8s 1 9s- 



Then the line integral is expressed by 



J «.CP.«(i-^)».-d J .--J o( P>Q )^ + ^)„ x £l 



ds 



L o L o 



= 4tt a(Q) G(P,Q) n x ^- ds 



L o 



where s is the length along L_ and n = 8x/8n. Then we obtain 



= - J J a(Q) G(P,Q) d S Q + j- J a(Q) G(P,Q) n x |^- ds 



" 4 ^0 J J, 







>(x',y') G(P,Q) dx'dy' (21) 



■z'-O 



If iwe assume the disturbance velocity is so small that the function <£>(x,y) 

 is a negligible second order contribution, the velocity potential can be 

 expressed by the source distribution on the hull surface accompanied by 

 the line distribution. This yields 



J J G(P,Q) O(Q) d S Q + ±- J 



G(P,Q) a(Q) d s n + ±- I G < P »Q) a (Q> n „ S~ ds ( 22 ) 



x ds 



L o 



One can determine the source density a(Q) so as to satisfy the boundary 

 condition on the hull surface. Brard named this kind of boundary value 



14 



