Fig. 2. 



Systems with Propagated Coupling. 435 



excited shall increase with time the, resulting coefficient 

 r must be negative ; the motion will then increase until 

 the equations (which are only approximate) no longer hold 

 o-ood and would need to be modified. A much more com- 

 plete treatment is to take the disturbance as itself arising 

 from a mechanical system for which the 

 full equations can be written down and 

 then to solve the equations for the two 

 systems simultaneously. 



As an example, consider a telephone 

 membrane placed at an adjustable dis- 

 tance x from the coil which is connected 

 to a battery. The current at any instant 

 is C, and the displacement of the mem- 

 brane if-. An alteration of C will alter 

 y and vice versa. If a chance alteration 

 is produced, will it persist? 

 The equations including a mutual action proportional to 



the current and to ~ are, 



The-mbrane 



dt 



4+*) C + "t= E ' 



( 



;C+ ('\S +r ! +K >-°' 



which on elimination of ;/ gives 



4-(rL + Rm)^-f(KL + rR-a 2 )~ + RK JC = EK. 



f T d " 



dt B 



Putting d ;. r R 7 K 



dt * m L m 





the auxiliary equation is 



f -f (a + b)P + (c + ah -/)f 4- be = 0, 



or e 8 +?p+*f+i=o, 



where g and j are essentially positive, whilst h may be either 

 positive or negative. 



If the roots are f 1? f 2 , f 3 , then 



fi + 6 + 6=-^ 



