of the Principle of Stationary Phase 55 



Now tq—'^~ ^ co^ 6 o^x{6 — ix) sin ix, 



when 6 f^ fM and -^r^^oo^iw fx .{6 — ^y. 

 Hence, as ;)^; — > 0, 



Lde -1 



and 



/, 



sin '^ c^x 



\ /-n'tauM-M) (• . 1 rf^") _1C0S 



/•n'tanM-M)( . i rft'l _1C0S , 



Now, as ?i -> CO , ?i (tan /x — ^) -> i (2a)^ and so the limiting 

 range of integration is of finite length, 



1 d9 

 Moreover, \/{2x sin fi) . ^^ -^ > — l as ?i — > go ^vhen % is ^■ero, 



1 df) 

 that is, when d ^ fi. But, when 6 -> 0, the limit of \/{2x sin A*) • %- ^ 



is 



— {2 sin fi (sin /jl — /xcos At)|"^/(1 — cos /a), 



a^icZ, a.9 n—>cc, the limit of this is not — 1 hit — 2 \/(^) ; and so 

 we cannot infer that 



•«(tunM-M)f . if^6l) _icos , f' _icos , 



|V(2.-sm;.).X-^^~j-% -^i^^X^^X-j^^X \sin^^^' 



where b is Lim •?? (tan yu, — /x). 



The evaluation of the approximate formula for Jn{^) in the 

 circumstances under consideration consequently seems to require 

 more elaborate analysis than is afforded by the methods contained 

 in this paper. 



