176 Lord Rayleigh on Melhods for detecting small Optical 



in any direction <£ is thus independent of R altogether. 

 This procedure would fail if R were discontinuous for any 

 values of 0. 



Resuming the suppositions of equation (31), let us now 

 further suppose that the aperture extends from fj to £ 2 , 

 where both f^ and £ 2 are positive and f 2 > f 1# Our expression 

 for the vibration in direction^ becomes 



sinT[cosp{Si(0 + 0)? + Si(0-0)f} 



+ sin ^{201(01) -Ci(^ + 0)f-Ci(^-0)f}]^ 

 + cosT[cosp{Ci(<9-<£)£-Ci(0 + <£)£} 



+ sin,>{2Sityf)-Si(0 + 0)£ + sin(0-<£)ff]f*. (31) 



SI 



We will apply this to the case already considered where 

 f 2 = 5OO, £i# = 7r; and since we are now concerned mainly 

 with what occurs in the neighbourhood of (/> = 0, we may 

 confine <\> to lie between the limits and \9. Under these 

 circumstances, and putting minor rapid fluctuations out of 

 account, we may neglect Ci(0 + (f>)^ 2 and equate Si(0 + 0)f 2 

 to \ir. A similar simplification is admissible for Si((/>£ 2 ), 

 Ci($f 2 ), unless (f>/0 is very small. 



When <£ = 0, (34) gives 



sinT[cos / 9{7r-2Si(7r)}+sin /0 {21og(500/7r) + 2Ci(7r)n, 

 in which 



7r-2Si(7r)=--5623, Gi(7r)=-0738, log (500/tt) = 5-0699. 

 Thus for the intensity 



1(0) = [ - -5623 cos p + 10-2874 sin p] 2 . . . (35) 



If p = 0, we fall back upon a former result (*3162). [f 

 P = Jtt,I(0) = 47-3. 



Interest attaches mainly to small values of p, and we see 

 that the effect depends upon the sign of p. A positive p 

 means that the retardation at the first aperture takes place 

 on the side opposite to that covered by the screen at the 

 second aperture. As regards magnitude, we must remember 

 that p stands for an angular retardation /cp, or 27rp/\ ; 

 so that, for example, p = j7r above represents a linear retar- 

 dation X/8, and a total relative retardation between the two 

 halves of the first aperture equal to A/4. 



The second column of Table IV. gives the general ex- 

 pression for the vibration in terms of p for various values 

 of <j>/6, followed by the values of the intensity (I) for 

 sin p = + 1/10 and sin p = + l/\/2. 



