434 Dr. E. H. Barton on the Equivalent Resistance and 



where S, T , T l5 &c, are functions of the time. A relation 

 between the T's is next established so that the subscripts are 

 replaced by coefficients. The value of H at the surface of the 

 wire is equated to AC, where A is a constant. This leads to 

 Maxwell's equation (13) of art. 690. The magnetic permea- 

 bility, //,, of the wire, which Maxwell had treated as unity, is 

 now introduced by Lord Rayleigh, who thus obtains in place 

 of Maxwell's (14) and (15) the following equations : — 



^=^* + i^-^ + -;;- + ii^^--a? + ---->--' (2) 



AC-S = J + ^ + g:?;g +.... + lit2 , |t<n ,^+....j > (o) 



where a, equal to //R, represents the conductivity (for steady 

 currents) of unit length of the wire. 

 By writing 



»(«) = ! + «+ pr^-+ — + 1 2 [ / tiW 2 + - • W 



equations (2) and (3) are then transformed as follows 

 ctt ~~ dt 



d$ A dC* ,( d\ dT 



-ty^dtj-dt' • • • w 



C = -^|).f: .... (6) 

 we have further 



5-^ (7) 



I- dt [n 



Lord Rayleigh then applies equations (5), (6), and (7) to 

 sustained periodic currents following the harmonic law, where 

 all the functions are proportional to e ipt , and obtains 



E=R'C+^L'C, (8) 



R / and 1/ denoting the effective resistance and inductance 

 respectively to the currents in question. The values of R' 

 and \J are expressed in the form of infinite series. For high 

 frequencies, however, they are put also in a finite form, 

 since, when p is very great, equation (4) reduces analytically 

 to 



*W=F7 = V' (9) 



£ V7T •*• 



* AC is printed here in Fhil. Mag., May 1886, p. 387 ; but appears to 



be a slip for A — . 

 v dt 



