ARTIFICIAL ELECTRIC LINES WITH MUTUAL INDUCTANCE. 211 



where 



770 = r/Zio. (41) 



Since the second terms in (40) and (39) are usually small, and since 

 a is also sufficiently small to make sinh a essentially equal to a, the 

 two equations reduce to nearly the same form in respect to co if 



Q/A = iCaco^. (42) 



Replacing A in (42) by its value from (14) with neglect of 77- in (15) 

 w^e obtain 



^ " 1 + i>cwy4' 



which compared with (14) shows that (42) is approximately satisfied 

 when 



M = L/4: = ii/2. (42) 



A substitution of this value of M into the exact equation (38) gives 

 in a careful approximation 



sinh a = I Via, CO I 1 - f + ^ (iCoV + I U~C,w) I . (43) 



If now w^e expand the hyperbolic sine into 



sinh a = a -\- a^/6, 

 and replace the a in a^ by the first term of (43), we obtain 



a = IVLC, CO I 1 - 3 + ^ (lC^co^ + I UC.}u:) \ ■ (44) 



The approximation of equation (44) shows that even when rj. is as large 

 as . 5 and ivith LCoui^ as large as 1 the introduction of a mutual inductance 

 of the value given in {4'2) makes the attenuation constant of the artificial 

 lumpy line of the same form as the attenuation constant a of the smooth 

 line, and that the two attenuation constants can be thus made to agree 

 over a wide range of frequencies. 



