Forced Vibrations of Electromagnetic Systems, 375 



very rapidly, and cause the current to be practically confined 

 to near the boundary, and we have a simplified state of things 

 in which the resistance varies inversely as the area of the 

 boundary, which may, in fact, be regarded as plane. The 

 resistance now increases as the square root of the frequency, 

 and must therefore, as said, be infinitely great at the front of 

 a wave, which is also clear from the fact that penetration is 

 only just commencing. 



But for many reasons, some already mentioned, it is far 

 more probable that the wire is a dielectric. If, as all physi- 

 cists believe, the aether permeates all solids, it is certain that 

 it is a dielectric. Now this becomes of importance in the 

 very case now in question, though of scarcely any moment 

 otherwise. Instead of running up infinitely, the resistance 

 per unit area of surface of a wire tends to the finite value 

 4*wi- This is great, but far from infinity, so that the 

 attenuation and change of shape of wave at its front pro- 

 duced by the throwing back cannot be so great as might 

 otherwise be expected. 



Thus, in general, at such a great frequency that conduction 

 is nearly superficial, we have, if /a, c, k, and g belong to the 

 wire, 



E/H=(4w$r + ^)*(47r£ + <2?)-*, . . . (281) 



if E is the tangential electric force and H the magnetic force, 

 also tangential, at the boundary of a wire. Now let B/ and 

 1/ be the resistance and inductance of the wire per unit of its 

 length. We must divide H by 47r to get the corresponding 

 current in the wire, as ordinarily reckoned. So 47rA -1 times 

 the right number of (281) is the resistance-operator of unit 

 length, if A is the surface per unit length ; so, expanding 

 (281), we get 



where p l9 p 2 are as before, in (208). Here n/2-7r = frequency. 

 Disregarding g, and p 2 , we have 



R'or L / n=(i)"47r^A- 1 {B±B 2 p, . . (283) 

 where 



B = n{4 Pl 2 + n 2 )-* = nc{{4<7rk) 2 + n 2 c 2 }-*. 



When c is zero, R x and Un tend to equality, as shown by 

 Lord Rayleigh. But when c is finite, \Jn tends to zero, and 

 R' to 47r/Lti-A _1 , as indeed we can see from (281) at once, by 

 the relative evanescence of h and g, when finite. 



