Theory of Metallic Conductors. 419 



or denoting by x the depth R—r, and neglecting the square 



of -ff 



R p_ 



PR 



i. e. ultimately, remembering that R\L is at any rate a very 

 large number, 



. / x\ . . (R-x\ . . R, 

 =f ^ 1+ _j gin h^_^J:sinh r , 



P.'pR 



( 1+ S-" r . 



which, for any # equal to several Z/ and to a small fraction 

 of R, reduces simply to the exponential decrease e~ x/i/ , agree- 

 ing with Thomson's result. 



5. Hollow sphere. — Again, let there be no charges outside 

 the metal (nor in the cavity), i. e. cj) e = or const. Then we 

 have again (8) as the general solution of the differential 

 equation. In this case there is, of course, no reason for 

 rejecting any of the constants A, B. Both have to be 

 determined, from the given charge and from the integral 

 equation (I), which in the present case becomes 



L 2 o+ — I -, — = const., 



±ttJ r 



the value of! the constant being irrelevant. Substituting p 

 from (8), denoting the inner and the outer radii of the 

 conductor by R\ and R 2 respectively, and throwing the 

 constant term of the space integral upon the " const." on 

 the left hand, the reader will easily find 



A R,+L - 2 ^ 



= P - L , 



B R,-L 



the required ratio of the coefficients * . It is remarkable 

 that this ratio contains only the inner radius of the conductor. 

 The solution (8) becomes 



B( - r L R x + L r -=^\ 

 P = v (e '+3^6 *)..-.. (10) 



The remaining factor B can easily be determined from 

 the given total charge, but this need not detain us here. It 

 is interesting to notice that the particular law of distribution 

 depends only upon the inner and not on the outer radius 

 (the latter entering only through the factor B and being thus 



* For a full sphere, *. e. for Ri — 0, this reduces to A= — B, as stated 

 before. 



