604 Prof. W. M. Hicks on the 



If c denote its radius 



mass = 



m = 



= 3™' 



U = 



.ft 



' DC 



j 



E 1= 



2 



: 35 



7Tfl 2 C, 



E 2 = 



= 0, 





_ 1 __ 7TJL6 2 C 7 _ 



3 4 75 30 



Thus the external energy is less than one quarter the 

 internal. 



The only other state in which an approximate solution is 

 attainable with the present state of theory is the ring state 

 when the ratio b/a is not large, where b is the radius of the 

 cross section, and a that of the circular axis. Here 



m 



2ira X irb 2 = 2ir 2 ab 2 = % ire 



or ab 2 — fr~ c z . 



5ir 



For this case 



L = - z — { W^ \ = - V (say), 



4aa? , ( ( /' — a 



where F= 7 r^- — 5 , k 2 = 



and y denotes — - x flow through disk narallel to the plane 



of the ring whose rim passes through the point x, y. When 

 brought to rest, therefore, the stream function in the stationary 

 condition is given by 



where now V= 7— | L — - J, L denoting log e -y- . 



In general the velocity at the centre of the ring is less 

 than that of translation. In this case region (2) is also 



