204 BELL SYSTEM TECHNICAL JOURNAL 



The constant K-2. becomes — 1 and the total current, / = /i + I2, 

 is propagated in accordance with / = Iq€~'^^, where 7 — -sZY-y. 



The constant Ki, which is the ratio of the current in the sheath to 

 tliat in the wire, is of interest. It becomes 



Jo T, /^ ]•> — i^oo ^]] — ioo] /2*"^ 



T-=-f^i = ^—f = ^—f (^0 



1 I Z/22 ■'-'■22 



The approximate formulas for the case where wire and sheath are 

 contiguous are derived as follows: The arguments, xo and ,V2, of the 

 Bessel functions differ by only a small amount when the magnetic 

 sheath is thin. This situation is favorable to an advantageous use 

 of Taylor's series. Joixn), for example, can be expressed in terms of 

 Joiy^), its derivatives and the difference of the arguments in a Taylor 

 series as follows: 



, /o(.r) = My) + rJo'iy) + ^j /o"(v) + fi M"(y) + • • • > (38) 



where r = .v — _v (x-i, y2 being written simply, x, v, here, for con- 

 venience). Furthermore, Bessel functions are subject to recurrence 

 formulas,^ which enable us to express each of the derivatives occurring 

 in the series in terms of the function of zero order, its first derivative 

 and the argument. Therefore, by applying the recurrence formulas 

 to the Taylor series, we find functions U and V (see Appendix B) 



such that 



Mx) = UMy) + VJo'(y), (39) 



Ko{x) = UKoiy) + VKo'(y) (40) 



{U2, V2 being also written now, U, V). Differentiating (39) and (40) 



with respect to t, 



Jo'(x) = U'My) + V'Miy), (41) 



K,'(x) = U'Ko(y) + V'Ko'(y), (42) 



3t7 , dV 

 where U = — — > V = -z— • 



OT OT 



''The two recurrence formulas required are: 



J,/{Z) = ^' /(=) - /„ + ,(3), 

 J,/{z) = /„_,(=) - "/„(=). 



l*lie Bessel Functions ut" the second kind satisfy the same furmuius. 



