544 Dr. A. Russell on the Effective Resistance 



IV. Simplified Formulae for particular cases. 



1. With direct currents. 



From the next solution, by putting m equal to zero, we 

 find that the resistance R f / and the inductance hd with direct 

 currents are given by 



R rf =/,+ / , .... (95) 



and b ll 



L d = 2//, log- + | 



(c — try °6 2(c 2 — 6 2 j v 



This value for the inductance agrees with that found by 

 Lord Rayleigh"*. 



2. With low frequency currents. 



Substituting the appropriate approximate formulae (54) 

 and (55) for S c and T c in the equations (91)-(94), and 

 noticing that Y c =(mV/4)(l-HnV/192) approx., we find 

 that 



C =C 1 m 4 + C 2 m 8 , , . , . . (97). 



where 



_ d 2 c\b 2 -a 2 ) a^e c_ 



°i- I6(c 2 -Z> 2 ) 4^-^ l0g 6 J 



and 



C 2 - 2 2^ 4 2 # g 



^-^|6V(61/> 2 + 19^--42a 2 ) 

 .b(c -b) L +(^-a 2 )(5a 2 6 2 -a 4 ~7?> 4 )U 



a 2 £V log (c/b) C 4 ") 



~ 2 2 .4 2 .12y~& 2 ) 2 l 14 ^ 2 -" ~ 2 * + 3 ^ 2 j 



4 3 ( C 2 -& 2 ) 3 \ 1o s^j 

 + 4V^^a log 6/ ' 



-D='Dim , +;D i m«, (9$ 



* Phil. Mag. [5] vol. xxi. p. 381 (1886), or Russell's 'Alternating. 

 Currents/ vol. i. p. 53. 



