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



For under these circumstances the series representing the 

 amplitudes of the harmonics may be expected to converge 

 rapidly so that the influence of the harmonics on the ampli- 

 tude of the fundamental may, as a first approximation, be 

 neglected. We are thus justified in neglecting as a first 

 approximation the presence of these higher harmonics and 

 combination tones and shall retain, therefore, in the terms 

 with v 2 and v 3 only those parts involving the frequencies o)i 

 and co u . We thus see that the term ftv 2 has no influence on 

 the result. In considering, however, 



r 3 = {a sin cojt + b sin (o) u t + X) } 3 , 



terms of several frequencies occur, such as a>j, &> n , 3o)j, 3ft) n , 

 o) I +2a) II , (Oj — 2o) n , 2ft) I + &) II , ... etc., but only the terms 

 involving the frequencies w 1 and w n will be retained. 

 Hence we have 



^ 3 = Ja(a 2 + 26 2 ) sin cojt f |6(6 2 + 2a 2 ) sin (w n t + \). (5) 



It may here be noticed that b occurs in the coefficient of 

 sin cojt and a in the coefficient of sin (co n t + X). This funda- 

 mental fact in the non-linear treatment of our problem shows 

 the mutual influence of simultaneous vibrations, and it will 

 further be found that the presence of one oscillation makes it 

 more difficult for the other to develop. When more than 

 three terms are used in the series expansion for i a , as is 

 advisable when working on the lower bottom part or higher 

 top part of the ?'« — v g characteristic, then the presence of one 

 oscillation is, however, occasionally favourable to the develop- 

 ment of another oscillation. Such special cases will, however, 

 not be considered here. 



We now r proceed to substitute from (4) and (5) in (3) and 

 thus get an equation of the form 



A sin cjjt + B sin (a> n £ + A) -f C cos cjjt + D cos (a) n t -4- X) — 0, 



. . . (5a) 

 where A, B, C, and D are functions of the variables &) D o> 



a, b, but they also contain a, b, —r- , ~-?r . 



These expressions A,, B, C, and D contain terms of three 

 orders of magnitude, viz. : 



first order : 



w 4 a, co 4 b ; 



second order : 



aco^a, ya) 3 a s , a) d a .... etc. ; 



third order : 



2 . <2 da * 



xco^a, 7&> -T- .... etc. ; 



but we only retain the first two orders of magnitude. 



