Forced Vibrations of Electromagnetic Systems. 361 



other hand, the distortion produced by conductivity depends 

 on <r, and vanishes with it. There is some utility in keeping 

 in g, because it sometimes happens that the vanishing of k, 

 making p= — a, leads to a solvable case. We can then pro- 

 duce a real problem by changing the meaning of the symbols, 

 turning the magnetic into an electric field, with other changes 

 to correspond. 



32. The steady Magnetic Field due to f constant. — Let /be 

 zero before, and constant after £=0, the whole medium having 

 been previously free from electric and magnetic force. All 

 subsequent disturbances are entirely due to /. The steady 

 field which finally results is expressed by (206), by taking 

 p=0 ; that is, hy has to mean Airk, and q = 4,ir{kg s )^, by (207). 

 To obtain the corresponding internal field, exchange a and r 

 in (206), except in the first a/r. The same values of k x and q 

 used in the corresponding equations of E and F give the final 

 electric field. The steady magnetic field here considered 

 depends upon g, and vanishes with it. 



33. Variable state ichen p x = p 2 . First case. Subsiding /. — 

 There are cases in which we already know how the final state 

 is reached, viz. the already given case of a nonconducting 

 dielectric (§§ 21, 22), and the case <x = in (208), which is 

 an example of the theory of § 4. In the latter case the 

 impressed force must subside at the same rate as do the dis- 

 turbances it sends out from the surface of/. Thus, given 

 f=f Q e~ pt , starting when £ = 0, with/ constant, the resulting 



electric and magnetic fields are represented by those in the 

 corresponding case in a nonconducting dielectric, when multi- 

 plied by e~ pi . The final state is zero because / subsides to 

 zero ; the travelling shell also loses all its energy. But there 

 are, in a sense, two final states ; the first commencing at any 

 place as soon as the rear of the travelling shell reaches it, and 

 which is entirely an electric field ; the second is zero, produced 

 by the subsidence of this electric field. There is no magnetic 

 field to correspond, and therefore no "true" electric current, 

 in Maxwell's sense of the term, except in the shell. 



34. Second case, /constant. — But let the impressed / be 

 constant. Then, by effecting the integrations in (206), we 

 are immediately led to the full solution 



H = f/± r e -!"->(i+ *yi_ i\ +e -, l(1+pt) ^ 



Zftvr L v P r J\ pa/ rap 



+ same function of —a \, • • (209) 



where the fully represented part expresses the primary wave 

 out from the surface of/, reaching r at time (r—a)/v ; whilst 



