PROPAGATION OF PERIODIC CURRENTS 543 



which can be written 



K/ ^ d-'\ Y'KdF 



where 



and 





7 = V(s + i-coL)(G + ^coC). • (12: 



Thus K is the characteristic impedance and 7 the propagation con- 

 stant of the transmission system composed of the overhead wire with 

 ground return; it is to be noted that G = G° + Y' , in accordance 

 with (8), and hence that G is in general an admittance — not a pure 

 conductance. 



If we define/' by the equation 



then (10) becomes 



Y'K dF 



7(7'-!^]^=/'. (H) 



which is formally the same as equation (25) of the text. There, 

 however, it is assumed that the term {Y'Kly)dF/dx is negligible; 

 probably this is usually the case but circumstances may arise where 

 it is not negligible. Its inclusion, however, introduces no formal 

 modification of the analysis. 



The foregoing derivation has been given in detail because prior 

 derivations known to the writers have not been entirely satisfactory. 

 Their chief defect has been that no explicit consideration was given 

 to the finite conductivity of the ground (except that it produces a 

 tangential component /„). In the derivation given above, the effect 

 of ground conductivity is expressly recognized and in the final equa- 

 tion appears implicitly in the values of K and 7. These parameters, 

 it will be observed, are experimentally determinable, and are the only 

 parameters besides Y' appearing in the final differential equation. 



A formula for the quantity G" occurring in equations (6) and (7) 

 can be derived by application of Gauss' theorem, as follows: Let Er 

 denote the radial component of the total electric force at the surface 

 of the wire, dS a differential element of the surface of the wire, and 

 0" and € the conductivity and specific inductive capacity of the medium, 

 which is homogeneous and isotropic by assumption. Then the leakage 



