OgiZvie 



We do not know the conditions at infinity yet, but let us assume that 

 the condition on $| is similar to that on ^q, i.e. , the gradient of 

 ^1 should be bounded. 



This problem is identical to the $q problem, and so we can 

 represent its solution outside of some circle by another series like 

 the one in (2-51). We have not determined yet what the coefficients 

 of the increasing terms are like, and so we allow two nnore arbitrary 

 terms (the first two terms in the following): 



$,(x,y;z) =a,x +p,y + 6, log r + Ti,tan' ^ + A,o + Aji^os_e 



■ B|| sin 9 I A|2 cos 29 Bjg sin 29 , .^ j.^> 



All terms must be the same order of magnitude if r = 0(c). 

 Assunning that order to be € , we have: 



«|,Pi = 0(€); 6,, Ti,, A,o= 0(€^); A,, , B,, = O(e^); etc. 



With this information in hand, we combine the first two terms 

 in the inner expansion and then we obtain the outer expansion of the 

 two-term inner expansion: 



4>(x,y,z) -- Ux 0(1) 



+ ilotan-' ^ + Aoo+ Q'|X + P,y 0(€) 



. Aqi cos 9 , Bqi sin 9, c, 4.^4.-'yj.A r>.iJ^\ 



+ —^ + — "J + 6, log r + Ti, tan -^ + A,« . 0(€ ) 



r r I *= 'I X '^ 



(2-54) 



First we can keep just the first two orders of magnitude and match 

 them with the two-term inner expansion of the two-term outer ex- 

 pansion, given in (2-45b). Using (2-52), we see that everything 

 adready matches except for the terms QrjX + P^j ^^'^ ^h® integral in 

 (2-45b). For these to match, we require that: 



--■ ^^--ht"^- <^-"' 



Physically, this means that the $| problem should have had as the 

 condition at infinity: 



706 



