Inductance of Two Parallel Wires. 259 



and with the above value of A, 



E 2^f J (*q) k Ko(Jta) l 



4" ife LJo^ifea) /t/uKo'^J' * l j 



and the effective self-induction is the real part of the ex- 

 pression on the right. The effect due to variability of current 

 in the section, deduced from the first term, is in accord with 

 the result obtained by Lord Kelvin * and otherwise by Lord 

 Rayleighf. Maxwell and Heaviside have also given other 

 forms of this solution. 



Transformation of Bessel Functions to a new origin. 



We have by the integral formula for the function finite at 

 infinity, 



K ?l { i(c + r) } = e-^^cosh^ cos jj ^ j ( j )m 

 Jo 

 But , 



e' { J = J (c) +■ 2 J, (e) | s. + l — L J , . (21) 



by an ordinary definition of Bessel functions of the first 

 kind whose order is an integer. 

 Writing z=— tcP, 



e- tCCOBh ^=J (c) + 22 (-i) n J" (c) cosh n<f>. (22) 

 i 



if the series be convergent. 



Thus 



]LC* • c + r ) 



/»00 



= ^.c- lrcoBh ^coslin0.{J o (c) + 2 2 {-b)*J s (c) cosh sd>] 



Jo 1 



= JoW f e-^^coshH^-f 2 (-0*J«(«)i «-»«*♦ / C0Bhn ±i* \ d s 



Jo i Jo (. + cosh n — .<</> j r 



Lj (c)K,(*r) + f (-0 Jf J^){K B+s (tr) + K n _ s (tr)} J f23) 



s = l 



where K„_,(«') is identical with K s _„ («r). 



* Presidential Address to Inst, of Elect. Engineers, 18S9. Math, and 

 Phys. Papers, vol. iii. p. 402. 



t Vide Phil. Mag. 1886, where a complete practical solution may be 

 found. 



