The Theory of Optical Images. By Lord Rayleigh. 463 



an equal contribution to I r . It will come to the same thing 

 whether, as indicated in (24), we integrate the sum of the series 

 from to v, or integrate a single term of it, e.g. the first, from — co 

 to 4- oo . We may therefore take 



lo = - — k-~ du - - ; . . . (25) 



T 2 f +00 sin 3 w 2trrv> , '. 



I = - — „- cos du. . . (26) 



V J _ „ 16 2 V 



To evaluate (26) we have 



T+ °° sin 2 u cos sm , f + °° 1 rf / • o x , 

 dw = — — . (sin- u cos sic) du, 



J -co w 5 J -« « ^ 6 



and 



— (sin 2 « cos s«) = — s sin si', 



+. 1 + f sin (2 + s) tt + ^-? sin (2 - «) 16 ; 



so that by (15) (s being positive) 



/+ " sin 2 it, cos &u , _ j s * 2 + s + 2 — s ) 

 _* ^~~ :7r l 2 + "iT" 4- J'' 



the minus sign being taken when 2 — s is negative. 

 Hence 



2 



I r = ^fl -51), orO, . . . (27) 



according as »* exceeds or falls short of r ir. 



We may now trace the effect of altering the value of v. When 

 v is large, a considerable number of terms in the Fourier expan- 

 sion (23) are of importance, and the discontinuous character of 

 the luminous grating or row of points is fairly well represented in 

 the image. As v diminishes, the higher terms drop out in suc- 

 cession, until when v falls below 2-7T only I and i! remain. From 

 this point onwards I x continues to diminish until it also finally 

 disappears when v drops below ir. The field is then uniformly 

 illuminated, showing no trace of the original structure. The case 

 v = ir is that of fig. 120, and curve iii. shows that at a stage when 

 an infinite series shows no structure, a pair of luminous points or 

 lines of the same closeness are still in some degree separated. 

 It will be remembered that v = nr corresponds to e = £\/sin a, 



2 h 2 



