1889.] Conductors in the Neighbourhood of a Wire. 5 

 Eliminating E and F from equations (3) and (4), we get 



{FJ '(«») + GV(«a)} lh-n£z*A 



= ^-AJ '(.2»)-^(BJ '(. 2 '«)+OI '( < 2'a)) (7), 



{PJ '(, K 6)+GI '( IK &)} 



" (7 



DI '«6)-^l-(BJ '(^) + CIo'(^)) (8). 



We can substitute for B and C from equations (1) and get four 

 equations from which we can eliminate F, Gr, A, and D ; as, however, 

 the result is lengthy, we shall only solve it for the particular case 

 with which we are concerned, when qa, q'a, q'b, and q"b, ica and /c&, 

 are all small. We get, substituting in the terms multiplied by v'—v, 

 the approximate values of the Bessel's functions for small quantities 

 of the variable 



{(m^ — rfi) T f N T // \ (m 2 — /e 2 ) T , , \ t / \ 1 

 ^ '- J (t«:a)J (ma) — ± L J {tKa)J {ina) > 

 jun k J 



f (m 2 -^ 2 ) Io (^)T Q '( m >ft)_0^-« 2 ) l '( lK a)l ( in l) \ 



L fJLTl K J 



{m 2 — n 2 T , N T // \ t// nTw 2 — /c 2 T/ x 

 l (</c<x)J (ma)-l (t/ca) J (ma) 



(V — i/) m 2 m 2 — w 2 t ; / > 6~1 "1 



— * ^ ^ aJ (ma) log - V 



v k fxn a J J 



— J («kS)I / (m* &) J '(t/c&)I (m&) > . 



fin k J 



In this equation the approximate values of the Bessel's functions 

 have been used only in the term multiplied by v'—v. 



This equation simplifies very much, since ica and icb are very small, 

 and therefore approximately 



J (iJca) = 1, J '(ika) = —jjika, I (iha) = log rpika, 

 and I '(Jf<z) = 1/ika. 



Making these substitutions, the above equation reduces to 



2 _ t P 2 / 11171 1 Jo( ma ) 1 Io( m ^) \ 1 + v '~ v m v 



v 2 \ m 2 —n 2 a J '(ma) m?—n'%b I \tn'b) J log(6/a) ' 



