166 Prof. J. C. Maxwell on Electric Induction. [Feb. 15, 



that of the currents in the sheet, then the electromotive force in the direc- 

 tions of x is 



dt dx* 



where $ is the electric potential* ; and by Ohm's law this is eq ual to <ru, 

 where a is the specific resistance of the sheet. 

 Hence 



dF cty , 

 all — — ■ — - — — i . 

 dt dx 



Similarly, J. '\ (g) 



dt dy ' 



Let the external system be such that its magnetic potential is represented 

 dP 



by ~-j£> tnen ine actual magnetic potential will be 



V=-|(P + P), (7) 



and 



F= ^ (p ° +p) ' G= ~^ (P(>+p) • h= °- • • • (s) 



Hence equations (6) become, by introducing the stream-function $ from 

 (1), 



dy dtdy K ^ } dx~' 



-<r^ = d * (V I P\ d ^ 

 dx dtdx K + } &y' 



A solution of these equations is 



acj>=- ^(Po-f P), constant (10) 



Substituting the value of in terms of P, as given in equation (4), 



: <») 



The quantity -—- is evidently a velocity ; let us therefore for concise- 

 ness call it R, then 



dz dt dt v ' 



24. Let P ' be the value of P at the time t—r, and at a point on the 

 negative side of the sheet, whose coordinates are x, y, (z— Rr), and let 



Q=j o "P 'Jr. . (13) 



At the upper limit when r is infinite P ' vanishes. Hence at the lower 

 limit, when r=0 and P ' = P , we must have 



P °=§ +B ? ' ^ 



* " Dynamical Theory of the Electromagnetic Field," Phil. Trans. 1865, p. 483. 



