﻿con 



in 



and its Relation to the Quantum Theory. 859 



nucleus might tend asymptotically to the value infinity *. 

 Even if this difficulty were non-existent, tiie introduction of 



/IB 



an extra frequency of vibration a /- — by setting Dj^fcO 



would lead to more complicated motions, an additional 

 quantum number, and a negative energy content of smaller 

 absolute value, none of which are characteristic of the 

 normal state of the atom. 



The values of p^\ and G { are determined by equating to 

 zero the coefficient of iv S/ ' 2 in equation (13). We thus obtain 



— ^ 2 -f 5 1= =3[^ 01 cos B£ + *01 cos 2Bt cos Bt] = 3p 0l cos Bt 



+ -015[cosB£ + cos3B£], 

 whence for a periodic solution 



p 01 = — -005, s 1 =Ecos (Bt-€ 2 ) -'001875 cos 3B*. 



The constant E may be set equal to zero, since we already 

 have an arbitrary term in cos Bt (viz., the term S = ur /2 cos Bt 

 .-responding to the unperturbed motion). The first terms 

 the expansion of p, C, and 5 in terms of w are therefore 



C = '875000--973125w, 



p=[- -005000 + -010000 cos 2Bt]w, 



s=(- -001875 cos 3B£> 3/2 (14) 



Determination of Higher Order Terms in w. 



Making use oE the relations given in (14) we can equate 

 to zero the coefficients of w 2 and iv 5 / 2 in equations (12) and 

 (13) respectively, and so determine p 2 , C 2 , and s 2 . Knowing 

 p 2 and s 2 we can determine third-order terms, and thus 

 continue the process to any desired degree in w. The 

 calculation of the first-order terms given above in detail is 

 typical of. the method used in computing the terms of higher 

 order. With the aid of the trigonometric reduction formulas 

 which express cos' 1 B^ as a sum of linear trigonometric 

 terms, the type forms obtained by equating to zero the 



* Cases, however, are sometimes found in dynamical theory where 

 the Poisson and secular terms (in which the time appears explicitly) 

 combine in such a way as to yield a conditionally periodic motion. Bohr 

 has shown that when the perturbing potential has axial symmetry, the 

 motion may be regarded as conditionally periodic if we consider only 

 first-order terms in the perturbing field (Quantum Theory of Line 

 Spectra, pp. 53-6), but this approximation is not sufficient in the present 

 problem owing to the large amount of mutual action between electrons 



3 K 2 



