142 BELL SYSTEM TECHNICAL JOURNAL 



line of equations (1.31), we see that /(.r) — because there are no direct 

 leakage paths. We may also write 



lo + O 



From this and similar equations it follows that 



v{x) = cosh .V7ov(o) — ° Zi{o) 



To 



—x8 ~i1o —x6' 



e e — e 



*^['-=r--— 1> 



2\_ jo + d yo-e J (1.37) 



i(x) = —Jo sinh xyo Z~ v(o) + cosh xyoi(p) 



n xyo ~3:9 —xyo —xB~ 



2 1 To + e 



2 L To + ^ To - ^ 



7.5 Results for Infinite Uniform Line — Impressed Field 



When the line extends from .v = to .v = =c and the impressed field is such 

 that the voltages and currents remain finite at x = =o , the appropriate solu- 

 tions may be obtained from the results of §1.4 by a limiting process. The 

 condition that v{x) remain finite suggests that the coefficient of e^ be zero 

 in the expression (1.32) for v{x). This gives a relation between v{o) and 

 i{o) which must be satisfied: 



v{o) = Zoi(o) - [ e~^'m - Zoimd^ (1.38) 



If the impressed field varies exponentially with .v expression (1.34) gives 



v{o) = Zoi{o) - (r + dir\\ - Zot) (1.39) 



Expressions for i)(.v) and i(.r) may be obtained by using relations (1.38) 

 and (1.39) in (1.32) and (1.34) respectively. As these are somewhat 

 lengthy we shall state only two which follow from (1.39). 



v{x) — e"^ vio) 



+ K^"'' - e-^'mv + nr\\ - Zot) 



- (r - 0/)-'(x + Zo7)\ 



1 -xr ^^-"^^^ 



i{x) = Z7 e ' vip) 



+ \Z-\e-'^ + ."^V)(r + Qir\\ - Zot) 



- \Z-\e-''^ - e^'U){V - Bir\\ + Zot) 



