Bessho 



satisfies the above water surface condition can be expressed as 



.IT /-»oo k(z+z')*ik(w'* u') 



4^(P.Q) =7^-;^-f;i» JJ^ 



dk de 



2 



k cos 9 - g + M-i cos 6 



{A. 4) 



•e PH(x.y,z), Q= (xV,y',z'), Q h (x' ,y' , -z') , r(P , Q) = PQ 

 w = X cos 9 + y sin 9, w = x' cos 9 + y' sin 9. Hereafter, we 



where 



eind w = X cos 9 + y sin 9, co' = x' cos 9 + y' sin 9. Hereafter, we 

 will call this the fundamental singularity. This solution approaches 

 the following values asymptotically: 



^<^'«::^rT^{nPTw*;7F:K?}' <^-=> 



x»x' •"'1=^^'-."' r(P,Q) 



6) 



o/r> n^\ gr r 9 sec 0{(z+z') +i(w*S")} 2^ ,q ,. 



S(P,Q) » •=• Im \ e sec 9 d9, (A. 



x«x« ^ *i-Tr/2 



By considering the integral 



JJ[4>v(Q)S(P,Q) - <MQ)Sv(P,Q)] dS(Q) 

 about a point P in the interior of the fluid, we have the expression 



<KP) = Cr [<}>vS - 4>sj ds. 



Since <j> and S satisfy condition (A. 3) on F, we have, finally, 



^MP) =yj [<i)^{Q)s(P,Q) - 4>(Q)s^(P»Q)] dS(Q) 



-■^C [<KQ)S,.(P,Q) - (|),,(Q)S(P,Q)] dy', (A. 7) 



where L is the curve on which F cuts S. 



When the water motion is due to a pressure distribution over 

 the water surface, we have 



♦(P) = -^ jy P(Q)S,,(P,Q) dx' dy', {A.8) 



566 



