357 



The last condition is obtained by taking Bernoulli's 

 equation on the siirface, assuming that gravity can be 

 neglected, and then neglecting square terms. 



Equation (4.1) is satisfied by taking 



4 = A A 



* R • 



2* 



This is equivalent to assuming a source of strength A A at 



the center of the sphere. We can then satisfy equations 

 (4.2) and (4.3) by reflecting this source in the bottom 

 and in the free surface. It turns out that the image with 

 respect to one boundary must be reflected in the other 

 boundary and this process carried out on the successive 

 Images leads to an infinite sequence of images: Let 



(4.4) i = A^A(^ + $') 



where f is the potential due to the images. It can be 



evaluated by a method similar to that in Report 37. IR. 



After f has been found we get the kinetic energy of 



■) 

 the water by evaluating f 



ip/* 



i* 



over all boundaries. Because of conditions (4.2) and 

 (4.3) this reduces to the integral over the sphere. 

 Using equation (4.1) we have that 



(4.5) E = -g p / $ II ds = A J $A^dco = A^A J $dco 



sphere 



2 

 where dco is the element of solid angle so that ds = A dco. 



By the Mean Value theorem 



^ J$' dco= $'(0) 

 where # (0) is the vsilue of the potential $ at the center 



