564 



BELL SYSTEM TECHNICAL JOURNAL 



The foregoing formulas for p and o- are exact subject to certain 

 restrictions on the impedance function Z{iw) which are satisfied in 

 the case of periodically loaded lines. Their useful application to the 

 problem under consideration depends, however, on the following 

 approximations. 



First it will be assumed that the line is approximately equalizeti, as 

 regards absolute value of steady state received currents in the neigh- 

 borhood of the impressed frequency a)/27r. By virtue of this assump- 

 tion, which is more or less closely realized in practice, the ratio 

 A{ci)+\)/A{o}) may be replaced by unity in the integrals (15) and 

 (16), and in equations (17) and (18). It is further assumed that the 

 function 



Bi(.i + \)-Biu>)-\B'{w) 



admits of power series expansion, so that 



FJ\)=sin[(/;oX)- + (/i.X)'+ . .], 



(19) 

 (20) 



/'n"=:^:3^B(«) = ^5(")(a)). 



where 



;, « = 



n ! f/o)" 



By virtue of ihc foregoing p and c are gi\cn by 



P=^ r" Vsin[/'X-(/;3X)^-(//6X)^ .] ■ cos [(y»2X)=+(//4X)^+ . 



: " f ',^sin [/'X-(7;:,X)'-(//5X)-> . .] • sin [(/;,X)= + (/;4X)' + 



TT.'o A 



(21) 

 (22) 



Now if llu' line is \cr\- long tlie integrals (llj antl 1,12) may be 

 replaced !>>■ llic api)i(i\inialioiis 



P-~ f '?^i" [/'X- (//3X)^'] ■ cos (/;.,X)-, 

 c=~ / ' sill |/'X-(/;:,\)''| • sin (/;oX)-. 



m) 



(24) 



In (itlu-r words we ri'lain only the leading terms in the e\|)ansion 

 of tile function 



B{w + \)-Bioi)-\B'(w). 



The jiistilicalion for this procedure depi'nds on argununis r~inniar to 

 those underl>ing the IVinciple of Siationar\- I'JKise (see Watson, 

 Theorj' of Bessel Kiuutions, p. 229). Kuniierniore the u|)per limit 



