Stability of the Steady State of Forced Oscillation. 707 

 o£ the values of R and <£, 



(41) 



where c — \ sec a (1+ 3 tan 2 a), 2n + l = 2zs'm cc. . (42) 



When n and 2 differ so little that a is approximately 90°, 

 the character of the expansion changes, and an independent 

 investigation is required. This will be given in a subsequent 

 paper. 



LXX. On the Stability of the Steady State of Forced 

 Oscillation. By Andrew Stephenson *. 



1. TN the case of a simple system, that is a system in 

 X which the restoring force is exactly proportional 

 to the * displacement,' the motion under a periodic force 

 is made up of a definite oscillation isochronous with the 

 generator and an independent free motion of amplitude and 

 phase determined by the initial conditions. If the system 

 is subject to kinetic friction, the free element is gradually 

 damped out and the motion approaches asymptotically to the 

 state of steady forced oscillation. Although in practice the 

 spring of a system varies to a certain extent with the 

 amplitude, the results obtained for the ideal simple system 

 are in general agreement with actual phenomena, and 

 are tacitly assumed as being practically of universal appli- 

 cation. From observation, however, of the behaviour of a 

 system such as the simple pendulum when resonant to a 

 force of nearly its own period, it is natural to question the 

 validity of this assumption in certain cases, even for moder- 

 ately small amplitudes ; and it is our object here to seek out 

 the circumstances in which the simple rules of the approximate 

 theory do not apply. 



When account is taken of the variation in the restoring 

 force, the equation of motion is to a first approximation 



x' + Zkx' + p 2 (l — c% 2 )x = acos (qt + e). . . (i.) 

 There does not appear to be any practicable method of 



* Communicated by the Author. 



