516 
MR. j. j. THOMSON ON THE VIBRATIONS ON A VORTEX RING, 
front when they are nearest together) is deflected towards the direction of motion of 
the other vortex A B through an angle whose circular measure is 
m'a' 2 cos a sin 2/3_ 2m'a' 2 sin 3 eviv(v — iv cos e) 
KC 3 AC % 3 
where m, m are the strengths of the vortices A B and 0 D respectively, and a and a 
their radii, k is the relative velocity of the two vortices, viz.: 
(v~-\-w 2 —2viv cos e ) 1 
The direction of motion of A B is deflected from that of C D through an angle 
whose circular measure is 
mo? cos /3 sin 2 a 2 me? sin 3 evw(iv—v cos e) 
ACC 3 ac 4 c 3 
The radius of the vortex C D is increased by 
m'a'~a cos a cos 3 (3 m'a' 2 a sin 3 evw 2 
kc? ac 4 c 3 
The radius of the vortex A B is diminished by 
mu?a’ cos [3 cos 3 a me?a' sin 3 ev~vj 
ACC 3 AC % 3 
The velocity of the vortex C D is diminished by 
mm' cos a cos 2 /3 ,,, 
a * log 
2 7T O. 
KC 
2 a 
c 
mm?a' 2 sin 3 evv? 
2 r TraK?c z 
2 a 
e 
The velocity of the vortex A B is increased by 
mm'a 2 sin 3 ewv 2 1 2a' 
“27 raW ° 8 ' V 
where e, e are the radii of the cross sections of the vortices C D, A B respectively. 
The kinetic energy of the vortex C D is increased by 
2irpmm'd 2 a 2 v cos a cos 3 (3 2tt pmm‘a' 2 a 2 sin 3 e.vho 2 
f»3 
KC° 
AC 4 C 3 
The kinetic energy of A B is diminished by the same amount. 
With the help of these results we may find the way in which two vortices affect 
each other in all cases. 
