Mr. 0. Heaviside on Electromagnetic Waves. 203 



_^ = _jc ^ 



dz dz dt 



8 being the permittance and K the conductance of the dielec- 

 tric per unit length of circuit, whilst R" is a " resistance- 

 operator," depending upon the conductors, and their mutual 

 position, which, in the sinusoidal state of variation, reduces to 



R"=R'+I/4 



dt 



where R' and 1/ are the effective resistance and inductance 

 of the circuit respectively, per unit length, to be calculated 

 entirely upon electromagnetic principles. It follows that the 

 fully developed sinusoidal solution is of precisely the same form 

 as if the resistance and inductance were constants. Disre- 

 garding the effect of reflexions, we have 



Y = Y e~^ sin (nt-Qz), 



due to V sin nt impressed at z = ; where P and Q are 

 functions of R/, U, S, K, and n. 



Now if H'/Ij'n is large, and leakage is negligible (a well- 

 insulated slowly worked submarine cable, and other cases), we 

 have 



P = Q=(±RSrc)*, 



as in the electrostatic theory of Sir W. Thomson. There is 

 at once great attenuation in transit, and also great distortion 

 of arbitrary waves, owing to P and Q varying with n. 



But in telephony, n being large, P and Q may have widely 

 different values, because Rf/Un may be quite small, even a 

 fraction. In such case we have no resemblance to the former 

 results. If W/h'n is small, P and Q approximate to 



r ~2LV^2St/' H v'' 



where v r = (L'$)~i. This also requires K/Sw to be small. 

 But it is always very small in telephony. 



Now take the case of copper wires of low resistance. 1/ is 

 practically L , the inductance of the dielectric, and v' is prac- 

 tically v, the speed of undissipated waves, or of all elementary 

 disturbances, through the dielectric, whilst R' may be taken 

 to be R, the steady resistance, except in extreme cases. Hence, 

 with perfect insulation, 



P„ * Q=2, 



2L u v 



