KENNELLY. — ARTIFICIAL LINES FOR CONTINUOUS CURRENTS. 101 



where 6 represents the true semi-sectional hyp. -angle. Similarly, the 

 true value of the surge-resistance, corrected for lumpiness, is 



T — r 'coshAa = r o 'cosh0 = r 'yl + (A-a') 2 ohms. (5) 



We now obtain from (3), (4) and (5) the true attenuation-constant a of 

 the artificial line, the true surge-resistance r , and the true hyperbolic 

 angles A.a, la, and La subtended by a half-section, a section, or the whole 

 line, respectively. These various quantities also define the actual line 

 (Figure 4) which the artificial line imitates, after being corrected for 

 lumpiness. The actual line of distributed leakance which is electrically 

 equivalent to an artificial line, after correcting the latter for lumpiness, 

 may be defined as the "imitated line." 



As an example, consider an artificial line of m — 5 sections, as 

 shown in Figures 5, 6, and 7, with a total nominal length L = 500 km. ; 

 so that each section has a nominal length I = 100 km., and a nominal 

 semi-length X = 50 km. The conductor-resistance of each section 

 is / = 500 ohms, corresponding to a nominal linear conductor-resist- 

 ance of 5 ohms per km. The leak of each section has a resistance 

 R' = 4000 ohms or a conductance of 0.00025 mho (0.25 millimho), 

 corresponding to a nominal linear leakance of 2.5 mieromhos per km., 

 or a linear insulation-resistance of 400,000 km. -ohms. The nominal 

 attenuation-constant of the artificial line will be, by (1), a' = 0.0035355 

 hyp. per km. The nominal hyperbolic angle subtended by a half- 

 section, a section, and the whole line, will be respectively = Xa' = 

 0.17678, la' = 0.35355, La' = 1.7678 hyps. The nominal surge- 

 resistance will be by (2) r ' = 1414.2 ohms. We must now find the 

 corrected values for these quantities corresponding to the imitated 

 line shown in Figure 4. 



W T ith reference to formula (3), we find in tables of hyperbolic func- 

 tions 2 that the angle whose hyperbolic sine is 0.17678 must be Xa = 

 0.17586 hyp. ; which is the true angle of a semi-section of the artificial 

 line, corrected for lumpiness. The true angle subtended by a section 

 will be la = 0.35172 hyp., and by the whole line 1.7586 hyps. The 

 true attenuation-constant of the artificial line, or the natural attenua- 

 tion-constant of the imitated line, will be a = 0.0035172 hyp. per km. 

 The true surge-resistance by (5) r = 1436.13 ohms. In other words 

 the artificial line will behave externally in all respects, after the steady 

 state has been attained, as though it were an actual smooth line of 

 distributed leakance with these corrected constants. The correction 



2 The best tables probably are " Tafeln der Hyperbelfunctionen und der 

 Kreisfunctionen " by Dr. W. Ligowski, Berlin, Ernst & Korn, 1890. 



