Lines in Telephonic Transmission. 325 



which takes its minimum value - 



A= 1-639 («' 2 «VW-^ .... (40) 

 for 



/=f.=*-2857, 



(o= x/4 = '0)547, , 

 and therefore : — 



For a given total weight of copper in cable and load the 

 attenuation of a given frequency is made a minimum by placing 

 2/7 of the copper in coils having a resistance equal to 3/7 of 

 the line resistance and spacing them it ^7/3 = 4-80 per uniform 

 line wave-length, i.e., irjsin- 1 V§Jl = 4*401 per actual wave- 

 length on the loaded line. These proportions make 



«= 1-058, 



n= -9171. 



d=2-028[" 



y 



\ w°p J 



V =0-695 (*IJ .... (42) 

 a \p~ww / 



K = (, «(^y.f • • • ^ 



W,= 1.759(^)r . . . (4.) 



The last formula, giving the total weight o£ copper in a 

 ]ine of length I and attenuation e~ Al , shows the relative im- 

 portance of the different factors. The weight increases as 

 the 2' 14 power of the range. Open wire circuits also increase 

 in weight somewhat faster than the square of the length, but 

 cable weights vary as the cube of the range. Loading, there- 

 fore, presents the greatest possibilities upon long cable 

 circuits. The weight is evidently comparatively independent 

 of the frequency. The attenuation coefficient comes in 

 approximately inversely as the first power. Of the two 

 specific weights w> that for the cable is far more important 

 than that for the coils w. Thus the total weight AV/ will be 

 doubled by changing w to 2'(>4 w, or w 1 to 11*3 u J . This 

 shows the comparative importance of the coil weight. 



In practical engineering, costs must be substituted for 

 theoretical copper weights, (37) (38) being replaced by the 



Phil. Mag. S. 6. Vol. 5. No. 27. March 1903. Z 



