339 



llhe vertical momentum, i, at the time of minimum 

 size is twice the vertical momentum at the time of maxi- 

 mum size, that is, 



(1.22) 3 = 2s^. 



The minimum radius of the bubble, a, is the root 

 near zero of the equation: 



(1.23) a^ + ka"^^"**"^^ + 3i^/2a^ = 1. 

 If we put 



(1.24) I = k^/^^-^^ 6 = SiVse 

 and 



(1.25) 5^ = ^u, 

 then u is the root of the equation: 



(1.26) u = u^""'" + 6. 



A graph of u for various values of 6 and f is given in 

 Figure 7. 



The non-dimensional distance the bubble moves down- 

 ward is given approximately by the formula: 



(1.27) Ab = 3i(I^ + ^ Lp^Iq) 

 where 



a-3/2 ^ 



(1.28) I^ = P^ 



\ll - s.^ - ka-^^^-^-^^ - 3i2/2a^ 

 and 



a-1/2 da 



(1.29) Iq = / 



\/l - a^ - ka-3(*^-^) - 352/2a5' 



