29 



313 



m, = io,t + â,, (o, = = (A- = 1,2, 3) (8t)| 



where we have denoted by E" the energy of the undisturbed system expressed as 

 a function of the /'s, which is given by the expression (76) for Since this ex- 

 pression contains /.^ and /., only in the combination + /g, cu^ will be equal to co^, 

 which means that the system is degenerate as it was already mentioned in § 2. If 

 we assume, however, that F is no longer equal to zero the motion of the system 

 will be perturbed; the coordinates x, ij, z of the electron may still be expressed as 

 a function of the /'s and the iv's by means of (77), but the /'s will no more be 

 constant during the motion and the w's will no more be linear functions of the 

 time. The rates of variation with the time of the Fs and ly's will according to 

 Jacobi's fundamental theorem, mentioned in § 1, be represented by a set of canoni- 

 cal equations 



"f'^lf. (t= 1,2,3) (81) 



at owk (it ol), 



where E is the total energy of the perturbed system expressed as a function of the 

 /'s and w's. We may write E in the form 



E = + 



where is the energy which the system would possess if the perturbing forces 

 vanished suddenly and which, as mentioned, depends on the /'s only, being given 

 by the expression (76) for «j, while E^ is the so called "perturbing potential", i. e. 

 that part of the potential energy of the system which is due to the perturbing 

 force, and which corresponds to the term Fer cos in (73). By means of (77) we 

 tind for E\ expressed as a function of the /'s and w's, 



E' ^ Fez ^ Fe2:D.cos2n(^lWi-\- w^), (82) 



where the quantities Ü. with neglect of small quantities of the order / are given 

 by (78). 



Owing to the fact that the trigonometric series for does not contain a term 

 which is independent of the w's, we may simply proceed in the calculation of the 

 perturbations in the following way'), by putting 



/, = + //. , w, = ivl -f- , (A- = 1, 2, 3) (83) 



where I'i wl represent the solutions of the equations (81) for F = 0, and where 

 /J. and wl contain only small quantities proportional to F and to higher powers of 

 F. For ll and wl we have 



') It maj' be observed tliat. hy applying to the quantities h and ij'a a so called infinitesimal 

 contact transformation, tiie results of the following considerations contained in the formulae (8.")i and 

 (86) might have been deduced in a way which, from an analytical point of view, is more elegant. Com- 

 pare J. M Burgers. Het atoommodel van Rutherford-Bohr (Haarlem, 1918), where a treatment of this 

 kind has been used in the discussion of a number of problems concerning perturbed atomic motions 



