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LXVI. Intelligence and Miscellaneous Articles. 



CONTRIBUTION TO THE THEORY OF THE MOTION OF ELECTRICITY 



IN SUBMARINE AND UNDERGROUND TELEGRAPH-WIRES. BY G. 



KIRCHHOFF. 



A SS TIMING- that the induction-effects produced by alterations of 



-£*- the current-intensity may be neglected against the influence of 



the charges of an underground telegraph-wire, Sir W. Thomson has 



referred the propagation of the electricity therein to the same laws 



as the conduction of heat. Gr. Kirchhoif develops this relation in 



connexion with the equations developed by Helmholtz respecting 



the components of the current-density (u= — A -^-, &c, where X de- 



ox 



notes the conductivity) and of the electrostatic moment dependent 



on the dielectric polarizability* (oL=—Jt—, &c), if <p denotes the 



electrostatic potential, which is a function of x, y, z and consists of 

 three parts, arising : — first, from the free electricity in and upon 

 the conductor; secondly, from dielectric polarization; and, finally, 

 from the double electric layer at the boundary surfaces of hetero- 

 geneous conductors. From the calculations, which cannot be given 

 in abstract, it follows that 



<f> = 6^(0 cos (nt + a,z) + C sin (nt-\-az)) + e-^(B cos (nt— olz) 



-f D' sin (at — as)), 



which equation represents two passages of waves in opposite direc- 

 tions along the z axis of the wire, in which the height of each wave 

 as it moves forward diminishes correspondently to the value /3. The 



period for cj> according to the time is — ; /3 and a are given by the 



equations ■ ™ i_ 2A 



fia, ~ 



9i 



n/j. 



A lPl 2 log^ 

 Pi 

 where p l and p 2 are the iuternal and external radius of the gutta- 

 perch sheath, A and A t the conductivities of it and the wire, 

 47rju=l + 47r& the constant of dielectricity of the gutta-percha. 



Therefore the velocity of propagation of the waves increases with 

 the conductivity of the gutta-percha, simultaneously with which 

 their height diminishes as they travel onwards. 



If the conductivity of the gutta-percha A=0, then becomes 



=li= \/-. 



UfX 



Vi ] °g- 

 Pl 



* Crelle's Journal, vol. Ixxii. 



