13 



297 



^ d^, 



so thai the e((uation.s (27) may be written in tlie simple form 



27:0^1 = \ (1 + £ cos ç^-) dé = 4' ~\' ^ sin 4' -\- n , 



\ dô 



-h £ COS 4< 



n / N ,1 , -1 £ -|- I s' sin -[- COS C^- I 



(30) 



According to the defniition of angle variables, an arbitrary constant may be added 

 to the values of w^ and w.^. In the present case the additional term - is written 

 on the right side of the first of the above equations in order to obtain a (inal 

 formula which is as simple as possible. 



In order to obtain now the coefficients Cri.Tj in the expansion (21), we might 

 proceed by directly applying (1-i), but the calculation can be made shorter by 

 observing that the mechanical system under consideration possesses symmetry round 

 the nucleus and that as a consequence of this all coefficients Cr,,T2 in (21) will be 

 equal to zero except those for which r.^ = 1^). This means that the expression 

 re'V'e^-"""» will be a function of only and may be expanded in a simple Fourier 

 series. In fad from (29) and from the second of the equations (30) we have 



reiçe^^iU.. ^ .7^1 + ^ cos ^).^r ' + ^^^^ ^ + ^os 4 \ 



^ ' ^'^ 1 + £ cos ^ (31) 



= X r^s i e' sin (/' -\- cos <p) , I 



and this is, according to the first of the equations (30), a function ol only. Now 

 the coefficients A. in the series 



e + is' sin ^ + cos ^ = l'^^e^ÄiTa., ^32) 



are easily obtained by evaluating, according to Fourier's theorem, the single definite 

 integral 



-\- is' sin (/> J- cos ô) e 2»tiTu;, ^y^,^ 



which is simply changed into an integral over d' because we have from the first 

 of the equations (30) 



dw^ = (1 + ecos^) ^ , 



so that 



^" » + is' sin -f cos (1 ^ s cos ç/,)e-'"'-ç'- '-'-"in c' rfç'.. (33) 



M See N. Bohb, Ioc. cit. Part I, p. 3.'!. 



