164] 



SPHERICAL VORTEX. 



265 



general velocity -u parallel to the axis, past a fixed spherical surface 

 r = a, is given by 



(vi). 



The two values of \^ agree when r=a; this makes the normal velocity 

 continuous. In order that the tangential velocity may be continuous, the 

 values of d-^/dr must also agree. Remembering that or=rsin0, this gives 

 A=-%u/a 2 , and therefore 



.................................... (vii). 



The sum of the strengths of the vortex-filaments composing the spherical 

 vortex is 5ua. 



The figure shews the stream-lines, both inside and outside the vortex; 

 they are drawn, as usual, for equidistant values of ^. 



If we impress on everything a velocity u parallel to #, we get a spherical 

 vortex advancing with constant velocity u through a liquid which is at rest at 

 infinity. 



By the formulae of Arts. 160, 161, we readily find that the square of the 

 'mean-radius' of the vortex is fa 2 , the 'impulse ' is 27rpa 3 u, and the energy is 



