260 VORTEX MOTION. [CHAP. VII 



The only variable part of this is the term -^o/t&amp;lt;r s 2 ; this shews that to our 

 order of approximation the stream -lines within the section are concentric 

 circles, the velocity at a distance s from the centre being a&amp;gt; s. Substituting 

 in Art. 161 (3) we find 



The last term in Art. 160 (8) is equivalent to 



fi 8ra o 7 ) ,-~\ 



{ log V-*} ............... (m) - 



in our present notation, m denoting the strength of the whole vortex, this is 

 equal to 3m 2 CT /47r. Hence the formula for the velocity of translation of the 

 vortex becomes 



&quot; 7/ &quot; m/ - 1 &quot; &quot; .(i v )* 



163. 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 result of Art. 147 

 (3). Thus, let us suppose that we have two circular vortices 

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

 same for both, the two rings will advance, on the whole, in the same 

 direction. One effect of their mutual influence will be to increase 

 the radius of the one in front, and to contract the radius of 

 the one in the rear. If the radius of the one in front become 

 larger than that of the one in the rear, the motion of the former 

 ring will be retarded, whilst that of the latter is accelerated. 

 Hence if the conditions as to relative size and strength of the 

 two rings be favourable, it may happen that the second ring 

 will overtake and pass through the first. The parts played by 

 the two rings will then be reversed ; the one which is now in 



* This result was first obtained by Sir W. Thomson, Phil. Mag., June, 1867. 



