328 H - NAGAOKA; DIFFRACTION PHENOMENA 



whence by expanding J b (2x sin to) in powers of 2x sin cu and integrating, 

 we obtain 



J M + Ji 2 (x) = 2 2*»+i[/7(w);|»//(«+l) **" ( ^ } 



This series converges very slowly for vaines of x little over unity ; 

 in place of the ascending series, we may conveniently employ the 

 following semi- convergent series, which can be easily arrived at from 

 the corresponding expansions of J -{x) and J x (.x). Thus 



T s . , l/i 1 . . n cos 1x ôsin 2x , \ 



TTx \ 8x* 4x 32x 2 r 



1 v ' ;:■# \ 8a; J 4 a* 3'2 x- / 



whence 



^) + ^=A( 1+ ^_^_^ + ) (B) 



The above series is rapidly convergent, and can be conveniently 

 used for values of x greater than the first root 3^=3. 8317 of J 1 (a;)=0 ; 

 at the last mentioned value of x, the number obtained for Jà 2 (x)+Jî\x) 

 will be accurate to the fourth decimal place. 



In the neighbourhood of the inflexion points of the first set, the 

 value of y remains nearly constant ; we can thus expand J<?(x) + Ji\x) in 



Maclaurin's series. Denoting the roots of J 1 (x) = by x , x lt x. 2 , and 



putting y n =Jo(x n ) , ç = x—x n , we shall obtain the following series 

 for y in the neighbourhood of the point x n , y„ 



,=/.w{i + . + * -4ri+£^-m -*)i+ i <°> 



Of these three expressions, we shall have occasion to use th e 

 form (B) most frequently, as it expresses the nature of the curve 



