142 THE INTERFEROMETRY OF 



telescope now contains two images, the first due to rays (K) entering it directly, 

 the second due to rays (L) reflected into it by the mirrors of the interferometer. 

 Suppose the object seen lies at infinity like a star, that its two images are made 

 to coincide by adjusting the angle a, and that the achromatic fringes have 

 been brought into the field by adjusting the micrometer displacement N. 



Now let the angle a be changed by A a until the two images of an object M 

 at a measurable distance d coincide. Displace the micrometer mirror by AN 

 until the achromatic fringes are restored to their former position. Let b be 

 the effective distance apart (ac or bd) of the paired mirrors in the direction 

 right and left to the observer or transverse to the impinging rays (L), and 

 finally 5 the angle at the apex of the triangle of sight on the base b i.e., the 

 small angle between the present rays KL. Then 



(2) d = b cotgs = b cotg 



(nearly) by the laws of reflection. Hence from equation (i) 



(3) d = bR/ANcosi 



Here 2 bR is the area of the ray parallelogram of the interferometer (abdc, 

 fig. 88). Using the constants of my apparatus, let 2 = 45, R= 10 cm., 6 = 200 

 cm., AN=io" 4 cm., the latter being the smallest division on the micrometer. 

 Hence 



d = 2ooXio/io- 4 Xo.7i =2.8Xio 7 cm. = 28o kilometers 



or about 1 70 miles, is the limit of measurement of the apparatus. 

 Again, from equation (3) the sensitiveness d(AN)/8d, since 



(4) dd = (d 2 cosi/bR)d(AN) 



is inversely proportional to the square of the long distance d and the area of 

 the ray parallelogram 2bR. Thus with the above constants, if d is 2 kilometers, 

 d (AN) = i o- 4 cm., 



8d = (d 2 cosi/bR) d (AN) 



Thus an object at about a mile should be located to about 30 feet. Per fringe 

 of mean wave-length X, moreover, since dd = \d z /2bR, the placement should be 

 about 6 meters at 2 kilometers. I have stated the case, of course, merely for 

 the interferometer, not for subsidiary optical appurtenances, nor for measure- 

 ment by angular fringe displacement. 



74. Theory. To account for these phenomena theoretically the equations 

 of displacement interferometry are available ; for the center of ellipses of these 

 and the central member of the achromatic fringes correspond to the same 

 position, AN, of the micrometer mirror. In fact, the fine white slit image 

 which produces the spectrum when observed through the spectroscope is the 

 central achromatic fringe when the spectroscope is removed. We have, there- 



