36 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The resistance of the line, at and beyond the point P, measured to 

 ground is 



M f p = z coth 8 ohms, (10) 



or 



B * = B *^0 ohms. (11) 



Single Line Grounded at Distant End. 



If the line, instead of being freed at B (Figure 1), is grounded at B, 

 its resistance at A is 



B gA = z tanh 6 ohms. (12) 



At any point P, angularly distant 8 hyps from B, the line resistance 

 beyond P, measured to ground, is 



R gP = z tanh 8 ohms, (13) 



or 



*'' = ^£E&J ohms - (14) 



The potential at P, in terms of the potential u A at A, is 



u P = u A & ^- Q volts. (15) 



The current-strength at P, in terms of the current-strength i A enter- 

 ing the line at A, is 



. cosh 8 



tp = t *^hd amperes - (16) 



For example, consider a line, AB, Figure 1, of L = 100 km., with a 

 linear resistance r of 20 ohms per wire-km., and a linear leakance g of 

 20 X 10 -6 mho per wire km. (20 micromhos per km.), corresponding 

 to a linear insulation-resistance of 50,000 km-ohms. The attenuation- 

 constant of this line is a = 2 X 10~ 2 hyp. per km. by (1), and the 

 hyperbolic angle subtended by the line is = 2 hyps, by (2). The 

 surge-resistance of the line is z = 1000 ohms by (3). Then the re- 

 sistance offered by the line at A, when freed at B, is, by (4), 



R fA = 1000 coth 2 = 1000 X 1.037315 = 1,037.315 ohms, 

 and when grounded at B, by (12), 



M gA = 1000 tanh 2 = 1000 X 0.964026 = 964.026 ohms. 



