392 Lord Rayleigh on the Self-induction 



Another condition is afforded by the consideration that, on 

 account of the symmetry, H must vanish when # = 0, or H =0. 

 Within the insulator, whose permeability we take to be unity, 

 w=0, so that by (29) j3=j3 a , and 



Hence from (32), by equating the values of H a , 



4*29 = i** +A+ B (37) 



y p dz 



Now by (35), (36), 



4tt//,C ^ + g~ w6 



my e mh — e- mW 



so that 



In (38) the first term represents a part of the effective self- 

 induction, and contributes nothing to the effective resistance. 

 This self-induction, per unit length, is simply 4:7ra/y, and is 

 independent of p. The second term, being neither wholly 

 real nor wholly imaginary, contributes both to self-induction 

 and to resistance. If, separating the real and imaginary parts 

 of the right-hand member of (38), we write 



-Z^=B'C + I/.*>C; . . . . (39) 



then Ttf represents the resistance and 1/ the self-induction of 

 length I of the conductor measured parallel to z. 

 From (34) 



m-^/(i^). (1+0 = ^(1 + 0, • • (40) 

 if we write for brevity 



"**) <«) 



The general expressions for W and 1/ are somewhat compli- 

 cated. If the rate of alternation be slow, p, and with it m 

 and q, are small. In this case (38) may be written approxi- 

 mately 



__ dW __ . p 4tt<2 pO/, . 4:7r/npb 2 \ m 

 dz ~~ %P y by\ % 3p J ' 

 so that 



*'-f. («) 



-wc- 



