368 ADAMS— THE PROPAGATION OF [December 2, 



., LK+RS . . 



P=-^L^> (20) 



which are the well-known solutions for the free vibrations of a uni- 

 form line for long waves. 



The differential equation of which (17) is the solution is obtained 

 by passing to the limit in the difference equation (5). We thus get 



d'^V d-V dV 



Equations (3) and (4) on passing to the limit give: 

 dV dC 



For the second problem, that of a periodic impressed electro- 

 motive force applied to one end of a line, the other end being earthed, 

 we have to solve equation (5) subject to the conditions: 



k = o, Vk^Ee'''^ 



(21) 

 k = 11, Vk = o. 



The resulting solution may of course be applied to a closed circuit 

 with the periodic force Ee^"^ introduced in it at any point. After 

 the free vibrations have been damped out, the solution will be 



where A and B are arbitrary constants and a and fi are given by 

 (10). Determining A and B by means of (21) we get 



sm 2(71 — k^O ^ . , . 



K = :-^ ^~ Ee'^' . (22) 



'■■ sm 2nU ^ ' 



6* is a complex angle defined by (6) and (9) if v is written for p 



