170 Lord Rayleigh on Methods for detecting small Optical 



aperture is symmetrically situated with respect to the 

 geometrical focus. This restriction we will now dispense 

 with, considering first the case where fi = and f 2 ( = £) is 

 positive and of arbitrary value. The coefficient of sin T in 

 (7) becomes simply 



Si{(* + *)£} + Si{(0-*)f}. . . . (19) 



In the coefficient of cosT, Ci{(0+£)fi}, Ci{(0-£)?i} 

 assume infinite values, but by (15) we see that 



CS{(* + *)f,}-Ci{(0-*)fi}=log 6 



e-<$> 



Ci{(tf-*)f}-Ci{(0 + *)f}+log 



(20) 

 (21) 



so that the coefficient of cos T is 



+ 4> 



The intensity I at angle <f> is represented by the sum of 

 the squares of (19) and (21). When (/> = at the centre of 

 the field of view, I = 4(Si#f) 2 , but at the edges for which it 

 suffices to suppose cf>= + 0, a modification is called for, since 

 Ci{(0 — <£)£} must then be replaced by 7+ log | (0— <j£>)£ \ . 

 Under these circumstances the coefficient of cos T becomes 



7+log(2<?f)-Ci(20f), 



and 



I-{Si(2^)} 2 + { 7 +log(2^)-0i(2^)p. . (22) 



If in (22) f be supposed to increase without limit, we find 

 I = i7r 2 + {log d£}\ (23) 



becoming logarithmically infinite. 



Since in practice f , or rather k%, is large, the edges of the 

 field may be expected to appear very bright. 



As may be anticipated, this conclusion does not depend 

 upon our supposition that £i = 0. Reverting to (7) and 

 supposing (/> = #, we have 



sinT[Si(26>f 2 )-Si(2(9f 1 )] 



+ cosT[Ci(2^f 1 )-Ci(2^ 2 )+ log (&/&)], . (24) 

 andI = oo, when f 2 =00 - If ?i vanishes in (24), we have 

 only to replace Ci(26Si) by y+lo$(20fi) in order to 

 recover (22). 



We may perhaps better understand the abnormal increase 

 of illumination at the edges of the field by a comparison 

 with the familiar action of a grating in forming diffraction 



