Self-induction of Wires . 131 



wherein J 2 and K 2 are derived from J x and K x as the latter 

 are derived from J and K . So the left side of (21) will 

 become 



__ £3f§ J^ = _ Ps _ _ t / 2 4) 



p 2 S 2 «3 a 4 p2 a ± S 2 



The inductivity of the sheath is now of no importance. Being 

 on the outer edge of the magnetic field, the thinness of the 

 sheath makes its contribution to the magnetic energy be 

 diminished indefinitely. 



Again, in important practical cases, the resistance of the 

 return is next to nothing in comparison with that of the wire. 

 Then put /o 3 = in (21). This makes the left side vanish, and 

 then we sweep away the denominator on the right side, and 

 get the determinantal or differential equation 



Q _ J 1 (s l a l )K (s 2 a l )-(p 1 s 1 /p 2 s 2 )J (s 1 a l )K 1 (s 2 a l ) j ^ . +K _ 



Although we may have the return of nearly no resistance 

 and yet of low conductivity (as in the case of the Earth), yet 

 it cannot be quite zero without infinite conductivity, which is 

 what is here assumed. The result is thai; we shut out the 

 return conductor from participation, except superficially, in 

 the phenomena. (24) will result from the condition /o 2 r 2 = 0, 

 or r 2 = 0, at r = a 2 ; that is, no tangential current, or electric 

 force, in the dielectric close to the sheath. If there could be 

 any, it would involve infinite current-density in the sheath. 

 As it is, there is none, and the return-current has become a 

 mere abstraction, to be measured by the tangential magnetic 

 force divided by 47r, and turned round through a right angle 

 on the inner boundary of the sheath. In a similar manner, if 

 we make the wire infinitely conducting (or of infinitely great 

 inductivity either) the wire will be shut out. Then the mag- 

 netic and electric fields are confined to the dielectric only, 

 and we shall have purely wave-propagation, unless it be a 

 conductor as well. 



Now, with the return of no resistance, let the dielectric be 

 nonconducting and the wire non-dielectric, or Ci = 0, k 2 = 0. 

 The most important simplification arises from the smallness of 

 s 2 a 2 . For we have 



—sl=fi 2 cp 2 + m 2 . 



If the length I of the line is a large multiple of the greatest 

 transverse length a 2 we are concerned with, m 2 is made a 

 small quantity — very small when the line is miles in length, 

 except in case of the insignificant terms involving large mul- 

 tiples of 7T in m=mrlL Again, {fi 2 c)~^ is the speed of light 



K2 



