and Differential Double Refraction. 349 



Equation (8) shows that the order increases toward the 

 violet end of the spectrum, and equation (9) that the number 

 of bands per unit distance in the normal spectrum increases 

 as the wave-length diminishes. We have also from equation 

 (7), if N is constant and d and X vary, 



• • (10) 



and also 



Bx_ 



N 



l 





Bd~ 



d 'U A+ 



3B 



X 



...) 



8N 

 Bd 



-K^S 



c 



■sxA. 



:ti) 



In equation (10) the second member is a positive quantity, 

 and hence if d increases X must increase, and consequently 

 the position of a band of any order must be shifted toward 

 the red end of the spectrum when the length of the path in 

 the crystal is increased. The direction of displacement pro- 

 duced by increase of d is taken as the positive direction. In 

 equation (11), if we take X constant (say for one of the 

 sodium lines) we have the number of bands passing this 

 point from the violet toward the red end of the spectrum for 

 any increase in the path. Furthermore, since the right- 

 hand side of the equation increases as X diminishes, we shall 

 have more bands coming into the violet end of the spectrum 

 than disappear at the red end, and hence the number of 

 bands visible will increase with d. 



It follows at once from the above interpretation that we 

 can determine the order of the plate for any wave-length by 

 counting the number of bands passing this point when d 

 is diminished or increased, until new coincidences in a 

 different ratio obtain throughout the spectrum. 



Thus we have 



and 



-BX X*\ + X ' ' 7 

 -BX \*\ \*'")' 



The first equation gives the intervals — BX in wave-length 

 for the same increment SN in the order, for path d throughout 

 the spectrum, and the second equation similarly the in- 

 crement SN 7 in the order for a path d', so that we may have 

 a coincidence of bands throughout the second spectrum at 

 the same intervals BX. 



