514 Mr. C. V. Raman on the Maintenance of 



vibrating bodies is the behaviour of a system when subject 

 to periodic forces which tend merely to alter its " spring " or 

 restitutional coefficient, and do not directly tend to displace 

 the system from its position of equilibrium. In discussing 

 this problem, it is assumed that this periodic change in 

 "spring" is of definite frequency and is imposed from 

 outside the system. One principal point for inquiry is the 

 relation that must exist between this frequency and that of 

 the free oscillations of the system in order that the latter 

 may be permanently maintained in vibration in the presence 

 of dissipative forces. Lord Rayleigh has investigated the 

 particular case in which the frequency of the imposed force 

 is double that of the oscillations of the system, and has shown 

 that under suitable circumstances the oscillation may be 

 maintained. The question arises whether there do not exist 

 other frequency-ratios which would permit of the maintenance 

 of the oscillations of the system. Mr. Andrew Stephenson 

 attacks this problem by pushing to a higher degree of 

 approximation the analytical method employed by Hill in 

 discussing certain problems in the Lunar Theory, and by 

 Lord Rayleigh in working out the particular case of double 

 frequency. His analysis (published in the Quarterly Journal 

 of Pure and Applied Mathematics for June 1906) leads to 

 the result that the oscillations of the system may be 

 magnified or maintained under suitable circumstances, if 

 the frequency N of the imposed variation of spring stands to 

 the frequency N 2 of the oscillation in the relation 2 :r where 

 r is any positive integer. When r is unity, we evidently 

 have the case of double frequency referred to above. 



From an acoustical point of view, the possibility of 

 maintenance is that which possesses the greatest interest. 

 Mr. Stephenson's treatment is purely mathematical, and his 

 paper does not explain in terms of ordinary physical ideas 

 why it is that the oscillations should admit of being main- 

 tained with the specified frequency-relations, nor does he 

 offer any experimental evidence in support of such a pro- 

 position. It is necessary therefore to discuss the problem 

 in its physical bearings before any results of experiment can 

 be rendered intelligible. 



If the oscillations of the system are to be maintained in 

 the presence of dissipative forces, a continued supply of 

 energy from without into the system is clearly necessary, and 

 we have therefore to show that with the assumed relation 

 between the frequency of the imposed variation of spring 

 and that of the oscillations of the system, it is possible for a 

 finite amount of energy to pass into the system during every 



