105 



Any redistribution of charge (norroal velocity) on the 



surface of the bubble, and any alteration in the shape of 



the bubble, due to its proximity to the surface, will give 



a contribution to this v;ork v/hich is of higher order in l/h, 



and neglicible if h is several ti:aes a. The work (4) is to 



be compared with the electrostatic energy e'^/2a of the 



bubble when it is, spherical and at an infinite distance 



from the plane. Thus, to the first order in l/h we have 



for given V, ^ , 

 dt 



Kinetic energy in „ ,._ a x /Kinetic energy J 

 presence of surface " '*""5K'^ ^ I ^^ absence of / (5) 



^ surface / 



where the upper sign is to be used for a free surface, the 

 lov;er sign for a rigid surface. 



The motion of the bubble in the presence of the 

 surface is accordingly given to the first order by multi- 

 plying the first term of (1) of Appendix 1 by the factor 

 (1 +-^r)» sii^d Integrating the resulting equation. This 

 equation may be v/rltten simply 



(R is used rather thfiin a for the radius of the bubble^ since it 

 will be convenient to use a(t) for the radius in the absence 

 of perturbing influences.) The form of the function f(R) 

 cGimot depend on h, since both the total energs'-, and uhe 

 I form of the potential energy f\«iction, are independent of h. 



