31 



f = 2'Cr„..,r, COs2n-(.(î-i«, + . . . T,(üs)i + CT„...r,}, (31) 



where the summation is to be extended over all positive and negative entire values 

 of the t's, and where the oj's are the above mentioned mean frequencies of oscilla- 

 tion for the ditferent q's. The constants Cr„...T^ depend only on the a's in the equa- 

 tions (18) or, what is the same, on the /'s, while the constants Cr„...T depend on 

 the a's as well as on the ß's. In general the quantities rjWj-f ... zsWs will be 

 different for anj' two different sels of values for the r's, and in the course of time 

 the orbit will cover everywhere dense a certain s-dimensional extension. In a 

 case of degeneration, however, where the orbit will be confined to an extension of 

 less dimensions, there will exist for all values of the a's one or more relations of the 

 type nijCuj -)- ... msWs = where the m's are entire numbers and by the introduc- 

 tion of which the expression (31) can be reduced to a Fourier series which is 

 less than s-double infiinite. Thus in the special case of a system of which every 



orbit is periodic we have —^^... = — = oj, where the x's are the numbers 



which enter in equation (23), and the Fourier series for the displacements in 

 the different directions will in this case consist only of terms of the simple form 

 Ct cos27: {rcut -{- Ct}, just as for a system of one degree of freedom. 



On the ordinary theory of radialion, we should expect from (31) that the 

 spectrum emitted by the system in a given state would consist of an s-double in- 

 finite series of lines of frequencies equal to riö>i+... -{- Tscos. In general, this 

 spectrum would be completely different from that given by (26). This follows al- 

 ready from the fact that the cu's will depend on the values for the constants a^, ... as 

 and will vary in a continuous way for the continuous multitude of mechanically 

 possible states corresponding to different sets of values for these constants. Thus 

 in general the cu's will be quite different for two different stationary states 

 corresponding to different sets of n's in (22), and we cannot expect any close rela- 

 tion between the spectrum calculated on the quantum theory and that to be ex- 

 pected on the ordinary theory of mechanics and electrodynamics. In the limit, 

 however, where the n's in (22). are large numbers, the ratio between the oj's for 

 two stationary states, corresponding to iij^ = n'^ and n^. = n". respectively, will tend 

 to unity if the differences n^ — - n^ are small compared with the n's, and as seen 

 from (30) the spectrum calculated by (1) and (22) will in this limit just tend to 

 coincide with that to be expected on the ordinary theory of radiation from the 

 motion of the system. 



As far as the frequencies are concerned, we thus see that for conditionally 

 periodic systems there exists a connection between the quantum theory and the 

 ordinary theory of radiation of exactly the same character as that shown in § 2 

 to exist in the simple case of periodic systems of one degree of freedom. Now on 

 ordinary electrodynamics the coefficients Cr,..., t^ in the expression (31) for the 

 displacements of the particles in the different directions would in the well known 

 way determine the intensity and polarisation of the emitted radiation of the 

 corresponding frequency rj^i -f . . . zsOJs. As for systems of one degree of freedom 



