Currents in Concentric Cables. 63 



If this succession of charges equivalent to a permanent 

 alternating current 27rScos«o£ be continued for a long time, 

 the term involving e - ^ 2 becomes very small and we have 



4? .f.fii 2 cos cot + o) sin cot 



s=oZ cos it) 5— — 5-71 . 



The potential at A is 



S . 

 — sin 

 coc 



wt 2 1 + ~ ) -\ — cos cot z — 1-f ^r ; 



_oo\ co- / coc _«, a> \ w J 



the potential at B is 



— smwlS -1' 1 + ^-y + —cos g>*2( — IV — (l + ^-s-l 

 <wc _ QB V \ co z J coc _«, w \ or / 



It may be seen that the electrical actions in the two cases 

 discussed are identical so far as the portion AB of the con- 

 ductor in the first case is concerned. 



Writing Ly=7rc, Jjp = irr, 7rS = A, and a / j!Las6, so that 



IT 



sinh 2/3tt + sin 2Bir 



-t \ 1 + 4/37 ^" cosh 2/3tt - cos 2/3tt 



[this expression I shall refer to as/i (/?)]; 



(» "2 



V ' fi e ' 4 \ _1 _/Q sinh 2/37T — sin 2/37T 

 ZW^ W) ~ cosh 2/3tt- cos 2/3tt 



[this expression I shall refer to as/ 2 (/3)] ; 



£ , ., N • / . i 4 \ -1 r> cos /37T sinh y37r + sin Bit cosh /37r 



2( — 1)» 1+j^i) = 2Bw- j— - = — -^ — 



_«, 7 \ 4/3/ cosh 2/37T— cos 2^7r 



[this expression I shall refer to as/ 3 (/3)]; 



v / i\i & (1 ** \ -1 a /-. cos Bit sinh /37r— sin /37r cosh Bit 



S ( — 1) oo"4 1+ Tail =2P 7r u^5 ^*o " £ — 



_«» 2/3- \ 4/3 / cosh 2/3tt — cos 2/37T 



[this expression I shall refer to as/ 4 (/3)]. 



In the particular case where B is integral 



JO + w)~ l= iw- i 1 + &T -*«**■. 



