Mass canned forward by a Vortex, 605 



ring shaped. To find the condition that it may be singly- 

 connected it must be possible to find a point on the axis 

 where the flow is zero. Write 



Near the axis, that is k very small, it is easy to show that 



At a point or! bifurcation d/x/dx = 0, 



1 fa v , .. . dX dk ^ T A 



_ aK X fh tC / 9 . 9 \ il • 



also -=- " = o~ ' * r=: r^C^ + F) near the axis, 



rt<2? £ x ^a 



whence it easily follows that 



Hence 



For this, and considerably larger, values of bja, the ex- 

 pressions for p£, U will still hold with great approximation, 

 and we shall be able to obtain some information as to the 

 singly connected states. For bja = '0116 } «/ = 0, and the 

 boundary has a node at the centre. The two configurations 

 require separate treatment. 



Singly-connected configuration. — Here 6/a>*0116. The 

 equation to the boundary is given by % = 0. It cuts the axis 

 at a point given by 



O 9 9 / — " 



i 2 +f=a 2 {j-— 



\2/3 



V' 



L — i<2rrr, 





L<6'53318 : 





->-01163. 



a 





Wtt&T-'}- 



With given b\a, U is calculated. The equation to the 

 boundary can then be written 



X=iVa.(f) 3/2 



a~\aV) ~\L-i) 

 = AX 23 (say). 



