42 PROCEEDINGS OF THE AMERICAN ACADEMY. 



COMPOSITE LINES. 



First Case. Sections of the same Attenuation-Constant and of the 

 same Surge- Impedance. 



If a line AB (Figure 5) of L x km. is connected to a line CD of L 2 km., 

 and each has the same attenuation constant a, and the same surge- 

 resistance z ohms (conditions which imply the same linear constants), 

 the line angles will be 9 X = L x a and 6 2 = L 2 <j- hyps respectively. 

 Then, if we free the composite line at D, the resistance at A is 



R f = z coth (6 X + 6 2 ) ohms, (36) 



while, if the composite line be grounded at D, the resistance at A is 



B g = z tanh (0 X + 6 2 ) ohms. (37) 



B'C ft p 



Figure 5. Composite line with sections of the same attenuation- 

 constant and surge-resistance. 



Reciprocally, freeing and grounding the composite line at A, we get 

 resistances U f and R g at D, respectively the same as in (3G) and (37). 

 It is evident, then, that the composite line differs in no way, except 

 in length, from either of the component sections. The angle subtended 

 by the whole line AD is the sum of the component section line-angles. 



Second Case. Sections of different Attenuation-Constant but of the 

 same Surge- Impedance. 



If a section CD (Figure 5) of Z 2 km. be connected to a section AB of 

 L x km., and their respective linear constants r 2 , g 2 , and r x , g x are such 

 that their attenuation constants a x , a 2 differ ; while their surge-resist- 

 ances z are the same, we assign the angles subtended by the sections 

 X = L x a x and 2 = L 2 a 2 hyps. The angle subtended by the whole 

 line will then be X + 2 , as in the preceding case. That is, except 

 for a disproportionality between the section-angles and their line- 

 lengths, two sections of different attenuation-constant, but of the 

 same surge-resistance, connect together like two sections of one and 



