Brard 



(^)^^Art(m,t') = (^)^^ r„,(V) [i-F*(t')] 



= (u) ^ ^oi(^') j F*(r') [H(t'-r')-l]dT', 



and 



(^) Ay*(m,t') = (J) ^ [Sy^Cm.+co) - Sy*(m,t')] . 



When (w/U)^/ is arbitrarily given, the differences are: 



Ar,(T,',t') = r,j(77') 



y (34) 



Ay/m,t') = g)^^ S7*(m,+co) - j g)^^ ^ Sy*(m, t'-r')dr' 



In the cases "bg," when (w/U)^- = 0, (Lq/U)^, = 0, (u/U)^' = fooC* ') ^^^ t' >0, 

 and in the case "bj," when (u/U) ,./ = 0, (w/U)^/ =o and (Lq/U)^< = fo2(t') , we 

 have similar formulae. 



In the general case, f ^ o( t ' ) , f ^ j( t ' ) , f g jC t ' ) being arbitrarily given for 

 t ' > 0, we get: 



2 t ' 



r(r,',t') = 2 r^iCT,') r foi(r') ^ F*(t'-r')dT' , 



i=0 



2 |. t' 



7(m,t') = y Jfoi(t') 7i(m,0+) + | ioi(^') A ^^i^'"' t'-T')dT' . 



i=o I -^0 ^t ^ 



y (35) 



5. Hydrodynamic Forces Due to the Velocity Potential 

 (case of par. 4) 



5.1. Definition of the Hydrodynamic Forces 



When w/u and LqAJ are small, there are no strong eddies due to separation. 

 Therefore the set of forces acting on the body is purely the sum of the follow- 

 ing sets: 



(i) (Jg) due to gravity (weight of the body, hydrostatic pressures), 



(ii) (?^) due to the inertia of the body. 



844 



