338 



Here, a and a, denote the smallest and largest roots, 

 respectively, of the denominator. Graphs of I, and !„ 

 for the most frequently occxirring values of k and y are 

 given in Figures 1 and 2. 



The vertical momentum at the time of maximvim size is 



^i-i^> ^1 = ^ -W^h - I ^i^A) ^ ^;^(i5 ^ I ^^^i^e) 



where 



(1.16) I3 = /*! a^/^^l - a^ - ta-»(v-t)' da, 

 (1.16) 1. = /"^ a'/2^1 - a^ - ta-*('^-l>' da. 



■4 



^O 



(1 17) I = Z'^-'- a ^a 



^ ^ ^ i \/l - a^ - ica-^^^-^^' 



(1.18) I 



^ y^l al^/2 da 



The maxim\3m radius of the bubble, a-,, is the root 

 near 1 of the equation: 



(1.19) a^ + ka"^^*^'^^ + 3s^/2a^ = 1. 



A good approximation to a, is given by the formula: 



(1.20) a^ = l .^- (Y^ + 1)1.2 .3 32^ 



Bubble Quantities at Time of Minimimi Size 



The time, t, from the beginning of the bubble motion 

 to the stage of minimum size is twice the time to the 

 stage of maximum size, that is, 



(1.21) t = 2t-|^. 



