﻿196 Dr. J. W. Nicholson on the 



With a positive W, the motion is not real. Thus W must he 

 negative and the path necessarily extends to infinity. A 



critical value of r is \J _ w , and the other is infinity. 



The phase-integral for p 1 is 



pi dr 



*J 



v -2mW 



which is infinite, but nevertheless independent of W. For 

 writing 



it becomes 



2m W 



= 2/^(1-^ 



A finite integral is secured, — Epstein's procedure, Joy 

 instance,— by. using the phase-integral not for p r , but for 

 Pi~~ {pi)r=x, which in the same way yields 



again independent of W. Now ft is quantized, or expressed 

 definitely in terms of integers already, from the phase- 

 integral for the momentum p 2 . The phase integral for pi 

 can only, in this case, lead to another expression of similar 

 type for j3 t but to no expression for W. It is not at all clear 

 that the two expressions for f3, also, can both be valid 

 simultaneously. 



This possibility has hitherto apparently been overlooked 

 by authors in this subject. 



No case has, however, been noticed in which W is inde- 

 terminate for a finite path. One very important conclusion 

 is that the whole investigation is valid for a negatively 

 charged atom with a distant electron. 



We proceed now to discuss the possible existence of 

 definite paths with a positive total energy and infinite 

 extent, for a single electron around a nucleus of charge ve, 

 situated at the origin. This is Epstein's problem, which he 

 treats as only two-dimensional. The energy equation is 



•Z y, 2 



C r r z sin- 6 ) r 



