136 Prof. C. V. Raman and Mr. Ashutosh Dey on the 



frequency of the field, but is a submultiple of it. To fix our 

 ideas, we may assume the free vibration of the wire when it 

 divides up into r segments to have nearly the same period 

 as the field, that is T/r. The period of vibration of the 

 wire, as a whole, is therefore T. Experiment shows that the 

 forced vibration having the period T/r may be unstable, 

 giving place to a vibration with period T. To explain this 

 result, we may examine the effect, according to our equations, 

 of superposing a small vibration of period T upon the ordinary 

 forced vibration, if any, of period T/r. If by , cy 2 , &c, be 

 neglected, there is no component in the impressed force 

 having the period T, and the initial disturbance assumed 

 would die away in the ordinary course. It is not possible, 

 therefore, to obtain the resonance of submultiple frequency 

 with uniform fields of force. With non-uniform fields, the 

 additional terms hy , cy 2 , &c, have to be taken into account, 

 and it may readily be shown, on expanding the product 

 F(y ) f(t) in a series of sines, that there would be a term of 

 period T in the expansion which would, under certain 

 circumstances, be capable of magnifying the assumed dis- 

 turbance continually till it assumes a large amplitude. For 

 example, we may take r = 2, and the initial disturbance to 

 be, say, 



. 2irt 

 7 sin -7jpr. 



The product 



to/o.aicosl nji e 1 J 



would contain a term 



; • 2rrf ^ITt \ 



oa-^ry sin -= cos J — ^ e x h 



which, on being expanded, is seen to include a component 



^ 6a, 7 sin e 1 cos-~- . 



This is proportional to the assumed disturbance, has the 

 same period, and has a phase in advance of it by 90°. It 

 would therefore tend to magnify the assumed disturbance of 

 period T till the latter reaches a considerable amplitude. 

 An explanation of the phenomenon noticed is thus possible 

 for the case r = 2, in which no part is played by the com- 

 ponent of y having the same frequency as the field. For 

 the cases in which r=3 or 4, &c, we have to proceed to a 

 higher degree of approximation by taking into account, not 



