CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNESS.' 323 



inductive to a system of currents determined by <f> = Az. And, if this system be 

 excited in an outer shell of the series, it excites the corresponding system in an inner one. 



Similarly, if - ^ Z 5 ^ c> were spherical harmonica of negative degree, we 



should prove that the system of currents of the type <f>, excited in an inner shell of 

 the series, would generate by induction a system of the corresponding type in any 

 outer one. 



Of Shells of Finite Thickness. 



41. If the superficial currents induced and maintained in a shell be very small, or 

 the currents per unit of area finite, the inductive effect of the shell itself is 

 inappreciable compared with that of the original or inducing system. 



Suppose, then, we have a series of similar, similarly-situated, and concentric shells 

 successively enclosing one another, so as to form one solid shell of finite thickness. 

 Let each be separately self-inductive to the currents excited by the external field. 

 If the thickness of the solid shell, though finite, be small, we may, without great 

 error, neglect the inductive effect of any inner shell of the series of which it is 

 composed upon the outer ones. 



Let then S be any shell of the series, fl the magnetic potential on the outer surface 

 of S due to the whole field outside of it ; then we shall have, by (30) and (31), if f! 

 vary according to the simple harmonic law, 



dtl . XQ . ., 



= A - sin \t, A being a constant ; 



<>l' <T 



and, if p be the phase, 



'/< _ XQ 



dv <r 



t 



Let c denote the linear dimensions of any shell, and let c 1( c. 2 be the values of c at 

 the outer and inner surfaces respectively of the solid shell. Then 



; dv 



li = -- , 



and these equations become 



dfl , dfl . XQA . ., A x \, 



- = h = A sin A/ = A - sin \t, 



dc dv a K 



dp^ X 



dc K 



or 



fli f! 2 = AX [ - dc sin \t, 



2 T 2 



