﻿Plane, < 'ylindrical, and Spherical Current' Sheets. 203 



fi + S/3, we have the surface-condition 



1 d h d£l 1 _ J_d . dflj 

 h x li 2 dt 8 f/y ~~ Ma^ * 3 ^7 



rf« 1 Cch 2 da J </£ 1 ( )ch x d& J 



But Laplace's equation which is satisfied by Xl x H 2 may be 

 written 



do\hJ h du ) + d/Ahfii dp) + rfy VM 8 d 7/ 



Hence it is possible to eliminate differential coefficients 

 with respect to a, /3 from the surface-condition if, and only if, 



Cc oc h 3 (11) 



Hence the thickness of the sheet at any point must be 

 inversely proportional to the distance between two neighbour- 

 ing surfaces of the family 7 = constant at that point. In the 

 case of an ellipsoidal sheet this leads to the well-known con- 

 dition that the thickness must be proportional to the perpen- 

 dicular on the tangent-plane, or that the sheet must be a 

 shell bounded by similar and similarly situated ellipsoids. 



Writing 



B =sSv ( 12 > 



the surface-condition now becomes 



dt\hh 2 dry J- dt\hi/i2 dyj- 2 dy\ hj 3 d^ h lj J { ° } 



Synthetic Determination of the Images in a Plane Sheet. 



8. In order to illustrate the synthetic method, consider its 

 application to the well-known case of a plane-current sheet. 

 Let the induced currents be excited by a magnetic pole placed 

 at the fixed point (# , y , z ) on the positive side of the sheet 

 whose strength is f (t) an arbitrary function of the time. 

 Then 



and the surface-condition satisfied by XI' the potential on 



