TKAXS MISSION LI Mi EQUATIONS 157 



We may interpret these expressions by Sylvester's theorem. For 

 example, 



<n) = E iV(f.) 





■ 1 X7" - e 

 + 



-X^^°-x-~t) <"'^ 



21,7^7=^1^' +X7^^IJJ 



2 



where ^(i"',) is given by (2.18). 



When the line extends to ti = » and the sources and end conditions 

 satisfy suitable conditions we have the relation 



v(o) = Zoi(o) - f: e"''[Bnp - 1) - ZoCv%p - 1)] (2.28) 



;>=1 



When the impressed lield is of the form (2.25) this becomes 



vio) = ZoUo) - (/ - e-'I)~\Bi° - ZoCv°) (2.29) 



provided that the inverse matrix exists. Expressions for v{n) and i{n) 

 in such an infinite line may be obtained by using (2.28) or (2.29) in (2.24) 

 or (2.26). 



Applying Sylvester's theorem to (2.29) gives 



r(o) = t .V(r,) {^ - -^. + Sr^\^ A (2.30) 



r=l \Xr — Xr Ar — e 2(Xr — 6 ) / 



The last term within the braces may be replaced by 



2BCv° 



{X7^ - Xr){K^ — e"") 



2.6 Derivation of the Properties of an Infinite Line 



We shall consider a symmetrical section line which is specified by the 

 equations 



v{n + 1) - Av{n) - Bi{n) 



(2.10) 

 i{n + 1) = -Cv{n) + A'i{n) 



From these equations and the relations A- — BC — I, AB — BA', A'C = 

 CA of (A4.6) it follows that 



v{n + 1) + z;(w - 1) - 2Av(n) 



(2.31) 

 i(n + 1) + i{n - 1) = lA'iin) 



