as applied to Mierometric Observations. 455 



and is a maximum when 



4:7rac . \nrac 

 - w =tan- sr j 



but the most visible fringes correspond to the early maxima. 



This form agrees exactly with the formula given by 

 Mr. Michelson for a uniformly illuminated segment of a 

 straight line perpendicular to the slits. We see, however, 

 that, provided the conditions stated be fulfilled, it is applicable 

 to a source of any shape. 



The most general form of the fringes is given by 



9 + 2 ( v i + v 2) — const., 

 and therefore consists of lines parallel to the locus of middle 

 points of chords at right angles to- the slits. These will be 

 straight lines in the case of a rectangular, circular, or elliptic 

 source. Here, however, a new difficulty presents itself. For 

 the rectangular source i^ — r 2 will be constant, whatever chord 

 perpendicular to the slits we may select. Fringes will there- 

 fore appear and disappear as a whole. 



But for a circular or elliptic source, v x — v 2 varies as we 

 pass from chord to chord. Thus the maxima will be invisible 

 for some chords when they are most visible for others and 

 conversely. Hence, whatever be the distance between the 

 slits, it appears at first as if we might always expect a mottled 

 appearance. 



But in the case of a circular or elliptic source the length of 

 the chord varies extremely slowly near the centre and there 

 fringes will be visible, the length of the chord being practic- 

 ally constant. The mottled appearance, on the other hand, 

 will predominate as we approach the sides. 



6. Consider now case {b) and let the dimensions of the 

 source be so small that, for any point sufficiently close to the 



centre of the image ^~ is a small angle throughout 



the range of integration. 



[For points not near the centre of the image the illumina- 

 tion will be very small and the appearances are comparatively 

 unimportant.] 



For a point distant < ^7-7 from the centre of the image, we 



may put, as in previous reasoning, 



sin ~&3T^ + t7 ) Sln Zx"(- P+M ) 



" =1 



1 



2irk , . '2tt1i , N 



21 i 



