Ill 



since d, = ^ 4l 

 ^ r dt 



Po Poo - d r ida 



e f "^ r dt ^ dt 



2 r'^ 



a, da ) 



(23) 



We shell not be interested in the spherically 

 syiiir.ietrlcal part of p J if we anticipate the result 

 to be proved below that R is sp'jierically symr.ietri- 

 cal and re:neraber that p(R,t) is constant over the 

 surface of the bubble, we have from (20) 



r, {r> A i-^ R r^P^l _L_ spherically 



Pja, «,t) : - R^-^_- -t- gyn^jsetrical (24) 



L- -Ir-a terrns 



(lii) In the equation || - = / .|^_ j let us 



(£) 



/r»R 

 Insert ,__ 





(25) 



b.a obtain, as in Appendix 4^ 



d^ - 2 da >;a; 

 a 



^5^ = - 2 da J'^ . f 2B' .. , ,„„, 



^^ -K St ^^ ^ [j5^ " ^^J (27) 



To obtain the differential equations for the R , 

 v/e first insert (18) into (26) and solve these equations for 

 A^ , B, , etc. In this v;ay the expression (17) for <ff can 



