106 Proceedings of the Royal Society of Edinburgh. [Sess. 
where (p and p' are for the moment quite arbitrary, except that they must 
not be chosen in the region of 
The various coefficients in equations (30) to (35) now reduce to 
_ 1)2 _ 7^2 /COS (2p +!)(/>- COS (/) 
^ ~ir+ ^ ^ • 
A' = etc. 
B =0 = B' 
r = _ V i cos (2p + !)(/) 
^ [l^(p + J)’^ + cos p 
(57) 
(58) 
and the equation for X becomes 
X4-X2(A2 + A'2-2CC') + A2A'2 + C2C'2-2AA'CC' = 0 . . (59) 
and therefore 
±2A = A-A'± x/(A + A'}2_4C0' .... (60) 
Now 
. . , _ ^/COS (279 + 1)(^ _ ”^/COS ( 2 ^' - 
~ p)H^ cos p ^ cfP cos p' 
^ ^ cos 2pp ^ ^ cos 2qp' 
= - 7t) - P'{p' - 7t)] 
= + (^1)* 
It is evident that a first condition for stability must he that the real part 
in the expression for X, viz. A — A', must vanish, but this involves 
or 
p = p' 
p-\- p' = 7T 
( 62 ) 
No matter, therefore, what relation exists between h and I, this arrange- 
ment is not stable for all disturbances of the type (55) and (56). 
Further, suppose the vortices in the g-row be not disturbed, so that 
Q = 0, and those in the y>-row be displaced in any arbitrary manner, then 
B' = C'-0 
and the equation for X becomes 
X4-X2(A2+ A'2)+A2A'2 = 0 
X= ± A ( 
X = ± A I 
( 63 ) 
* See equation (67) in Appendix. 
