EXPERIMENTS WITH THE DISPLACEMENT INTERFEROMETER. 3 



Since the angles a and b are invariably small, the cosines may be expanded, 

 so that 



(3) 



is nearly true, or 

 (4) 



*= ,d tan 6 



4- 2 



Since 6 = 28 is exceedingly small (but a few seconds) , 



(5) 



Finally, if a is also sufficiently small, which will usually be the case, and D 

 is zd or zd', so that the distances to the far screen, F" and F'" ', are measured 



from the mirror N, 



(6) x = 4D8 



One may note in passing that the distance over which the N ray travels from 

 coincidence is 



d sin b 



(*+} 



6) \ cos a/ 



cos (a b) 

 whereas the M ray travels over 



x*=d' (tan (a+6) tana) 



where ^=^i+^ 2 . Hence, for small angles, the N ray travels over 3 times 

 the distance db of the M ray, the total being $db. Thus the angle of devia- 

 tion B is measured by x, apart from any other consideration, except that the 

 distance D is very large and therefore invariable and the sensitiveness is 

 twice as large as in the case of single reflection. 



To test this result in its practical aspects, a millimeter micrometer was 

 installed at the end of the pendulum, at a distance of 51.5 cm. from the axis. 

 The two images traveled in opposite directions, in steps, from end to end of a 

 30 cm. scale, while the micrometer was moved forward i mm., successively, 

 eight times, as follows: 



With the exception of the first and last readings the steps of x are 7.4 cm. 

 and equidistant. Hence 



0=' i/5i -5 = 7-4/4^ 



