110 



and, in general, the k*^ derivative of (^" at the 

 orlcin Is of order h" ~ . Thus, we have 



It ^ J _n '» 



^2 = ^' ^2-^t §1' ^2 = =° (1^^ 



(11) To order l/h% the pressure at the boundary 

 of the bubble is 



p(R,t) = p,(a,t) -r (R,/h 4- R^/h^) .||£. 



^, 5 Pi. ^. 5P, 2. 



-^ —Z -T"X-^ P, (a,t)/h^~^ ^-f-p,/H (20) 



2har h^dr ^ 



Also, to the first order in l/h, 



p(R,t) ^ p^(a,t) -f 4ra^-^^ 



R, dp 

 = Po^—HT- -S <21) 



where R is the spherical average of R, o Since 

 at the boundary of the bubble 



dP^ - !22 u- da ^Pq 

 Ht" " dt 'T"S€ "aT" * 



120) and (21) ci"^e 



p^(a, ^,t) = _|^ 14|fl.| (22) 



Into this we nay insert p from (14) which is. 





