.0 K(z+z ) [^"^1 



+ 7rcoK I f(z,,t)e 1 jQ(Kr)dZ|^ continued 



4^1 + ^2 



where 



+ TTcoK r f(zi,t) e^^^"*"^!^ J(.(Kr)dz, 



The potential 4^, corresponds to an axial distribution of simple sources 

 together with an image distribution above the free surface z = 0. To 

 emphasize this fact •we write ij^i i^ the form 



H 1 



4^1 = i j ^f(-|z;L|,t)[r2 + (z- zi)2]"^dzi [58] 



From the conventional slender -body theory of aerodynamics, we may ex- 

 pect this potential to satisfy the boundary conditions [55] and [56] on the 

 body to leading order in R.. In fact, differentiating with aspect to r and 

 neglecting terms which are of order R or R cos in the neighborhood 

 of the body r = R, we have 



d^, r H - 3 



'1 



"87 



= - i f |-f(-Ui|,t) r[r2 + (z- zi)^]'^ dz^ 



J _ll dt II 



H - 



- -i^f(-|z|.t) f r[r2 + (z- zj)^]" dz [59] 



24 



