14 BELL SYSTEM TECHNICAL JOURNAL 



integrals become independent of 5 and approach the Hmits 



(X - \')ds' = I, 



«y — c» 



r (m - i^')ds' = - , 



J- 00 C 



whence 



rl + ioill -j-J^ I ^ E. 

 twc 



Finally assuming that the impressed electric intensity E = 0, and 

 introducing the relation 



we get 



i^Q=-l^L 



r + icoZ + J-^, ) / = 0, 



which is the telegraph equation. 



Besides its formal theoretical interest the foregoing derivation of 

 the telegraph equation admits of some deductions of practical impor- 

 tance. These deductions, which are rather obvious consequences of 

 the analysis, may be listed as follows. 



1. The telegraph equation, as derived above, applies with accuracy 



only at points at some distance from the physical terminals of 

 the line. 



2. The accuracy of the telegraph equation in formulating the phys- 



ical phenomena decreases in general with increasing frequency. 



3. The telegraph equation is the first approximate solution of an 



integral equation. The first approximate solution decreases in 

 accuracy with decreasing wave length of the propagated current. 



4. While the telegraph equation indicates a finite velocity of propagation 



of the current along the line, it is based on the assumption that 

 the fields of the currents and charges (as derived from the potential 

 functions $ and A) are propagated with infinite velocity. 



5. As a consequence of (4), the telegraph equation does not take 



into account the phenomena of radiation, and in fact indicates 

 implicitly the absence of radiation. 



The Coil Antenna 



An important example of the type of problem to which the fore- 

 going analysis is applicable is the coil antenna. To this problem 



