Forced Oscillations of a New Type. 5L7 



It is thus seen that the 2nd term in the expression for the 

 displacement is from a physical point of view of very great 

 importance, though its amplitude may be small compared 

 with that of the 1st term which is much the larger part of 

 the oscillation actually maintained. It is not developed or 

 brought out in the final expression for the magnified or 

 maintained motion as given by Stephenson, and the impor- 

 tance of: the part played by it in magnifying or maintaining 

 the nmtion is therefore not made evident in his paper. 



It seems well to consider a few nuinerical examples. If 

 the frequency of the imposed variation of " spring " were 

 60 per second, the oscillations of the system would be main- 

 tained (under suitable circumstances) if the frequency of its 

 free oscillations were nearly equal to 30 or 60 or 90 or 120 

 or 150 and so on, the degree of approximation to equality 

 necessary increasing as we proceed up the series. The fre- 

 quency of the maintained oscillation would be exactly 30 or 

 60 or 90 and so on. But in the case of the oscillation of 

 frequency 60, the motion would be very approximately repre- 

 sented by a periodic term of that frequency plus a small 

 constant. In the case of the 3rd type in the series, the 

 maintained oscillation would be represented by a, periodic 

 term of frequency 90 plus a small term of frequency 30. 

 In the 4th case, the motion would be represented by a 

 periodic term of frequency 120 plus a small term of fre- 

 quency 60 and so on, the 2nd term in each type being less in 

 frequency than the 1st term by 60, which is the frequency of 

 the variation of spring ; in the case of the 2nd type in which 

 the frequency of the first term is itself 60, the 2nd term 

 naturally assumes the form of a constant as stated above. 

 The significance of this is that an oscillation when inaintained 

 by a variation of spring of the same frequency has for its 

 mean point, not the equilibrium position of the system, but 

 one slightly displaced to one side thereof. This result is no 

 doubt somewhat paradoxical, but there is nothiug absurd in 

 it, inasmuch as we are here dealing with motion under vari- 

 able spring in the presence of dissipative forces. The 

 restoring forces at the points of maximum displacement on 

 either side of the equilibrium position may be equal in 

 magnitude and opposite in sign despite the fact that the§e 

 displacements from the equilibrium position are themselves 

 slightly different, and a steady oscillation about a displaced 

 mean point is therefore possible. It is readily seen that in 

 this case it is the slight displacement of the mean point of 

 the normal oscillation that enables a surplus of energy to 



Phil. Mag. S. 6. Yol. 24. No. 142. Oct. 1912. 2 M 



