The Theory of Optical Images. By Lord Rayleigh. 465 



that the field becomes uniform. Under this kind of illumination, 

 the resolving-power is only half as great as when the object is 

 self-luminous. 



These conclusions are in entire accordance with Abbe's theory. 

 The first term of (30) represents the central image, the second 

 term the two spectra of the first order, the third term the two 

 spectra of the second order, and so on. Kesolution fails at the 

 moment when the spectra of the first order cease to co-operate, 

 and we have already seen that this happens for the case of perpen- 

 dicular incidence when v = 2tt. The two spectra of any given 

 order fail at the same moment. 



If the series stops after the lateral spectra of the first order, 



A W = "{l + 2cos 2 -p}, . . (31) 



showing a maximum intensity when u = 0, or ^ v, and zero intensity 

 when u = ^ v, or § v. These bands are not the simplest kind of 

 interference bands. The latter require the operation of two spectra 

 only; whereas in the present case there are three — the central 

 image and the two spectra of the first order. 



We may now proceed to consider the case when the incident 

 plane waves are inclined to the grating. The only difference is 

 that we require now to introduce a change of phase between the 

 image due to each element and its neighbour. The series repre- 

 senting the resultant amplitude at any point u may still be 

 written 



!E1 + &in ( u + v ) c -imv . sin (u - v) c + imv 



u U + V U — V 



g n(«+ 2 «> ,_,». 



u 4- 2 v 



For perpendicular incidence m = 0. If 7 be the obliquity, e 

 the grating-interval, \ the wave-length, [i = *J ( — 1 )], 



mv j 2-7T = e sin 7 / X (33) 



The series (32), as it stands, is not periodic with respect to u 

 in period v, but evidently it can differ from such a periodic series 

 only by the factor e imu . 



The series 



e -imu s i n u e - im(u + v) s i n ( u _|_ v } 



u u 4- v 



e - im (u - v) s i n ( w - v) e ~ im(u + 2 v )sm(u + 2v) . 



_| 1 . _ ,-j- .... (o4) 



u — v u + Iv 



