116 



the bubble center, where 



da 



dt 



4h2 dt '^ 



whence, to or-ier l/h , 



d_ 

 dt 



[a3 £lvl , ^_4 

 L dt J Ah2 



a da 5 



dt2 



(ttfj 



Adding (59) and (Ao) gives 



(AO) 



1_ d_ 

 -2 dt 



.3|.(»pt3v 



5 -?[ 



^-'0^1 ^'(If) j '*^' 



Comparison of (31) an5 (Al) shows that to this accuracy (z -f z^) 



(1) 2 

 satisfies the same differential equation as R^ '/h • 



