550 Dr. A. Eussell on tlie Effective Resistance 



To deduce the resistance R; of the inner tube we write a 

 for b, and a x for c in the coefficient of m 4 , and noticing that 

 the area of the cross section is now 7r(ct 3 — a x 2 ), we get 



For a given cross sectional area 7r(a 2 — a x 2 ) we see from 

 (114) that the coefficient of m 4 diminishes as a^ increases. 

 Hence, at low frequencies, making the inner conductor hollow 

 diminishes the skin effect. Similarly for a given cross 

 sectional area, the larger the outer tube the smaller the 

 effective resistance of the tube. 



From (103) we see that at high frequencies the resistance 

 E of the outer tube is given by 



_ pm sinh m (c — b) V-+ smm(c-b) \/2 



2irb s/2 coshm(c — b) \ y 2— cosm(c — b) s/2 



Hence the resistance R> of the inner tube is given by 



pm sinh m(a—a{) \/2 + sin m(a—a-^) \J2 ( ^^r\ 



xxi — ■ — = — . ( X. X. ) 



2ira V- cosh m(a—a 1 ) V- — cosm(a — a 1 ) \/2 



The difference, therefore, between the effective resistance of 

 a thin and a thick inner tube having equal outer radii is 

 very small at high frequencies. 



VII. The Impedance of a Concexteic Main. 



If the length of the main be very long compared with its 

 diameter and the insulation resistance of the dielectric be 

 very high, the assumption that the current flow is linear is 

 permissible. In this case, if e be the potential-difference 

 between the mains at a point at a distance x from the 

 alternator, we have* 



"be dz 



and — 5- = 7t +K^-, 



on ft at 



where K and S are the capacity and the insulation resistance 

 between the mains per unit length. In these equations R 

 and L have the values found earlier in the paper. Hence 

 we have to substitute these values in the well-known expres- 

 sions for the impedance of the main. 



* Russell, ' Alternating Currents/ vol. ii. p. 458 et seq. 



