46 PROCEEDINGS OF THE AMERICAN ACADEMY. 



hyps. The sending-end resistance offered by the line at D will then 

 be, by (4) and (42), identical with that found previously at A. The 

 conductance of the leak will, by (17) and (43), be the same as that 

 found from A. Finally, the resistance of the DO line-branch will, by 

 (18) and (44), be identical with that of the AO branch (905. 149"). 

 This completes the T of the composite line. 



We may infer from the above reasoning, and it may be readily dem- 

 onstrated formally, that when a composite line is composed of sections 

 differing in linear constants, but having the same surge-impedance, the 

 angle subtended by the whole line is the same at either end, and 

 whether the distant end be freed or grounded. Consequently the 

 equivalent fl and T of the composite line will be symmetrical. That 

 is, the two leaks of the l~] are equal and the two line branches of the T 

 are equal. 



Conversely, it follows, from equations (21) to (24), that any com- 

 posite line made up of sections differing in attenuation constant, but 

 with the same surge-impedance, may be replaced by an equivalent 

 single line of uniform attenuation and linear constants. 



Third and General Case. Sections with Different Surge- Impedances. 



Let a section AB of 100 km. (Figure 7) be connected to a section CD 

 of 300 km., and let their respective linear constants be as follows: 



n = 20 ohms/km. ; g x = 20 X 10~ 6 mho/km. 

 r 2 = 10 ohms/km. ; g 2 = 2.5 X 10~ 6 mho/km. 

 from which 



ai = 0.02 hyp/km. ; B x = 2 hyps ; z x = 1,000 ohms ; 

 a 2 = 0.005 hyp/km. ; 2 = l.S hyps ; r 2 = 2,000 ohms, 



so that the surge-resistance of the two sections are unequal. It follows 

 that the angle subtended by the composite line will differ at the two 

 ends, and will also differ according to whether the distant end is freed 

 or grounded. 



Equivalent I~l. 



Let us ground the end A 2 of the composite line A 2 D 2 (Figure 7). 

 Then by formula (12), the sending-end resistance at B of the section 

 BA grounded, will be 



R gB = %\ tanh di ohms (45) 



= 1,000 tanh 2.0 = 964.026,5 ohms. 



