99 



the differential equation which results for R^ will bo 

 homoceneous when ^>1, so that the solution which satisfies 



^« " °* ^i^ ' ^ at t=0 (start of motion) (lO) 



will be simply R^^ ^5 0. Tlae same will be true when Jts 0, 

 since the second terra on the right of (7) is proportional 

 to R . Thus v/e have only the equation for /C s 1 to con- 



sider: this equation is 



0) 



1 dB, da ^ 2B, /-> \ 



r*a 

 or, using (0) and the relation {^^^ = -P^-^ 



2- CO 0) 



a d R; ' . 3 da dR , ^ - ,,,, 



This integrates to 



(12) 



The equation 



CO r, 



R = a -t- gR , cos k? 

 moans that to the first order in g the bubble is displaced 

 upv/ard by the amount gR^ , v;ithout change of size or shape. 

 If one is interested only in the relation (12), a much 

 simpler derivation can be given. Consider the momentum 

 of the water contained in a large vertical cylinder of 

 radius \ and length L >^ ^ , with center at the origin. 

 This monentujn is 



1? '^ ^f/T'^ dT z ^ ffn<fdS -/o/^fdS 



bubble cylinder 



