204 ADVANCED ELECTRICITY AND MAGNETISM. 



L is the inductance of unit length of the line and L-Ax is the 

 inductance of the elementary circuit abed. Therefore we have: 



di de 



Equations (i) and (2) contain the two unknown dependent 

 variables e and i, and it is necessary to eliminate one to get an 

 equation involving the other alone. By differentiating equation 

 (i) with respect to / arid equation (2) with respect to x we get: 



d?e d z i d*i d?e 



But 3 -r. = ',. , ; therefore we get: 

 dx'dt dt-dx 



d?e __ I d z e 

 dt* ~ CL dx* 





By differentiating equation (2) with respect to / and equation 

 (i) with respect to x and eliminating as before, we get: 



dtt i dH 

 dt* ~~ CL dx 2 



The general solution of these equations establishes little besides 

 the simple idea of travel, and if we introduce this idea at the 

 start we can get a clear idea of the electric wave very much as we 

 obtained a clear idea of a wave on a string by the discussion at 

 the end of Art. 115. 



Another point of view. Imagine current to be distributed over 

 a transmission line in any arbitrary manner, the current i at 

 any given point ad of the line (outflowing current in one wire, 

 returning current in the other wire) being represented by the 

 ordinate i of any given curve CC, Fig. 140, and suppose this 

 current distribution as a whole to travel to the right in Fig. 140 

 at velocity V. Such a traveling current distribution would 

 produce a voltage distribution over the line such that : 



e = LiV (5) 



