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BELL SYSTEM TECHNICAL JOURNAL 



that containinj? cosines and obtain a relation between the angles in- 

 volved. We square each equation of a pair, and add them to obtain 

 a relation between the magnitudes of velocities and currents, the im- 

 pedances, and the frequencies. From these it follows that ly is a 

 constant. By means of these relations the equations derived from (13) 

 may be reduced to a form where the only variables are Eg, Vm and dg, 

 and the constant, /„, appears only as a divisor of Eg. These equations 

 are then squared and added to give an equation which determines Vm- 

 The final solution takes the form 



<Pm = ^d, 



(21) 



/, = ^ [Z„,a;,„Z.a;,,]'/2, 



Vm = ^ ZdCx^rlZgOJ,, i — COS (sPm + (ffi) 



sin^ {ip„, + ip,i) 



n 



/./ = 



ZmO^d 



V 



'here 



Z I 



COS a = ^r-^sin (^„ + ifg), 



and the sign in (25) is so chosen that 



and 



TT „ TT 



+ 0d= a + 7r±2. 



(22) 



, (23) 

 (24) 

 (25) 



(26) 



(27) 

 (28) 



where the same sign is to be taken for 7r/2 as in (25). 



The nature of the variation represented by (23) is shown in Fig. 1, 

 which is taken from the accompanying experimental paper. ^ Here the 

 amplitude, Fm/co„,, of the plate displacement is plotted against the 

 generator voltage. Eg, for the case of exact resonance and for one 

 involving a slight departure from resonance. 



Let us interpret these results physically. The phase angles in (21) 

 depend only on the physical constants of the system and the frequen- 

 cies of the oscillations. This equation, therefore, determines at what 

 frequencies oscillations may occur provided the other conditions are 



