351 



(3.14) 



i3 = f^ a^/2/rTirn~^^(7:^ ^^^ 



I^ = p^ a'7/2^/, . ^3 _ i^-3(r-l)- ^^, 



^3 = r 



a '^ da 



/l - a^ - ka-^C^-1)' 

 «ll/2 



I = /I a^*'" da 



^6 



/l - a=5 - ka-5C^-l)' * 



These eqiiations are the same as (1.14)-(1.18) . 



The momentum can be used to give us an approximation 



to the energy equation (3.6) which takes Into account the 



fact that b does change . We shall neglect the terms 



2 2 3 3 

 a L {Op ^'^'^ ^ ^ 'Pz which are of second and third order, 



respectively, as compared to the first order tena siLp-.. 



Now, since 



b = 3s/a^, 



equation (3.6) becomes 



(3.15) a^a^d + aL^p^) + a^ + ka"^^*^""^^ + 33^/2a^ = 1. 



If s^ were known as a function of a, this cotild be inte- 

 grated: ^ 



n a^'^^(l + aL(pT/2)da 



(3.16) J , _ /■ . = t. 



1^1 - a^ - kk-5(^-l) - 3s2/2a^' 



The vertical distance the bubble moves can be fovmd 

 from this formula since 



6 = 33^*^ 



