Forced Vibrations of Electromagnetic Systems. 147 



(67) can be finitely decomposed into separate solutions of the 

 form (70) in the case of perfect insulation, when (67) takes 

 the form 



C= ^ 1-6 ~ 2 ^ + Tl 6 " ? ' 2cosm ' ? ^ sinX ^ ' (71) 



where q = s 1 = s , by the vanishing of s 2 hi (64). 



Therefore (70) represents the real meaning of (71) from 

 t = to l/v, provided vt > z. But on arrival of the wave Ci at 

 B, V becomes zero, and doubled by the reflected wave that 

 then commences to travel from B to A. This wave may be 

 imagined to start when £ = from a point distant I beyond B, 

 and be the precise negative of the first wave as regards V and 

 the same as regards C. Thus 



V*=^- qt Jo{l[W-zY-vH^y . . (72) 



expresses the second wave, starting from B when t = l/v, and 

 reaching A when t = 2l/v. The sum of Cj and C 2 now ex- 

 presses (71), where the waves coexist, and C x alone expresses 

 (71) in the remainder of the circuit. 



The reflected wave arising when this second wave reaches 

 A may be imagined to start when t = from a point distant 

 21 from A on its negative side, and be a precise copy of the 

 first wave. Thus 



^=t e ~ t ' J '{i m+ " r ~ ,lH1 ' ii }' ■ ■ (73) 



expresses the third wave ; and now (71) means Cx 4- C 2 + C 3 in 

 those parts of the circuit reached by 3 and Q x + C 2 in the 

 remainder. 



The fourth wave is, similarly, 



C 4 =^ e -^J {|[(4Z-^-,¥] }, . . (74) 



starting from B when t = 3l/v, and reaching A when t — U[v. 

 And so on, ad inf.* 



* It is not to be expected that in a real telegraph-circuit the successive 

 waves have abrupt fronts, as in the text. There are causes in operation 

 to prevent this, and round off the abruptness. The equations connecting 



L2 



