ELECTR/C CIRCUIT TIIRORV 357 



Let us apph' this formula to the case of a line of lent^th s, with unit 

 e.m.f. directh- applietl at .v = .v, and line short cireuited at .v = i. Ke- 

 ferrinii to equation (244) and putting; X=/xi = /i2=l we get 



^ l cosh7(^-» )^ 1 



A sinh 75 Av(p) 



as the operational formula of the probleiu. This can l)e written as 



A, = (Cp+G) -^"^^ = 1 (281) 



7 suih 75 ^x{p) 



where in the general case, 



^ = \/{Lp+R){Cp + G). (282) 



The values of 7 for which Zx{p) vanishes are the roots of the trans- 

 cendental equation 



sinh 75 =0 



excluding zero. These roots are infinite in number: Let ■ym be the 

 m*^'' root; then 



^^=^^, m = l,2, ... 00. (283) 



The corresponding values of pm are then gotten by solving (282) for 

 p and writing 7 = 7m. 



The explicit solution of the operational equation (281) is then 



A.(i) = ^ + V ( C+ f ) '^"f ""^^""' .": 



Zx{0) ^4w V p„J djm , 



•^Tm J-r- cosh JmS 

 dpm 



(284) 



1 , X"^ ,^1 ^ /. X cosh JmX 



Zx{0) ^^-^ d-^m 



dpm 



Let us apply this to the non-inductive, non-leaky cable in which 

 L = G = i) and 7 = VRCp, so that 



A„ = 7./i?C=-^-^^, 



and 



djm _ -KC 

 ^'"#. ~ 2 ' 



