WAVE PROPAGATION 553 



and corresponding to the \alues of a taken above 



r = 0.2 and 0.02 in round numbers, 

 while 



4 

 e = sin-i ~^ = 63° 30' approximately. 



For both cases, therefore, we can employ, in calculating J = P-\-iQ, 

 the approximate formulas, 



P = ~ ^~~r. cos e + ^ cos 2ef.6728+log -W^e sin 26, 



8 3-^2 lb \ rj lb 



(2= -0.0386 + ^ log f-W^^r cose. 



For X = 10~'- and r = 0.2, this gives 



J = 0.369+i 1.135 

 and 



Z;2 = 4a;(0.369+i 1.135). 



The foregoing assumes that the only return conductor is the ground. 

 If, however, an equal and opposite current flows in the rail we must 

 subtract from the foregoing mutual impedance, the mutual impedance 

 between rail and telephone line; that is, the mutual impedance Z 2 

 between the telephone line and a conductor at the surface of the 

 earth. For this case 



p" = Vh^-^-x^ = 4.12 X 103 



e = sin-i^^ = 76° 

 T7 



cos 9 = 0.242, r = 0.184 for X = 10-'2. 



The corresponding value of J is 



/ = 0.378+i 1.165 



and the resultant mutual impedance between railwa>- and parallel 

 telephone line is, 



Z'i2 = 4a;f0.369-0.378+i (1.135-1.165) ) 

 = 4co(0.009-* 0.030). 



The very large reduction in mutual impedance, due to the current 

 in the rail, is striking. 



