47 



motion, we get Na- = x/,-, where Xi,...Xs aie llie numbers entering in lornuiia 

 (23). Comparing a given perturbed motion ot the system with some undislur])C(i 

 motion of which it may be regarded as a small variation, we get therefore from 

 (28), with neglect of small quantities proportional to the square of the intensity 

 of tlie external forces, 



UEdi=^xknk, (47) 



where the /'s are calculated with respect to a set of coordinates in which a separa- 

 tion can be olitained for the perturbed motion, and where oE is the dilTerence 

 between the total energy of the undisturbed motion and the energy which the 

 system would possess in its perturbed state, if the external forces vanished suddenly 

 at the moment under consideration, and which in the above calculations was 

 denoted by a^. Now the energy E of the undisturbed motion is determined com- 

 pletely by the value of / = IxiJk- If therefore the perturbed motion is all the 

 lime compared with a neighbouring undisturbed motion of given constant energy, 

 it follows directly from (47), that, with neglect of small quantities of the same order 

 as the square of the external forces, the integral on the left side, taken over an 

 approximate period of the perturbed motion, will remain unaltered during the per- 

 turbations through any time interval, however long. 



Before proceeding with the applications of the equations (46) which apply to 

 the case of a constant perturbing field, it will be necessary to consider the effect 

 of a slow and uniform establishment of the external field. Let us 

 assume that, within the interval < / < <!/ where ä denotes a quantity of the same 

 order as "/;., the intensity of the external field increases uniformly from zero to the 

 value corresponding to the potential Q. Since the variation in the perturbing field 

 during a single period will only be a small quantity of the same order as /.-, we 

 see in the first place that the secular variations of the constants a.^, ... Us, ßo, ■ ■ ■ ßs, 



with the same approximation as for a constant field, will be given by a set of equa- 



/ 

 tions of the same form as (46), with the only difference that '/' is replaced by ^ '/'. 



Moreover it may be shown that in these equations the quantity a^ may be con- 

 sidered as constant, just as in the equations which hold for a constant perturbing 

 field. In fact the total variation in «^ at any moment / will be equal to the total 

 work performed by the external forces since the beginning of the establishment 

 of the perturbing field, and will therefore be given by 



where the expression on the right side is obtained by partial integi-ation; but, since both 

 terms in this expression are of the same order of magnitude as /«j, we see that the 



7' 



