Coupled Circuits used in Wireless Telegraphy. 571 



system that is analogous to the " natural period " of the 

 primary electric circuit, we have simply to place the bob of 

 the second pendulum on the beam and then measure the 

 period of the first by observation in the usual way. Similarly 

 the natural period, as we shall call it, of the second pendulum 

 is determined. 



Returning to equations I. above, if we eliminate X or 6 2 

 between them we find that either 6 X or 6 2 will satisfy the 

 differential equation 



{{w+tf)(w+^)- Pl p^}d = o, 



which is identical in form with equation IV., § 2, which is 

 satisfied by the variables in the coupled circuit system. 



It is evident that the " coupling " of the pendulum system 

 or the value of sin i/r for the latter is equal to 





(M + mJfl + m^ 



Hence all the variables either of the mechanical or of the 

 electrical system satisfy the differential equation 



{cos 2 tD 4 + (/*i 2 +^ 2 )D 2 +^W}* = 0, . (IV.) 



where fi x and fi 2 are the natural frequencies, and t/t the 

 coupling angle in either case. 



4. In order to follow up this analogy, it is necessary to 

 know fully the details of the motion in either system arisino' 

 from analogous initial conditions. We shall therefore ob- 

 tain the solution when for the electrical system the initial 

 conditions are 



Y 1 = E, V 2 = 0, d = 0, C 2 = 0, when* = 0, 



the usual initial conditions to a disturbance in a Marconi 

 transmitter, and when for the mechanical system the initial 

 conditions are 



0, = E, 2 = O, <?! = (), 2 = O, when£=0. 



Let us put a for ^ and o for \x 2 in equation IV. § 3, 

 and it becomes 



{cos 2 <fD 4 + (a + b) D 2 + ab}^ = 0. 



Proceeding with the solution of this differential equation 

 in the usual way by obtaining the factors of the operator 



