CONTEMPORARY ADVANCES IN PHYSICS 697 



The vibrations are described by the following formula (I have put 

 Ai = Aj = 1 and Bi = Bj = 0, which simplifies without injury to the 

 generality of the result) : 



y = yi + yi = fA^) cos i-KVit + /j(x) cos iirvit, (202) 



which is easily transformed thus: 



y = C cos {lirvit — a), (203) 



in which 



C = fi' + Jr + Vifi COS 2ir{vi - v,)t, (204) 



and a = a constant not important for our purpose. 



Here we have a vibration in which the amplitude at any fixed point 

 varies with time; the square of the amplitude is the sum of a constant 

 term and a sine-function of time, and the frequency of the sine- 

 function is the difference between the frequencies of the two coexisting 

 modes of vibration. 



Identifying the square of the amplitude with the density of electric 

 charge, we see that this charge-density varies at each point with the 

 frequency {vi — Vj). We might therefore expect radiation of this 

 frequency. 



Now the vibration-frequencies Vi and Vj corresponding to the modes 

 of vibration, that is to the Stationary States i and j having energy- 

 values Ei and Ej, are respectively Eijh and Ejlh. 



If therefore — to speak in a vague but suggestive fashion — the linear 

 oscillator were simultaneously in two Stationary States, their energy- 

 values being Ei and Ej, then the square of the amplitude of the oscilla- 

 tions of ^ would be fluctuating at each point of the "imaginary 

 string" with the frequency {Ei — Ej)/h; and if this squared amplitude 

 were to be identified with charge-density, then the system might be 

 expected to emit radiation of the frequency {Ei — Ej)lh. 



Transition between two states would then signify coexistence of the 

 two states .^^ 



We proceed a step further in the development of this idea, by forming 

 the following integral: 



xC-dx = I xf^dx + I xfj'dx 



00 •-' - 00 1/ — 00 



r ^00 ^ (205) 



+ pj xf,J)dx cos27r(.i - p,)t. 



This integral represents the electric moment of the supposed distribu- 



" I should again recall that in the picture we have, not two distinct coexisting 

 modes of vibration of the same elastic string; but the fundamental (and solitary) 

 modes of vibration of two distinct elastic strings. 



