-35- 281 



appendix I. The least squares solution yields the following 

 formula • 



(3.14) Ab = 19s (1 - 62s^). 



A graph of (3.14) is drawn in figure 7, and is compared with 

 the tabulated values. The formula (3.14) is seen to be ■very- 

 close to the tabulations. 



5. The correction due to the free surface . 



For the sake of couple teness, we shall include some 

 remarks concerning the effect of the free water surface. The 

 latter exerts a repellent force on the bubble, aiding the 

 downward force of the sea bed. A complete discussion would 

 require a knowledge of the potential function 4* s^^ci of the 

 corresponding values for f , f, and fg. A careful examina- 

 tion of the previous argument, however, shows that only the 



first terms in the expansions of f^ and 2 in powers of 



° ab 

 a were used. In appendix II these first terms are obtained by 



successive reflections. The results are as follows: 



If C is the depth of the bubble below the free sirrface, 



and X is set equal to .il-i- , then the expansion of f begins 



with 



C+B 



^^'^^i 



3fo 

 and the expansion of -^ begins with 



-Ar G(x) 

 2b'^ 



where the formulas for P(x) and G(x) are given in appendix II. 

 A short table of their values follows: 



