62 THE INTERFEROMETRY OF 



where e is the distance apart of the two parallel halves of the grating G"0, 

 OG'. Hence n vanishes with e, or the fringes become infinitely large. Lateral 

 displacements are here without signification, as stated above. 



If the grating G' is rotated over an angle <p, figure 43, and e = bp where 26 

 is half the virtual distance apart at the grating G' of the two corresponding 

 rays impinging upon it (Chapter II, fig. 26), the rotation of the grating per 

 fringe is thus 



_ X cos i 

 ^ = 6 i -cos (0-f) 



or n (above) passes through zero as <p or b decreases from positive to negative 

 values. If b is considered variable there is a wedge-effect superposed on the 

 interferences. 



It is this passage of n through zero that is accompanied by the rotation of 

 the fringes, as above observed. 



In case of two independent gratings, GG and G'G" (G'G" to be treated as 

 consisting of identical halves, OG' and G"0), nearly in parallel, fringes may 

 be modified by rotating G'G" around the three cardinal axes passing through 

 the point of symmetry 0. The rotation of G'G" around an axis normal to 

 the diagram is equivalent to the fore-and-aft motion of G'G' when mirrors 

 are used (fig. 26, Chap. II). The rotation around OT in the diagram and nor- 

 mal to the face of the grating requires adjustment at the mirrors around a 

 horizontal axis to bring the spectra again into coincidence. This is equiva- 

 lent to rotation around G"OG'. Both produce enlargement, and rotation of 

 fringes is already explained. 



Let the grating G'G" be rotated over an angle <p into the position g'g", figure 

 45. Then the angle of incidence at the second grating, d, on one side is 

 increased to 6"=6-}-<f> and on the other decreased to 6 1 =d <p. In such a 

 case the diffracted rays are no longer parallel. If 6' and B" are two angles of 

 diffraction on the right and on the left, 



whence 



sin #"+sin B' = 2 sin <p cos 6 



or if 6 is the mean value of Q' and 6" 



B = (p cos 0, nearly. 

 Similarly, since sin = X/D, for i = o, 



sin 6' -sin 6" = 2\ (i -cos <p)/D 



Hence only so long as <p is very small, are the rays appreciably parallel on 

 rotating G'G" around O normal to the diagram; but this is usually the case, 

 as <p = o is aimed at, and fringes are thus seen in the principal focus. 



To the same degree of approximation is it clear that on rotating the grating 

 into a position such as og" the rays emerge parallel to IT, figure 43. 



The next question at issue is the rotation of fringes with fore-and-aft mo- 

 tion, or rotation around an axis normal to the diagram, as shown in figure 26, 



