132 BELL SYSTEM TECHNICAL JOURNAL 



amplification ratio for one round trip is equal to the complex quantity 

 A J, where ^ is a function of the gain only and J is a function of co 

 only, being defined for all values of frequency from to co . 



Let the disturbing wave /o(0 be applied anywhere in the circuit. 

 We have 



/o(/) = 7^ f " F(ic^)e'-'do: (13) 



or 



/o(/) =^.{ Fiz)e''dz. (13') 



^T^-^ Js+ 



The wave traverses the circuit and on completing the first trip it 

 becomes 



/i(/^ = T- r"^c<ta;)F(ico)e*'-Wco (14) 



= 7^ f iv{z)F{z)e''dz. (14') 



After traversing the circuit a second time it becomes 



f.{t) = 7f^. r Fiv^e^'dz, (15) 



and after traversing the circuit n times 



In{t) = ^- r Fiv^e^^dz. (16) 



Adding the voltage of the original impulse and the first n round trips 

 we have a total of 



Sn{t) = ZMt) ^^. f F{l+w+ "■ w^)e^'dz. (17) 



The total voltage at the point in question at the time t is given by 

 the limiting value which (17) approaches as n is increased indefinitely ^ 



s{t) = Y.Mi) = Hm ^. f Sn{z)e''dz, (18) 



i!:=0 n^-x -^TT' J g+ 



where 



S^ = F + Fiv + Fw"" + ■ ■ ■ Fiv" = ^ ■ (19) 



1 — ty 



* Mr. Carson has called my attention to the fact that this series can also be 

 derived from Theorem IX, p. 49, of his Electric Circuit Theory. Whereas the 

 present derivation is analogous to the theory expressed in equations (a)-(f) above, 

 the alternative derivation would be analogous to that in equations (/)-(//)• 



