98 



provided there are no obstacles or free surfaces 

 nearby. 



(11) The bovindary condition which p^ must satisfy 

 Is determined by the fact that p(R) is a fixed 

 function of the volume V of the bubble, indepen- 

 dent of g. Thus, to the first order in g 



p (R,t) = p^(a,t) -f- c^S-^^.-h gp, (a,t) 



= p,{a,t) ^g/||-') ^^^ 



from ivhich 



p. (a#6;,t,) = 



3 Po 



rrj. 1 ^y.Vpo(a,t) .. 



4 7ra^',as/ 

 na t 



(Hi) Since 



dR - ./ac^^ 



/ r*R 



we have 



dR, Ao/^„R;VA/ , 2B, cos 6 , 



2 |a Ai, 2B_,cos_£, . (g) 



a dt "/ „r-^ «a ~ 



The procedure is now to substitute (6) and (7} in (5), and 

 to use (0) to express A^, B^ , in terms of the coef- 

 ficients in the spherical harmonic expansion of R^ « If we 

 let ^ 



R, s X R, ^(t) p. (cos^) (9) 



