KENNELLY. — ARTIFICIAL LINES FOR CONTINUOUS CURRENTS. 107 



The ratio of local maximum resistance just before a leak to the local 

 minimum resistance just after the preceding leak is : 



fl'n+i.g sinh(2n + 1)0 

 R f a>n sinh (2 n- 1)0' 



When n is increased indefinitely, this ratio becomes: 



R\ 

 7?' 



°- = £ 20. 



(24) 



(25) 



Receiving-End Resistance. Far End grounded. 



The receiving-end resistance, or resistance which the artificial line 

 appears to offer, as judged by an observer at the far end, from the 

 received current to ground and the impressed emf. at the sending 

 end, is: 



Ri = r sinh L 2 a = r sinh 2ra0 ohms. (26) 



In the case of Figure 5, fl z = 1436.1 sinh 1.7586= 1436.1 X 2.81602 

 = 4044.2 ohms. The received current to ground at the far end will 

 therefore be 100/4044.2 = 0.02472 ampere. 



Voltage. Far End grounded. 



The emf. at the nth junction in terms of the emf. e m impressed on 

 the rath junction is: 



sinh2n0 ._, /r> _ N 



e n = e m a volts, (2/) 



sinh 2 mv 



or, in terms of the current i to ground at the far end, it is ; 



e n = I r sinh 2 n6 volts. (28) 



Consequently, the voltages at successive ascending junctions are pro- 

 portional to the hyperbolic sines of the angles of those junctions. 

 The emf. at the nth leak is: 



sinh (2n — 1) 6 , /0 > 



*-- rinh(2»-l)» . V ° ItS ' (29) 



in terms of the emf. e m at the rath leak; or 



(n _ IoTo sinh(2n-l)fl = w ^ (2 n _ 1)$ yoltSj (3Q) 

 cosh0 



