146 



On the Vibrations of a Vortex Ring, fyc. [Dec. 8 y 



ture. The cross section is supposed small compared with the aperture^ 

 so that e is small compared with a. 

 Thus the time of vibration is — 



c i we 2 -, 2a i 



2a* e ' 



and the motion is stable for all such displacements. 



In the second part of the paper the action of two vortices, which 

 move so as never to approach nearer than a large multiple of the 

 diameter of either, upon each other, is considered, and the following 

 results among others obtained : — 



If e be the angle between the direction of motion of the vortices, 

 c the minimum distance between their centres, v the velocity of trans- 

 lation of vortex (i), w that of vortex (ii) ; a. and (3 angles given by — 



W COS ac = V COS j3, 

 oc+(3 = e. 



m and m' the strength of vortices (i)'and (ii) respectively, a and b their 

 radii; h the relative velocity of the vortices, viz., V v 2 + iv 2 — 2viv cos e ; 

 then, in the standard case when the vortices are moving in the 

 same direction and (I) first passes through the points of intersection 

 of their directions of motion, we have the following results : — 



The direction of motion of I is deflected towards the direction of 

 motion of II through an angle whose circular measure is — 



m'b^a cos a sin 2/3 



The direction of motion of II is deflected in the same direction 

 through an an°;le — 



marb cos /3 sin 2a. 



The radius of vortex (i) is increased by — 

 m'b~a cos a(l + cos 2(3) 



The radius of vortex (ii) is diminished by — 



ma~b cos /3(1 + cos 2a) 



W 



The effects for all circumstances of motion, whether the vortices are 

 moving in the same or opposite directions, may be summed up in the 

 following rule : — 



