24 



field, which is of the same kind as the relation, expressed by (8) and (10), between 

 the energy and frequency of a periodic system of one degree of freedom. As it will 

 be shown in Part II in connection with the physical applications, this condition 

 is just secured if the stationary states in the presence of the field are determined 

 by the conditions (22), and it will be seen that these considerations offer a means 

 of fixing the stationary states of a perturbed periodic system also in cases where 

 no separation of variables can be obtained. 



In consequence of the singular position of the degenerate systems in the 

 general theory of stationary states of conditionally periodic systems, we obtain 

 a means of connecting mechanically two different stationary states 

 of a given system through a continuous series of stationary states without 

 passing through systems in which the forces are very small and the energies in all 

 the stationary states tend to coincide (comp, page 9). In fact, if we consider a given 

 conditionally periodic system which can be transformed in a continuous way into 

 a system for which every orbit is periodic and for which every state satisfying 

 (24) will also satisfy (22) for a suitable choice of coordinates, it is clear in the first 

 place that it is possible to pass in a mechanical way through a continuous series 

 of stationary states from a state corresponding to a given set of values of the n's 

 in (22) to any other such state for which •/inj^-\- ...-{- xgUg possesses the same value. If, 

 moreover, there exists a second periodic system of the same character to which the 

 first periodic system can be transformed continuous!}', but for which the set of x's 

 is different, it will be possible in general by a suitable cj'clic transformation to 

 pass in a mechanical way between any two stationary states of the given condi- 

 tionally periodic system satisfying (22). 



To obtain an example of sucli a cyclic transformation let us talte the system consisting 

 of an electron whieli moves round a fixed positive nucleus exerting an attraction varying as 

 tlie inverse square of the distance. If we neglect the small relativity corrections, every orbit 

 will be periodic independent of the initial conditions and the system will allow of separa- 

 tion of variables in polar coordinates as well as in any set of elliptical coordinates, of the 

 kind mentioned on page 21, if the nucleus is taken as one of the foci. It is simply seen that 

 any orbit which satisfies (24) for a value of n > 1, will also satisfy (22) for a suitable choice 

 of elliptical coordinates. By imagining another nucleus of infinite small charge placed at the 

 other focus, the orbit may further be transformed into another which satisfies (24) for the 

 same value of n, but which may have any given value for the eccentricity. Consider now 

 a state of the system satisfying (24), and let us assume that by the above means the orbit 

 is originally so adjusted that in plane polar coordinates it will correspond to n^ = m and 

 «2 = n — min (16). Let then the system undergo a slow continuous transformation during which 

 the field of force acting on the electron remains central, but by which the law of attraction 

 is slowly varied until the force is directly proportional to the distance apart. In the final 

 state, as well as in the original state, the orbit of the electron will be closed, but during the 

 transformation the orbit will not be closed, and the ratio between the mean period of 

 revolution and the period of the radial motion, which in the original motion was equal to 

 one, will during the transformation increase continuously until in the final state it is equal to 

 two. This means that, using polar coordinates, the values of x^ and Zg in (22) which for the 

 first state are equal to z^ = Zg = 1, will be for the second state Zj = 2 and Zg = 1. Since 

 during the transformation n-^ and n^ will keep their values, we get therefore in the final state 

 I = h {2m + in — m)) — hin + m). Now in the latter state, the system allows a separation of 



