122 PROCEEDINGS OF THE AMERICAN ACADEMY. 



It is possible, by known methods of substitution, to derive combina- 

 tions of resistance and leakance that shall replace a given T or IT ; as, 

 for instance, a combination like that shown in Figure 19. All such 

 conductors must manifestly be either graphically symmetrical about 

 a vertical through their centre 0, or must be reducible to such symmetry. 

 In general, these combinations are unnecessarily complex and have 

 little practical interest. From this standpoint, a multiple-section arti- 

 ficial line like that of Figures 5, 6, and 7 may be regarded as a complex 

 substitute for the simple T of Figure 11, or the simple II of Figure 12. 



It may be observed, however, that the total leakage of current to 

 ground in corresponding Figures is the same for a smooth uniform 

 line, its equivalent T, equivalent IT, or equivalent 5-section artificial 

 line. On reflection, this proposition is almost self-evident. 



9- ft 



Figure 19. — Complex substitute for an actual line of distributed 



leakance. 



As an instance of the use of substituting equivalent T's for sections 

 of actual line, consider the case represented in Figure 20, of a uniform 

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

 resistances of 2 = 2 a ohms, at uniform intervals of I kms. Required 

 the equivalent smooth line. 



First substitute uniform T's for the sections of uniform line, as in 

 Figure 21, by formulas (79), (80), and (SI). Then load the T's by add- 

 ing a to each end, as in Figure 22. Finally replace the loaded T's by 

 their equivalent lengths of smooth line, as in Figure 23, using formulas 

 (82) and (83). We deduce by this process the following results: 



• i •> / • i •> i / h & coth Aa 

 sinh Aa' = sinh \a.y H , (89) 



To 



cosh Aa' = cosh Aa/l + (rtanhAa , (90) 



r 



