64 THE INTERFEROMETRY OF 



around these two axes is equivalent to a rotation around a single oblique 

 axis, and the fringes will therefore in general be arranged obliquely and 

 parallel to the oblique axis. 



Thus if <pv and <ph are the angles of rotation of the grating (always small) 

 around a vertical and a horizontal axis, respectively, and if %' is the angle of 

 the interference fringes with the horizontal edge or axis of the spectrum 



j. / <?V 



tan x = - 



<fh 



so that if <pv = o, x' = o; if <ph = o, x' = go. This recalls the result obtained 

 above for the interferences of two coarse grids. In other words, for a rotation 

 of grating around a vertical axis (parallel to slit) the fringes of maximum size 

 will be horizontal (Chapter II, fig. 21), because the adjustment around the 

 horizontal axis remains outstanding and the residual fringes (large or small) 

 are therefore parallel to it. For a rotation of grating around a horizontal 

 axis, the fringes of maximum size will be vertical (Chapter II, fig. 22), for the 

 vertical adjustment is left incomplete. When both adjustments are made, a 

 single fringe fills the whole infinite field, and this result follows automatically 

 if but a single grating is used to produce the fringes, as in the original method 

 (Chapter I). 



To deduce equations it is convenient to regard both gratings as trans- 

 mitting and to suppose one of them to be cut into independent but par- 

 allel halves, either by a plane through its middle point and parallel to the 

 rulings (vertical axis of rotation), or by a plane through the same point and 

 normal to the rulings (horizontal axis of rotation). The parallel halves of 

 the grating are then displaced along the normal, e, to both. 



27. Case of reflecting grating. Homogeneous light. The results exhibited 

 in figure 43 for transmitting gratings are shown in figures 47 and 48 for the 

 combination of one transmitting grating G and one reflecting grating G' (the 

 adjustment used in Chapter II), for which the direct path-lengths of rays 

 were computed (cf. figs. 23 and 24, Chapter II). The path-differences 

 obtained were inadmissible. It is now necessary to completely modify the 

 demonstration. 



In figure 47 the rays are shown for the case of complete symmetry of all 

 parts, gratings at G and G' vertical and parallel, opaque mirrors at MI and 

 Ni, telescope or lens at T. The incident ray I at normal incidence is diffracted 

 and reflected into Y, X, T, and Y', X f , T, respectively; the incident ray I' at 

 an angle of incidence di into YI, Xi, etc., and Y'i, X'i, etc., respectively; both 

 at a mean angle of diffraction dd (nearly) to the right, corresponding to di. 



The angles of diffraction (di=*o) are 0i, and 2 ; the double angles of reflec- 

 tion, therefore, 8 = 6* 6 it on both sides; the double angles of the grating G' 

 with the mirrors MI and A/\, symmetrically, <r= 0i+0 2 - 



The normal from the point of incidence at G and at G' , N, and n makes 

 angles 5/ 2 with Y and A', respectively, on both sides. The method of treat- 



