SYSTEMS WITH NUN-LINEAR REACTANCE 437 



(35) now becomes 



a ^ \ - ^ — • (48) 



From (34) it is evident that when Z^ is finite the constant current of 

 the generator frequency, and so also the threshold voltage, are greater 

 than when it is infinite. Thus the presence of current of the sum fre- 

 ([uency makes the conditions for oscillation more exacting. As we 

 approach the limiting imjiedance ratio, where the powers approach 

 the ratio of their frecjuencies, the threshold v'oltage approaches infinity, 

 and the probability of oscillations approaches zero. 



The relative powers at the difference frequency and at the mechan- 

 ical frequency are now given by 



P = , "%.. , • (49) 





The presence of a finite impedance at the sum frequency increases this 

 ratio over that of the frequencies. For the limiting condition of oscilla- 

 tions it approaches infinity, the amplitude at the difference frequency 

 then becoming infinite and that at the mechanical frequency remaining 

 finite. 



From these results it appears that proportionality between power 

 and frequency is a limiting case which occurs only under the conditions 

 which are most and least favorable for the existence of oscillations. 

 We should, therefore, expect to find it only under the favorable condi- 

 tions where the transformation of energy is from a higher to a pair of 

 lower frequencies. 



Effect of Other Frequencies 



In the interests of simplicity the above treatment was limited to the 

 case where all but four frequencies are suppressed by high impedances. 

 Such a limitation is not, however, essential to the production of 

 oscillations. In fact, as many as desired of the series wzw^ + wcom may 

 be produced by the proper choice of impedances and the use of high 

 enough voltages, provided, of course, the apparatus can withstand the 

 stresses involved. In general, the presence of certain frequencies will 

 be favorable to oscillations and that of others unfavorable. 



Summary 



By way of summary, then, it is possible to maintain a movable con- 

 denser plate in sustained oscillation by applying to the condenser an 



