142 144.] CIRCULAR VORTICES. 171 



We can now make use of (43) to estimate the magnitude of 

 the velocity -, of translation of the vortex. By (51) T is of the 



order ??i 2 log e, and -7- is, as we have seen, of the order . Also 



x x is of the order e. Hence the second term on the right- 

 hand side of (43) is, in this case, small compared with the first, 

 and the velocity of translation of the ring is of the order m log e, 

 and approximately constant. 



An isolated vortex-ring moves then, without sensible change 

 of size, parallel to its (rectilinear) axis, with nearly constant 

 velocity. This velocity is small compared with that of the fluid 

 in the immediate neighbourhood of its circular axis, but large 



compared with , the velocity of the fluid at the centre of 

 the ring, with which it agrees in direction. 



A drawing of the stream-lines due to a single circular vortex 

 is given by Thomson*. 



144. If we have any number of circular vortex-rings, coaxial 

 or not, the motion of any one of these may be conceived as made 

 up of two parts, one due to the ring itself, the other due to the 

 influence of the remaining rings. The preceding considerations 

 shew that the second part is insignificant compared with the first, 

 except when two or more rings approach within a very small 

 distance of one another. Hence each ring will move, without 

 sensible change of shape or size, with nearly uniform velocity in 

 the direction of its (rectilinear) axis, until it passes within a short 

 distance of a second ring. A general notion of the result of the 

 encounter of two rings may, in particular cases, be gathered from 

 the theorem of Art. 130. 



Thus, let us suppose that we have two circular vortices having 

 the same rectilinear axis. If the sense of the rotation be the 

 same in both, the two rings will advance in the same direction. 

 One effect of their mutual influence will be to increase the radius 



* On Vortex-Motion, Trans. E. S. Edin. 1869. Copied in Maxwell, Electricity 

 and Magnetism, plate XYIII. 



