106 



If we set 



R(t) = a(t)-^ 1 R,(t)-h|3^Rjt)^ . ... (7) 



the first order part of (6) is just 



— a /<!&).„ da dR, r df p _ R . , „ da d'^a 



= 2 R, §^ (8) 



since R^«0 at the start of the motion, which we take at 

 t = 0, the solution of (8) is 



How H,/h -^ is simply the time interval by which tha 

 curve of R against t lacs behind the curve of a against t 

 for a fixed radius. The value of this quantity after one 

 complete cycle represents the amount by which the period 

 is shortened because of the surface. Thus, we have to 

 the first order in l/h 



T (h) s T (^) (1^^) (10) 



v;here "a is the time average of a, and v/here the upper sign 

 is for a free surface, the lower for a rigid one. 



When there are tv/o or more plane surfaces near 

 the bubble, their effects on the motion are not simply addi- 

 tive, even in the first order. In some cases the combined 

 effect can be calculated rather easily by the image methods 

 of electrostatics already used. For exQ:aple, consider a 



