234 Prof. Gr. N. Watson on Bessel Functions oj 



Since we have just proved that 



2k7T 



cos {n(w — &mw)\ dw = 2kJn(n), 



o 

 it follows that in no sense can 





7T 1 



cos {n(w — siniy)j dw 

 o 



be said to be an approximation to J„(n) . 



The fact is that Dr. Airey has made an error of the 

 same nature as one which occurs in the course of Lord 

 Rayleiglr's work ; but, as Lord Rayleigh's error appeared in 

 onlv a single line of his work, his final result was not 

 vitiated. 



Lord Rayleigh's analysis was as follows : — 

 "If z = n absolutely, we may write ultimately 



1 C* 

 n (n) = — 1 cos {n(w — sin w)\ dw 



IT J 



i r°° 



= — I cos \n(w — sin io)\ dw 

 Jo 



-urn 



= T (1)2-13 -bi-hi-i." 



cos ~-dw = — ( - Y 1 cos a 3 du 



[Of course, at the present time, most mathematicians 

 would employ a notation which has been introduced in the 

 last few years and would replace the symbols = in the 

 second and third lines by ~.] 



For the reasons explained above, 1 cos{n(w — sin w) } dw 



C" Jo 

 is not an approximation to i cos {n(iu— sin w)} die. But, 



.° 

 in tJw range 0<_w^7r, it is permissible, J or purposes of 



approximation^ to replace w — sin w by \w z ; though it must 



be stated that it is somewhat difficult to give a completely 



rigorous proof of this. And, since I cos (nw*/6) die is con- 



.1 

 vergent, it is easy to prove that it is an approximation to 



I cos (7iw z /6) dw when n is large. Consequently, Lord 



Jo 



Rayleigh's work may be made strictly accurate by replacing 



