Theory of Spectrum Emission. 55 



It remains to average F over a period of the undisturbed 

 motion and to substitute the result into (11). 



Let the plane of the undisturbed (osculating) orbit make 

 with the equatorial plane (i. e. the plane through 0, perpen- 

 dicular to the axis) the angle i ; this will be the inclination 

 of the orbit. Further, let co be the longitude of the peri- 

 helion, counted from the ascending node Q, (the line of 

 nodes being the intersection of the orbit plane with the fixed 

 equatorial plaue), and 6 the angle between the instantaneous 

 radius vector and Oft. Then, if e be the eccentricity, the 

 equation of the orbit is 



1= ^'[l + ecosO?-©)], • • • • ( 15 > 



where p= nir^O = const., and our previous rj is related 

 to ?', 6 by 



cos 7] = sin i . sin 6 (16) 



The disturbing function (13) becomes, by (15) and (16), 



F = 6.[l + ecos(6>-&>)] 3 . [3 sin 2 i sin 2 0-1], 



Wher6 iWm'y , 4 /aW\» , m 



b =*(-y>-)= b ^[yr)- • • • (17) 



— 1 f 2 * 



W hence, the required average F= — j i^t/0, 



- 7r Jo 

 Z= p.sin 2 i[l + |6 2 (l + 2sin 2 a J )]-^(l + |6 2 ). (18) 



Notice that this mean value of the disturbing function 

 contains four of the elements of the undisturbed orbit,, 

 to wit, its parameter through p (appearing in 5), its 

 inclination i, eccentricity e, and the longitude of the peri- 

 helion co. Of these only the three first can be " quantized " 

 (i.e. fixed in terms of integers and Ji), while the last, co, will 

 retain its freedom of assuming any value between and 2ir.. 

 This feature, most immediately conditioned by the absence 

 of radial symmetry or isotropy (replaced by axial symmetry) 

 will give rise to diffuse lines, i. e. spectrum lines of finite 

 breadth. The only orbits which will give rise to ideally 

 sharp lines will be those for which e . sin i, the coefficient of 

 the non-quantizable term in (18), vanishes, i. e. all circular 

 orbits, whatever their inclination, and all equatorial orbits, 

 whatever their eccentricity. This will become more clear 

 presently. 



Let us now quantize the three elements^, ?, e by the usual 



