mat 

 n 



1 



Triad e Generator toith Tiro Degrees of Freedom. 715 



kedly different possibilities arise. For example, the 

 ■ange of values of co 2 2 for which a 2 is possible is now 

 'epresented by OL and is seen to extend heyond the 

 resonance position co 2 2 = a) l 2 . In the same way the ampli- 

 tude 1>q is now possible for all values of co 2 2 greater than 

 OA. Hence a region AL for co 2 exists in which both 

 modes of vibration are separately "possible." We thus 

 must have recourse to a consideration of the conditions of 

 stability in order to decide which mode of vibration will 

 actually be present for any given conditions. 

 Now in order that 



a/ = a 2 , b s 2 = be stable, 

 we must have according to (14) 



E I ^ 2 + E„(2ao 2 -&o 2 )>0 1 



y. . . (18 a, b) 

 and a 2 (2a 2 — b 2 ) > 0,j 



where the second condition is the more stringent one. 

 Hence for a 2 only to be stable, we must have 



a 2 > ^b 2 , 



and similarly for b Q 2 only to be stable we must have 



b 2 > h< { 2 . 



These conditions are represented in fig. 5 for -^ON. 



The vertical SP is so chosen that SR = HQ and the vertical 

 TW such that TU = UV. Hence, though we found pre- 

 viously AL as the region where a 2 as well as b 2 were 

 separately "possible" ', we may now further conclude that 

 the common region where a 2 as well as b 2 are separately 

 stable is given by the smaller distance PW only. 



Which one of the two possible and stable oscillations 

 a smo)it or b sh) ((out-\-\) will be attained in the region 

 PA\ ? The answer to this question, which must also include 

 the explanation of the hysteresis effect, is given by (9), and 

 will be seen to depend on the initial conditions. For let us 

 see what happens when co 2 2 is gradually brought from a 

 small value such as represented by OH, through resonance 



to a big value represented by OX. (We again assume 

 to be given by ON.) * 2 



First, when &) 2 2 = OH, only the first mode of vibration is 

 possible and stable and we therefore have 



a s 2 = a 2 , b s 2 =Q. 



