326 



MOTION OF THE GAS SPHERE 



and it is more convenient to use a series expansion in powers of s/a. 

 For image points inside the sphere, s is one of the lengths Cn which are 

 less than a, and an expansion in positive powers of Cn/a gives 



^^ = i[ 



cos ^ + ^ Pi(cos d) + 



+ i 



(?)>■ 



(cos d) + 



where the P, (cos 6) are the Legendre polynomials and c n is positive for 

 points above the origin. For image points behind the plane, s is one of 

 the lengths bn (bn > a), and an expansion in powers of a/hn gives 



(fz = — 



1 + 2 f- Pi(cos d) + 



On 



-fe) 



Pi(cos d) + 



Terms of degree higher than the first in powers of (cn/a) or {a/hn) 

 are neglected for simplicity in applications which have been made of 

 expansions in zonal harmonics, the justification being that these terms 

 involve increasingly higher powers of the fraction a/2h, which is always 

 less than 3^. Integration of ip and (p cos B over the sphere for the series 

 of dipoles then gives 



(8.48) 





dS = 27ra3 



.n = 



v?3 COS ddS = — a^ -\- — a^ 



o o 



n = \ J 



where the Dn, dn are defined by Eq. (8.47). 



On substituting these expressions and Eq. (8.46), the expression 

 (8.45) for the kinetic energy may be written 



T = 2Trpoa'{l + 



"(^h"'^-'H^ 



+ ~PoaKl + mU' 



where the quantities /„, /i, /2 are the bracketed series in Eqs. (8.46), 

 (8.48). The equation of energy including the potential energies of 

 hydrostatic buoyancy and internal energy E{a) of the bubble is then 



(8.49) 



27rp„a3(l +/J (^J - Arp^a^frU ^]'^') + ^ p.a'(l + ^hW 



+ -- p^ahjz =Y - E{a) 



