r-a=0-5f.- . . . (60) 



Self-induction of Wires. 277 



r — a 2 surface). Therefore 



1 d 2 G = m 2 dY 



Sp dz 2 ~~ Sp — cZe 



which establishes the equivalence. 



Particular attention to the meaning of the quantity V is 

 needed. It is the line-integral of the radial force in the 

 dielectric from r = ai to r=a 2 . Or it may be defined by 



8Y = 27ra l a = Q, 



if Q be the charge per unit length of wire. But it is not the 

 electric potential at the surface of the wire. It is not even 

 the excess of the potential at the wire boundary over that at 

 the inner boundary of the return. For, as it is the line- 

 integral of the electric force from end to end of the tubes of 

 displacement, it includes the line-integral of the electric force 

 of inertia. It has, however, the obvious property of allowing 

 us to express the electric energy in the dielectric in the form of 

 a surface-integral, thus, ^Vcr per unit area of wire surface, or 

 2 VQ per unit length of wire, instead of by a volume integra- 

 tion throughout the dielectric. Hence the utility of V. The 

 possibility of this property depends upon the comparative 

 insignificance of the longitudinal current in the dielectric, 

 which we ignore. It may happen, however, that the longi- 

 tudinal displacement is far greater than the radial ; but then 

 it will be of so little moment that the problem could be taken 

 to be a purely electromagnetic one. We need not use Y at 

 all, (58) being the equation between e and C without it. It 

 is, however, useful in electrostatic problems, for the above- 

 mentioned reason. Again, instead of V, we may use a or Q, 

 which are definitely localized. 



The physical interpretation of the force — dY/dz, in terms 

 of Maxwell's inimitable dielectric theory, a theory which is 

 spoiled by the least amount of tinkering, confusion and 

 bemuddlement immediately arising, is sufficiently clear, espe- 

 cially when we assist ourselves by imagining the dielectric 

 displacement to be a real displacement, elastically resisted, or 

 any similar elastically resisted generalized displacement of a 

 vector character. When there is current from the wire into 

 the dielectric there is necessarily a back electric force in it 

 due to the elastic displacement ; and if it vary in amount 

 along the wire, its variation constitutes a longitudinal electric 

 force. 



(58) being a differential equation previously, let in itm 2 be 

 a constant. Then It/' and R 2 " may be thus expressed : — 



H/^RZ + L/p, BJ' = BJ + LJp, . . . (61) 



