Field of Circular Currents. 

 and by the well-known integral 



1 



379 





/-eco /id 



U = 2tt I I e-^rJ (Xp)J (\r)d\dr, 



Jo Jo 



which by the relation 



J (Xr) 



1 d 

 X'd?= 



l^iO)), 



u 



g-A* 



= 2ira i ^— — J (Xp) Ji(\a) dX. 

 Jo X 



(4) 



The magnetic potential of a circular current is easily 

 derivable from the above result by simple differentiation 



<f> = -^ = 2ira f <r^J (ty) J,(Xa) dX, . (5) 



".y Jo 



and the function which represents the magnetic lines of 

 force is 



■f = -p^ = 2irapi e-^J^XpyJiiXaydX. . (6) 



OP Jo 



These formulae were already obtained in my former 

 paper ; the present investigation aims at the development 

 of formula (5) for expressing the magnetic field due to 

 different coils. 



By a simple differen iation of (5) with respect to p and ?/, 

 we obtain y- and ^-components Y and P of the magnetic 

 field due to a unit current : 



Y = -^ = 2ira j A<T X * J (\/>) Ji(Xa) dX, 



P = -1^= 2?ra jXe-^J^Xp) J^XaJdX. 

 cp J 



From the addition theorem, we obtain 



J (Xo) Ja(Xa) = - f" Jl - ( ^(a-pcos(9)^, 

 ^Jo K 



J 1 (Xp)J 1 (Xa) = - (" J o [Xll)cos0d0, 

 •""Jo 

 and 



(7) 



