188 Lord Rayleigh on the Theory of Optical Images, 

 The rth terms of (35) are accordingly 



J{/-([l + » + ,] + [l-m-,]) 



+ «-'»([l + m-«] + [l-m + «])h 

 or for the original series (32), 



^ d ^**»([l + ^+,] + [l_-«_ f ]) 



+ /«.-.)» ([ - 1 + m _ s ] + |- 1 _ m + J ,-] ) } t (41) 



For the term of zero order, 



■w 



A.rf-= 5 ;r"([l+»] + [l-m]). . . (42) 



From (41) we see that the term in e* m+ vanishes unless 

 (in + s) lies hetween +1, and that then it is equal to 

 tt/v . e Km+s)u ; also that the term in e km ~ s> vanishes unless 

 (m — s) lies between +1, and that it is then equal to 

 w/v . 0*< w *~*> tt . In like mariner the term in e mw vanishes unless 

 m lies between ±1, and when it does not vanish it is equal to 

 ir/v . e tmu . This particular case is included in the general 

 statement by putting s = 0. 



The image of the grating, or row of points, expressed by 

 (32), is thus capable of representation by the sum of terms 



where si = 2tt/v, s 2 = 47t/i>, &c, every term being included for 

 which the coefficient of u lies between +1. Each of these 

 terms corresponds to a spectrum of Abbe's theory, and repre- 

 sents plane progressive waves inclined at a certain angle to 

 the plane of the image. Each spectrum when it occurs at 

 all contributes equally, and it goes out of operation sud- 

 denly. If but one spectrum operates, the field is of uniform 

 brightness. If two spectra operate, we have the ordinary 

 interference bands due to two sets of plane waves crossing 

 one another at a small angle of obliquity *. 



Any consecutive pair of spectra give the same interference 

 bands, so far as illumination is concerned. For 



JH $ e iu[m+2rjr/v] ^_ e iu[m+2(r + l)n/v]l — z!L cos Z[^£tw[m + 2(r +$)*•/»] 



vl y v v ' 



of which the exponential factor influences only the phase. 

 In (43) the critical value of v for which the rth spectrum 



* Enc. Brit, "Wave Theory," p. 425. 



