Forced Oscillations of a New Type. 515 



complete period of the variation of the spring, and that this 

 is just sufficient to balance the loss by dissipation during the 

 same time. This again would only be possible, if, assuming 

 that an oscillation of any particular frequency was maintained, 

 the product of the imposed variation of spring into the 

 displacement, which may be regarded as the impressed part 

 of the force tending to restore the system to its position of 

 equilibrium, had a periodic component of the same frequency 

 as that of the oscillation proposed to be maintained, and this 

 should also be of a suitable magnitude and phase. This is 

 evident from the general principles of resonance, since 

 components of the restoring force having any other frequency 

 would obviously be comparatively ineffective in maintaining 

 the vibration. 



The condition referred to above, it may readily be shown, 

 is satisfied in the case of double frequency. Thus, putting 

 N and Nj equal to nj2iT and n x \2ir respectively, and assuming 

 that the variation of spring is represented by — 2an 1 2 smnt 

 and the displacement <f> at any instant is equal to 

 Ccos (/tf/2 + ex), it is seen that the product of the variation 

 of spring and the displacement has a periodic component 



— a.Cn 1 2 sm(nt/2 — € l ) which has the same frequency as the 

 oscillation proposed to be maintained. It is readily shown 

 that the work done by this part of the force acting on the 

 system in a time equal to the period of the variation of spring- 

 is aj4zrii 2 nC 2 t cos 2e, and that this is equal to the loss by 

 dissipation in the same time if /cn = an 1 cos 2e x , 2ku 1 being the 

 dissipation coefficient. This result, it will be seen, is precisely 

 equivalent to that obtained in a different manner by Lord 

 Rayleigh (Scientific Works, vol. ii. p. 192). 



Taking now the general case discussed by Stephenson, we 

 may first assume that a motion of frequency rN/2 is main- 

 tained by the variation of spring — 2an{ 2 sin nt. If cf> the 

 displacement at any instant is put equal to C r cos (rnt/2 + e r ), 

 where C r is a constant, it is readily seen that the product of 

 the variation of spring and the displacement which may be 

 termed the impressed part of the restoring force is 



— 2aG r n! 2 s'mnt cos (rnt/2 + € r ), and this has obviously no 

 component of frequency r~N/2 and cannot therefore maintain 

 an oscillation having that frequency. The expression for the 

 work done by it in a time equal to the period of the variable 

 spring will be found equal to zero. The inference is that, if 

 the oscillation is actually maintained, some important term 

 in the expression for the displacement at any instant has 

 been omitted by us to be taken into account. This we now 

 proceed to find directly from physical considerations. 



