126 PROCEEDINGS OF THE AMERICAN ACADEMY. 



to their equivalent smooth-line sections by formulas (87) and (SS), we 

 obtain \a' = 0.20766 hyp. or la' = 0.41532 hyp. and r '= 1206.45 ohms, 

 as in Figure 28. The apparent conductor resistance of a section of 

 loaded line is larj = 501.06 ohms, or 1.06 ohms in excess of the total 

 actual resistance. The apparent total leakance of a section is 0.34425 

 millimho, or 0.00272 millimho in excess of the total actual leakance. 



It may be observed by comparing Figures 20-23 and 24-28, or 

 formulas (91) and (97), that if loads are applied at assigned uniform 

 distances along a smooth line, a leak load Y will produce the same 

 equivalent attenuation-constant as a resistance load 2 in the conductor, 



if - = r 2 ; that is, if the resistance I/7 of a semi-leak be a third pro- 



portional to the resistance a of a semi-conductor-load, and the surge- 

 resistance of the unloaded line. In other words, the attenuation-con- 

 stant of the loaded line will be the same, whether the loads are inserted 

 in series, or applied in derivation, provided that a: r :: r : 1/7. The 

 •surge-resistance of the loaded line will not, however, be the same in 

 these two cases. The surge-resistance will be less with leaks than 

 with series coils. The two values have the unloaded surge-resistance 

 as their geometrical mean. 



In all cases of direct-current lines, loads, either in series coils or in 

 leaks, necessarily increase the attenuation-constant of the line. With 

 alternating-current lines, this limitation is removed. 



Summary of Conclusions. 



Every artificial line composed of similar mid-leak sections, carrying 

 either continuous or alternating currents in the steady state, may be 

 reduced trigonometrically to its equivalent smooth line, and recipro- 

 cally. The resistance, current, and voltage at the various junctions 

 and leaks along the line are simple hyperbolic functions of their angles. 



Every smooth line in the steady state, carrying either continuous or 

 alternating currents, may be externally completely replaced by one 

 and only one T, or single-section mid-leak artificial line; or by one 

 and only one II, or single conductor with equal terminal leaks, and 

 reciprocally. This proposition has numerous implications in telegraphy, 

 telephony, power transmission, and distribution. 



