Lord Rayleigh on Maintained Vibrations. 233 



in which e, being equal to tan -1 (B^A^, is also definite. On 

 the other hand, as is evident at once from the linearity of the 

 original equation, there is nothing to limit the amplitude of 

 vibration. 



These characteristics are preserved however far it may be 

 necessary to pursue the approximation. If A 2m+ i, B 2JW+1 , 

 may be neglected, the first m pairs of equations determine the 

 ratios of all the coefficients, leaving the absolute magnitude 

 open ; and they provide further an equation connecting p 

 and a, by which the pitch is determined. 



For the second approximation the second pair of equations 

 gives 



A — " B i t> — * A i 



whence 



0= P sin (pt + e ) + -^— 2 cos (3p* + e)j . (9) 



yp — n 6 



and from the first pair 



tane={n 2 -/ +;? ^ 2 }^( a + «p), . (10) 

 while p is determined by 



(n'-^)»- (M ,_f V) , =,»-«y. . . . (11) 



Returning to the first approximation, we see from (8) that 

 the solution is only possible under the condition that « > tcp. 

 It a — fcp, then p = n ; i.e. the imposed variation in the " spring ' ' 

 must be exactly twice as quick as the natural vibration of the 

 body would be in the absence of friction. From (7) it appears 

 that in this case e = 0, which indicates that the spring is a 

 minimum one eighth of a period after the body has passed its 

 position of equilibrium, and a maximum one eighth of a period 

 before such passage. Under these circumstances the greatest 

 possible amount of energy is communicated to the system ; 

 and in the case contemplated it is just sufficient to balance the 

 loss by dissipation, the adjustment being evidently indepen- 

 dent of the amplitude. 



If a < tcp, sufficient energy cannot pass to maintain the 

 motion, whatever may be the phase-relation; but if a > tcp, 

 the equality between energy supplied and energy dissipated 

 may be attained by such an alteration of phase as shall dimi- 

 nish the former quantity to the required amount. The altera- 

 tion of phase may for this purpose be indifferently in either 

 direction; but if e be positive, we must have 



p 2 = n 2 — */ {a 2 — /e 2 p' 2 j- ; 



