72 On the Laws of the Double Refraction of Quartz. 



this supposition is at least very nearly true in the neighbourhood 

 of the axis. It is probable, however, not only that (7 varies with 

 0, but that it becomes different in equations (1) and (2) ; that 

 is to say, it is probable that the following equations 



(17) 



_ 7? _ r" 

 d? ~ dz* d**' 



in which C f is a little different from (7, would be more correct 

 than those which we have assumed. Indeed Mr. Airy's experi- 

 ments seem to indicate that C' is greater than C ; for he found, 

 as we have already said, that the ratio of the axes of the little 

 ellipse described by a vibrating molecule is somewhat different 

 for the two rays, being more nearly a ratio of equality for the 

 ordinary than for the extraordinary ray. Now if we set out 

 from equations (17), instead of (1) and (2), and proceed in all 

 respects as before, we shall arrive at the formula 



-) sin $.* = (I 8 ) 



C' 



instead of formula (8). The quantity -^- will be greater than 







unity, if C' be greater than C, and the value of A* will be greater 

 than before. This seems to be the explanation of the -difference 

 between the ratios observed by Mr. Airy. 



It may be proper to state briefly the considerations which led 

 to the foregoing theory. Beginning with the simple case of a 

 ray passing along the axis, the first thing to be explained was 

 the law of M. Biot, that the angle of rotation varies inversely as 

 the square of / or of A. Now it was remarked by Fresnel, who 

 first resolved the phenomena of rotation into the interference of 

 two circularly polarized waves, that the interval 8 between these 

 waves, at their emergence from the crystal, must be inversely as 

 /, if the angle of rotation be inversely as the square of I. This re- 



