113 



I.e., ^ 0)\ / 



1 d A3 dRA. j „ ^. 5 d / J da 



which when integrated by parts gives 



^ =i|«"al + 5^/ -laf)" c^' 



Ei'^.t 1 (><t, - a^oj ^ I f ^ / //|X,,„ ,33, 



t /t' 



dt« / 



'0 Jo vrv 



we conclude that as far as the second order in 

 l/h, the effect of the surface Is to change the period of 

 the raotlon (first order) and to shift the center of the 

 bubble a distance Ti /Yi toward the surface. This latter 

 displacerient consists of a periodic part and a rionotonic 

 part. The periodic part represents a sucking of the bubble 

 back and forth by its inage: the velocity at the origin 

 doe to the inage is — e/4h = — a •gr/'^li J the first term 

 of (32) is three times this because the bubble acquires a 

 dipole moment just sufficient to keep the pressure constant 

 over its surface. T].ie laonotonic second term of (32) is 

 negligible compared to the first in the limit of small 

 velocities, although not for the velocities encountered 



in explosions. This term, like (12) of Appendix 4, con- 



3 



tains a in the denominator, and thus illustrates tlie in- 

 stability of deviations from spherical syr.Kietry during the 

 contraction. Tlie ratio of the monotonlc part of the velocity 



i .^4-2=- I <i^s to the surface, to the gravity torm g ,4^-'. of 

 h dt dt 



