance unless A = 0 when the equilibrium is neutral. Now the 
two series in (38) when summed become 
Z2 cosh2 
llTT ZP 
T 
. ( 45 ) 
