

410 Prof. Henry A. Rowland on Gratings 



Let J n be a BessePs function. Then 



cos(Msin0) = J o (w)+2[J 2 (i*) cos 2 0+J 4 (w) cos 4 </> + &c], 

 sin (m sin $) = 2[J X (?0 sin (/> + J 3 (w) sin 3 </>+ &c.]. 



e -i«sin<p _ cos ( w s i n 0) _ 2 * s i n ( w sin 0) . 



Hence the summation becomes 



£— ib/xa Q n 



x [J (5/*a 1 ) + 2(— i3 x (bfia^) sin <? x n + J 2 (bfia{) cos 2<? 1 n— &c.) 

 S<( x [J (fyu,a 2 ) + 2(—iJ 1 (bfjLa 2 ) sin £ 2 ™ + J 2 (fy^ 2 ) cos 2^/1 — &c.) 

 X [ J (bfia s ) + &c.) ] 

 x[&c.]. 



CW I. — Single Periodic Error. 

 In this case only a and a x exist. We have the formula 



^n-\ e -ipn = e - 



. pn 

 sm 2 



• P 



sin --, 



Hence the expression for the intensity becomes 





m-^-r- - 



sin 



2 J 



sinK 



b/j,a + ei 



sm 



smn 



bfia — ei 



. bfia —ei 

 sin-^ — ■- 



+ &c. 



J 



As ?i is large, this represents various very narrow spectral 

 lines whose light does not overlap, and thus the different 

 terms are independent of each other. Indeed, in obtaining 

 this expression the products of quantities have been neglected 

 for this reason because one or the other is zero at all points. 

 These lines are all alike in relative distribution of light, and 

 their intensities and positions are given by the following 

 table : — 



