102 PROCEEDINGS OF THE AMERICAN ACADEMY. 



has in this case diminished the nominal attenuation-constant and 

 hyperbolic line angles by 0.52 per cent, but has increased the surge- 

 resistance by 1.55 per cent. The linear conductor-resistance of the 

 imitated line, Figure 4, will be ar = 5.051 ohms per km. The linear 

 leakance of the imitated line will be a/r = 2.44989 X 10~~ 6 mho per km., 

 corresponding to a linear insulation resistance of 408,320 km.-ohms. 



Figures 5, 6, and 7 are diagrams of the voltage and current distribu- 

 tion over the artificial line above defined, for the respective cases of 

 line grounded, freed, and grounded through 750 ohms, at B, the dis- 

 tant end. The steady impressed emf . at the sending end A is assumed 

 as 100 volts in each case. Conductances are written in millimhos. 

 All of the numerical work on these diagrams was carried out by the 

 ordinary formulas of Ohm's law, and inspection will show that the 

 arithmetical results are consistent. The various formulas given in 

 this paper admit, therefore, of being checked by reference to these 

 diagrams. 



Artificial Line freed at Far End. (Figure 6.) 



S 'ending-End Resistance. 



The sending-end resistance of an artificial line at the nth junction; 

 i. e., the resistance offered to ground by the line, at and beyond the 

 nth junction, is 



Rf = r coth L 2 a = r coth 2 nO ohms, (6) 



where Z 2 is the length of the line in km. reckoned from the far free end. 

 When the sending-end resistance is measured at A, Figure 6, so as 

 to include the whole line, L 2 = L, and n = m. As L 2 increases from 

 to oc, coth L 2 a diminishes from oc to 1. Thus, in Figure 6, with r = 

 1436.1 ohms, and m=5; or L = 500, L 2 a = 1.7586 = 2 m,6, 

 coth 2ra<9 = 1.0612, and R f = 1436.1 X 1.0612 = 1523.99 ohms, as 

 indicated at A. In the case of a smooth actual line, such as is shown 

 in Figure 4, L 2 may be varied continuously between and L kms. ; 

 but in an artificial line, L2 can only be varied in steps of 2 6. That 

 is, formula (6) applies to all points of the imitated line, but only to 

 the junction points of the artificial line. 

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



, cosh (2 n -1)0 _ r cosh(2n -1)0 

 Kn 'f- r ° s i n h (2 n - 2) 6 ~ cosh 6 sinh (2 n - 2) 6 ° hmS " (7) 



