100 



where "ri is the outward noiTOal in each case. As L-^*^ 

 th& second surface Intecral roes to zero. If we take the 

 oricin of coordinates at the instantaneous center of the 

 bubble, the evaluation of the first surface integral is 

 simplified: v;e have R^ "^ at the c^ven Instant, so that 

 to the first order in c 



" ? "3" ^ ^^ -W ^^^^ 



hy (0). How the time I'ate of change of 1,1 is equal to the 

 total gravity force acting on the water in the cylinder, 

 plus the total pressure force on the ends, plua the momen- 

 tum entering across the surface. In the limit L — * <=« the 

 last contribution vanishes, and the first tv/o give 



^^-^■^ -- 4^-V« '"' 



Equating (13) to the time integral of (14) gives (12) o 



At the raaxJ.raura of the first expansion the value 



dP ^'^ dfl 



of ^^r— can be obtained by substituting the value of S& 



from Appendix 1 into (12): disregarding gas pressure, the 



result turns out to be 



^*^, 



o 

 ax 



The value of gR at this stage is obtained, as to order of 

 magnitude at least, by multiplying half of (15) by To/2. 



