124 PROCEEDINGS OF THE AMERICAN ACADEMY. 



r o = ro\ ( tanh \a -\ — J ( coth Xa -\ j ohms ; (94) 



= r o y 1 + — coth la + (—) ohms ; (95) 



r ' sinh la' 

 r sinh la ' 



(9G) 



Thus, if a uniform line of attenuation-constant a = 0.0035172 

 hyp. /km., and surge-resistance r = 1436.13 ohms, has a resistance 

 % = 200 ohms, inserted at intervals of 100 kms., required the cor- 

 responding constants of the loaded line. Here, as indicated in Figure 

 21, cr = 100 ohms and Xa = 0.17586 hyp. If we compute the equiva- 

 lent T's of the sections of unloaded line, we find p' = 249. 9S5 ohms 

 and R' = 4000.215 ohms. The hyperbolic corrections for these lengths 

 of sections are thus only 0.015 ohm in conductor-resistance and 0.215 

 ohm in leak-resistance. Adding on the loads to the ends of the T's, 

 we have, as in Figure 22, p' = 349.985 ohms and R' = 4000.215 ohms. 

 Using formulas (82) and (83), we obtain for the equivalent smooth 

 line Xa' = 0.20766 hyp., la'= 0.41532 hyp., and r ' = 1709.54 ohms. 

 The apparent conductor-resistance of the loaded line is, therefore, 

 r 'la' = 710.06 ohms, or 10.06 ohms more than the actual resistance 

 of conductor and loads. The apparent total leak r '/la' — 4116.2 

 ohms, or 116.2 ohms in excess of the actual total leak. 



As an example of the use of substituting equivalent TI's for sections of 

 smooth line, consider the case represented in Figure 24 of a uniform 

 line of attenuation-constant a, and surge-resistance r , loaded with 

 uniform leakances of T mhos at uniform intervals of I kms. Required 

 the constants of the equivalent smooth line. 



First divide the leakage conductances into equal parts 7 = T/2, as 

 in Figure 25. Then substitute for the unloaded line sections their 

 equivalent II's by formulas (84), (85), and (86), as in Figure 26. 

 Next add on the terminal leakances 7 to the pillars of the II, as in 

 Figures 27. Finally, deduce as in Figure 28, by formulas (87) and 

 (88), the equivalent smooth line. 



We also obtain by this process the following relations : — 



, , ,, * u w *■ u\ i/l + yr coth A a 

 tanh b' = tanh Aa' = tanh Aa } — . x 



1 + yr tanh Aa 



(97) 



rj = ° ohms. (98) 



V (1 + yf tanhAa) (1 + yr coth Aa) 



