the Steady State of Forced Oscillation. 711 



disturbance, the instability must be readily perceptible if only 

 the friction be sufficiently small. 



The approximate analysis does not enable us to decide 

 whether the variation continues periodically, or whether 

 the motion tends to settle down into the steady state again ; 

 this is not a matter of any consequence^ however, as the 

 condition of perfect freedom from small disturbance cannot 

 be realised in practice. 



3. The case of instability that comes next in importance is 

 that in which fxi/q is somewhere in the vicinity of 2. The 

 solution of (v.) for this relation is given in the paper 

 previously quoted, but the result is not directly applicable to 

 the present question, as it is no longer permissible to neglect 

 the terms of argument 3qt in Q. 



It is easily found that to a second approximation 



Q = — y=. i COS qt ~^~* C0S ty) > 



and therefore for the initial motion of y following a small 

 disturbance 



y" + 2ki/ + /Wl — 2a 2 cos 2^ + jz« coslqtjy = 0, 



where /^ 2 = fi 2 (l — 2a 2 ). The necessity for retaining the 

 small term in cos ±qt is evident on the general ground that a 

 spring variation of double the natural frequency is much 

 more powerful in forcing an oscillation than a variation of 

 equal frequency. 



Putting y = e~ Kt z we assume for the solution 



z=+a^ 2 cos'{(4:q+p)t + €}+a- 1 cos : {(2q+p)t + €} + a cos{pt + e} 



+ a 1 cos((2g'-— ^)* + e}+a 2 cos {(4g-?>K + e} + » 



and find on substitution approximately 



a ±2 = — g* 2 a±i, 



a =a 3 (a 1 + a_i), 



a 1 

 Hence 



3B2 



f=LM^-^-W)-^ 



