109 



and the first and second order ports of this are 



■1^ = -f- (1=) 



9 4>, a4o g^> 



d t ' d r 



"aT^ - dT—dr -^(V<^,)= -^- (16) 



As in Appendix 4, we use three additional relations to 



inte^jrate these: 



(i) Since V (^^= , the expansion of <t in 

 spherical harEionlcs must have the form 



(17) 



^ ^ .HUM 



v;here tno terms inA , B , C , represent 



n n n 



the field produced by the inace of the bubble in 

 the surface. Tlie potential d) " at the oricin due 

 to this image is ^T -ct-, correct to the second 

 order in l/h, where e is given by (3) and where 

 the upper sic" is for a free surface, the lov;er 

 for a risid one. Similarly, the gradient of the 

 iraa^Q potential at the origin is, to the second 

 order in l/h 



dV ^ . - 6 



a z / " ■♦" 4 h"^ 







