﻿Binding of Electrons hy Atoms. 199 



we have $ = in the critical position (perihelion, in the 

 usual terminology), and 



mve 2 1 /m 2 v 2 e 4 + 2mW8 2 

 cos< t>=--pr/'\/ ' —p- -=cos77 (say), 



when r=oo. 



What is required for the correct evaluation of the phase- 

 integral is a continuous variable which shall change in one 

 direction, — and thus give a definite integral, — as r goes to 

 infinity and returns, the sign o£ p x being automatically taken 

 into account, — or the sign of pi— (piXc when (pi) m is not 

 zero as in a parabolic path. The new variable cj> has this 

 property, and ranges from zero to 2tt as r goes through its 

 changes. We have denoted its value, when r=co, by r) 

 above, where rj is evidently an obtuse angle. 



The phase-integral for jt^ alone would be 



r2 V 



n 3 h=\ 



xrr 2mve 2 B" 1 

 dr \/ 2mW H S; 



(the square root being properly interpreted in different 

 regions) where 



- = ~^r -f- 7^ V m Ve 4 + 2m W/3 2 cos # 

 r (3 1 p z 



= ™^ + ~ VmW + 2mW/3 2 cos 77, 



and we find 



dr= T . *? + ** - v , £, p VZmW + mVeS 

 (cos 9 — cost;) 4 // 



V 



n TTr 2vme 2 /3 2 q . , 



2m W + ^ = £ sin <£. 



r r p. • 



If the integration were continuous throughout, — as as- 

 sumed by Epstein, — we should thus have 



n z h= j3 \ 



sin 2 (f> d(j> 

 J (cos<£— cost;) 2 



sin 2 $ d$ 



(cos<£— cosr?) 2 ' 

 which is an infinite integral, as would be expected. 



