f 



103 



' :: on tho free surface (2a) 



Alternatively, we must have 



n • \7 <6 2 on the ricid surface (2b) 



where "n is the unit normal to the surface. It will b e 

 convenient to picture <p as the electrostatic potential 

 due to a charee (and, if necessary, a set of multlpoles) 

 located at the initial center of the bubble, in tho pi'-esence 

 of (a) a grounded conductinc plane, or (b) a plane boundary 

 of a medium of dielectric constant zero. The total "charce" 

 of tho bubble is 



e : - ^_ // „" „ r?^ ri'.n „ 1 dV ,-^ 





bubble 

 surface 



v;here V is the volume of the bubble. As is well Imormf the 



field due to the charges induced on the plane ir. case (a) is 



the same as that produced by an oppositely chari^ed mirror 



ima^e of the charged bubble: the linos of force (fitr«'ani 



lines in the hydi'odynainlcal problem) accordingly look as 



shovm in Picure 6 (a). Similarly in case (b) the field 



produced by the polarized dielectric is the same as that 



produced by en image v/ith the sane si^Ji of charge, end 



the lines of force arc as sho'«m in Figure 6 (b). Nov/ the 



kinetic energy of the hydrodynariical problem ia 



s 



e - ^^ 



-f-<:^<^) dr. 



entire 

 fluid 



