172 Lord Ray Leigh on Methods for detecting small Optical 

 notation #£=500. Thus 



(19) = Si{5OO(l + 0/0)} + Si{5OO(l-0/0)}, . (26) 

 and 



(21) = Ci^5OO(l-0/0)}-Ci{5OO(l + 0/0)j 



+ log (1+0/0)- log | 1-0/0 | . . . (27) 



For the purposes of a somewhat rough estimate we may 

 neglect the second Ci in (27) and identify the first Si in (26) 

 with ^7r for all (positive) values of 0/0. Thus when = 0, 

 I = 7r 2 ; and when 0= oo , 1 = 0. 



When 0/0 = 1, we take 



(26) =Jw\= 1-571, (26) 2 =2-467. 

 In (27) 



Ci]5OO(l-0/0)}= 7 + log 500+ log (1-0/0), 

 so that 



(27) =7+ log 1000 = 7-485, (27) 2 = 56'03 ; 



and 1 = 58-50. 



For the values of 0/0 in the neighbourhood of unity we 

 may make similar calculations with the <-iid of Grlaisher's 

 Tables. For example, if 0/0 = 1 + '02, we have 



500(1-0/0; = +10. 

 From the Tables 



Si(+ 10) = + 1-6583, Ci(+ 10) = - -0455, 



and thence 



I(-98) =31-13, 1(1-02) = 20-89. 



As regards values of the argument outside these units, we 

 may remark that when x exceeds 10, Si(<r) — \ir and Ci(#) 

 are approximately periodic in period 27r and of order x" 1 . 

 It is hardly worth while to include these fluctuations, which 

 would manifest themselves as rather feeble and narrow 

 bands, superposed upon the general ground, and we may 

 thus content ourselves with (25). If we apply this to +10, 

 we get 



I(-98) =30-98, 1(1-02) =21-30; 



and the smoothed values differ but little from those calculated 

 for + 10 more precisely. The Table (II.) annexed shows 

 the values of I for various values of 0/0. Those in the 2nd 

 and 8th columns are smoothed values as explained,, and they 

 would remain undisturbed if the value of 0f were increased. 

 It will be seen that the maximum illumination near the 

 edges is some 6 times that at the centre. 



