310 MR. S. H. BURBTTRY ON THE INDUCTION OP ELECTRIC 



The, Effect of Resistance. 



21. If cr be the specific resistance per unit of area of the material of which a shell 

 is composed, the components of electromotive force must be, by OHM'S law, cru, cry, crw, 

 where u, v, w are the component currents per unit of area. That is, (a-/h) U, (<r/h) V, 

 and (<r/7i) W, where h is the thickness of the shell, and U, V, W the components of 

 superficial current. We see then that a-/h stands to U, V, W in the same relation as 

 cr to u, v, w. And the heat generated by resistance per unit of time and unit area of 



the sheet is ha- (u z + v 2 + ?) or | (U 2 + V 2 + W 8 ). 



22. Let dF/dt, dG/dt, and dfL/dt denote the time variations of vector potential. 

 Let y be the electrostatic potential. Then the law of decay of the system, if left 



to itself to decay uninfluenced by induction, is expressed by the equations 



<r TT dF d^f 



<ru = - U = 



ii dt dx 



a- T , dG dy 



crv =: V = 



h dt dy 



& . rfH dy , . , 



a-w = - W =-- . (A) 



h dt dz 



These conditions must be fulfilled with some value of y or other at every point 

 within the substance of the shell. If cr, the specific resistance, be constant, we obtain 

 by differentiation 



i ,dv dw\ 

 ; ~*~ dy dz] ~ 



at every point within the substance of the shell. 



23. If the currents decay as closed currents, their variations being given by that of 

 a current function, this requires v z y = at every point within the substance of the 

 shell. 



But we have also 



7 dF . dG dH , dy 



at every point on the surface. 



Therefore, if the currents decay as closed currents, S? is the associated function 

 above defined to dF/dt, dG/dt, dR/dt, which we denote by ty. The shell being of 

 very small thickness, it matters not whether the distribution whose potential is / be 

 on the inner or the outer surface. In either case dF/dt + d\(i/dx, &c., are the 

 tangential components of the time variation of vector potential. 



Generally an arbitrarily given system of closed currents on a shell of given form 



