62 



lines by means of (1), allow of the same degree of approximation as the fixation 

 of the energy in -the stationary states of the original perturbed periodic system. 

 If// is larger than l//, however, the stationary states will in general not be as well 

 defined as for the original system, and from relation (1) we may therefore expect 

 that the components will be diffuse, although, as long as « remains small com- 

 pared witli unity, the width of the components will remain small compared with the 

 displacements from their positions in the presence of the first external field alone. 

 Only when ii becomes of the same order as unity, the simultaneous effect of both 

 perturbing fields may be expected to consist in a diffusion of the lines of the un- 

 disturbed periodic system; unless of course the secular perturbations due to the 

 simultaneous presence of both fields are still of conditionally periodic type, as it 

 may happen in special problems. In certain cases the second external field will not 

 only give rise to small displacements of the original components but also to the 

 appearance of new components of small intensities proportional to p.'^. This occurs 

 if for the original perturbed periodic sj'stem, due to some pecularity of the motion, 

 some of the coefficients Cr, tj, . . . tj_j in the expressions (65) for the displacements 

 of the particles as a sum of harmonic vibrations, corresponding to certain com- 

 binations of the numbers r, tj, ... ts i, are equal to zero, while in the presence 

 of the second external field these coefficients are small quantities proportional to fi 

 (Compare Part I, page 34).^) In the preceding considerations it has been assumed 

 that the perturbed system in the presence of the first external field is non- 

 degenerate. In case, however, this system is degenerate, it is obviously impos- 

 sible, by a direct application of the principle of the mechanical transformability of the 

 stationary states, to determine the alteration in the energy in the stationary states 

 of the system, which will be due to the presence of a second external field small 

 compared with the first field; because, as mentioned, the stationary states of the 

 system, in the presence of this field only, will be determined by a number of condi- 

 tions which is less than the number s of degrees of freedom, and that consequently 

 the cycles of shapes and positions, which the orbit will pass through in these states, 

 will not be completely determined. For the calculation of the energy in the stationay 

 states it will therefore be necessary to consider the secular perturbing effect of the 

 second external field on these cycles. In the special case where the secular jjerturba- 

 tions due to the first field are simply periodic, it will in this way be seen that the 

 problem of the fixation of the stationary states in the presence of the second ex- 

 ternal field, by means of the method exposed in this section, may be reduced to 

 the problem of the fixation of the stationary states of a system of s — 2 degrees of 

 freedom. If, as in the applications considered below, s is equal to 3, this problem 

 allows of a general solution, and we must therefore expect that in this case the 



') As regards the degree of definition with which the positions of the new components will be 

 determined, we must be prepared to find that tlie frequencies of these components are only defined 

 with neglect of small quantities proportional to X/j.. Compare the detailed discussion of the example 

 in § 5 on page 97. 



