6 Prof. J. J. Thomson. On the Effect produced by [May 2, 



(see also " Electrical Oscillations on Cylindrical Conductors," 

 ' London Math. Soc. Proc.,' vol. 17, p. 320), 



or, 



v v 2 I \m 2 -w 3 aJ '(mfl) m 2 — w' 2 & I '(m'&)/ log(6/a) / 



The nature of the solution of this equation will depend upon the 

 magnitudes of na and rib. 



Case I. na and rib both small. In this case 



J (ma)/J '(m<z) = — 2/ ma and I (m'&)/I '(m'&) = m'b log fym'fc. 



Making these substitutions, equation (9) becomes 



„ v p 2 { log 7 m'b a 1 "I m 



m 2 — = 1- 8 ;. . + «-;—-- — tt~7 \ \ (10). 



v v* I log (6/a) 27ra^ log (o/a) J 



Since na. is by hypothesis small, (xj27ra 2 p is large compared with 

 unity, and unless log (b/a) is very great, it will be much the largest 

 term inside the bracket, so that (10) may be written 



v' v 2 27r<x 2 j3 log (b/a) 



m== f\/ {^Wjplog (b/a)}' + 

 This represents a disturbance propagated with the velocity 



„{(i^£) log (&/.)} 



and dyiug away to 1/e of its original value after traversing a 

 distance 



,2n 



This case, which is that of slowly alternating currents, was solved 

 many years ago by Sir William Thomson. 



Case II. na large, rib small. This is the case of rapidly alternating 

 currents travelling along a wire which is surrounded by a substance 

 whose conductivity is so small that 4 Trjm'pb^ia' is a small quantity. 



In this case, since J '(ma) = tJ (mu), equation (9) reduces to 



m %JL = £ 1 1_ A 1 logy <*'b 1 _ 

 v L na log (Oja) log (b/a) J 



