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XX. On a Peculiar Property of the Asymmetric System. 

 By Andrew Stephenson *. 



IF an asymmetric system is subject to a direct periodic 

 force the equation of motion is of the form 



x + Knx + m 2 (1 + x/a)x = bq 2 cos qt. 



a being finite and b and tc small, the steady motion is 



Q 2 

 x = b-^- — , cos (qt+rj), 



n 2 — q 2 V2 



when terms of the second order are neglected. This motion 

 may, however, be unstable if q approximates to 2n f . Putting 

 q = 2(n+p) and bq 2 ~3cn 2 , we find that there is instability if 

 c 2 >(/ca) 2 for the range of frequency for which 



\p\<in^/(cla) 2 -K?. 



The steady state of motion is then given by 



1 r 2 1 r 2 



x = —- — \-r cos{(n+p)t — «} + ^ -cos 2{(n+p)t — u} — c cos 2(n+p)t r 



where 



p = ±kn\/(c/a) 2 — fc 2 , \Jc\ being < 1, 



r 2 = %ca(l - k)Vl - (ica/cf, 

 and 



sin 2u — Ka/c, where 7r/2>|a| >7r/4. 



Thus if c sufficiently exceeds /ca, e. g. = 2/ca, the amplitude is 

 large, being of order V c. Any small deviation from this 

 motion may be analysed into two simple motions one of 

 which is reduced and the other undisturbed (so far as terms 

 of order c are concerned) . 



The asymmetric system then is sensitive to direct periodic 

 force of double the natural frequency and of intensity 

 exceeding a certain limit proportional to the motional 

 resistance. 



If asymmetric oscillations exist within the molecule there- 

 would follow the possibility of monochromatic fluorescence 

 with a frequency of emission half that of incidence. 



September, 1910. 



* Communicated by the Author. 



f " On the Stability of the Steady State of Forced Oscillation," Phil 

 Mag. December, 1907. 



