160 Mr Mayall, On Current Sheets, especially [Feb. 26, 



These conditions with equation (3) are necessary for the 

 solution of the problem, and they are also sufficient; for the forms 

 assumed for Oj, fig each contains an arbitrary constant, so that 

 we have three equations to determine these and <I>. 



We proceed now to the examination of some simple cases. 



(2) Suppose the surface a=a^ is an infinite plane, e.g. the 

 plane of xy, using Cartesian co-ordinates, and let us further suppose 

 that this plane is made of a conducting substance whose specific 

 resistance is the same at every point. Then we have 



ds^ = daf + dy^ + dz^, 



and equation (3) becomes 



(d^^ d^^\ d d ,^ ^. ,,. 



'^(s?+^)=-*s;("+"«> w- 



Let P be the potential due to a distribution of density <I> over 

 the sheet, then 



27r dz , 



"~ dz I 



P being here the value of the potential on the positive side of the 



dF 



sheet, hence if ft^ = — -~ (4) becomes 



27r dz Kdx" + dyV ~ dt dz'^ '^ '^' 



or since V^P = 



(T d ± d d' , jy j-jv 



27r~d7^did?^'^°^' 



and this is satisfied if 



a dP ^ d p p. 

 27r dz dt^ '^ '^' 



which is the equation given by Maxwell, having for a solution 



p=j-{..,.(.+|i)}. 



We can, however, find a solution in a different way, for suppose 

 Vl^ = A/P^J^^ (qr) e-^^ cos m0^ 



n^ = A^e^P^J,,, (qr) e"?^ cos to<^ i (5), 



n^ = A/P^J„, (qr) e+?^ cos mcf>) 



