158 



g^ X -.U43U g^- .08625/(a^ -a) j^ 

 (a-i)ji= -.3583 g„- .C266i/(a^-a) j^ 



(69) 



The relation between the pairs of arbitrary constants appearinj In the alternative 

 expressions for /5 is obtained immeaidtely from (69), for if 



and g, and j are expressed in terms of g and j By (69) we have, on comparing coefficients of 

 9 and j_ in the resulting identity, 



Conversely 



Bo = - '""S^ *o - -3583 Aj/2 



j^yj = - .08625 A^ - .0266 A , 



A^ = 1.393 B^- 18.76 B^^j 



, , = - U.516 e„ + 23.22 B,,, 



(70) 



(M) 



Since the initial velocity of the deviation is zero, we have 



^2.1 = '• K.1 - 



^2.1 = {^^'i^V.^TrfTTi ii^^'--)i-i]} 



_ 1 



2 1/2 a' 



2 /( 2 - 2 cr) as a - 1. 



Hence the above conditions require 



*0 = 1 ^"-^ *l/2 = °' 



that is, the value of /3 during the first expansion is given by g, in Table I. 



The coefficients B^^, B^^^ in the alternative expression for /3, are obtainable from (70), 

 and the coefficients In the corresponding expression f or /9, , .ire thus'obtained by simply changing 

 the sign of B^^^ ^^ '" ('!)• The coefficients for the succeeding expanding and contracting phases 

 are then obtained by successive applications of equations (58) - (61) and (70), (71). Their values 

 for the first two pulsations of the bubble are given in Table II. 



It will be seen that the values of A^ correspond to the successive values of /3 at a » 1, and the 



values of B„ to the successive values of i6 at a = a . 

 o ' m 



The 



