155-156] IMPULSE AND ENERGY. 249 



For instance, in the case of a single vortex-pair, the strengths of 

 the two vortices being m, and their distance apart c, the impulse 

 is 2wc, in a line bisecting c at right angles. 



The constancy of the impulse gives 



= const., 2wy = const., 



(2). 

 y-) = const. 



It may also be shewn that the energy of the motion in the 

 present case is given by 



When %m is not zero, the energy and the moment of the 

 impulse are both infinite, as may be easily verified in the case of 

 a single rectilinear vortex. 



The theory of a system of isolated rectilinear vortices has been put in a 

 very elegant form by Kirchhoff*. 



Denoting the positions of the centres of the respective vortices by 

 (* i&amp;gt; yi)&amp;gt; ( X M 2/2)1 an d their strengths by m lt m, 2 , ..., it is evident from 

 Art. 154 that we may write 



dx l _ dW d(/ l _ d 



dx. dW dy 9 dW 



m&amp;lt;&amp;gt; ji j-- , m 2 FT = -j j 



dt tfym &quot; dt dx 



where W= - 2m 1 m 2 log r 12 



if r 12 denote the distance between the vortices m lt m 2 . 



Since T7 depends only on the relative configuration of the vortices, its 

 value is unaltered when x lt # 2 ,... are increased by the same amount, whence 

 3d W jdxi = 0, and, in the same way, ^,dWjdy l = Q. This gives the first two of 

 equations (2), but the proof is not now limited to the case of 2m = 0. The 

 argument is in fact substantially the same as in Art. 154. 







Again, we obtain from (i) 



/ dx dii\ ( dW dW\ 



2m (x -=- t -- = - 2 jc -- -11 --s- 



-=- + t/ -f- ) = - 2 (jc --T -11 --s- 

 dt J dt) \ dy J dx 



or if we introduce polar coordinates (r lt ^j), (r 2 , ^ 2 ), ... for the several vortices, 



* Mechanik. c. xx. 



