168 Lord llayleigh on Methods for detecting small Optical 

 represented by 



J=f^ (*) 



Jo v 



J can be expressed by means of the Si-£unction. As may by 

 verified by differentiation, 



J = Si(2^)-7 7 - 1 sin 2 ^ .... (10) 



vanishing when rj — 0. The places of zero illumination are 

 defined by?7 = n7r, when w=l, 2, 3, &c; and, if y assume one 

 of these values, we have simply 



J=Si(2^) = Si(2n7r) (11) 



Thus, setting n=l, we find for half the light in the central 

 band 



J = Si(27r) = i<7r--15264. 



On the same scale half the whole light is Si(oc), or ^tt, so 

 that the fraction of the whole light to be found in the central 

 band is 



1- 2X ' 15264 =1-097174, . . . (12) 



IT 



or more than nine-tenths. About half the remainder is 

 accounted for by the light in the two lateral bands imme- 

 diately adjacent (on the two sides) to the central band. 



We are now in a position to calculate the appearance of the 

 field when the second aperture is actually limited by screens, 

 so as to allow only the passage of the central band of the 

 diffraction pattern. For this purpose we have merely to 

 suppose in (8) that 6^=w. The intensity at angle <\> is thus 



4 [ Si( «±* .).<-=♦,)]'. . . (.», 



The further calculation requires a knowledge of the function 

 Si, and a little later we shall need the second function Ci. 

 In ascending series 



1 r B 1 ^ 5 1 v 1 



SiM=*- 3J7273 +5 1.2 .3.4iT7EO + ' (14) 



1 1 X 2 1 <T 4 



CiO) =r+2 ^g O 2 ) - 2 1T2 + 4 1.2.3.4 " ' " ' ' ; (15) 



7 is Euler's constant '5772156, and the logarithm is to base e. 

 These series are always convergent and are practically 



