704 PllOCEEDINGS OF THE AMERICAN ACADEMY. 



" practical " system and use the kilometer (or the mile) as the unit of 

 length : 3 



e = ^^coshXia — /cosinhZi a volts (U) 



i = /cosh Zi a — — sinh Zi a amperes (15) 



where L^ is the distance w measured from A in kilometers, 



a = wTMwrm ^^^ (16) 



and Zo = J'^-^tJbL ohms (17) 



the vector, or complex quantity, a is commonly called the attenuation- 

 constant of the circuit. It comprises a "real " part ai called the real 

 attenuation-constant, and an " imaginary " part Jag called the wave- 

 length constant, by the relation 



The vector impedance z^ is called the " initial sending-end impedance " 

 of the line, and sometimes the " surge-impedance." 



r = the linear resistance ohms per kilometer. 



/ = the linear inductance henrys per kilometer. 



g = the linear leakage conductance . . mhos per kilometer. 



c = the linear capacity farads per kilometer. 



Since equations (14) and (15) hold for any moment of time, they 

 must apply to the final steady state. We may easily derive from them 

 the following steady-state formulas : 



1. When the distant end of the line at B is free, 



2. When the distant end of the line at B is grounded, 



Ib = — • , J Ia = — : — r-r- amperes (21) 

 Co smh La Zq tanh La 



E^= E Eb= a volts (22) 



2 " The Distribution of Pressure and Current over Alteriiatiug-Current Cir- 

 cuits," by A. E. Kennelly. Harvard Engineering Journal, 11)05-11)00. 



