90 PROPERTIES OF CIRCULAR CURRENTS. 



By the same reasoning as above (731), we find that the maximum 

 action corresponds to the case in which the resistance of the coil is 

 equal to that of the external resistance.* 



736. ACTION OF A CIRCULAR CURRENT OUTSIDE THE Axis. 

 The potential of a circular magnetic layer of unit density (366) at a 

 point at a distance y from the axis, may be expressed by a series 

 developed in even powers of 7, of the form 



(28) P=2T(/ +/ i y+/ 2 / + ...), 



in which the coefficients / ,./i,/2> ... are functions of the distance x 

 of the point in question from the plane of the current. 

 If a be the radius of the circle, and we put 



the series is always converging when y<u. 



If / ', //, / 2 ', ...... / ", //' ... are the successive differentials of 



the coefficients in respect of x, the potential V at the same point of a 

 uniform shell of the same contour whose magnetic power was equal 

 to unity (368), or the potential of unit current along the circular 

 contour, is then 



The expressions for the components of the magnetic action of 

 the current are 



x- -= 



Y= = 



The values of the coefficients are 



du 

 f n =u-x, fn = -r- 



(3) , _i_ d*u y = _l_ ^ 



f* = "(2.4.6)*'^' ~ (2.4.6)2'^' 



* AYRTON and PERRY. Journal of the Society of Telegraph Engineers, 

 Vol. vii., p. 297. 1878. 



