MOTION OF THE GAS SPHERE 311 



, (Pa , „ da dbo , d^^ 



— bo ho — ■ h a = 



'dt' dt dt ^ dt' 



d^a da dbs d^bz ^ 



— 263-— +3- — ; — [-a — — = 



df" dt dt dt^ 



as can be verified by substituting the expressions for B2 and B^ in terms 

 of 62 and 63 given by Eqs. (8.39). 



Penney and Price have considered the solution of equations of this 

 type, and they showed numerically in a special case that departures 

 from sphericity represented by 62 (deformation into a spheroid) were 

 very much greater at or very near the minimum than anywhere else. 

 This is, of course, in agreement with the observed behavior and there 

 seems no reason to doubt that the instability of bubble contraction is 

 perfectly consistent with noncompressive theory. These writers have 

 also examined other cases of the first order perturbation theory, for 

 which their paper should be consulted. 



If only the first power of U is included, the second of Eqs. (8.40) 

 gives Herring's formula for the bubble rise; 



(8.41) U = %{ a'dt 



=1/- 



as B2 is of order U^. Retention of U^ terms in the first of Eqs. (8.40) 

 gives 



d^a , S/daV U^ P, 



a h- — I = — — oz 



dt^ 2\dtl 4 po 



which, together with Eq. (8.41) for U, is equivalent to the equations 

 used by Taylor, as developed in section 8.5. The formula for the rise 

 is, however, correct only for the first order in U, the term 

 {^/^g)f{B'i/a^)dt being neglected to obtain it. 



A completely consistent theory, correct for terms of order C/^, should 

 from the foregoing include departures from sphericity as determined by 

 the coefficient B2. Ward has computed the corresponding value of 62 

 in the expansion of R{t), using the result from Eq. (8.39) that 



= 1 f ^ 



a^ J a 



{dt 



and computing Bi by numerical integration from the third of Eqs. 

 (8.40) with Comrie and Hartley's values for a, da/dt, etc., for zj = 3.0. 

 The results of this approximate calculation show that 62 remains small 



