120 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Thus, the T of Figures 13, 14, and 15 has p' = 1014 ohms and R' = 

 509.99 ohms. Hence by (82), sinh = 0.99707, from which the semi- 

 angle 6 = 0.S793 hyp., which is also the semi-angle Xa of the equiva- 

 lent smooth line. Again, r = 1436.1 ohms by (S3). These are the 

 constants for the line simulated by the T. 



Instead of a T, or conductor with a single central leak, we may sub- 

 stitute for any actual smooth uniform line a conductor with two equal 

 terminal leaks, as shown in Figure 12. Such a conductor may be 

 called a U for convenience and brevity. In Figure 12, the values to 

 be assigned to the conductor-resistances r" and leak resistances R" R" 

 ohms, in order to replace a smooth line of length L kms., semi-length 

 X kms., attenuation-constant a hyp. per km., and surge-resistance r 

 ohms, are: 



ohms, (84) 



ohms, (85) 



or a" = ^ = — ■ mhos. (86) 



v To 2 r 



That is, the conductor resistance r" must be equal to the receiving-end 

 resistance of the imitated line when grounded, and each leak must be 

 the square of the surge-resistance divided by the sending-end resist- 

 ance of half the imitated line grounded. 



Thus, with L = 500 kms., X = 250 kms., a = 0.0035172 hyp. per 

 km., r = 1436.13 ohms, La = 1.7586 hyps., Xa = 0.8793 hyp., we 

 have r" = 1436.13 X 2.81602 = 4044.2 ohms, and R" = 1436.13/ 

 0.70607 = 2034.05 ohms. These values have been used in Figures 

 16, 17, and 18 to construct the H there indicated. It will be seen by 

 comparing these Figures with 5, 6, 7, and with 10, 11, 12, respectively, 

 that the external distributions of resistance, conductance, voltage, 

 current, and power are the same for all. 



Consequently, any smooth uniform line in the steady state, carrying 

 either continuous or single-frequency alternating currents, may be 

 completely replaced, so far as concerns all external conditions, either 

 by one equivalent T, or by one equivalent II. Either of these forms of 

 equivalent conductor may be selected for replacing the line, according 

 to convenience. 



Conversely, any given U may have its hyperbolic angle and surge- 

 resistance determined ; that is, its equivalent smooth uniform line can 

 be determined by the following formulas : 



