21 



tion of the moving particle, and for q, we take the angle between the plane through 

 the particle and the centres and a fixed plane through the latter points, or, in 

 closer conformity -with the above general description, some continuous periodic func- 

 tion of this angle with period 2-. A limiting case of this problem is a Horded by 

 an electron rotating round a positive nucleus and subject to the effect of an addi- 

 tional homogeneous electric field, because this field may be considered as arising 

 from a second nucleus at infinite distance apart from the first. The motion in this 

 case will therefore be conditionally periodic and allow a separation of variables in 

 parabolic coordinates, if the nucleus is taken as focus for both sets of paraboloids 

 of revolution, and their axes are taken parallel to the direction of the electric force. 

 By applying the conditions (22) to this motion Epstein and Schwarzschild have, 

 as mentioned in the introduction, independent of each other, obtained an explana- 

 tion of the effect of an external electric field on the lines of the hydrogen spectrum, 

 which was found to be in convincing agreement with Stark's measurements. To 

 the results of these calculations we shall return in Part II. 



In the above way of representing the general theory we have followed the 

 same procedure as used by Epstein. By introducing the so called "angle-variables" 

 well known from the astronomical theory of perturbations, Schwarzschild has given 

 the theory a very elegant form in which the analogy with systems of one degree 

 of freedom presents itself in a somewhat different manner. The connection between 

 this treatment and that given above has been discussed in detail by Epstein.^) 



As mentioned above the conditions (22), first established from analogy with 

 systems of one degree of freedom, have subsequently been proved generally to be 

 mechanically invariant for any slow transformation for which the 

 system remains conditionally periodic. The proof of this invariance has 

 been given quite recently by Burgers^) by means of an interesting application of the 

 theory of contact-transformations based on Schwarzschild's introduction of angle 

 variables. We shall not enter here on these calculations but shall only consider some 

 points in connection with the problem of the mechanical transformability of the 

 stationary states which are of importance for the logical consistency of the general 

 theory and for the later applications. In § 2 we saw that in the proof of the mechanical 

 invariance of relation (10) for a periodic system of one degree of freedom, it was 

 essential that the comparative variation of the external conditions during the time of one 

 period could be made small. This may be regarded as an immediate consequence of the 

 nature of the fixation of the stationary states in the quantum theory. In fact the answer 

 to the question, whether a given state of a system is stationary, will not depend 

 only on the motion of the particles at a given moment or on the field of force in 

 the immediate neighbourhood of their instantaneous positions, but cannot be given 

 before the particles have passed through a complete cycle of states, and so to speak 



M P. Epstein, Ann. d. Phj-s. LI, p. 168 il916). See also Note on page 29 of the present paper. 

 2) J. M. BüBGERS, loc cit. Versl. Akad. Amsterdam, XXV, p. 1055 (1917). 



