﻿198 Dr. J. W. Nicholson on the 



whence 



£ = 27T + p 3 = v 1 ^ 71 *)^' 



these integers being thus additive, in the usual way. 

 The phase-integral for p 1 is 



j dr^/2 



n 3 k = \ dr\/ 2mW + 



/3 2 



if we seek to quantize ^ as it stands. The limits would then 

 be a positive value of r and infinity, for half the path, and 

 the integral would be infinite. But it is clearly necessary 

 to suppose that when the electron is at infinity, out of range 

 of action of the nucleus, it should not be subject to a quantum 

 relation, so that (j»i) r =« is not affected by the rule, and only 

 the variable part 



P1-O1X0 



is so affected. Yet this question of quantizing p ± presents 

 some difficulties in whatever way it is suggested that it 

 should be effected, and we consider that Epstein's discussion 

 of the matter is very incomplete and not logically justifiable 

 in its mathematical procedure. We shall thus consider 

 various alternatives which may give a finite phase-integral. 



Now the actual r-path is not a passage from r = oc (say) to 

 r = ao and back, and the phase-integral is not twice the 

 defiuite integral between these limits. The electron goes 

 from a limiting radius to infinity, and back to the same 

 radius elsewhere, and the passage through infinity distin- 

 guishes this phase-integral from those which occur in the 

 other coordinates. 



We must, of course, also remember that the sign of p x 

 depends upon the part of the path concerned, — whether the 

 electron is departing or returning. The critical value of r 

 is the positive root of 



1 _ mve 2 + \/m 2 v 2 e 4 +2mWj3 2 _ 1 

 r ~{P~ ~ ~u 



Writing, generally, with a new variable <f> 



1 _ mve 2 _ AiiW + 2 

 r ~W~ ~ V ~gi 



— lilt/ c. I 'V I' 1, v £■ l^ —II' YT kj JL . 



r Jgi = ~ a ( sa ?> 



1 mve* /m 2 v 2 e± + 2mW6 2 



V & ~ cos 9> 



