30 



over that, if the system allows of a separation of variables in an infinite continuous 

 multitude of sets of coordinates, the total energy will be the same for all motions 

 corresponding to the same values of the I's, independent of the special set of 

 coordinates used to calculate these quantities. As mentioned above and as we have 

 already shown in the case of purely periodic systems by means of (8), the total 

 energy is therefore also in cases of degeneration completely determined by the 

 conditions (22). 



Consider now a transition between two stationary states determined by (22) 

 by putting rt,. = n'^ and n^. = n^' respectively, and let us assume that n[, ... n[, 

 n", ... n" are large numbers, and that the differences n',. — nl are small compared 

 with these numbers. Since the motions of the system in these states will differ 

 relatively very little from each other we may calculate the difference of the energj- 

 by means of (29), and we get therefore, by means of (1), for the frequency of the 

 radiation corresponding to the transition between the two states 



Ä s 



. = !(£'-£") = l-^^,(7;_/p =^co,(n,-nl), (30) 



1 1 



which is seen to be a direct generalisation of the expression (13) in § 2. 



Now, in complete analogy to what is the case for periodic systems of one degree 

 of freedom, it is proved in the analytical theory of the motion of conditionally 

 periodic systems mentioned above that for the latter systems the coordinates 

 qi, ■■ ■ qs, and consequently also the displacements of the particles in any given 

 direction, may be expressed as a function of the time by an s-double infinite 

 FouBiER series of the form: 



of 9,.. From (5*) equation (28) follows at once, and it will further be seen that by introducing (5*1 in 

 (3*) we get the result that ever}' of the q's, and consequently also any one-valued function of the qs, 

 can be represented by an expression of the tj'pe (31). 



In this connection it may be mentioned that the method of Schwarzschild of fixing the stationary 

 states of a conditional!}' periodic system, mentioned on page 21, consists in seeking for a given system 

 a set of canonically conjugated variables Qi, . . . Q^, Pi, . . . P^ in such a way that the positional 

 coordinates of the system gi, . . . q^ and their conjugated momenta p^, . . . p^, when considered as func- 

 tions of the Q's and P's, are periodic in everj' of the Qs with period 2-, while the energy of the 

 system depends only on the P's. In analog}' with the condition which fixes the angular momentum in 

 Sommerfeld's theory of central systems Schwarzschild next puts every of the P's equal to an entire 



multiplum of ~. In contrast to the theory of stationary states of conditionally periodic systems based 



on the possibility of separation of variables and the fixation of the Ps by (22), this method does not 

 lead to an absolute fixation of the stationary states, because, as pointed out by Schwarzschild himself, 

 the above definition of the P's leaves an arbitrary constant undetermined in every of these quantities. 

 In many cases, however, these constants may be simply determined from considerations of mechanical 

 transformability of the stationary states, and as pointed out by Burgers (loc. cit. Versl. Akad. Amsterdam 

 XXV p. 1055 (1917) Schwarzschild's method possesses on the other hand the essential advantage of 

 being applicable to certain classes of systems in which the displacements of the particles mav be 

 represented by trigonometric series of the type (31), but for which the equations of motion cannot be 

 solved by separation of variables in any fi.xed set of coordinates. An interesting application of this to 

 the spectrum of rotating molecules, given by Burgers, will be mentioned in Part IV. 



