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THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



and the outward flux through a closed contour surrounding the charge, 

 Fig. 3b, is 



^ = -9^0 + g(^o + 27r) = lirq. (21) 



The flux from a set of charges is additive, so that equation (21) is general, 

 when q is interpreted as the total charge inside the contour. 





Fig. 4 — Narrow closed contour surrounding an arc whose accumulated charge is q{z). 



Consider now a charge distributed continuously on a contour (C), and 

 let q{z) be the total charge on the arc extending from Zo to z. If we surround 

 this arc by an infinitely narrow closed contour, Fig. 4, we can pass from 

 Zo to z on the enclosing contour in either a clockwise or a counterclockwise 

 manner, by traversing respectively part 1 or 2 of the contour, on one or 

 the other side of the enclosed arc of C. The flux leaving the enclosing contour 

 through part 2 is 



^2=^2(Zo) -^2(Z), (22) 



where ^2 is the stream function in the region on the corresponding side 

 of (C). Similarly the flux leaving the enclosing contour through part 1 is 



<I>i=^i(z) -^i(zo), (23) 



where ^1 is the stream function in the region on that side of (C). Since the 

 total flux, *i 4- ^2 , is given by (21) we see that the stream function is 

 discontinuous across the line charge, and the amount of the discontinuity is 



[^1(2) -^i(zo)] - [^2(2) -^2(zo)] = 27rg(z). (24) 



