468 Mr. MacCuttacu on the Laws of 
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 
altar ae Cc’ 
ke — sal @—0 )sin o k=% (18.) 
instead of formula (8.). The quantity = will be greater than unity, if C’ be greater 
than C, and the value of &, will be greater than before. This seems to be the expla- 
nation of the difference between the ratios observed by Mr. Airy. 
It may be proper to state briefly the 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 \. Now it was remarked by Fresnel, who first resolved the 
phenomena of rotation into the interference of two circularly polarized waves, that 
the interval § between these waves, at their emergence from the crystal, must be in- 
versely as J, if the angle of rotation be inversely as the square of /. This remark sug- 
gested* to me the idea of adding, to the equations of the common theory, terms con- 
taining the third differential coefficients of the displacements ; for it was evident that 
such additional terms would give, in the value of s*, a part inversely proportional to /. 
It was also evident, that the third differential coefficient of € should be combined with 
the second differential coefficients of », and the third of » with the second of &, in 
order that, after substitutions such as we have indicated in deducing formule (5.) 
and (6.), the sines or cosines might disappear by division, and that thus the value 
of s’ might be independent of the time, as it ought to be. This kind of reasoning 
led me to assume the equations 
Ce ao Ce aN dn 
— = —" or — 9 
dt? dz dz 
dn =, dn D ae 
dt? dz” dz 
(19.) 
, (20.) 
for the case of a ray passing along the axis of quartz ; and then, substituting in these 
equations the values of the differential coefficients obtained by differentiating the 
formule 
ep cos} Fost—2) b, n= + psin f2r(s-2)h, 
*«<The singular relation between the interval of retardation [6] and the length of the wave [/] seems 
to afford the only clue to the unravelling of this difficulty."—Report on Physical Optics, by Professor 
Lloyd; (Fourth Report of the British Association, p. 409). It was in reading this Report, that 
Fresnel’s remark, about the relation between 6 and J, first came to my knowledge. 
