144 Proceedings of the Koyad Society of Edinburgh. [Sess. 
appear at rest relative to these axes. The component speeds may now be 
written when the vortices are disturbed as 
dj drj . d%_ drf 
dt ’ dt ’ dt ’ dt' 
■■■ 
+ -^v( a - a ) ■ ( 21 ) 
di ‘ + i J 4- = (f - O tan 2 ™ + 4 1 + sec 2 ™) - id tan 2 ™ + 2 cosec 2 ™Y 
2a \ 2a/ \ 2a 2a/_ 
dt 
dt 32a 2 
2U „ s 
From which, by separating real and imaginary terms, it follows that 
32a 2 
dt; 
7 TK 
~di 
32 a 2 
dr] 
7 TK 
dt 
32 a 2 
dt' 
7 TK 
dt 
32 a 2 
drf 
7TK 
~dt 
tan 2 ^ + 2 cosec 2 — ) - W 1 + sec 2 — ) 
2a 
o 7ra\ 64U 
2a/ K7T 
(a - a)>7 
7ra 
2a 
2 dt; / , 9 7ra\ 9 7ra , n 9 7ra\ , 64U/ \ / 
= w 1 + sec 2 — - 7i ( tan 2 — + 2 cosec 2 — + (a - a)ri 
dt \ 2a/ \ 2a 2a/ K7r 
This is a system of linear equations with constant coefficients. 
The solutions are consequently in general of the form 
(£, y, y') = 
i 
and the equation to determine the X’s is 
32a 2 
A, 
X. 
tan 2 —, X, 
2 a 
0. - Y. 
, o 7TCL 
tan 2 - , 
2a 
A, 
tan 2 — , 0, tan 2 - , 
2a 5 2a’ 
\nt 
1 
Y 
0 
X 
A 
= 0 
where 
v , 9 7ra , 0 9 7ra 64U/ \ 
X = tan 2 — + 2 cosec 2 — (a — a) 
2a 2a kit 
( 22 ) 
(23, 
(24) 
(25) 
(26) 
(27) 
Y -t , o 7 rot 
= 1 + sec 2 — . 
2 a 
On expanding (27), it follows that 
A 2 [~A 2 + 2tan 2 ™ (Y-X) 
L 2a 
or 
A 2 = 0, and A 2 = 4 — ^ L --— (a - a) tan 2 ~ 
K7r 2a 
( 28 ) 
