CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 313 



Of course we may make cr vary instead of, or as well as, h. But it is most con- 

 venient to keep cr constant, or the shell of uniform material. 



If h be so chosen with <r constant, it harmonises the equations (A) 



(A); 



dt dx d* 



- V _^ 



h ' ' dt" dy 



a <m_<ty 



h ~ dt "" ./: 



and the shell so formed is, with the given current function <f>, self -inductive. 



It follows from the above that if a shell be self-inductive to a system of currents 

 denoted by U, V, W, then if the components of electromotive force due to the 

 external field be at every point on the surface proportional to the components of those 

 currents, they will induce in the shell a system of that type. 



We see further that, if < do not satisfy the required condition, the shell cannot be 

 made self-inductive, however we may choose cr/h. 



27. The constant K determines the rate of decay of the currents with the time. 

 Since K = cr/Qh, we see that K varies directly as <r, and inversely as h ; that is, if the 

 thickness of the shell be increased at every point in the same ratio, K is diminished in 

 that same ratio. Also K varies inversely as Q, and Q depends on the forms of S and <. 

 As an example, if S be a sphere of radius a, and <f> a spherical surface harmonic of 

 order , then, as is well known, 



4ira 2n + 1 <r 



(j = r r , and K = 7 - . 

 2n + I 4va h 



28. If now a conducting shell S be placed in any varying magnetic field, and if the 

 system of currents induced on it by variation of the field be always self-inductive, the 

 state of the system at any time, t, can be found, if the law of time variation of the 

 external field be given. 



For let fl be the magnetic potential of the external field, II that of the induced 

 system. Then we have 

 (1.) Due to induction alone 



(2.) Due to resistance alone 



rfft 



Therefore, generally, 



from which fl can be found as a function of t, if ddjdt be given 



MDCCCLXXXVIIL A. 2 S 



