17 



which, in connection witli (1), have been found to give I'esults in convincing agree- 

 ment witli experimental results about line-spectra. Subsequently these conditions 

 have been proved by Ehrenfest and especially by Burgers') to be invariant for 

 slow mechanical transformations. 



To the generalisation under consideration we are naturally led, if we first consider 

 such systems for which the motions corresponding to the different degrees of freedom 

 are dynamically independent of each other. This occurs if the expression for the 

 total energy E in Hamiltons equations (4) for a system of s degrees of freedom 

 can be written as a sum E^ -\- ...-{- Eg, where Eu contains qi; and p^- only. An 

 illustration of a system of this kind is presented by a particle moving in a field 

 of force in which the force-components normal to three mutuallj' perpendicular 

 fixed planes are functions of the distances from these planes respectively. Since in 

 such a case the motion corresponding to each degree of freedom in general will 

 be periodic, just as for a system of one degree of freedom, we may obviously ex- 

 pect that the condition (10) is here replaced by a set of s conditions : 



h = J Pkdqk = nkh, {k = 1, . . . s) (15) 



where the integrals are taken over a complete period of the different q's respectively, 

 and where Hj, . . . Hs are entire numbers. It will be seen at once that these condi- 

 tions are invariant for any slow transformation of the system for which the in- 

 dependency of the motions corresponding to the different coordinates is maintained. 

 A more general class of systems for which a similar analogy with systems of 

 a single degree of freedom exists and where conditions of the same type as (15) 

 present themselves is obtained in the case where, although the motions corresponding 

 to the different degrees of freedom are not independent of each other, it is possible 

 nevertheless by a suitable choice of coordinates to express each of the momenta p^ 

 as a function of qk only. A simple system of this kind consists of a particle moving 

 in a plane orbit in a central field of force. Taking the length of the radius-vector 

 from the centre of the field to the particle as 7, , and the angular distance of this 

 radius-vector from a fixed line in the plane of the orbit as q^, we get at once 

 from (4), since E does not contain q^, the well known result that during the motion 

 the angular momentum p^ is constant and that the radial motion, given by the 

 variations of p^ and q^ with the time, will be exactly the same as for a system 

 of one degree of freedom. In his fundamental application of the quantum theory 

 to the spectrum of a non-periodic system Sommerfeld assume^ therefore that 

 the stationary states of the above system are given by two conditions of the form : 



^1 = ]Pidqi = "i/î, h = \P2dq2 = n.,h. (16) 



J) J. M. Burgers, Vers). Altad. Amsterdam, XXV, pp. 849, 918, 1055 (1917), Ann. d. Phys. LII. p. 195 

 (1917), Phil. Mag. XXXIII, p. 514 (1917). 



D. K. D. Vidensk. Selsk. Skr., n.iturviilensk. og nwthem. Afd., S. Række, IV. 1. 3 



