Theory of Spectrum Emission. 59 



Of these only the circular orbits (?i 3 =rc 1 =0), for which 

 <j= — 1 ? will give rise to ideally sharp lines, forming series 

 of the type 



The remaining elliptical orbits (n 3 >0) will give rise to 

 more or less broad lines; the greater the eccentricity, the 

 broader the lines. The part of// responsible for this effect is 



9 2 . 2 

 — j e- sn:r a>, 



ran o in a therefore, for the different individual atoms, from 



9e 2 

 Oto -~. 



All equatorial orbits (i = 0) are given by 



and the corresponding value of g being independent of the 

 perihelion longitude, 



rfO,.)-!^-;-^,. . . (E) 



all these orbits will give rise to sharp lines, no matter what 

 their eccentricity. For the sub-class of circular orbits 

 (;i 3 =z;i 2 = 0) we shall have g = l } so that the corresponding 

 lines will form the series 



Compare this with (M c ). 



All circular orbits (e = 0), of any inclination, are given by 



"3 = 0. 



The corresponding value of g being 



#(*,0) = l-§sm 2 z=^ ^T , . . (0) 



independent of the perihelion, all these orbits will give rise 

 to sharp lines, of which, in general, a plurality will enter 

 into one group. (Thus, for instance, if ft=ra 1 + 7i 2 = 2, we 

 shall have, for + 2, 1 + 1, 2 + 0, the three possibilities 

 g= — i or — $ or +1, and for n = 3,g= —J, — J, + £or +1, 

 so that even the group 2, 3 would contain many " circular " 

 components, so to call them shortly. Which of these are to 

 be rejected as being more or less " improbable" is a further 



