210 Prof. Gr. M. Minchin on the Magnetic Field 



From (14) we can derive an expression for the conical 



angle subtended at any point, P, in space by a circle, i. e., for 



the magnetic potential due to a current coinciding with the 



circle. It is well known that this conical angle is numerically 



equal to the component of the attraction, perpendicular to the 



plate, at P due to a uniform circular plate coinciding with the 



aperture of the circle — a result which is evident from the 



principle that the current can be replaced by a magnetic 



shell, or thin plate, the upper and lower surfaces of which 



are, of course, of opposite signs. But the resultant potential 



of these two indefinitely close plates is the difference between 



the value of Y in (14) and the value which (14) assumes when 



z + Az is substituted for z ; that is, the magnetic potential at 



dV 

 P due to the current is — — . Az, and the strength of the 



CLZ 



magnetic shell is m . Az, which is i, the current in the circle ; 

 so that the magnetic potential is i multiplied by minus the 

 differential coefficient of the right-hand side of (14) with 

 respect to z. 



Denote the function tt + KK/ — KE^ — EK^ by the symbol 

 A e , and for simplicity in the differentiation with respect to z 

 [x being constant) write (14) in the form 



^ = -^ + P B+(a 2 -^)|. . . . (16) 



Now 



a fcfc . J* OjHj k » y^ 



-y = sin a ; -p = - sm c/ ; 



az p az p 



dp . a d6 1 Q 



(17) 



and, regarding P as determined by the coordinates (z, x) 

 instead of (Jc, 6) , we have 



but 



therefore 



d _dk d dO d 

 iz dz ' dk dz ' dd' 



<iA_ K— E sinflcosfl 



~dic = ~T~ ~~E Q > 



dA_ Kk l2 sm 2 0-E 



dd ~ A' e ; 



dA 1 „., ft 



