KENNELLY. — BUILDING UP VOLTAGE AND CURRENT. 703 



Equation (2) states that the descending gradient of voltage in 

 abvolts per linear centimeter at any point of the line is always 

 equal to the current strength at that point multiplied by the con- 

 ductor resistance per centimeter plus the time-rate of change of the 

 current into the conductor inductance per centimeter. In other words, 

 the voltage per linear centimeter is equal to the momentary linear Ir 

 drop, plus the linear back e. m. f of self-induction. 



Ei^uation (3) states that the descending gradient of current along 

 the line is always equal to the local voltage multiplied by the linear 

 dielectric conductance, plus the linear capacity into the local time-rate 

 of change of the voltage, i. e. the linear leakage current plus the linear 

 charging current. 



Differentiating (2) and (3) with respect to x the length of line, 



,, ., abvolts per cm. , , 



we get e" = — zi' = ^jz-e (6) 



cm. ^ ^ 



... , . absamperes per cm. 



i" = -ye'=yz-i ^^T^ <^') 



By (6) the curvature (or gradient of gradient) of the voltage at any 

 point along the line is always yz times the voltage at that point. By 

 (7) the curvature of the current along the line is also yz times the local 

 current. The complete solutions of the second-order differential equa- 

 tions (6) and (7), for any point distant x from A, are known to be : 



e = Eao^xVyz — IK/" ?,\ri}a ws/yz abvolts (8) 



i = I cosh x\^z — E^ - sinh x's/yz absamperes (9) 



where E and / are the instantaneous impressed voltage, and current, 

 at the sending end A (Figure 1), respectively. 



For an impressed e. m. f. which is sinusoidal, or simply harmonic, in 



y and z are thus plane- vector constants. 



We have hitherto kept the equations within the absolute C. G. S. 

 magnetic system of units. We may, however, transfer them to the 



