REVERSED AND NON-REVERSED SPECTRA. 81 



Equation (22) may be integrated, and if C is an experimental constant, 



There remains the equation for crossed rays or achromatic conditions, 



i8o- 2 



or 



(24) e = 



in which there should be no motion of fringes throughout the spectrum, but 

 for secondary reasons. 



It is now possible to consider the above results on the increase of the range 

 of displacement within which interferences are visible, with the dispersion of 

 the grating. In this case the equations in d\/de, viz, Nos. 20, 22, as well as 23 

 and 24, may be consulted. Since sin d = \/D, all of them involve the dis- 

 persion i/D, where D is the grating space. In the case of No. 24 the range 

 of displacement should be indefinite, since the locus of fringes is stationary in 

 the spectrum. It is found to be exceptionally large, but limited by special 

 diffraction. Equation (20) is cumbersome, but otherwise similar to equation 

 (22), which may be treated first. The displacement of any given fringe in 

 wave-length increases with/= i/D, the number of lines per centimeter. If a 

 fringe travels between any two wave-lengths X and X' and if D is large relative 

 to X, equation (23) shows that approximately 



The range of displacement should therefore be roughly proportional to the 

 square root of the dispersion, and one is not at once at liberty to conclude that 

 the uniformity of wave-trains is enhanced by dispersion. 



In fact, if D is not large compared with X, as in the higher orders of dis- 

 persion, the full equation must be taken. Unfortunately the data of Chapter 

 I, 25, table 12, which are the most complete, do not easily admit of compu- 

 tation in full. I have compared them both with e = e \/\/(D-\-\'), in which 

 the ratios of e observed and computed run up with D, regularly, from i to 7 ; 

 and with 2de/d\ = C(2D+\}/(D+\} 3J2 , in which the regular change of ratios 

 for the same D (as D decreases) is again from i to 7. In other words, the 

 observed values of e varying with D decrease enormously faster than coeffi- 

 cients of this type, as they should. In view of the equation 



it follows that 



de 



dD 2 



and the comparison of the e observed in table 12 with this coefficient is there- 

 fore crucial. 



