464 Transactions of the Society. 



e being the linear period of the original object and a the semi- 

 angular aperture. 



We will now pass on to consider the case of a grating or row 

 of points perforated in an opaque screen and illuminated by plane 

 waves of light. If the incidence be oblique, the phase of the 

 radiation emitted varies by equal steps as we pass from one 

 element to the next. But for the sake of simplicity we will 

 commence with the case of perpendicular incidence, where the 

 radiations from the various elements all start in the same phase. 

 We have now to superpose amplitudes, and not as before inten- 

 sities. If A be the resultant amplitude, we may write 



A(u)= shlU + Sin ^ + $ + Sin ^ "" ^ + . . . . 



= A + A x cos f- . . . + A cos — + . . . . (28) 



V ' r V 



When v is very small, the infinite series identifies itself more 

 and more nearly with the integral 



1 C + M sin u 7 . 7T 



an, viz.— . 

 v J _„ u v 



In general we have, as in the last problem, 



I n+<* 8 { nu . 2 r a sin u 2irru 1 .... 



A = - — du; A„ =- — —cos— du; (29) 



•y J — oo w r v J _„ u v ' v ' 



so that A = ir /v. As regards A r , writing s for 2 7rrjv, we have 

 A r = I f + "sm(l+^ + sin(l- g )^ = ? 



the lower sign applying when (1 — s) is negative. Accordingly, 



A(u) = - < 1 + 2 cos — — + 2 cos + V (30) 



the series being continued so long as 2 it r < v. 



If the series (30) were continued ad infinitum, it would repre- 

 sent a discontinuous distribution, limited to the points (or lines) 

 m = 0, « = + v, u = + 2v, &c, so that the image formed would 

 accurately correspond to the original object. This condition of 

 things is most nearly realised when v is very great, for then (30) 

 includes a large number of terms. As v diminishes the higher 

 terms drop out in succession, retaining however (in contrast with 

 (27) ) their full value up to the moment of disappearance. When 

 v is less than 2ir, the series is reduced to its constant term, so 



