that is 



of equal order and argument 

 ["(/)-« sin (7i(/)) . /; (<^) ^0 = 3* 2 - ^ ?i - * r (|)/35 + o {n " '), 



45 



and so 



^ 8 ( f'^ sini/r r'-^ sin^/r ] 



(6f) 



-r 



??7r j 



</> ~ I/2 (</)) sin (7?</)) . f/<^ + (/i-2) 





secon 

 r such 



sin-^lr 



Now, by the second niean-vahie theorem, there exists a number a 

 exceeding nir such that 



liTT "^^ 



dyjr 



1 



(nirf 



sin^p• d/\jr I < 2(mr) •', 



and so we have at once that, when n is large and real, 

 ^» (n) = ^T—^ ^ + (n - -^ ), 



which is the result to be established. T(3 obtain a closer approxi- 

 mation by these methods would necessitate some very tedious 

 integrations by parts. 



3. We next consider the approximate formula for Jn (n). It 

 is immediately deduced from the Bessel-Schlafli integral that 



j:/(7i)=- sin (9 . sin 71 (^ - sin (9) . c?^ 



TT Jo 



Now we get, on integrating by parts, 



rsinhde-^^^ + ^'^'^'^^dd 



Jo 



~ nJo 1 



sinh 6 d 



^-n(d + sinhd)^^0 



1 



nj 



+ cosh 6 ' dd 



^r e-''Ud = 0{n-% 

 2njQ 



