115 



In conclusion, It will be shown that the Rg 

 satisfying (31) can be represented as a sum of the 

 displacement due to the pressure gradient produced by the 

 Image, and the masn motion produced by the image. The 

 pressure gradient due to the image is 



3Plm 



= ,_i,j;p,a) -p^ + 4P'j + 0(i,) (37) 



Equating the time rate of change of the normal momentum 

 (compare (13) and (14) of Appendix 4) to the buoyant effect 

 of this pressure gradient would give a value z for the 

 coordinate of the bubble in the z-dlrection, satisfying 



d 



3 ^t-J 3 -Pi— 



dt 



In combining this with (37) we may eliminate p(a) by the 

 differentiated energy equation (5) of the text— i.e., by 

 Eq. (19) of Appendix 2 with c set equal to infinity — viz., 



.2„ , a,on2 



p'^'-po-rte+^l?)] 



o 



The result is, to order l/h , 



Now as was stated above in the paragraph following Eq. 

 (33) » the velocity field due to the image would, in the 

 absence of the bubble itself, cause a velocity of motion 

 dz /dt in the z-coordinate of the water at the position of 



