104 PROCEEDINGS OF THE AMERICAN ACADEMY. 



At the nth leak, including the same, the sending-end resistance is 



D , . cosh (2 n - 1) 6 r cosh (2 n - 1) 6 , /tA 



R ^ = r ° sinh2nfl = cosh0sinh2n0 ° hma ' (8) 



When the number of sections of artificial line becomes indefinitely 

 great, the two immediately preceding expressions respectively be- 

 come: 



R '«>f = r °' *' = ^T8 ° hms ' 



and R'i x = r o 'e~ = ° , n ohms, (9) 



h coshtf ' 



where e is the base of Napierian logarithms. 



The ratio of the sending-end resistance at and excluding the 

 (n + l)th leak to that at and including the nth leak is 



R'n+i,f cosh(2n+ 1)0 nm 



R't, n " cosh (2 n- 1)6' ( ' 



This is the ratio of the extreme sending-end resistances, when ascend- 

 ing from one leak where it is a local minimum, to the next higher leak 

 where it is a local maximum. When the artificial line becomes indefi- 

 nitely long, this ratio tends to the limit e 20 . 



Voltage. Far End Free. 



The voltage e at the far free end of the artificial line, Figure 6, 

 will be: 



cosh 2 m6 cosh i 2 a 



where e m is the voltage impressed on the rath junction, or sending end. 



If the voltage e n should be impressed on the line at the nth leak, the 



formula is 



e„ cosh 6 



e ° = iTT^ "TTZ volts - ( 12 ) 



cosh (2n — 1) 6 



Thus, if e m = 100 volts, and ra = 5, as in Figure 6, 2 md = 1.7586 

 hyps, and cosh 2 md = 2.9883 ; so that e = 100/2.9883 = 33.46 volts. 

 The voltage at junction (n) is 



e n = to cosh 2 n6 = e m — r-r — 2 volts. (13) 



cosh 2 ra 6 



