24 Prof. H. Nagaoka on the Potential and 



Thus the potential of a circular current is given by 



where ra= ^ — . 



ZCOi 



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

 given by Hicks and Minchin, but it leads to the same result 

 (A"). The process of reduction from the expressions given 

 by the above-mentioned authors is more laborious. 



For the convenience of calculation the following values 

 of 7(a)— ^A. ^^^ tabulated for X=lj 2, 3 : — 



Thus ^i>7(a)>^2? showing that a is a purely imaginary 

 quantity. In addition to this 



2 Jacc\l 

 ^2-^3=:^=2(^-- I 



27 



r i^z-, = — -TZZ, 1^2—^3; 



^' '''"A iax " \ax 



-^2=1 ^rf- ■ = tL ie.,-ei). 



"^ "'^ A 'lax 4iax 



In practical calculation the most painstaking part is the 

 evaluation of a, for which we put, following Schwarz "^ 



\A1---g3 V7W— ^2— v^gi — ^2 VtW— ^3 ^j^ 

 where 1— v/^' 



1- V^' 



* Schwarz, Formeln und Lehrsdtze zum Gebrauche der elliptischen 

 Functionen, p. 71, Berlin, 1893. 



