174 Lord Rayleigh on Methods for detecting small Optical 



When = oo, 1=0. When = 0, 



I=[7r~2Si7r] 2 = -3162. 

 When = 0, 



I= [i 7r _Si(27r)] 2 +[log(500/7r)-fCi(27r)] 2 = 25-51; 



^o that the brightness of the edges is now about 80 times 

 that at the centre o£ the field. The remaining values of I in 

 Table III. have been calculated as before with omission of 

 the terms representing minor periodic fluctuations. 



Hitherto we have treated various kinds of screening, but 

 without additional retardation at the plane of the first 

 aperture. The introduction of such retardation is, of course, 

 a complication, but in principle it gives rise to no difficulty, 

 provided the retardation be linear in over the various parts 

 of the aperture. The final illumination as a function of 

 can always be expressed by means of the Si- and Ci-functions. 



As the simplest case which presents something essentially 

 novel, we may suppose ihat an otherwise constant retar- 

 dation (R) changes sign when = 0, is equal (say) to 4-/0 

 when is positive and to — p when is negative. Then (3) 

 becomes 



+ p + 0f)i0 + I " sin (T -p + 6£)dd 



r° sm(T+p+0f)i0+ r 



~ . rn r smvt . 1— cosutl /0 ~ N 

 = 2 sin 1 cos p — s-s -f sin p ^ . (30) 



reducing to (5) when p = 0. This gives the vibration at the 

 point f of the second aparture. If f = 0, (30) becomes 

 20 cos p sin T, and vanishes when cosp = 0, for instance, 

 when the whole difference of retardation 2p = 7r, or (reckoned 

 in wave-lengths) -JX. 



The vibration in direction behind the second aperture 

 is to be obtained by writing T + 0£ for T in (30) and inte- 

 grating with respect to f . This gives 



« • mf'?*. i j- f sin0£ . 1 — cos 6P\ 

 2 sin I | a£ cos 0£ U cos p — ~- + sin p ^ > 



o tC it- • j.f f sin0f . 1— cos0|n /Q1 . 

 -I- 2 cos I I d% sin 0f -j cos p — —■ -f sin p ^ ? W , (31) 



and the illumination (I) is independent of the sign of 0, 

 which we may henceforward suppose to be positive. 



If the second aperture be symmetrically placed, we may 



