'910.] LONG ELECTRIC WAVES ALONG WIRES. 369 



in (6). Putting = 6' -{-id", we get as the real part of (22) 



K = —f^ W ^ {^2(2"-«9" cos ivt + 2ke') 



'^ 2 (cosh a^nv — cos \nv ) ^ \ > / 



+ ^-2(2»-'oa" cos (y/ - 2k&) ~ i'"«" cos (i'/ + ^7i& (23) 



_ 2/'6>') - ^-2''»" cos (i^/ - 4;/^' + 2/^^')}, 



which together with 



4(sin- 6' cosh- ^ — cos^ & sinh- ^") = v^ L'S' — R'K' (24) 

 — 4sin2^'sinh2r = v(L'i^' + i?'5'') (25) 



gives the complete solution. 



Now on passing to the limit as before, we can replace sin 6 by 6, 

 L' = Lhx, etc., and we find 



2e'=—Qhx, 

 2e"=Phx, 



where 



P, Q= :^- {{iJ'D + R'){v'S' + K') ± {RK- v'LS)y^, 



We thus have 



4nd" = 2Pl, 4n6' = — 2Ql, 



2kd" = Px, 2kO' = — Qx ; 

 (23) thus reduces to 



77 PI 



V= Ec-''' cos {yt - Qx) + ^ 



where 



2(cosh 2PI— cos 2^/)5 

 X {^^^ cos {yt + (7.1- + (/)) — e-^^ cos (i// — g.r + (^)} , 



sin 2 ^/ 

 tan 9 = 



^-2«_cos2e/' 



which is the solution for this case as given by Heaviside/ except that 

 leakage is here considered and the real impressed force is E cos vt 

 instead of E sin vt. 



^ " Electrical Papers," Vol. 2, p. (i2. 

 Princeton University. 



