552 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



other they are connected with the plates of a condenser of capacity 

 K . The conditions are then 



F=0, x = l, 

 (9) 



Applying these to the solutions (3), (7), (8), we have 

 A cos fi,l + B sin /zZ = 0, 



Eliminating A/B we obtain 



J 



(I i) /itan^=^-, 



Jtt-o 



a transcendental equation to determine /*. This has an infinite 

 number of roots, which may be real or complex. When these are 

 determined, X is determined for each root by the equation (5). 

 Thus we find that there are an infinite number of possible periods 

 for the free vibration, corresponding to the n periods for a system 

 with n degrees of freedom. The equation (11) corresponds to the 

 determinantal equation of 241 (10). The ratio B/A is determined 

 by (10). The determination of the absolute value of the co- 

 efficients depends on the initial conditions. 



Having found an infinite number of particular solutions, any 

 root fjL 8 being distinguished by its suffix, the general solution is 



(12) F= 2eV (A s cos fax + B 8 sin fax), 



where we sum for all the roots. If the potential is initially given by 



(13) V=F(x\ t = 0, 

 we must have 



(14) F(x) = S (A 8 cos fax + B s sin /JL S X). 



The problem to be solved is then that of developing an 

 arbitrary function of a? in a trigonometric series of the form (14) 

 where the fa's are the roots of a certain transcendental equation, 

 namely (n). The problem is in general of considerable com- 

 plexity, and we shall content ourselves with referring to Heaviside, 

 who has treated it at great length. 



If there is no condenser at # = 0, but the circuit is open, 

 equation (n) is 



tan jbl = oo 



