Quantum of Action, 161 



total number, and for systems having the energy 2hv the 

 fraction belongs to the second order of small quantities. 

 It follows therefore that in all but a very small number of 

 cases the energy emitted must be equal to hv, and if an 

 emission is accompanied by the ejection of an electron, its 

 kinetic energy cannot exceed this value. 



A very important consequence of hypothesis (3), to which 

 I wish to draw attention here, is that any emission of energy 

 of frequency v to the aether must be equal to 



nhvj 

 where n is an integer. Such considerations as those in the 

 last paragraph lead to the conclusion that n must be unity 

 for frequencies in the visible range of the spectrum and for 

 higher frequencies. This result appears in the theories of 

 Bohr and Ishiwara as a special assumption which cannot 

 be deduced from the other hypotheses they adopt. 



Any form of Quantum Theory must involve assumptions 

 intimately connected with physical quantities having the 

 dimensions of an action, and the most important of these 

 is angular momentum. The hypothesis (3) given above 

 makes the angular momentum of a system equal to 



nh 

 2^ 

 (where n is an integer), even in the case of an electron the 

 orbit of which is elliptical *. This is not inconsistent with 

 Bohr's hypotheses, and coincides with his views in the case 

 where the ellipse degenerates into a circle. The assumption 

 (3) lays certain restrictions on the possible values of the 

 eccentricity of the electron orbit. This can be shown as 

 follows : — Let q x represent the distance of the electron from 

 the nucleus and q 2 its angular distance from a suitably 

 chosen fixed line. Then we have for the equation of the 

 ellipse 



B 



and therefore 





11 1 + e cos q 2 ' 



p 2 e . 



and 



dqi 



27T 



2 f sin 2 q 2 dq 2 



* That there is a real relationship between angular momentum and 

 the constant h was first noticed by Nicholson some years ago (Monthly 

 Notices of R. A. S. lxxii. p. 679 (1912)). 



