250 VORTEX MOTION. [CHAP. VII 



Since W is unaltered by a rotation of the axes of coordinates in their own 

 plane about the origin, we have ^dW/d6 = 0, whence 



(iv), 



which agrees with the third of equations (2), but is free from the restriction 

 there understood. 



An additional integral of (i) is obtained as follows. We have 





Now if every r be increased in the ratio 1 -f e, where c is infinitesimal, the 

 increment of W is equal to 2er . d W/dr. The new configuration of the 

 vortex-system is geometrically similar to the former one, so that the mutual 

 distances r 12 are altered in the same ratio 1-f e, and therefore, from (ii), the 

 increment of W is e^" 1 . 2%w 2 . Hence 



2wr 2 -r = 2w 1 w 2 .............................. (vi). 



, Ctt TT 



157. The results of Art. 155 are independent of the form of 

 the sections of the vortices, so long as the dimensions of these 

 sections are small compared with the mutual distances of the 

 vortices themselves. The simplest case is of course when the 

 sections are circular, and it is of interest to inquire whether this 

 form is stable. This question has been examined by Lord Kelvin*. 



Let us suppose, as in Art. 28, that the space within a circle /* = , having 

 the centre as origin, is occupied by fluid having a uniform rotation , and that 

 this is surrounded by fluid moving irrotationally. If the motion be continuous 

 at this circle we have, for r<.a 



*=-K( a -H') .................................... (i), 



while for />, 



i/r=-fa 2 logc0- .................................... (ii). 



To examine the effect of a slight irrotational disturbance, we assume, for 

 r<a, 



and, for r>a, .................. ( n 'i)> 



\jf = - a 2 log - + A cos (sB <rt) 



where s is integral, and o- is to be determined. The constant A must have 

 the same value in these two expressions, since the radial component of the 



* Sir W. Thomson, " On the Vibrations of a Columnar Vortex," Phil. Mag., 

 Sept. 1880. 



