Murthy 



j> V 2 (*G£ - h&g) + ZiaV *G 



L 3 



I— <J» V 2 (*G£ - GV^) + 2i<r V*G dr. 



dV + 



+ 4 



where the line integrals are taken in the clockwise direction (viewed 

 from above) along L 3 and in the counter-clockwise direction along 

 L^ in order to keep the domain of integration to the left, bearing in 

 mind that the positive side of the element of area d£drj is along the 

 z-axis i. e. below the free surface. This is in accordance with the 

 usual convention. 



It is important to note that the above substitution for ^t 

 from (V. 4) is only valid for potentials of 0(5, 13 , 8a and a ) . It is 

 not valid for the potential of 0(5/3 ). 



It can be shown that the line integral along L-. also vanishes 

 under the assumed radiation conditions. 



We have then, 



iS 



*(x,y,z> -^-JJ (g -§*-*-§£.) as ; 



2 r 



+ — ~ & (*G, -G*£+— — *G) dT, (V. 11) 



4*g { * V 



4 



We may now contract the surface / , surrounding the ACV 

 on the water surface to the actual boundary of the craft composed of 

 the immersed hulls S 1 and S and the internal free surface S . 

 The curve L, will then tend to the curve Ln in the limit. 



222 



