20 



sj'slem at a given moment, when the as are known. For special systems it may 

 occur that the orbit will not cover the above mentioned s-dimensional extension 

 everywhere dense, but will, for all values of the a's, be confined to an extension 

 of less dimensions. Such a case we will refer to in the following as a case of 

 "degeneration". 



Since for a conditionally periodic system which allows of separation in the 

 variables q^, . . . q^ the p's are functions of the corresponding q's only, we may, 

 just as in the case of independent degrees of freedom or in the case of quasi- 

 periodic motion in a central field, form a set of expressions of the type 



h =]Pk {qk, «1, • • • as) dqu, {k = \,...s) (21) 



where the integration is taken over a complete oscillation of qu. As, in general, the 

 orbit will cover everywhere dense an s-dimensional extension limited in the charac- 

 teristic way mentioned above, it follows that, except in cases of degeneration, a 

 separation of variables will not be possible for two different sets of coordinates 

 q^, . ■ ■ qs and q[, . . . q's, unless q^ = f^^ {q[), . ■ ■ qs= fs{q's), and since a change of 

 coordinates of this type will not affect the values of the expressions (21), it will be 

 seen that the values of the /'s are completely determined for a given motion of the 



system. By putting j, /;. 1 .-> roo^ 



h = nkh, (k = 1, . . . s) (22) 



where /jj, . . . n« are positive entire numbers, we obtain therefore a set of condi- 

 tions which form a natural generalisation of condition (10) holding 

 for a system of one degree of freedom. 



Since the /'s, as given by (21), depend on the constants «j, . . • a« only and 

 not on the /9's, the as may, in general, inversely be determined from the values 

 of the I's. The character of the motion will therefore, in general, be completely 

 determined by the conditions (22), and especially the value for the total energy, which 

 according to (17) is equal to «j, will be fixed by them. In the cases of degenera- 

 tion referred to above, however, the conditions (22) involve an ambiguity, since in 

 general for such systems there will exist an infinite number of different sels of 

 coordinates which allow of a sepai-ation of variables, and which will lead to different 

 motions in the stationary states, when these conditions are applied. As we shall 

 see below, this ambiguity will not influence the fixation of the total energy in the 

 stationary states, which is the essential factor in the theory of spectra based on (1) 

 and in the applications of the quantum theory to statistical problems. 



A well known characteristic example of a conditionally periodic system is 

 afforded by a particle moving under the influence of the attractions from two 

 fixed centres varying as the inverse squares of the distances apart, if the. relativity 

 modifications are neglected. As shown by Jacobi this problem can be solved by a 

 separation of variables if so called elliptical coordinates are used, i. e. if for q^ and 

 q^ we take two parameters characterising respectively an ellipsoid and a hyperboloid 

 of revolution with the centres as foci and passing through the instantaneous posi- 



