PROPAGATION OF PERIODIC CURRENTS 



499 



We are now prepared to write down the generalized telegraph equa- 

 tion, which is obtained by eUminating Vi, • • • F„ from equations (2) 

 by aid of (8); it is: 



sy//y = t ( w,; £^ - Zj, ) /,, (i = 1, 2 • • • n). (10) 



This system of n equations, which constitutes the generalized telegraph 

 equation, will be written as: 



where 



Wii/i + W12/2 + • • • + niuih = 0, 



W72l/l + ^22/2 + • • • + monin = 0, 

 ninJl + Wn^h + • • • + ninnh = 0, 



(11) 



(11.1) 



mjj = Zjj + Z,j - Wjj 



dx^ 



(11.2) 



Equations (11) are a system of n homogeneous equations; a finite 

 solution for the currents /i, • • • In therefore necessitates the vanishing 

 of the determinant of the system, that is, 



mil m\2 niiz 

 W21 W22 moz 



ni2n 



m-ni mni ninz 



nin 



(12) 



In order to solve this equation the operator d-/dx^ is to be replaced by 

 7^, which is equivalent to the assumption that the n currents /i, • • • /„ 

 involve the variable x only through the common factor exp (7^). 

 With this substitution, equation (12) is of the wth order in 7^ and its 

 solution gives, in general, 2n values of 7, namely, 71, 72, • • • jn and 

 — 71. — 72, ••• — In- The general solution of equations (11) is 

 accordingly of the form 



/;■ = E {Aj,e-y^' - Bj.e-^'--), 



(i = 1, 2 



n). 



(13) 



The potentials are then determined from (8) and (13) in terms of the 

 parameters of the system, the propagation constants, and the arbitrary 

 constants of integration Ajk, Bjk- 



By means of the relations (11) obtaining among the currents, it is 



