Forced Vibrations of Electromagnetic Systems. 441 



Jo*(Jo*— yiGrox) 4ttK 1 ^ 1 



— 3/1 47rK 1 J 0l (J i— yiGro-c) 



7T6\£ 



and, by comparison with (317) we see that it is now the same 

 as if the inner tube were non- existent. That is, when it is 

 situated at a nodal surface of E due to impressed force in the 

 outer tube, and there is therefore no current in it (except 

 transversely, to which the dissipation of energy is infinitely- 

 small), its presence does nothing, or it is perfectly trans- 

 parent. 



It is clearly unnecessary that the external impressed force 

 should be in a tube. Let it only be in tubular layers, with- 

 out specification of actual distribution or of the nature of the 

 medium, except that it is in layers so that c, k, and fi are 

 functions of r only ; then if the axial portion be nonconduct- 

 ing dielectric, the J 0r function specifies E and allows there to 

 be nodal surfaces, for instance J 0a = 0, where a conducting 

 tube may be placed without disturbing the field. Admitting 

 this property ab initio, we can conversely conclude that e in 

 the tube at r = a will, when J 0a =0, make every external cylin- 

 drical surface a nodal surface, and therefore produce no 

 external disturbance at all. 



54. Now go back to § 51, equations (317) (318). There 

 are no external nodal surfaces of E in general (exception 

 later), We cannot therefore find a place to put a tube so as 

 not to disturb the existing field due to e in the tube at r = a. 

 But we may now make use of a more general property. To 

 illustrate simply, consider first the electromagnetic theory of 

 induction between linear circuits. Let there be any number 

 of circuits, all containing impressed forces, producing a 

 determinate varying electromagnetic field. In this field put 

 an additional circuit of infinite resistance. The E.M.F. in 

 it, due to the other circuits, will cause no current in it of 

 course, so that no change in the field takes place. Now, 

 lastly, close the circuit or make its resistance finite, and 

 simultaneously put in it impressed force which is at every 

 moment the negative of the E.M,F. due to the other circuits. 

 Since no current is produced there will still be no change, or 

 everything will go on as if the additional circuit were non- 

 existent. 



Applying this to our tubes, we may easily verify by the 

 previous equations that when there are two coaxial tubes, 

 both containing impressed forces, we can reduce the resultant 

 electromagnetic field everywhere to that due to the impressed 

 force in onp tube, provided we suitably choose the impressed 



