88 



Since V In the non-compresslv© approximation varies inversely 

 as r , $ will die off very rapidly as v/e (zo av/ay from the 

 bubble, and the chief contribution to the double integral 

 in (13) will come fron values of (r, - ct) very close to a. 

 Therefore, there can hardly be a very large error intro- 

 duced by usinc the non-compressivo approximation to v in 



the evaluation of this integral. Also, we may note that 



da 

 if .j-S^c, most of the variation in P during the integration 



on t is due to t?ie change in the arginent (r, - ct), and 



vei»y little to the change in the arg\:jment t. Thus, we 



may sot 



P(r, = ct,t) f^ F(r, - ct,U)-ht P(r, -ct,0) (16) 

 and gauge the adequacy of the approximation by the smallness 

 of the effect of the second term. We have 



fp(r - ct,t)dtdr,«^ J / F(r,0) ^ dr^-hj [§^) P(r,0)^ dr. 



,t)dtdr,«^ / /F(r,0) dr ^r +/ /^^) P(r,0)- 

 /a / r^ ^ /a / r^ 



si/ (r-a) P (r,0) dr - ^ /(r-a) P(r,0)dr (17) 



Using (14) the first of these integrals becomes just 

 — ®(a,0) . The second can be evaluated using (14) and (15) 

 with the non-corapressive approximation to v. The details of 

 this will not be given here and we shall merely state the 

 result. It is that the effect on the motion of the second 

 term of (17) is rather less than that of the first during 

 the first contraction of the bubble. Doth terms are largest 

 for the stages near that of niinlmun radius, although the 



