Ogilvie 



this term represents a distribution of vortlclty extending to infinity 

 both upstream and downstream. Thus, It leads to a discontinuity 

 across the y = plane, even upstream. The second term must 

 compensate for this behavior, since there can be no discontinuities 

 In the region x < 0. 



2) The second term In brackets on the right-hand side of 

 (2-41) must be considered carefully with respect to the branches of 

 the square-root function. With a bit of effort, one can show that, 

 as r -♦ , Its contribution Is: 



31^>(l-itan' XXl+OU")), < tan' 2<„; 

 lk)(3 - |tan-' i) ( 1 + OU'')) , ,r< tan-' i< Eir . 



Combining this result with the previous one, we find that the 

 Inner expansion of a llftlng-llne potential function can be written as 

 follows: 



jL r \ v(o de d^ 



'-s -'o [(x-er + y' + (z-;n- 



for < tan' ^< 2w. (2-42) 



X 



Inner expajislon of the dlpole-llne potentleil : An Integration 

 by parts with respect to Z, transforms these Integrals Into an 

 appropriate form so that one can let r -^ and thereby obtain the 

 first terms In the desired expansions. Typical terms In the second 

 and third sums of (2-40) have the following Inner expansions: 



1 r' [yH(U^xXa)] d; _ rjtizLlxMz) 1 n ^ 0,^2 j T (^.^j. 



Note the occurrence of the logarithm of € 1 



Inner expansion of the outer expansion: In order not to con- 

 fuse the picture, I shall make more assumptions now, namely: 



704 



