186 Lord.Rayleigh on the Theory of Optical Images, 



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

 represent a discontinuous distribution, limited to the points 

 (or lines) u — 0, u= +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?; is less than 27r, 

 the series is reduced to its constant term, so 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. Reso- 

 lution fails at the moment when the spectra of the first order 

 cease to cooperate, and Ave have already seen that this 

 happens for the case of perpendicular incidence when v=27r. 

 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(«) = ^{l + 2cos^}, . . . (31) 



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

 intensity when u=^v, or fv. 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 inci- 

 dent 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 representing the resultant amplitude 

 at any point u may still be written 



sinw + sin(tt + tt) Q _ imv | sinju-v) £+imv 

 u u + v u — V 



An(u + iv) e 



u-\-2v 

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

 e the grating-interval, X the wave-length, 



mv/2Tr = e$m<y /\ (33) 



