-598 Prof. W. M. Hicks en the 



between the centre and the filament, and it will be ring- 

 shaped. As the energy diminishes, the aperture becomes 

 smaller, and the ratio of cross section to aperture larger. 

 The velocity of translation increases, but the velocity at the 

 centre increases at a greater rate, until a state is reached at 

 which the two become equal. In this case the acyclic 

 boundary just loses its ring form and its section has a lemni- 

 scate form. As the energy still further increases this boun- 

 dary cuts across the straight axis of the vortex and the volume 

 of region (2) will ultimately diminish, until the vortex itself 

 closes up into the spherical aggregate, when it entirely dis- 

 appears. Thereafter it is the actual rotational portion alone 

 which is propagated through the surrounding fluid. 



In determining the energies of the three portions it is 

 most convenient first to find their values for the first and 

 second regions when in the stationary state. In this case the 

 stream functions % along the boundaries are constant. If 

 E. E' denote energies of the actual and stationary states 

 respectively 



where x~2- %i are tne constant stream functions along the inner 

 and outer boundaries of a region. 



"When the Tonicities are constant throughout the rotational 

 portions, w — const, for two-dimensional motion and =\x iov 

 three. 



For two dimensions co x area of section =ifi, 



E/ =^(% 2 -%i)+^lj%^%. 



For three dimensions co = \cc. codX = ^d/x = \j:dA = ~-.dV, 



'llT 



a — — x vol.= ■- — , 



17 IT 



2tt 2 u, CC 



Passing now to the case where the system is moving with 

 velocity L through the fin id at rest at infinity, let p, q denote 

 the component velocities in the stationary state, so that 

 /> + U, q are the actual ones. Then 



\pdxdy = momentum in stationary case = 0. 

 .'. E = E' +^ (vol. of region) U 2 . 



