18 



While the first integral obviously must be taken over a period of the radial motion, 

 there might at first sight seem to be a difficulty in fixing the limits of integration 

 of g,. This disappears, however, if we notice that an integral of the type under 

 consideration will not be altered by a change of coordinates in which q is replaced 

 by some function of this variable. In fact, if instead of the angular distance of the 

 radius-vector we take for q.^ some continuous periodic function of this angle with 

 period 2-, every point in the plane of the orbit will correspond to one set of 

 coordinates only and the relation between p and q will be exactly of the same 

 type as for a periodic system of one degree of freedom for which the motion is of 

 oscillating type. It follows therefore that the integration in the second of the 

 conditions (16) has to be taken over a complete revolution of the radius-vector, 

 and that consequently this condition is equivalent with the simple condition that 

 the angular momentum of the particle round the centre of the field is equal to an 



entire multiplum of ~^. As pointed out by Ehrenfest, the conditions (16) are 



invariant for such special transformations of the system for which the central 

 symmetry is maintained. This follows immediately from the fact that the angular 

 momentum in transformations of this type remains invariant, and that the equa- 

 tions of motion for the radial coordinate as long as p, remains constant are the 

 same as for a system of one degree of freedom. On the basis of (16i. Sommerfeld 

 has, as mentioned in the introduction, obtained a brilliant explanation of the fine 

 structure of the lines in the hj-'drogen spectrum, due to the change of the mass 

 of the electron with its velocity. ^) To this theory we shall come back in Part 11. 

 As pointed out by Epstein-) and Schwarzschild') the central systems con- 

 sidered by Sommerfeld form a special case of a more general class of systems for 

 which conditions of the same type as (15) may be applied. These are the socalled 

 conditionally periodic systems, to which we are led if the equations of motion 

 are discussed by means of the Hamilton-Jacobi partial differential equation'). In 

 the expression for the total energy £■ as a function of the q's and the p's, let the 

 latter quantities be replaced by the partial differential coefficients of some function 



*) In this connection it may be remarked that conditions of the same tj-pe as il6) were proposed 

 independently by W. Wilson' I'Phil. Mag. XXIX p. 795 iI915i and XXXI p. 156 (191611. but by him applied 

 only to the simple Keplerian motion described by the electron in the hydrogen atom if the relativity 

 modifications are neglected. Due to the singular position of periodic systems in the quantum theory 

 of systems of several degrees of freedom this application, however, involves, as it will appear from the 

 following discussion, an ambiguit}- which deprives the result of an immediate phj'sical interpretation. 

 Conditions analogous to (16j have also been established by Plasck in his interesting theory of the 

 "physical structure of the phase space" of systems of several degrees of freedom (Verb. d. D. Phys. 

 Ges. XVII p. 407 and p. 438 (1915. Ann. d. Phys. L p. 385. 1916. This theory-, which has no direct 

 relation to the problem of line-spectra discussed in the present paper, rests upon a profound analysis 

 of the geometrical problem of dividing the multiple-dimensional phase space corresponding to a system 

 of several degrees of freedom into "cells" in a waj- analogous to the division of the phase surface of 

 a system of one degree of freedom bj' the curves given by (10). 



-) P. Epstein, loc. cit. 



^) K. Schwarzschild, loc. cit. 



"*) See £ inst. C. V. L. Charlier, Die Mechanik des Himmels. Bd. I. Abt. 2. 



