﻿200 Dr. J. W Nicholson on the 



If we merely quantized over the finite part o£ the hyper- 

 bola, — another possible suggestion, — we should have 



n s k= fi ( f " + f * ) ,. *}»'* v d4> 



6 I Jo } 2n-n) (cos </> -cost;) 2 r 



=4' 



sin 2 </> dcf) 



(COS <j>— COST;) 5 



which is again infinite. 



The nature of the first infinity merits a remark, however, 

 for it is independent of rj and therefore of W. For 



d<f> 



cos?; 



j_C* sin 2 <t>d<l> r sin </> ~| __ C n cos <$> 



J (cosc/>— cos*;) 2 — |_cos<£— cost;J J cos<£ — 



r sin<i "1 C* d<b 



= \ — — — it— cost;! -r-t- . 



|_cos <p — cos 7] J f cos (p— cost; 



The principal value of the last integral is well known to 

 be zero, for all values of t;, so that the last term is zero. 

 Our equation would be 



^=-2^8+ r f* i, 



Lcoscp— cost; J 



where the principal value of the bracket must be taken, 

 i. e. it is to be interpreted as 



Lu { r :»+ r v r f» r }. 



I Lcos — cos t; J LCOS0— COS^Jjj+e J 



This becomes 



^ C e sin t; e sin t; J 2e 



which, though infinite, is an infinity independent of rj and 

 therefore of W. We have another aspect of the indeter- 

 niinateness of W for such paths. 



Our fundamental objection to Epstein's mode of integration 

 may now be introduced. He integrates />i— (pO^? an d not 

 pi, but this fact does not affect the question. For as (/> ranges 

 between and 27r, if p 1 — /(<£), we have p ] varying con- 

 tinuously with 6, and remaining positive, till 6 — r\. Then/?! 

 becomes —f(<f>i) when <f> = 27r — ^>i on the return journey 

 after <£ = 27r— -t;. Between (/>=?; and <£ = 27r — t;, the value 

 of r should be infinite, and p x changes from V^mW to 

 — v 7 2wiW, as in the figure. 



