40 THE INTERFEROMETRY OF 



where n is the number of fringes passing at wave-length X. This equation 

 is not obvious, since for constant X, the distance between G and G' ', measured 

 along a given ray (prolonged) for any position of M or N, is also constant. 

 The equation may be corroborated by drawing the diffracted wave-front, 

 which cuts off a length 2e sin 6 from d". 



Since sin 6 = \/D if D is the grating space, the last equation becomes 



n ze/D 



or per fringe 



a remarkable result, showing that the displacement of the mirror M per 

 fringe is independent of wave-length and equal to half the grating space. An 

 interferometer independent of X and available throughout relatively enormous 

 ranges of displacement is thus at hand. It will presently appear that it is 

 also independent of the angle of incidence at G. 



In case of the given grating and sodium light, 6= 19 37'. Hence if 8e is 

 the displacement per fringe, 



e = \/2 sin = 10^X88 cm. 



Actuating the micrometer at M directly by hand (this to my surprise was 

 quite possible without disturbing the fringes, except for the flexure of sup- 

 ports) , the following rough data were successively obtained from displacements 

 corresponding to 10 fringes: 



io 6 X5e = 65 95 90 80 60 cm. 



Without special precaution the fine fringes can not be counted closer than 

 this, so that the data are corroborative. 



If the incidence is at an angle i, as in figure 26, the rays entering at G 

 obviously leave at the same angle i (symmetrically) at G'. In other words, 

 rays enter and leave in pencils of parallel rays. The optic path of the com- 

 ponent N ray, /+g, is 



/z/cos 6" 



where h is the normal distance between G and G' and 6" the angle of diffrac- 

 tion on the left. Similarly the optic path of the M' rays c-\-d which meet 

 /+ g in r is 



e' 



where d' is the angle of diffraction on the right. If now M' is displaced to M, 

 c-\-d is changed to a+ d of the same length; but if the wave-front w is drawn, 

 it appears that the optic path of the M ray has been shortened to ze sin 6', 

 where e is the normal displacement of M' to M. Hence the path-difference 



h/cos 0'- 2 esin e'-h/cos 0" = X 



n being the ordinal number of fringes of wave-length X. Furthermore, since 

 h, i, 6', and 6" do not change, the displacement per fringe is as above 



de = \2 sin 0' 



