710 Dr. B. van der Pol on Oscillation Hysteresis in a 



and thus E T > 0, 



E n > 0. 



Further, the term a 2 introduced in (9), represents the 

 square of the stationary amplitude which would be obtained 

 when an oscillation in the first mode of vibration, i. e. with 

 a frequency co v alone was present. 'Similarly b is the 

 stationar}^ amplitude which would be attained if the system 

 vibrated only in the second mode of vibration. These ampli- 

 tudes a 2 and b 2 are obtained directly from (7 a) and (7 b) 



by putting __ = — = and are found to be 



••) ■> 9 \ 



ft'," (O 



1 w l 



4 7 ^f ~<V ' ft) 2 2 |7 



(9 a) 



by — q o • o 



However, a and b are not the only " possible " stationary 

 amplitudes as may be seen from (9) by putting 



^=0, f =0. 

 at at 



We thus have in general for the "possible" stationary 

 amplitudes a s and b s the two equations 



a s 2 (a$ — a 2 — 26 s 2 ) = 0, 



the four sets of solutions of which are 



(i.) a 2 = 0, b s 2 = 0, 



(ii.) a 2 = i(2b<?-a 2 ), V = i(2^o 2 -V), /11N 



' (11) 

 (iii.) a s 2 = a 2 . b 2 — Q>, 



(iv.) a/ = 0, b s 2 = b 2 . 



But we shall further have to investigate separately which 

 one of these four stationary solutions (11) will be attained 

 in any given circumstances. Therefore (9) would have to 

 be, solved, which is a difficult, if not impossible, matter. 

 However, in order to investigate the stability of each of the 

 four solutions (11) we may consider the effect of a small 

 forced change of amplitude from the stationary value due to 

 some disturbing cause, and investigate the tendency of the 



