THE AID OF THE ACHROMATIC FRINGES. 23 



spectrum ellipses, therefore, has no direct relation to the center of white light 

 fringes; for the latter occur only when the rays pass the plate normally. 

 On the other hand, when the white-light fringes are straight lines correspond- 

 ing to very oblique incidence of interfering rays, the spectrum fringes are 

 none the less perfect ellipses. 



It is finally necessary to account for the coincidence in adjustment of the 

 center of spectrum fringes and the achromatic fringes, as the latter overlie 

 the coincident white slit-images from which the superposed spectra are 

 produced by the grating. This is easily seen to be referable to the fact that 

 interferences with white light can only be visible if the light in the region 

 of interference, when analyzed spectroscopically, contains but few dark 

 bands. Since the number of bands in the spectrum is least near the center 

 of ellipses, and is further reduced on making them as large as possible, the 

 relation is obvious. In the case of strong, large achromatic fringes, a single 

 fringe virtually occupies the whole spectrum. The light is either white or black. 



The displacement of the center of ellipses with the angle of incidence for 

 a given adjustment may be computed from the original equation for centers 

 N =e (cos r-\-2 J?/X 2 cos r), where r is the angle of refraction for the incidence 

 i, e the thickness of plate, and B the dispersion constant. When N and e 

 do not vary it may be shown that (since n = A +.B/X 2 ) , 



d\_ X sinr cos i (2.B/X 2 ^ cos 2 r) 

 di 2B/\ 2 fj. cos 2 r (2+cos 2 r) 



To obtain an estimate 2B/\ 2 = 0.026 may be neglected as compared with 

 H cos 2 r and the equation given the approximate form (^ = 1.5) 



d\ X sin * cos i 



=- - = 10 X sin i cos *, nearly. 



di 0.026 2.5 jj. 



Thus, if t' = 45, d\/di = $\ or 0.09 X per degree of i, which is about 100 times 

 the distance of the sodium lines. If i = o or 90, the shift vanishes. 



10. Compensators. When the fringes are found they may be erected as 

 stated by rotating either pair of the diagonal mirrors of the ray parallelogram 

 towards or from each other, usually on a horizontal axis. The fringes may be 

 enlarged by rotating the paired mirrors on either end of the ray parallelo- 

 gram and restoring the fringes after each small step of rotation by displace- 

 ment AAT at the micrometer. But these processes are tedious and must be 

 very cautiously performed or the fringes are liable to be lost. The same re- 

 sult may be accomplished by the aid of plate-glass compensators, about 

 0.5 to i cm. thick, placed normally in each of the two interfering beams 

 and originally parallel and vertical. (See fig. 7, C and C'.) In addition to 

 rotation and enlargement, these compensators serve with further advantage 

 in equalizing the two beams in intensity. For this purpose it is merely 

 necessary to half-silver lightly the compensator in the stronger beam of 

 the ray parallelogram. If the fringes are more nearly vertical (between 



