61 



represented in Fig. 2 the cap was only 12 inches 

 from the surface, and was so close to a slanting 

 steel plate beneath it that the bubble almost 

 touched the plate at the time of its maximum size. 

 The correction to the simple theory necessitated by (i) is 

 estimated in Appendix 1, and that necessitated by (ii) in Section 

 k and Appendices 2 and 3. These quantitative calculations show 

 that, as one would expect, the corrections from factors (i) and 

 (ii) are negligible when a is near a , and tend anyway to 

 bix^aden the curve in time, as compared with the simple theory. 

 The correction from (iii) can hardly be appreciable in the 

 neighborhood of the maximum radius, since the bubble is observed 

 to be very nicely spherical at this stage. There remains the 

 factor (iv). The effect on the motion of proximity to a free 

 surface can easily be predicted qualitatively. The water between 

 the bubble and the surface can be more easily given a radial 

 acceleration when the surface is near than when it is distant. 



Consequently the stream lines tend to bend towar*d the surface, 



da 



with the result that for a given a and ^ , the kinetic energy 



of the water is less than the value (3) which applies in the 

 absence of a free surface. The potential energy is of course the 

 same function of a and p_ as in the absence of a surface. 

 Thus the effect of the surface is like decreasing the inertia of 

 a simple oscillator without changing the spring constant, and so 

 it decreases the period. In Appendix 5 this effect is worked 



