256] ELECTROMAGNETIC WAVES. 553 



and the roots are 



(2S + 1)7T 



(14) /**= ~2~~T 



The series (14) then becomes 



a Fourier's series, with the even terms omitted. If R = we see 

 by (5) that X is a pure imaginary, so that all the oscillations are 

 harmonic. 



The wave-length L is 



or the length of the wires is an odd number of quarter wave- 

 lengths. 



If on the other hand the capacity K Q is infinite, we get 



(17) /*tan/zZ = 0, 



which is the same as if we had considered the circuit closed at the 

 origin also, putting V= 0, for x = 0, from which (7) and the first of 



(10) would give 



sin pi = 0, 



(18) STT , 21 



*~T' Z = 7' 



The length of the line is then a multiple of a half wave- 

 length. The two cases correspond to the cases of an organ pipe 

 open at one end and closed at the other, or closed at both ends. 



These conclusions have been verified by experiment. The 

 above theory applies, for instance, to the experiments of Saunders 

 cited above. 



The method employed in this example is typical of the general 

 process, for the terminal conditions, of whatever nature, are given 

 in the form of an ordinary linear differential equation in the time, 

 involving the derivatives of V and /. Applying this to our 

 assumed solutions (3) introduces algebraic functions of X, so that, 

 eliminating by means of (5), we obtain an equation of the form 



(19) tan ^ =<(/*), 



where <f> is an algebraic function. The case we have considered is 

 the simplest case of this transcendental equation. 



