714 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the current Iq e— -^" arriving at B after the first passage, is partly re- 

 flected with inversion as from a discontinuity, and the rest of it passes 

 through the impedance c^ to ground without' further reflection. Let 

 m be the fraction of the current that goes through Zr- This coefficient 

 m may be called the transmission-coefficient of the impedance Zr. We 

 have 



m = -^ (53) 



a vector. Consequently the remaining portion of current 1 — m is 

 reflected with inversion, as from a discontinuity, and may be called 

 the reflection-coefficient of the impedance z,.. 



l-m = ^^-^ (54) 



Zo + Zr ^ ^ 



This portion of current retreats with its sign reversed. 



The current which flows through the receiving impedance Zr on first 

 passage is ?n /o e—^'^ amperes, and a current — (1 — m) I^ e—^"^ = 

 (m — 1 ) /(, e—^'^ returns to the sending end. It reaches that end in 

 the condition (m — 1) /oe~~^-^" and is reflected from the ground back 

 to B without inversion. On reaching B the second time, it will have 

 dwindled to (m — 1) /(, e—^J^'' amperes and subdivides. The portion 

 m, {m — 1) /„ €—'^La. passes directly to ground through c,., and the re- 

 mainder is reflected homewards as (w — 1)^ /q e^^^". The final current 

 strength built up in the receiving impedance Zr will be : 



/^ = m /„ c--'^" + m {m — 1) /„ e"^-^" + m (m — l)'^ Iq i~^^'^ + • • • (55) 

 = ??//oe-^«{l -f (m — l)e-2i'^ + {m — l)-e— '^-" + {m—iy^-^^'' -f • • • 



■^0 sinh La + Zr cosh La 



The final result of (56) is thus identical with the value of /» in (23). 

 The receiving-end-impedance with the impedance Zr at B is 



Zi — Co sinh La -+- Zr cosh La ohms (57) 



Following the above line of reasoning, we can readily understand 

 that every discontinuity in a circuit, such as a leak or a lump of im- 



