and Inductance oj a Concentric Main. 



'here 



and 



V2 S\/2x 



25 



13 



384^/2^ 128.i 



s/2+ b °~SS2x 



2h 



531 



. (18) 

 .. . (19) 



Vox 1 381 s/2% 1 

 Hence, since y = ber x + 1 bei x, we have 

 ber x — (A/\/x) e a cos ft, and bei x = (A/\Zx)e a sin ft. 

 To determine the values o£ A and b we notice that 

 ber x + l bei x = I (x\/i) , 



and thus, from equation (3), we see that A is l/\/27r, and 

 Z> is — 7r/8. 



We thus find that when x is large 



and 



ber a?= — - 1 — cos ft, 



^/'llTX 



bei x = 



Vi 



sm 



7T^ 



(20) 



(21) 



where a and /3 are given by (18) and (19), and b is — 7r/8. 



The series for « and ft are semi-convergent and a rigorous 

 mathematical justification of (20) and (21) is difficult. It is 

 easy, however, to verify that, if we only include the terms of 

 the series given above, (20) and (21) give the values of the 

 functions with great accuracy when x is greater than 5. The 

 values of a and ft are easily computed by the formulae 



* = 0-707105ff + (M)8839/j,— O'046/.r 3 , (22) 



and 



ye = 0'707105^-0-39270-0-0S839Ai'-0-0G25/.r-0-046/.i' 3 . (23) 

 Differentiating (20) and (21) we find that 



ber' x = ( -L _ 1 L-} ber x - (-L + — -*— + A ) bei x, . (24) 



and 



bei 



' x=(X + —L- + X) ber x + ( — - 1 ~ ) bei «. (25) 



Squariug equations (20) and (21), and adding, we get 



X: 



:ttx 



(26) 



2 2 



