AND THE ACTION OE TWO VORTICES IN A PERFECT FLUID. 
505 
Tlien 0 H, K 0, 0 A are parallel to our axes of x, y\ z respectively. The angle A G 
or K is e, and if F is parallel to the radius vector above referred to, F L is equal to 9. 
Then 
l— cos HF — cos 6 cos e 
m— — cos FK= — sin 9 
n— cos FA = cos 9 sin e 
The velocity y along the axis of z due to the vortex A B is by formula 3 given by 
yz=z±m (2a /3 A 0 —dp A x ) 
where m' is the strength of the vortex A B 
a _ l'f 8 ' dd 
" 0— 27tJ 0 (cd+ p'* + ^-2p'd co$A)f 
_ir^_ cos e.cie _ 
1 27tJ q (A 3 + jo' 2 + £ /3 —2 p'a cos 6f 
Now since the vortex rings never approach nearer than a large multiple of their 
diameter, a 3 will be small compared with p 2 -\~l' 2 , if we neglect small quantities of a 
higher order than d 2 /p' 2 -\-l' 2 , we find 
A == _i_. aJSWzffi 
° (/un |t4 G /3 +r 3 ) i 
. _ oaf p' 
1 "V+r 2 ) 1 
Let the coordinates of the centre of the vortex ring C D be f } 0, h, then for a point 
on the vortex ring 
x=f -f -al — f-\-a cos e cos 6 
y—am =—a sin 0 
z — an = h+a sin e cos 9 
3 T 2 
