146 Mr. 0. Heaviside on Electromagnetic Waves, and the 

 and current initially, insert a steady impressed force e at A. 

 Required the effect, both in Fourier series and in detail, 

 showing the whole history of the phenomena that result. 



Equations (36) and (37) are the fundamental connexions 

 of V and C at any distance z from A. Let R, L, K, S be 

 the resistance, inductance, leakage-conductance, and permit- 

 tance per unit length of circuit, and 



Sl = R/2L, s 2 = K/2S, q = si + s 2 , s = Sl -s 2l . (64) 



\=(wV-« 8 )* ( 65 ) 



It may be easily shown, by the use of the resistance ope- 

 rator, or by testing satisfaction of conditions, that the required 

 solutions are 



-»***£], . . (67) 



where m—jirjl, and j includes all integers from 1 to go ; 



whilst V and C represent the final steady V and C, which 



are 



xr ( sinmo-sA 



V =, (cosm ,- i ^^j, .... (68) 



n m e / . cosm^e\ / . nN 



°=t ( 8mm °* + s^> ■ ■ ■ ( 69 > 



where m 2 = — RK. 



Now if the circuit were infinitely long both ways and were 

 charged initially to potential-difference 2e on the whole of 

 the negative side of A, with no charge on the positive side, 

 and no current anywhere, the resulting current at time t later 

 at distance z from A would be 



°' = jV" J »{? (r! - cV)i }' ■ • < 70 > 



by §§ 7 and 8; and if, further, K = 0, V at A would be perma- 

 nently e , which is what it is in (66). Hence the C solution 



