791 



we have 



— A V ^^ _ i ^* 

 ^ Rl^ Hs 2 R 



= — — -VT^ -2< ??s ds 6s — 



R IR Ro 



s'n. 



These series are convergent, if the conditions for the ordinary 

 Fourier series are fulfilled. We can therefore calculate «« and /?s with 

 the help of the given formulae. 



§ 4. In the case ^ is a given function of the time, our equation 

 can also easily be solved. 



a. First if E is constant, we have 



I 

 E=RJ-\- H fij dw. 







The current J and ?/ can he divided into two parts, the one 

 depending on /, the other not ; we indicate those parts by the indices 

 1 and 2. For the first part we have 



= a» -^ -4 '- 



E = R, 

 therefore 



from which y^ can be determined if we take into account that yj 

 vanishes for x = and x = I. The determining of the second part 

 leads to the problem treated in § 3. The solution can be used in 

 order lo fulfill given initial conditions. If an initial value of J is 

 given, then y must fulfill at / :== a condition following from (3). 

 b. Further, we can consider the case E = E cos pt. 

 Putting L =zO, we can try the solution 



y — (f cos (pt -f ^) 

 J=I cos {pt + /?) 



where ff is a function of x. The first equation gives 



ÖV/} 



This equation can be solved by 



