252, 253] ELECTKOMAGNETIC WAVES. 531 



number of degrees of freedom, and there are an infinite number of 

 periods. The above problem corresponds to the lowest possible 

 frequency for a sphere, when the surface-density of the electrifica- 

 tion is a zonal surface-harmonic of degree one. For the general 

 treatment of oscillations in spheres and cylinders the reader is 

 referred to J. J. Thomson, Recent Researches in Electricity and 

 Magnetism. 



253. Waves on Wires. We now come to what is perhaps 

 the most important practical problem connected with electrical 

 waves, namely their propagation along wires, for upon this question 

 depends the theory of telegraphy and telephony. The subject has 

 been treated in great detail by Heaviside*, to whose papers the 

 reader is referred. 



We shall suppose that the direct and return conductors are either 

 cylindrical wires parallel to the X-axis, or concentric tubes. Let R 

 be the sum of the resistances of the two wires per unit of length. 

 Let K be the capacity of the pair of conductors per unit of length, 

 L their self-inductance per unit of length. Let the total current 

 in one wire be / and the difference of potential between points 

 on the two wires having the same ^-coordinate be V. All these 

 quantities are supposed measured in the electromagnetic system. 



We may describe the phenomenon as follows. When an 

 electromotive force is applied between any two corresponding 

 points on the wires, say by connecting them with the poles of a 

 battery, electricity of opposite signs flows out upon the surfaces 

 of the two wires, producing an electric field in the surrounding 

 space. The electrifications then move along the wires, causing a 

 current, thus producing a magnetic field. Both these fields im- 

 mediately begin to penetrate into the conductor, and are there 

 dissipated into heat. As the electric field, whose lines are in the 

 planes perpendicular to the conductors, rises from zero, it gives 

 rise to displacement currents in the planes perpendicular to the 

 conductors. The magnetic effect of these displacement currents 

 we shall ignore in comparison with those of the conduction 

 currents in the wire. We shall also ignore the penetration of the 

 currents into the conductors, the theory of which would lead us 

 too far for the present purpose. This we may safely do if the 



* Heaviside. "Contributions to the Theory of the Propagation of Current in 

 Wires." Papers, xx. et al. 



342 



