Theory of Spectrum Emission. 51 



elliptic and to hyperbolic (oscillatory) motions can be written 

 down with comparative ease. But details of this kind need 

 not detain us here any further. 



4. Large orbits. Let the nucleus still consist of two equal 

 centres, \ice, at a distance 2a apart, but let us confine our 

 attention to electronic orbits whose smallest diameter is 

 large us compared with this distance. That such and only 

 such orbits are physically interesting will become manifest 

 if it is remembered that while the dimensions of the spectro- 

 scopically relevant orbits are at least as large as 10 -8 cm., 

 the dimensions of the nucleus are, according to Rutherford's 

 scattering experiments, certainly smaller than 10~ u cm.,and 

 possibly much smaller than this. 



Such being the case, let r be the radius vector o£ the 

 electron (drawn from the midpoint of the binary nucleus), 



and let us assume that - is permanently a small fraction. 



This excludes, for instance, all the hyperbolic orbits 7} = const, 

 mentioned in the last section ; for each of these passes 

 through the axis right between the two centres (thus pushing 

 r below a). Of course, a multitude of other types of motion 

 is excluded by the assumption of small a/r, and what remains 

 are quasi-keplerian motions, with orbits (in our case of a non- 

 escaping electron) but slightly deviating from ellipses. 



The problem in hand could be treated by means of the 

 more general, rigorous, formulae of Section 3, with obvious 

 simplifications. In fact, since r 2 = x 2 -f r/ 2 , we have 



r- = a 2 (cosh 2 f cos 2 tj + sinh 2 f sin 2 y) 



or 



a 



r 



a 



— = sinh 2 f 4- cos 2 77, 



so that - large means large f, and therefore, sinh £= cosh f 



1.A 



2 5 



and r = i a e% 



while t, becomes simply the angle between the radius vector 

 and the axis. This introduces at once considerable simplifi- 

 cations into the formulae (9), and so on. 



But the present problem can be dealt with much more 

 simply by treating the atomic system as an ordinary system, 

 with a simple point-charge as nucleus, subjected to slight 

 'perturbations. For this purpose it is enough to write down 

 the disturbing function F and to find its average F for 



E 2 



