﻿Binding of Electrons by Atoms. 203 



hyperbola. We have the sum n\ + n 2 of the angular quanta 

 in place of his single integer. 



But this formula, with all the applications he makes to 

 characteristic 7 radiation, is not tenable, as resting on a 

 mathematical error. Its apparent success appeared at one 

 time to the writer to justify it as an empirical formula, in 

 spite of his independent investigation, outlined above, indi- 

 cating the impossibility of quantizing such orbits. Close 

 examination, however, of the calculations of 7 radiation and 

 so forth made it clear that they' were in several cases 

 illusory, and determined more by order of magnitude than 

 by the nature of the formula. 



There is one convincing argument against the formula, 

 however. It should give an emission spectrum for all values 

 of W], n 2 , n s and wi 1: m 2 , m 3 making 



— W(m l9 m 2 , m. d ) +W(n 1 , n 2 , n 5 ) 

 positive. This can be tested in great numerical detail on 

 the spectrum of a hydrogen atom, and the test fails entirely. 

 No spectrum line is found, — in the secondary hydrogen 

 spectrum, — in any of the assigned positions. Thus the 

 formula really fails as an empirical one. 



We have seen above that it must be replaced by 



J (cose/)- cos 77) 2 





•2/5 



m 6— sin 79 , , ■ , J 



, — - — <P + C0t 77 l0£ e < - 



cos 6 — cos 77 r &e ) . 



- sin 



which is logarithmically infinite. 



The attempt to obtain a finite phase-integral, in this 

 manner, in fact fails, and we must give up the hypothesis 

 that even the variable part of p 1 can be quantized for the 

 infinite path. 



It is not difficult to see that this conclusion is general for 

 an} r infinite path which is possible for an electron about a 

 physically existent atom, whose nucleus can always be 

 regarded, for the present purpose, as a superposition of free 

 charges and a set of doublets. We have demonstrated the 

 result for a single free charge, and previously for sets of 

 doublets. Further analysis of the more general case does not 

 seem necessary, and could readily be supplied by the reader. 



Our conclusion must be as follows : — 



A determinate and finite value of W cannot be obtained 

 for an electron moving about any atomic nucleus, if the path 

 involved takes the electron to infinity. 



