448 Mr. J. Larmor. A Dynamical Theory of [Dec. 7 f 



this closed surface which lies in that part of the field ; as we have 

 seen there is practically no displacement, anywhere else in the field 

 except at the conducting wire ; therefore to preserve the law of the 

 circuital character of displacement throughout the whole space, we 

 must suppose that this alteration is compensated by a very intense 

 change of displacement at the conducting wire. So long as the move- 

 ment of the plates continues, as long does this flow of displacement 

 along the wire go on ; it constitutes the electric current in the wire. 

 Now, in calculating the magnetic force in the field, which is the 

 velocity of the sethereal medium, from the change of electric displace- 

 ment, we must include in our integration the effect of this sheet of 

 electric displacement flowing along the surface of the perfectly con- 

 ducting wires, for exactly the same reason as in the correlative 

 problem in hydrodynamics, of calculating the velocity of the fluid 

 from the distribution of vorticity in it, Helmholtz had to consider a. 

 vortex sheet as existing over each surface across which the motion 

 is discontinuous. 



The next stage in this mode of elucidation of electrical phenomena 

 is to suppose, once the current is started in our non-dissipative 

 circuit, that both the condensers are instantaneously removed, and 

 replaced by continuity of the wire. We are now left with a current 

 circulating round a complete perfectly conducting channel, which in 

 the absence of viscous forces will flow round permanently. The ex- 

 pression for the kinetic energy in the field is easily transformed from 

 a volume integral of the magnetic force, which is represented by 



the velocity of the medium ( 77, ), to an integral involving the 



Clt 



current (/, g, Ti), which is in the present case a line integral round 



Cvv 



the electric circuit. The result is Franz Neumann's celebrated 

 formula for the electromagnetic energy of a linear electric current, 



