546 Dr. A. Russell on the Effective Resistance 



In power transmission cables a 2 = c 2 — b 2 . In this case the 

 greater the value of b the smaller the value of pi + p^Z + PsZ 2 - 

 In power transmission cables, when the frequency is not 

 greater than 50, the increase in the effective resistance of 

 the outer conductor due to the skin effect is negligibly 

 small. 



We find in a similar manner that 



T r /x a 4 4 

 L==L « 2-J92 m 



-^(\ 1+ X 2 f + X 3 p + X 4 P)m 4 , . . (102) 



where _ 19c 6 + 103c 4 6 2 -41£V + 3£ 6 



1_ 2 2 .4 2 .6 2 (c 2 -£ 2 ) 



146V(2c 2 -& 2 ) 



X s = 



2 2 .4 2 .3(c 2 -6 2 ) 2 ' 

 6V 



4(c 2 -6 2 ) 3 ' 



X *=2(^7' and ? = lo si' 



Heaviside * gives the formula as if the coefficient of m 4 

 were zero. The term —(/xa 4 /384)?n 4 can be deduced from 

 the formula given in Gray and Matthews' ' Bessel's 

 Functions/ p. 160, as the " self-inductance " of a cylindrical 

 conductor. 



Formula (102) shows that at low frequencies the inductance 

 diminishes as the frequency increases, the effect being more 

 pronounced the thicker the shell of the outer conductor. 



3. With high frequency currents. 



When ma is greater than 5 we can use the formulae (50) 

 to (53) for the ker functions and the corresponding formulae 

 for the her functions. Substituting these values in (89) and 

 (90) we get, after a little reduction, 



t>_ P m J~ ^ 1 3 "> 



2^ La/2 + 2ma + 8y/2m*a 2 J 



pm sinhm(c — b) \/2 + sin m(c— b) \/2 /inq\ 

 2irb\/2 coshm(c-b) \/2 — cosm(c — b) V2 



* ' Electrical Papers/ vol. ii. p. 192. [Dr. Heaviside has pointed out 

 to nie that he was only giving the iirst term correction to H-\-L<ai, and 

 thus it was unnecessary to give the coefficient of m 4 in (102).] 



