ARTIFICIAL ELECTRIC LINES WITH MUTUAL INDUCTANCE. 211 



where 



170 



= r//co. (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 = iCscol (42) 



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

 we obtain 



^~l+LC2coV4' 



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

 when 



M = L/4. = Xi/2. (42) 



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

 in a careful approximation 



sinh a = I Vie, c*^ I 1 - f + ^ (lC,o:'~ + § ^'^''''') \ ' ^^^^ 



If now we expand the hyperbolic sine into 



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

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



a = IV LC, CO I 1 - 8 + ^ [LC2o:'~ + 4 L^C.W) J • (44) 



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

 as . 5 and with LCiw- as large as 1 the introduction of a mutual inductance 

 of the value given in {42) 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 ivide range of frequencies. 



