REVERSED AND NON-REVERSED SPECTRA. 



61 



The case of a single grating, moreover, is given if the planes of the grating 

 GG and G'G" and their lines are rigorously parallel, the planes OG' and G"O 

 being coplanar. To represent the interferences of the two independent gratings 

 and with homogeneous light for the case of oblique incidence, it is necessary 

 to suppose the grating G'G" cut in two halves at 0, parallel to the rulings, 

 and to displace the parts OG' or OG" separately, normally to themselves. 

 Figure 43 shows that for normal incidence i = o, the displacement per fringe, 

 8e, would be 



X 



I COS 



or the fringes are similar to the coarse set of the present chapter. 



If the rays impinge at an angle i, figures 43 and 46, they will be parallel after 

 the two diffractions are completed; for it is obvious that the corresponding 

 angles of incidence and diffraction are merely exchanged at the two gratings. 

 Hence the homogeneous rays I', impinging at an angle i, leave the grating at 

 D'i and D'\ in parallel, at an angle of diffraction i, and the rays unite into a 

 bright image of the slit. If, however, OG' be displaced to OiG\, parallel to 

 itself, as in figure 44, the paths intercepted are 



6 



.and - -.cos (Q i) 



cosz 



cos* 



and the path-difference per fringe, therefore, 



X cos i 



i cos (0-i) 



which reduces to the preceding equation if i o. Hence a series of inter- 

 ference fringes of the color X must appear in the principal focus of the tele- 

 scope or lens, on either side of i = o. The theory of diffraction again annuls 

 the apparent path-difference between GG and G'G", 



44 



45 



As to the number of fringes, n, between any two angles of incidence i and 

 i' , it appears that for homogeneous light of wave-length X, 



_/i cos (0-t) _i cos (0'-&')\ 

 X V cos i cos i i 



