﻿398 Dr. C. V. Burton on 



28. In like manner, if T is the total brightness per unit 

 of vertical length in a strip whose breadth extends from 

 0=—7r to 6=— tt + cx, that is, from 7 = to y = a, we find 

 approximately 



T=jjJ (/3« 3 + /3 3 «) (5) 



If a. changes in value from a to cc + Sa, the corresponding 

 change in T is 



ST=|;-4(3/3a 2 + /3 3 )^ ( G ) 



O 7T W 



Suppose, then, that two narrow strips of illumination, such 

 as here concern us, can be unmistakably seen to differ in 

 strength when their total brightnesses are in the ratio 

 1 +n~ l : 1 + w" 1 , or in other words, when the wire HK, LM 

 (fig. 2) is so far laterally displaced that 8T/T = l/yi at one 

 side of it, with a corresponding contrary change at the other 

 side. Then for critical discrimination we have 



1 = f3fo * 2 + /3 3 )g« 

 n~ "«/3(« 2 + /3 2 ) > 



or . _ a 1 + /3 2 /* 2 



n 3+/37^ 7 ' 



29. Since the central bright band of the diffraction pattern 

 extends from 6= — it to = 7r, the maximum error in setting 

 the wire centrally upon the pattern, expressed as a fraction 

 of the width of the band, is 



^ = ^l + /3 2 /a 2 m 



2tt 2717t3 + /37« 2 ^ ' 



30. To make this expression a minimum, a and /3 must be 

 adjusted by trial until the consistency of setting appears to 

 be as great as possible ; there is, however, a theoretical 

 optimum relation between a. and /3 which, on a certain 

 assumption, is readily deduced. The assumption is that the 

 visibility of a faint elongated object of very small apparent 

 breadth depends on the product breadth X average bright- 

 ness, that is, in our ease, on the total luminosity per unit of 

 vertical length, represented by T of equation (5). TVe 

 suppose, in fact, that, for any one observer working under 

 given conditions, the number n defined in § 28 is a function 



