622 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX I 



Active Return Loss Of Intermediate Paths 



Let Lk = the loss from the west side of repeater C to the output of 

 repeater X which is the k th repeater to the west, plus the loss from the 

 input of the W-E side of repeater X back to the input of repeater C 



(Fig. 1). 



LetL, = 201ogio42*. (19) 



Now from Equations (26) and (27) from " Irregularities in Loaded 

 Telephone Circuits ": ^ 



hR 



0-"-L 



tk 



hu = ^f^A L Ak'^'-'^ COS^ 0y,, (20) 



;=i 



IqR l l^ 



^."(^•-Dsin^^y,, (21) 



where /u is 7' (from the paper) for the currents from the (fe + 1) th 

 repeater section from C (counting the adjacent repeater section as the 

 first) and hk is /" for the same section. 

 The total components will be 



y T — — 



j=l k=0 lk 



h = URl X E E \,k cos2 diu, (22) 



"lk N-\ _4 ^4(3-1) 



h = URl X E L -i^^ sin2 e^k, (23) 



\ i = \ k=0 lk 



where N = the total number of repeater sections so considered. 

 Assuming (as in the paper) that /i — I2 



J _ 7 -J^Rl /y (1 - Ak^-") _ ,24) 



^'-^'--liT^htk^K^-Ak^) ^^^^ 



By analogy, Equation (34) from the paper referred to may be used 

 for the distribution of the total current If, with 7i = /'; i.e., 



F = e-^'^''-'''K (25) 



Equation (42) from the paper may therefore be written 



S = Sn + Sw -h Sr - Sa - F{N, T) (26) 



where 



/ -^'-1 (\ — A I*"'') \ 

 &, + Wr) = 101og,.(£;L_l_L). (27) 



^ G. Crisson, October 1925, Bell Sys. Tech. Jour. 



