PROCEEDINGS OF SECTION A. 37 



showing, (1) that the amplitudes vary harmonically and have a constant 

 phase difference of —' that is, when one waxes the other wanes ; (2) that 



TT 



the oscillations also have a constant phase difference of o their frequency 

 being \ ((-n + W2)- 



From the values for / ?? given above, it wdll be seen that / ^ E, 

 ^ Vi ^ Pi 



the amplitude of the amplitude or the maximum amplitude of F2, is 

 independent of the coupling when a = h, but that this maximum 

 amplitude is only attained in the time t given by 



>2 ~ t'^l 



2 2 



and as the " triangle " is isosceles it is easy to deduce that 



_ cos v// T 

 ' ^ sin 1I//2 * 4 



2 TT ' . 



where T = —r=~ = the natural period of either circuit. 



\i a 



It will be seen from this expression for t that as the coupling 

 (sin i//) diminishes the time t required for V2 to attain its maximum 

 amplitude increases, and when the coupling is zero {i.e., when-*// = 0) 

 it requires an infinite time for the maximum to be reached. 



The above can be very effectively demonstrated by the model, 

 the coupling being regularly reduced by loading the beam, and, if 

 necessary, diminishing the masses of the bobs. 



From the remarks at the beginning of this paragraph we may 

 write the energies in the two circuits as being equal, at the time t, to 



^K^E^ cos2 ''^^1 t 



and \^\Ei^ sin^ ^ ^ '^^ t respectively. 



These expressions show that \ KxEP', the whole of the energy in 

 the system, passes from either circuit to the other and back to the 

 original one in the time t, given by 



W2 - wi . 



Is = 7r 



2 

 80 that ^,, which we shall call the period of the surges, is given (easily 

 by the triangle) by 



_ cos ^ T 



^' ~ sin 1///2 ' 2 

 where T is the natural period of either circuit. 



U' 



- » ^ -^ek * — - , ___ i 



