68 REVERSED AND NON-REVERSED SPECTRA. 



If G'G' is displaced to G\G\ over a normal distance e, or e/cos <p along the 

 line of symmetry GT, the virtual images G m and G n will be displaced to G' m 

 and G' n over the same normal distance e. This is obvious, since the quadri- 

 laterals ab and a'b' are rhombuses by the law of reflection, and hence the 

 perpendicular distances e between the (equal) sides all identical. 



If D 2 is the grating space of G', 



(1) sin 2 +sint = X/Z} 2 sin 0' 2 -{-sini' = X/.D 2 



or if i and i' are very nearly equal and both small, as in the experiment, 



(2) cos 6zdO= cos idi 



Again, in case of the displacement e of G f , the paths are shortened at G m 

 by e/ cos 2 , at Gn by e/ cos 0' 2 , resulting in the path-difference AP, or 



(3) AP = e(sec 02-sec 0' 2 ) 



Since 2 and 0' 2 are nearly the same, this may be adequately simplified by 

 differentiation. Putting 



(4) dd= 2 0' 2 = 2(a <p) AP = 2(a <p)e tan 2 sec 2 

 Hence per fringe, apart from sign, 



(5) = Xcos- 2 



2(0: ^>) sin 2 

 Thus, if 



X = 6Xio~ 5 a (f>= i = o.oi75 2 = 2O 



then 



6Xio- 5 Xo.88 



d = - =0.004, 4 cm. 



2X0.0175X0.342 



per fringe for each degree of arc of non-symmetry, a<p. 



The effectiveness of the fore-and-aft motion, according to this equation, 

 is evidence of a residual angle, a <p, of non-symmetry. This is not improb- 

 able, as my apparatus was an improvised construction, lacking mechanical 

 refinement. Further, the wedge effect due to a, which makes e m variable, 

 would be superposed on the interferences, and hence these could not be in- 

 creased in size above a certain maximum. This is also quite in accord with 

 observation. 



If a = <p, 18 = 2(0: <p) =o and 2 = 0' 2 ; i.e., the virtual images G m and Gn and 

 the diffracted rays are parallel and 8e = co . In other words, the fore-and-aft 

 motion has no effect. If a = o, /3 = 2<p; or if <p = o, fi = 2a. In either case de 

 is finite, and fore-and-aft motion is effective. If the mirrors and grating were 

 rotated in counter-direction so that tp is negative, de will depend on a-\-<p, and 

 the fore-and-aft effect will be correspondingly marked. Moreover, the inter- 

 ference will not in general appear in the principal focus, but usually suffi- 

 ciently near it for adjustment. 



If 5e g is the actual displacement of the grating G' in the line of symmetry, 

 8e a = 5e/ cos if>, so that the angle <p enters equation (5) again, but only to a 

 small extent. 



