472 Transactions of the Society. 



The application to a grating, of which all parts radiate in the 

 same phase, proceeds as before. If, as in (58), we suppose 



A (p) = A 4- . . • 4- A r cos sp + . • • ; . • (70) 

 we have 



A 



G^ cos sp dp; .... (71) 



— 00 



from which we find that A,, is 4 it 2, or 0, according as the ordinate 

 is finite or not finite at x = s. The various spectra enter and dis- 

 appear under the same conditions as prevailed when the object 

 was a row of points ; but now they enter discontinuously and 

 retain constant values, instead of varying with the particular 

 ordinate of the object-glass which corresponds to x = s. 



We will now consider the corresponding problems when the 

 illumination is such that each point of the row of points or of 

 the grating radiates independently. The integration then relates 

 to the intensity of the field as due to a single source. 



By (9), (10), (11), the intensity I 2 at the point (p, q) of the 

 field, due to a single source whose geometrical image is situated 

 at (0, 0) is given by 



\ 2 f I' 2 = {ff cos (px + qy) dx dy} 2 + { ff sin (px 4- qy) dx dy} 2 



= ff cos (pas' 4- Qy') dx' dy x ff cos (px 4- qy) dx dy 



4- ff sin (px 4- Qy') dx' dy X ff sin (px 4- qy) dx dy 



- Sfff cos { p (*' - x ) + e 0/ - y) } dx d y dx ' d y' < 72 > 



the integrations with respect to x', y, as well as those with respect 

 to x, y, being over the area of the aperture. 



In the present application to sources which are periodically 

 repeated, the term in cos s p of the Fourier expansion representing 

 the intensity at various points of the image has a coefficient found 

 by multiplying (72) by cos sp and integrating with respect to p 

 from p = — co to p = 4- °o . If the object be a row of points, 

 we may take q = ; if it be a grating, we have to integrate with 

 respect also to q from q = — oo to # = 4-co. 



Considering the latter case, and taking first the integrations 

 with respect to p, q, we introduce the factors c* 11 ?* 1 "!, the plus 

 or minus being so chosen as to make the elements of the integral 

 vanish at infinity. After the operations have been performed, 

 h and k are to be supposed to vanish.* The integrations are per- 

 formed as for (60), (64), and we get the sum of the two terms 

 denoted by 



2 ^ & n^ 



{h? + (x'-x±sy.}{k* + (y'- y y\ ' ' v ) 



* The process is that employed by Stokes in his evaluation of the integral inten- 

 sity, Edin. Trans., xx. p. i!17 (1853). See also Knc. Brit., « Wave Theory,' p. 431. 



