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 2mc, in a line bisecting c at right angles. 



The constancy of the impulse gives 



^mx = const., ^my = const.,) 



(2) 

 2m (x* + 2/-) = const. 



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

 present case is given by 



T=-pHWdxdy = -p?,m+ (3). 



When 2m 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 

 ( x u #i) 0% 2/2))'" an d their strengths by m^ w 2 , ..., it is evident from 

 Art. 154 that we may write 



dx. dW dy, dW 



m, f- = i- j wii -fr = -? * 

 1 dt dy ' 1 dt dx l ' 



M* i7 = i i m <L Jl = 



2 dt di/<> ' 2 flfo 



where W=- 2i 



7T 



if r l2 denote the distance between the vortices m lt m z . 



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

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

 ^dW/da: 1 = O t 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) 



or if we introduce polar coordinates (r lf 0,), (r 2 , 2 ), ... for the several vortices, 



Mechanik. c. xx. 



