22 



have got to know the entire field of force of influence on the motion. If thus, in 

 the case of a periodic system of one degree of freedom, the field of force is varied 

 by a given amount, and if its comparative variation within the time of a single 

 period was not small, the particle would obviously have no means to get to know 

 the nature of the variation of the field and to adjust its stationary motion to it, 

 before the new field was already established. For exactly the same reasons it is a 

 necessary condition for the mechanical invariance of the stationary states of a 

 conditionally periodic system, that the alteration of the external conditions during 

 an interval in which the system has passed approximately through all possible 

 configurations within the above mentioned s-dimensional extension in the coordinate- 

 space can be made as small as we like. This condition forms therefore also 

 an essential point in Burgers' proof of the invariance of the conditions (22) for 

 mechanical transformations. Due to this we meet with a characteristic diffi- 

 culty when during the transformation of the system we pass one of the cases of 

 degeneration mentioned above, where, for every set of values for the as, the orbit 

 will not cover the s-dimensional extension everywhere dense, but will be confined 

 to an extension of less dimensions. It is clear that, when by a slow transforma- 

 tion of a conditionally periodic system we approach a degenerate system of this 

 kind, the time-interval which the orbit takes to pass close to any possible con- 

 figuration will tend to be very long and will become infinite when the degenerate 

 system is reached. As a consequence of this the conditions (22) will gener- 

 ally not remain mechanically invariant when we pass a degenerate 

 system, what has intimate connection with theabove mentioned ambiguity in the 

 determination of the stationary states of such systems by means of (22). 



A typical case of a degenerate system, which may serve as an illustration of 

 this point, is formed by a system of several degrees of freedom for which every 

 motion is simply periodic, independent of the initial conditions. In this case, which 

 is of great importance in the physical applications, we have from (5) and (21), for 

 any set of coordinates in which a separation of variables is possible, 



= \(Pi9i 



+ ...+Psqs)dt=x,I,+ ...+xJs, (23) 



where the integration is extended over one period of the motion, and where 

 x^, . . . xs are a set of positive entire numbers without a common divisor. Now we 

 shall expect that every motion, for which it is possible to find a set of coordinates in 

 which it satisfies (22), will be stationary. For any such motion we get from (23) 



I ^ (x,n,+ ...+x,n,)h = nh, (24) 



where n is a whole number which may take all positive values if, as in the ap- 

 plications mentioned below, at least one of the x's is equal to one. Inverselj', if 

 the system under consideration allows of separation of variables in an infinite con- 

 tinuous multitude of sets of coordinates, we must conclude that generally every 



