The Theory of Optical Images. By Lord Rayleigh. 467 

 or for the original series (32), 

 JL | e i(m + s)u ([1 4. OT + s ] + [1 - m - s]) 



-f e<("»-*)»([l + m - s] + [1 — m -f s])} . (41) 

 For the term of zero order, 



A e imu = ~ e imu ([1 + m\ + [1 - to]). . (42) 



From (41) we see that the term in <f ( m + s ) M vanishes unless 

 (to -f s) lies between + 1, and that then it is equal to 7r/v.e^ TO + *) w ; 

 also that the term in e* C»— »)« vanishes unless (to — s) lies between 

 + 1, and that it is then equal to 7r/ < y.e i<Hl—s)M . In like manner 

 the term in c ima vanishes unless m lies between + 1, and when it 

 does not vanish it is equal to Trjv.c imu . 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 



7r/i?.{6 irou + «*(» + ■»)« 4. e^" 1 -^" + c i(™ + *2)" -J. } . (43) 



where s, = 2 ir/v, s 2 = 4 tt/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 represents 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 suddenly. 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 



^ J c iu [m + 2r7r/o] i c %u [m + 2 (r + 1) tt/oJ I _ TL^. cos 77 ^— e iu ^ m + 2 ( r +*) w /"3 

 V \ / v V 



of which the exponential factor influences only the phase. 



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

 appears is given by, when we introduce the value of to from (33), 



or, since (as we have seen) ~ — = — > — > v 44 / 



e (sin 7 ^ sin a) = + r X. . . . (45) 

 * Enc. Brit., 4 Wave Theory,' p. 425. 



