164] SPHERICAL VORTEX. 265 



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

 r=a, is given by 



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 ar=rsin0, this gives 

 4 */~2 an d therefore 



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

 vortex is 5u. 



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 



