S Dr. J. R. Airey on the Roots of Bessel and 



by the method employed by Lord Kelvin * in evaluating 



the integral 



/»« 



u =1 cos m \_x — (/"(wi)] d m ' 



Jo 



When x and n are great, the range of integration to be 

 considered in (3) is in the neighbourhood of the places where 

 x sin<£ — 71(f) is stationary with respect to cf>, i. e. within the 

 small range (f> x — 77 and (f>i + rj, where 



x cos (pi — n = 0, or cos0j=-. 



By Taylor's theorem, for small values of <j> — <f>i, 



• (6 — fa) 2 . 



^siri(p — n(j) = xsin<pi—n(p 1 — KT xs\n(j) 1 



_itp, eosf+ (ttf, sin ^... a (4 ) 

 Substitute 



^-« 2 =ilfi ( 4 ') 



Then 



/ 2 \i 



♦1 



and the limits of integration become ultimately —00 and 



Therefore (3) becomes, when 



, X COS (pi I 



and 



^rss^sin^! — w^ 1 = « (tan <^>i — ^>i), 

 J " ( * )=S KSE^)"J-- C0S ^-^-V + ^...)^- (5) 



^(ji^) i tT sin( " s+6 ' tS -^- ) ^ (6) 



"* Lord Kelvin, Proc. Roy. Soc. Feb. 1887; Lord Ravleisrh, Phil. 

 Mag. Dec. 1910. 



