TRANS.UISSIOX LINE EQUATIONS 141 



corresponding integral in some otiier way. Thus it may be advantageous 

 to use the formula 



j-r \ iT~i.ei 



-{e-^' - e-'"l){T - diV = e'' e''-'"' d^ 







-xF 



e 



ci + ~ (r - 07) + _ (r - dif + . . . 



(1.35) 



Two special cases of (1.34) are of interest. When the line is shorted at 

 both ends, v(o) — v(x) — 0, w^here -v is the line length, and 



i(o) = \ Z-' (sinh xr)-'[(c"'' - e"^'/)(r + dir\\ - Zot) 



- {e-^^ - e-''l){V - dI)-\\ + Zot)] 

 i(:c) = ^ Z7' (sinh .vr)-'[(/' - e^' I)(X - ei)-\\ + Zot) 



- {c"'' - e''l){V + dir\\ - Zot)] 



When the line is terminated in its characteristic impedance at both ends, 

 v{o) = —Zoi{o), v{x) = Zoi{x), and 



i{o) = h(I - e-''c-''')(T' + eir\Z7'\ - r) 



i(x) = -i (e~''' - e-'^'DiV - dir\Z-\ + r) 



The matrices occurring in the expressions (1.34) for v{x) and i{x) may 

 be computed by the first or second method described for the uniform line 

 in the absence of an impressed field. The second method involves the use 

 of expansions similar to 



(/r _ g-V)(r + BI)-\\ - Zot) 



(e'"' -e"'l){T' + dI)-\Z-:'\ -r) 



w^here the summations run from r = 1 to r = m and ^'(7^) is the transposed 

 of the square matrix N{yr) given by (1.23). In obtaining these expansions 

 by Sylvester's theorem, Zo in the first is replaced by T~ Z and Z7 in the 

 second by r'~ I'. 



If we assume that an impressed field acts upon the perfect transmission 



