CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNESS. 315 



and 



ft + ft = C (cos \t sin a sin Kt a) 



= C cos a cos \t a. . . ....... . (D) 



The internal field is therefore diminished in intensity in the proportion cos a : 1 , 

 and retarded in phase by /2ir of a complete period. 



Thia result agrees with that obtained by Professor LARMOR in case of a spherical 

 sheet ('Phil. Mag.,' January, 1884). 



31. The above results are obtained on the tacit hypothesis that the shell, what- 

 ever its thickness, is to be regarded for our purpose as a single shell in which all the 

 currents would decay pari passu, no allowance being made for variations along the 

 normal. On that hypothesis we may, if <r/h be finite, have finite superficial currents 

 in the shell ; and, as a consequence of their being finite, we have a finite difference 

 of phase, and intensity of the field diminished in a finite ratio, between the outer and 

 inner surfaces. 



This method cannot give accurate results except in the case of very thin shells. 

 Another way of treating the subject would be to regard the shell as made up of 

 a number of separate layers or subsidiary shells, successively enclosing one another 

 and separated, suppose, by non-conducting surfaces. Then we might apply the 

 formula (D) to each separate layer, and finally proceeding to the limit, make all the 

 functions vary continuously throughout the thickness of the solid shell. We will 

 consider the question in this aspect later. In the meantime we will point out certain 

 consequences which follow from the formula (D), when <r/h becomes infinite, and the 

 superficial currents infinitely small, namely, since cot a = /X, cos a = KJ^/(K Z -\- X 2 ), 

 sin a = X/ V /(K S + X 2 ). 



When a/A becomes infinite, K becomes infinite compared with X, Hence, cos a = 1 

 and sin a = a = X/K. 



The formula (D) becomes then 



ft = Ca sin \t ; 



or, since ft and a are infinitesimal, we may write 



rfn n w 



= C sin \t. 



da 



Again, 



X XQA 

 a= - = - ; 



and, writing dv, an element of the normal, for h, 



'' n =-CsinX^. 

 dv <r 



282 



