Ogilvie 



and the inner expansion of the two-term outer expansion is: 



^(x,y,z) ~ Ux + cif|(x,z) + — |y I o-|(x,z) + . . . . 

 0(1) 0(e) O(e^) 



I have taken my usual liberty of indicating unproven orders of mag- 

 nitude, I am not really assuming these orders of magnitude; I am 

 saying that one can prove that these are correct, and I display them 

 here now simply as an aid to the reader. 



Now consider the near field. Just as in the infinite -fluid 

 problem, one may stretch coordinates, y =€Y, and follow through 

 the consequences. This is effectively what I do, without writing the 

 change of variable explicitly. Thus, the Laplace equation yields 

 the condition: 



d), = 0, 



and so c^i must be a linear function of y. The same analysis as 

 used in the infinite -fluid problem. Section 2,11, leads to the con- 

 clusion that $1 is even more restricted than this. It must be a 

 constant with respect to y. Thus, let: 



^l(x,y,z)= A,(x,z). 



The two-term inner expansion is then: 



^(x,y,z) ~ Ux + A|(x,z). 



Matching gives the unsurprising result that: 



A,(x,z) = a|(x,z), (4-5) 



In other words, once again the inner expansion starts out simply as 

 the inner expansion of the outer expansion; it is not necessary to 

 formulate a near- field problem to obtain this result. 



The same arguments lead to the prediction that: 



$2(x,y,z) = A2(x,z) +U\(x,z)|y|. (4-6) 



Thus the three-term inner expansion is formed, and it can be matched 

 with the three-ternm inner expansion of the two-term outer- expansion, 

 yielding the familiar result once again that: 



772 



