Ogilvie 



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



<|>(x,y,z;€) ~ Ux +^ B,(x,z}e) |y I . 

 0(1) 0(€) 



On the other hand, the two-term inner expansion of the two-term 

 outer expansion is, from (2-13), 



<|)(x,y,z;c) ~ Ux +Q', (x,z;€). 



There is no linear term here at all, and it seems that we cannot 

 match the two expansions. 



It is a very comforting feature of the method of matched 

 asymptotic expansions that things go wrong this way when we have 

 made unjustified assumptions. Our mistake was this: When we 

 found that apparently B, = € UH^ = 0(c ) , we eliminated the possi- 

 bility that there might be a term which is 0(e) in the inner expan- 

 sion , Now we rectify this error. Once again, let $| be given by 

 (2-20), but suppose that both "constants" are, in fact, 0(e). The 

 body boundary condition immediately yields the condition that: 



B,(x,z;e) = 0, 



and so we have: 



*,(x,Y,z;€) = A,(x,z}€) . 



The inner expansion, to two terms. Is now given by: 



4)(x,y,z;e) ~ Ux + A,(x,z;c). 



When we match this to the inner expansion of the outer expansion, 

 we find that: 



A|(x,z;c) = Q',(x,z;c) = <^|(x,0 ,z;€). 



(See 2-11.) Now we have matched the expansions satisfactorily, but 



*This trouble would have been avoided if I had started by assuming 

 that the expansion is a power series in c , as many people do in such 

 problems. However, that procedure can lead to even greater diffi- 

 culties sometimes. 



686 



