AND THE LINES OF FORCE OF A CIRCULAR CURRENT. 



Thus, putting 



11=1 - . — or }iz=^lu) 



^ S 



Ü^A zfdu + z (a^-^) f ,, ^'' (12) 



Let 



then 



and 



^ (a)= 2(^''+a;-)-^^ 

 '6 ax A 



Ar '/ acr. 



Evaluating the last integral, we arrive at the result 



i2=-^^h^ z^% i L,a-w,~^ (a) \. (14) 



«2 — ^3 ^ ^ ' 



Thus the potential of a circulai- carrent or of a vortex ring is 

 given by 



yp=2;7-2 f '"'^ + iU a-to,~^ia)Y\ (A") 



L e^ — fig I (T fJi 



=:2;r -^ z — t — ^ ^ ^ ^ where v^=: 1 



The form of integral (A') is somewhat different from that given 

 by Hicks and Minchin, but it leads to the same result. The process 

 of reduction from the expressions given by the above mentioned 

 authors is more laborious. 



