Dr Searle, Experiments with a plane diffraction grating 89 



refracted front is independent of the particular ray. But, in the 

 case of a grating, the time of passage from an incident to a diffracted 

 front increases or diminishes by ir as the point of incidence of the 

 "rav" is moved from the centre of one opening to the centre of 

 the next. Here r is the periodic time of the vibration and i is a 

 positive integer. 



§ 4. Diffraction of a plane wave; general case. Take the axes of 

 X, y, z to coincide with the axes ON, OT, OL of the grating, as in 

 Fig. 1. Let J? be a point on the centre hne of the 

 qth. opening and let the coordinates of R be 

 0, qd, h. 



Let the direction cosines of the forward di- 

 rection OP^ of the incident beam be l-^, m^, n^, 

 and let those of the forward direction OP^ of the 

 diffracted beam of order i be l^, m.^^ n^. 



Through draw planes perpendicular to these 

 two directions. The distance of R from the first 

 plane, counted positive when the incident wave 

 front reaches before it reaches R, is 7yi-^qd + n^h. The distance 

 of R from the second plane, counted positive when the diffracted 

 front leaves before it leaves R, is 7n^qd + nji. If Vq is the velocity 

 of Ught and \q the wave length in a vacuum, and if yu^, /Xg are the 

 refractive indices of the media on the two sides of the grating, the 

 times corresponding to the two distances are 



/xj {mT<}d + nJi)lvQ and ijl^ {m.^qd + n^h^VQ, 

 and these differ by qir. Thus, since tVq = Xq, we have 

 1^2 {rn^qd + n^h) — /Aj {m-^qd + n^^h) = T qi^Q. 

 This result must hold good for all positions of R on the grating, 

 for which q is integral. We thus obtain 



fjL^m^ = [iiiniT iXo/d, (1) 



H2n2 = fJ-ini (2) 



These equations completely determine the directions of the 

 diffracted beams of order i. 



Let the incident and diffracted beams make angles e^, 63 with 

 OT and angles rj^, 172 with OL. Then 



cos ei = /%, cos 63 = W2, (3) 



cos r]■^^ = n^, cos ''72 = '^2 (^) 



Hence (1) and (2) may be written 



jLt2 cos €3 = fii cos ei =F iXJd, (5) 



1x2 cos rj2= [J-i cos t7j (6) 



