Ogilvie 



<^(x,y,z;c) ~ Ux 0(1) 



+ a|(x,z;€) 0(e) 



+ Qr2(x,z;c) +2 |y |o-,(x,z;c) 0(€^) 



111 i I |2 3 



+ Q'3(x,z;€) +2 |y |o-2(x,z;€) --^lyl (o^i^x ■*" ^^i^z ) ^Ce*) 



+ 0(c^). (2-13) 



Note that we have reverted to far -field variables. We must here 

 consider that y = 0(e) in order to recognize the orders of magnitude 

 as indicated above. 



Next we must find the inner expansion of the exact solution. 

 Substitute y = eY in the formulation of the problem. The Laplace 

 equation transforms as follows: 



The kinematic condition on the body is: 



± <^jjh^ - <^y =*= ^^i = on y = ± h(x,z), 

 which transforms into: 



«I>Y==^ €^(*x"x ■*■ *z^2^ on Y = ±H(x,z). (2-15) 



We assume that there exists a near- field asymptotic expansion of the 

 solution: 



N 



^x,y,z;€) ~ ) $n(x.Y,z;e), where $„♦! = o($n) as € "* 0, 



nsO 



for fixed (x,Y,z). (2-16) 



We could show carefully that: 



$o(x,Y,z;c) = Ux . 

 (Perhaps it is obvious to nnost readers.) We then express the con- 



684 



